- And good morning to you, everyone.
It's my absolute pleasure to
be back here today with you
for another three hours
of quantum physics.
I noticed after the lectures yesterday,
we had quite an active chat
and some of us had some
interesting discussions on Twitter,
so looking forward to day two.
Let's get to it.
What's on the road ahead today?
After reviewing some of
the main takeaway messages
from yesterday, I'll dig into the details
of what is the transmon qubit
how exactly does it originate?
And you notice that all
of the things we spent
diligent due time on yesterday
understanding the harmonic oscillator,
will very naturally extend
into the world of nonlinear
or anharmonic oscillators.
The harmonic oscillator really
serves as a founding basis
for so much of physics.
So, that's why I wanted to make sure
that we spent some good time reviewing it
so that everyone can
really get a both detailed,
but also intuitive sense of how it works,
recalling the quote by Henri Poincare.
And today, we'll see
that the transmon qubit
isn't too far away,
we're actually only one or two steps away,
conceptually, from understanding
how to create the qubit,
how to even calibrate it,
as well as how to take advantage
of some of its higher excited states.
That will lead us into the next step,
which is on the control of the qubit.
And I think a lot of you
have already gone quite far
in discussing the lab,
which is a very deep
dive into qubit control.
And here, I'll try to give
a bit more of an overview
of the conceptual ideas,
as well as grounding some
of the core features of that physics
in some of the math of quantum equations.
After that, we will take a leap
into discussing how is
all of what we learned,
related to actually reading out the qubit,
because nothing lives in Hilbert space.
At the end of the day,
you always have to
communicate between the world,
the macroscopic world in which we live
and the somewhat abstract
unitary operations gates
and so forth that we perform
in a quantum computer
and the link is measurement.
And how is that measurement exactly done?
We'll see that in our
superconducting quantum systems
that's carried through
using a type of oscillator
and we're back to oscillators.
So, you'll see that
coupling a transmon qubit
to an oscillator,
can allow us to do a very
beautiful projective measurement
of the Z quadrature of the qubit
and that is exactly
how all the experiments
you might have run in Qiskit run.
Going into more detail,
also, if you've touched on
things like Qiskit pulse,
you'll notice that
measurement is a very rich
and deep field of its own.
And we'll really dive
into some more details
on how do I now look at the coupling
between the qubit and its
measurement apparatus,
the readout cavity,
which mediates the interaction
between the observer and the bath
and the quantum system at
its core, the transmon qubit?
And that will lead us into
conditional pointer states.
And finally, as we close
out on the road today,
there'll be a lab, of course,
that follows all of the discussion.
Now, let's begin with a bit
of a takeaway on lecture one,
some of the core messages
that we can take home
and then build on.
I love this quote by Asher Peres,
"Quantum phenomena do not
occur in Hilbert space,
they occur in laboratory."
And I hope that given
the discussion yesterday,
you saw that a lot of what
I hope we can achieve here,
is to both focus on the
description of the phenomena,
as much as on their physical character
and providing a mental picture.
I think it was also Pauli or Heisenberg
who said that we can conceive
of so many great things
with our mind,
but not necessarily always immediately
understand them mathematically,
but often it's the heuristics
that leads the discovery.
And that leads me to maybe the first slide
which is an overview
slide of the connection
between where you sit today
at your laptop computer
and the abstract Hilbert space
which you control and manipulate
and can experiment on in the real devices.
And the link between these two is formed
by the following chain.
You might send some sort
of encoded information
as a cue object, for instance,
from your laptop into a laboratory.
This laboratory contains
a superconducting,
or rather, a dilution refrigerator
with some control electronics around it,
which houses the golden chandelier.
And there were some interesting
questions yesterday,
which I think have answers
summarized the discord channel
that discuss why it's actually golden
and why that's really important.
However, the quantum chip
sits at the very bottom
of this dilution fridge and
often looks something like this,
where each of these two
metal pads connected
by some sort of nonlinear
inductor, form a transmon qubit.
Now, all the squiggles between
them are, take a guess,
more oscillators, our bread
and butter ingredient,
hence you can tell my
love for oscillators.
The transmon qubits can be
connected by oscillators,
as well as these extra oscillators
that sit on the outside,
which are readout
oscillators or resonators.
We can model these
using lump element
electromagnetic circuits,
which form a very beautiful bridge
between simple, classical physics
you might have started
learning about in high school
and opening the door into
unveiling the quantum world,
in which the rich character
of quantum oscillators
and nonlinear quantum
physics and quantum optics,
can be exhibited
and they form an amazing testbed
for both academic research
and industry applications today.
