- [Zlatko] Where are we now?
The reason that I'm
spending some time here
giving you this very physical
picture of a physical device
is because I want to link
what usually is very abstract
of Hilbert spaces.
And bosonic and fermionic creation
and annihilation operators to
a very physical, real thing
which is what I work
with in the lab every day
and what you can now work with, of course,
through the cloud.
And the physics of it will
allow you to do some things
that otherwise you might not think about
if you are too
only focused on the world of
operators as we'll get to.
And so it's this very
interesting, simple structure
of two metal pads and an inductor here
that can be described by
several manifestations
of electricity, magnetic flux,
charge, voltage and current.
And by several, I really only mean four
and two more in number
such as the inductance
and capacitance.
That will be the basis of our
qubits, readout resonators,
amplifiers and really everything else.
So really want to explain
them in great detail.
The way we symbolize an inductor
is with one of these squiggly lines here.
So you'll see a lot of
pictures of inductors.
One key concept about quantum is energy.
Energy is quantized.
So what is the energy here,
at least at the classical level
of this electromagnetic circuit?
Well we said that basically
consists of inductive
and capacitive parts, right?
Things that are associated
with the magnetic fields here
and things associated with
the electric fields here
and to the charges.
It really follows from F equals ma
and the way that charge
and force in electric field
are related, that the rate
of change of energy is
of course the power.
And the power is the voltage
times current, right.
And this is now the energy that
is stored in the components
such as the capacitor or the inductor.
It could of course be the energy
delivered by the component
but we'll take the convention
that it's the energy stored
in the component or
delivered to the component.
And likewise, this is the power,
the instantaneous power
flowing into the component.
So that's power going in.
As earlier, of course, if you
integrate this equation here,
you will get that the energy is equal
to the time and grow
from the reference state
to time t of the power of d tau.
Technically there should be of course,
an energy at time zero here
or at time minus infinity, excuse me.
But we'll assume that that's zero.
I'll leave it as an exercise to you
because it's pretty easy
to change the variables
of integration here from
the integral of v of t,
i of t, dt.
from minus infinity to time t.
And these should be primes, excuse me.
Using the fact that say
phi dot is equal to d.
And this is don't have the
eraser here in PowerPoint
but phi dot the rate of
change of the magnetic flux
is equal to the voltage.
To calculate that the
energy and the capacitor is
one half c phi dot squared
or basically it's one half
CV squared, the voltage.
And the energy in the inductor
is phi squared over two L.
Or is the magnetic flux squared
divided by twice the inductance.
So the larger the inductance
for a given energy.
Well, we'll get to that.
The important point here is that
you can actually very equally
represent all of the energies,
not just in the magnetic
flux but also in charge.
You can swap the basis there's
almost a dual basis here.
But I won't really talk
about charge very much.
I'm going to almost exclusively focus on
describing everything in
terms of magnetic flux.
And that's motivated by the fact that
the nonlinear inductor
that we're really going
to be interested in
is the Josephson tunnel junction.
And it is described by magnetic flux.
It's not very convenient to talk about it
in terms of charge.
Now, let me summarize before
moving on what we just learned
because I've just covered a
lot of fundamental ground.
Things that are simple.
You may have seen a few times before
but at the same time,
they're really the four
basic fundamental
manifestations of electricity
that are universal.
They hold for all elements whether linear,
nonlinear, hysteretic
or anything else, right.
The electric charge
is related always
at a particular point or branch,
whether it's plus or
minus right is related
to the current by the rate of change here.
And equally the current
is related to the charge
by this integral.
Similarly in a very dual manner,
the magnetic flux which maybe we can
symbolize with these
magnetic flux arrows here
going through a loop of current,
is linked to the voltage
by the rate of change
or the voltage is linked to the flux
through the time integral.
And this I mostly give you for reference.
We also learned that
we can link all four
variables and form a loop.
This will be the loop of oscillations
by introducing elements such
as the inductor over here
or the capacitor over here,
which can link across the two branches.
So the capacitive element
that links charge to voltage
and the inductive element
links current to magnetic flux.
In other words if I run a
current through the inductor
it creates a magnetic flux.
Now it's a little bit more subtle
because in a Josephson tunnel junction,
you don't actually create magnetic flux.
The flux here describes kinetic energy
which is associated with the electrons
that has to do with superconductivity.
But to the earlier question
that someone asked,
we cannot ignore that level of detail
and it really at this level
doesn't change anything.
And I'll emphasize again
that all of these variables
are real numbers between
minus infinity plus infinity.
And I'll assume that both
capacitors and inductors
are only strictly positive.
And this slide gives us
really the foundation
of both quantum and
classical electrical theory.
Because it describes
how all of the variables
that describe the inductor
and capacitor here
such as the magnetic flux
that might be associated
with this inductor.
Or maybe the charge that
we might think of on
this inductor or the magnetic
flux of this capacitor
or the charge on this capacitor
are related to each other
as well as to the voltages
and currents and so forth.
The one piece that we are missing,
the link that's missing currently
is how do different elements
relate to each other?
And of course is you have
I'm sure seen before,
this is given by Kirchhoff's network laws
which are effectively a reduced version
of Maxwell's equations in
the conservation of current.
And since charge can't
be created or destroyed,
it means that a charge that is flowing
say from the inductor
to the capacitor
has to be accounted for correctly.
The charge leaving the
capacitors should change
the charge associated with the capacitor.
Let's call it Q sub C.
That Q sub C should decrease.
And now when the charge
has transferred over
to the inductor, the charge
associated with that inductor
call it QL should increase.
And I'm running out of space here
so I apologize for the little text.
And so Kirchhoff's current law tells us
that the sum of the currents into a node
such as this node n1 depicted
up here has to be zero.
Remember that q dot is
nothing but the current.
There we go.
The plus-minus sign is determined
by the relative orientation
of these little arrows.
We don't need to concern
ourselves with too much detail
on that level of the signs
because I can give you
an example here of...
Let's look at node one in the
circuit which is this node.
There is a current going into
this node due to the capacitor
and a current going in
due to the inductor.
Well, their