The oscillators, of course,
provide us with a sub-manifold
of the larger space of the
oscillator which we use
as the qubit two-state manifold,
or which we can represent
in the Bloch sphere,
what you learned about in first week
of the Qiskit summer school.
Now, to make a more concrete connection,
yesterday we drew this
picture of a transmon qubit
connected by two metal pads
and a little wire between them.
This wire is usually a nonlinear,
so called Josephson tunnel junction.
The amazing thing about
almost all of this,
is that you don't really
necessarily need to know
a whole lot about the details
of the superconductivity here,
you can take that as an abstraction
and really focus on some
of the physical effects
of what are the charges,
what is the charge
doing as it flows across
and what is the magnetic flux doing,
which is related to either
the integral of the voltage
or the magnetic field lines?
And we saw that both the transmon qubit
and the readout oscillator
will all have simple phenomena
of the sloshing of the electric fluid,
with charges back and forth
between different regions in space,
which are associated with time variations
in the charge and flux.
And because of the capacitance
and inductance of this circuit,
can create very specific
types of oscillations.
And in the quantum world,
in the world of low dissipation,
in the world of some non-linearity
and low temperature,
we can begin to find
that the energy spectrum
of these oscillators is quantized,
that you can create coherent
superpositions between them,
that if you take two
of these transmissions,
you can create entanglement,
you can create logical gates and so forth.
And that led us into an
interesting discussion
of the quantization of
the harmonic oscillator,
which I emphasize again,
is the bread and butter to understand
before leaping into all
of the next subjects
that we will do.
The harmonic oscillator, of course,
has a quadratic dependence
in terms of the position variable
which were identified as flux.
So, this parabola here
is the potential energy,
which goes like phi squared over 2L,
where phi was the integral of the voltage
with respect to a reference data from,
we used to call it time minus infinity,
but it's just some
initial reference state.
It's mostly a label, it could be T0
two sum time here.
Now, the important thing is that,
when we quantize this oscillator
and move into more of the quantum world,
we can notice that the harmonic
oscillator has energy levels
that are spaced exactly by
energy H bar omega naught,
where omega naught is
the resonance frequency
of the oscillator,
the very familiar one over
the square root of LC.
So, the geometry, through
the effective inductance
and capacitance it creates,
will set the resonance frequency.
And as I'll show in a minute,
the zero point quantum
fluctuations are also set
by both L and C.
When we move into the quantum domain,
we can describe this quantization
ladder of energy states
by Hamiltonian, that we can write
in the Eigen basis of the Hamiltonian
and I call these Fock states.
There was some number
of questions, I think,
yesterday about what exactly
do I mean by Fock states
and sorry, when I write N here,
I mean that this bottom
level is the zero Fock level,
the one Fock level the, two Fock level.
Why am I calling these Fock states?
Here, by a Fock state,
I mean that I have either
zero photons of energy,
which is the ground
state in the oscillator.
Now, that doesn't mean
that there's no energy,
there is zero photon particles.
That doesn't mean there's no energy,
it just means that the oscillators
in the ground state,
however, it still fluctuates,
as we saw yesterday.
The one state, it's the first Fock level,
or it's the one-photon excitation
and it's created by the A dagger operator.
So one, is equal to A dagger zero.
So, we can think of A as the
photon creation at operator,
two as the photon creation
operator created twice,
and so on
so, we've put two quanta of energy
or two photons inside the oscillator.
Now, taking the bridge from
the classical to the quantum,
I think for me is also one of
the next essential elements
and I'm very glad to see that,
I'm going to assume Bruno here,
like that as well.
And I think it's important to highlight
that even though quantum
is fundamentally richer
than classical physics,
we can still try to gain so much intuition
by bridging from the classical world,
into the quantum world
and trying to make a close analogy
between the two in our bridge,
because some things that can
be quite difficult, perhaps
to understand, such as what
exactly is the latter operator,
and the annulation operator
and why does it evolve
using an E to the minus i omega naught T,
can have very simple pictures
that can explicate that
in the domain of classical physics.
We saw that in classical physics,
we can think of this alpha
operator or alpha variable,
excuse me, a complex
action angle variable,
is a point in the phase-space
that simply rotates around the circle
and represents the harmonic evolution
and that's something very simple.
However, as we go into the quantum world,
all of that gets more complicated
or more rich, due to the
intrinsic uncertainty
between position and momentum.
We cannot identify in the quantum world,
the quantum state with a
single point in phase-space.
We can also make the
analogy between position
and momentum for the
somewhat abstract charge
and magnetic field.
And in our case, we can make the analogy
that these quantum
oscillators have position
that the magnetic flux,
that looks a lot like the
mechanical position of a spring,
as the spring oscillates up and
down from its rest position,
its equilibrium position,
the mass goes like the capacitance
and then the spring
constant of this spring,
goes like the inverse inductance.
The Hamiltonian in the
quantum world can be expressed
in two different bases, essentially.
The first basis we write
it in terms of the flux
and charge operators,
which directly parallels the
more standard description
in classical physics.
But it also has a very
advantageous representation
in terms of creation and
annihilation operators, A dagger A.
Now, I think there was a question I saw
in the discussion box here,
which was why did I drop
the one half yesterday?
This is a very good question
and it's somewhat more subtle
than it appears at first.
Typically, in measurements,
we can really only measure
differences in energy.
So, in a sense, absolute
energy isn't exactly
a measurable thing.
So, you could ignore the one half.
And if you look at the
equations of motion,
any constants simply drop out
from the equations of motion.
However, that's a little
bit too simplistic
in more general applications,
because you can, in fact,
take this your point back
and fluctuation energy,
if you do things such as
vary the frequency here
or look at more advanced
dynamical effects,
such as the Kashmir effect,
which we won't get into today.
The eigenstates of this operator
are very nicely described,
in terms of the energy
levels, the Fock basis,
which we explored in the previous slide
and can have a beautiful
representation in the flux basis
or the position basis, as
particular types of polynomials.
Here, the different curves
are staggered in energy,
so, the position between the zero curve
and first curve, or sorry,
the wavefunction corresponding
to the ground state
and the wavefunction corresponding
to the excited state,
which are the blue and
orange curves respectively,
I've separated by an energy difference
of H bar omega naught.
And these represent the
probability amplitude,
meaning that the square of
these gives you the probability
to find the circuit to have a
magnetic flux of this amount.
So, for instance, the ground state we see,
is localized around zero magnetic flux.
In the ground state,
you don't expect to have any charge
or magnetic flux, especially classically.
However, we notice that on a time basis,
there are fluctuations.
So, while the mean of the
expectation value is zero,
there are inescapable fluctuations,
which we call the zero-point
quantum fluctuations,
which give the root mean squared,
amplitude of the variants, or sorry,
of the magnetic flux fluctuating in time.
And so, this relationships here,
are going to be very, very key today
to determining all of the parameters
of the quantum oscillators
and the quantum qubits
or the quantum transmon.
Okay and maybe I will pause very quickly
and see Josie if there
are any questions yet.
- [Josie] Yeah, it looks like we have
a couple of questions here.
I think one of the most popular ones,
is how a single IBM quantum
computer uses roughly
as much energy as three hairdryers.
And I'm not gonna lie,
I'm pretty curious myself.
(giggling)
- [Zlatko] Yes, that's
an interesting question.
Yes, so that's an interesting question.
I guess if we go back to
the golden chandelier here,
let me scroll back.
Everything we talked about,
I think you remember from
the diagram yesterday,
all the quantum properties
are at microwave frequencies,
in the order of one to 10 gigahertz.
So, in terms of energy and temperature,
these are minute, minute scales.
They're very, very little quantities
of energies and photons.
However, as we mentioned,
to work in the quantum world,
you have to suppress fluctuations
of the thermal environment
in which this qubits are
embedded down here at the bottom.
Oops, let's look at that,
down here at the bottom.
And so we operate at
negative 273.114 Celsius.
Now, to get to that temperature
requires a lot of energy
because you have to extract
heat from this entire structure,
you have to extract the
constant flow of heat
from the room that the fridge is in.
And of course, you need to run
all of the electronics
associated with the fridge
and all of the other pipes and pumps
and vacuum systems and so forth.
And it's really all of
this auxiliary systems
that require all this energy.
It's not really the
quantum computer itself,
it's all the auxiliary stuff.
And, of course, all this stuff
is getting more efficient
and miniaturizing and
getting smaller and so forth.
So, hopefully, that gives
us enough perspective there.
- [Josie] Thank you, sorry,
I have to toggle the mute
between Zlatko, I'm a little delayed,
I'm sorry about that.
I think the next question
that looks like a lot
of people are asking,
is about the qubit electric dipole moment
and you guys can make
fun of me, that's fine.
I'm just an admin, I'm not a mentor.
But they're asking,
can we express this dipole moment
in terms of the circuit
parameters, LC, et cetera?
- [Zlatko] And this is
actually a kind of a debate
in the community that's been going on
for some time, is exactly,
should you even talk
about the dipole moment
of this transmon qubit?
And there're really two views,
although I think the picture
has been clarified a lot.
This transmon picture here,
let's get the pen,
it looks a lot like I have
some piece of metal up here,
connected to another
piece of metal down here,
through a wire and you have
current going through this wire.
So, you can try to model
this with some radiation,
sort of dipole field pattern
and talk about its dipole moment.
But, it's not really in that regime.
This transformer qubit,
it's relatively large,
in terms of the microwave approximation,
say this might be 200
microns across, for instance.
And so, if you look at
the microwave fields
or the radiation pattern, if
you want, of this transmon,
it has a quadrupole moment
and a sixth-order moment
and so forth.
And there are also many
different pieces of metal around,
so it's not a very pristine
picture of a dipole
and this picture is much more complicated.
So, this is actually
where something called the
energy participation ratio,
which we'll briefly talk
about, comes in later,
which allows you to bridge
this very-complicated
electrodynamics structure
and simplify it through just calculating,
how much of the energy of mode, excuse me,
is housed inside of this
electromagnetic element,
as opposed to all the other
magnetic fields around.
And so, you can talk about
an effective dipole moment,
the effective dipole moment
of the qubit, in a way,
is going to really be related
to the zero-point fluctuations,
again, because they will
tell us how much current
or charge is fluctuating
back and forth, on average,
between these two islands.
And perhaps it's worthwhile
to point out that
that number is going
to be very, very small.
On average, you'll see
that, even though over here
there are about 10 to 12 mobile
electrons or Cooper pairs,
the number of charges fluctuating across,
is going to be only about one Cooper pair,
typically for transmon qubit,
which is one in 10 to the 12,
it's a very small fraction.
So, it looks like a fairly small number,
but we can engineer
that to be, nonetheless,
a pretty large electric field.
So, the best thing for now,
is to think of the size
of that dipole moment
to be dictated by, effectively,
the zero point vacuum fluctuations
which are, of course,
given by the parameters of the circuit,
the square of H bar Z over two,
where Z is the effect of
impedance of, note here,
the square root of L over C, thank you.
- [Josie] Thank you that Zlatko
and you muted for just a second.
I think those are the
two biggest questions,
so I can let you resume, if you're ready.
- [Zlatko] Yep, perfect and all right.
So, yesterday, we saw that if you look
at the phase-space
functions of the oscillator,
where you can make a picture of analogy
between the classical world
where the energy here,
by the way, this curve here represents,
excuse me, a paraboloid in two variables,
the flux and the charge
and this is Q squared over
2C plus five squared over 2L.
So, these are contour
plots of constant energy.
The innermost one was H
bar omega naught over two,
it corresponds to essentially
the vacuum fluctuations.
The next one is the one
photon and so forth.
These are these so-called
Husimi Q functions
of the ground state, the excited state,
but basically they're representing
that we have a cloud of uncertainty,
as to where to find the artificial atom
where the harmonic oscillator is.
In the one stage,
you won't really find it in
the middle as its excited,
but you tend to find it around here,
so very much like the hydrogen
atom or any other atom.
There was a question which asked,
what happens as you go
to higher-energy level,
so I'll briefly answer that
question from yesterday.
It's essentially getting donut
and the donut grows.
And the radius of this donut growing,
as you go from the ground state
to the first excited state
to a second excited state,
to three photons of energy four,
grows like the square
root of the energy here,
or in other words, this grows like alpha.
And we can recall that the
Hamiltonian or the energy,
goes like alpha star alpha,
or the magnitude of this complex number.
Now, with all that introduction,
let us move into the transmon qubit.
So, what is the transmon
and how is it different
from the harmonic oscillator
we've so extensively discussed so far?
It's surprisingly similar
because the only thing
we're going to change,
is this element in the center here.
Rather than a linear inductor,
we're going to replace it
with a nonlinear inductor,
Josephson tunnel junction.
And this is what a Josephson
tunnel junction looks like
in practice, many thanks
to Luigi Frunzio here
for this image he's
taken from some time ago.
What you see is a microscopic image
of the order of about
100 nanometers across,
this is very, very, very small.
And so, this is one piece of metal
that you see in this picture here,
which has a width of
about, say 200 nanometers,
this is really nanolithography.
And there's another
piece of metal over here,
which also has a similar size
and the two pieces of metal
fall on top of each other.
We see that if I take a
cross-section picture,
over here, there's the
bottom piece of metal
and then there's the top piece of metal.
Now, between the two,
you'll notice that there
is a ridge, or not a ridge,
but rather there is some
sort of an oxide barrier.
It doesn't even matter, actually,
what the details of this
whole structure are.
The important thing is
that, however it's formed,
this weakling between
these two superconductors,
establishes a beautiful relationship
between the two terminals,
which I might label as, say
terminal one and terminal two
and we can describe the elements
by a potential energy
function, or an energy function
which only depends on the
change of magnetic flux
across the two terminals and nothing else.
Even though you can see that
the picture on the right
is very messy and there's
all kinds of little pieces
of metal floating around
and a lot of things are happening,
we could ignore all that detail
and just very robustly abstracted away
into a nonlinear inductor,
an inductor that has
potential energy function
that looks like this,
where the potential energy
of the Josephson junction,
so I've included an index J here
to denote Josephson junction,
I'll suppress it in the next slide,
depends on the flux difference
across its terminals,
is given by some energy scale,
which we'll call EJ, the
Josephson tunneling energy
and one minus the cosine
of the flux scaled
in terms of phi naught.
Phi naught is the reduced
magnetic flux quantum,
it's a somewhat universal constant, oops
and has a very small value
of about 10 to the minus 16 Webers.
Now, if that looks a bit
complicated, I agree,
we can actually break
it up into two pieces.
Let's simplify it, let's break
it up into a linear piece,
like harmonic oscillators,
plus a nonlinear piece,
which we can quantify the
amount of non-linearity.
So, if I separate it out
into two contributions using
the Taylor series expansion
and dropping constant terms,
we can see that the cosine breaks up
to first order into a quadratic term,
which goes like that magnetic flux squared
over phi naught, times EJ.
And this is nothing but the
energy potential landscape
of a linear inductor,
remember phi squared over 2L.
This phi squared over 2L,
allows us to make the identification
that the effective inductance
and zero bias of the
Josephson tunnel junction,
is related to EJ.
Or, if delete that and
write it a bit more clearly
and say that we can
make the identification
that EJ divided by phi naught squared,
is equal to one over LJ.
Or in other words, the
Josephson tunnel energy,
is equal to phi naught squared,
where that's the reduced
magnetic flux quantum
divided by LJ.
And so, we can really link the expression
for the Josephson tunnel energy,
which describes how
easy or difficult it is
for electrons or Cooper pairs to tunnel
across these two islands to a
linear effect of inductance.
Now, the nonlinear part is
what we haven't yet seen
and that's going to really
provide all the magic
for all of the transmon
qubit, the amplification,
the coupling to the readout and so forth.
And that's a quartic function,
it depends on the fourth
power here of the flux
and it has a coefficient of
four factorial, which is 24.
There are also higher-order terms
which go like the magnetic
flux to the sixth power,
we can ignore those
because we'll see that the
overall contributions here,
will be pretty small
of the nonlinear piece.
In other words, we want
nonlinear oscillators,
but we don't want them that nonlinear,
that they're too difficult to control.
And we can symbolize the purple part,
which represents the non-linearity
with the quartic non-linearity
with one of these symbols here.
So, let's look at the transmon qubit
from the same viewpoint which
we use to look at the supercut
of the harmonic oscillator.
The transmon qubit has a
potential energy function
that looks like negative EJ
cosine of phi over phi naught,
it's a funny cosine potential
and it has the exact same
capacitive of energy function
as a harmonic oscillator.
Now, the magnetic flux
quantum to remind us,
is H bar over 2E.
So, you see that it's very fundamental,
it's is formed from two
fundamental constants of nature,
Planck's constant for the
quantization of energy
or the quantum of action
and the electromagnetic
charge of a single electron.
There's a two here
because we're talking
about superconductivity,
so we're interested in pairs of electrons
and it has a value that's very, very tiny.
A Weber is the unit of magnetic flux.
The potential energy landscape
associated with the cosine,
looks like this,
where on the bottom axis here,
you can write out the
reduced magnetic flux
across the Josephson tunnel junction.
At the center it's zero and
it's periodic in pi or in here
in two pi.
And we have, the classically
forbidden region,
which is where a classical
particle can never be found
or live at a given energy,
shaded out in gray.
The cosine is being represented
by this oscillating
potential and the energy goes
between here, negative EJ and positive EJ,
'cause of drop the constant
term in the energy.
Now, somewhat crucially,
we can make the following approximation.
We can approximate the
potential energy landscape
of the Josephson junction to
first order as a quadratic,
that linear inductance.
And so in this picture,
it looks like this purple parabola, here.
A striking thing emerges,
which is that the parabola and
the cosine are very similar,
they're not very different actually.
If you notice, especially in
the limit of low excitation
and the limit where the
junction isn't highly excited,
but we operate near zero magnetic flux,
you'll notice the difference
between the purple curve
and the gray curve here is
almost indistinguishable.
And we're going to use that,
we're going to heavily,
heavily exploit that.
So, here is the classical Hamiltonian
of the Josephson tunnel junction.
If we apply the same
semi-classical intuition as we did
for the harmonic oscillator,
we can quantize its energy in, say units
of equal photon number.
Now, that's actually going to be wrong,
we should be quantizing the action,
but I won't get into that subtlety,
since we'll see that the
pictures are fairly close.
And writing out this
equation here on a 2D plane,
in terms of the position variable phi
and the charged variable
or momentum variable Q,
we can see that the contours
of constant energy represented
by these paths over here,
which represent the classical trajectories
that a particle will experience,
start out fairly circular
and then elongate as you get
to higher and higher energy.
Higher energy's indicated
by the brighter spots
in the region,
so the dark spot here is
sort of the deepest part
of the valley.
You can think of this as a mountain range.
So up here, if you're a cartographer,
you notice down here is very low
and each contour tells
you that you've climbed up
by another hundred meters
if you like hiking.
(giggling)
Here, we can identify the
superconducting, or sorry,
the magnetic flux point.
Let's say that the particle
transmon starts here.
Classically, it's described by point,
the trajectory it follows
as a function of time,
is no longer a circle, but an ellipse.
And that will lead to this
strange nonlinear behavior.
In the quantum world, we
simply will put hats on things.
Now, formally, we follow
the exact same procedure
for the classical circuit where
we write down the Legendre
and do a Legendre transform
to go to the Hamiltonian
by identifying the canonically
conjugate variable Q
and then we apply the RX
quantization procedure
with the commutators, but really,
nothing at all at this point has changed
between what we did yesterday
for the harmonic oscillator
and for the transmon.
The only thing that's changed,
is the potential energy function.
However, it doesn't affect, in any way,
any of the steps we took so far
because the basis we chose was
to use the position variable
as the magnetic flux.
And therefore, in identifying
the canonically conjugate
variable to that,
we didn't need to worry
about changing anything about the cosine.
That's a maybe more subtle detail
for the physicists in the audience.
If we expand the Hamiltonian
in the quantum world
into a linear piece
and a nonlinear piece
and drop terms that are
higher than fourth order,
we can, again, effectively approximate
the potential energy landscape here
using the quadratic part
which is this purple curve,
represented by the Q
squared and phi squared
and then we'll take kind of
the next order approximation
between the purple and the gray.
If you remember, from
the harmonic oscillator,
we can exploit the fact
that the beginning of
part of this Hamiltonian,
or the start,
is just a harmonic oscillator,
something very near and
dear and familiar, right?
We can exploit it's solutions
that we found in the first
part of this lecture,
as an efficient basis
in which to describe the nonlinear piece
formed by the second potential
energy function here,
which we'll call H nonlinear.
And if we use the simple
harmonic oscillator solutions,
you'll recall that several times,
we wrote down the position operator, phi,
is the quantum zero-point
fluctuations of the flux,
times A plus A dagger.
Now, we can plug that into this equation
and define a reduced variable,
which I'm running out of space here,
but let's write it as lowercase phi zpf,
which is simply capital phi ZPF,
stripped of its capital status
by the reduced magnetic flux quantum.
And for definition,
I'll use this asymmetric equal
sign symbol with two dots
and an equal sign.
You'll notice that also,
in moving from the box line
to the next line,
I made the substitution of
replacing the linear part
of the harmonic oscillator,
in terms of the annulation
and creation operators,
A dagger and A and
replacing the frequency.
So, everything we saw
about the harmonic oscillator so far,
is simply a plug and play
into the slightly more
non-harmonic equation
and harmonic equation.
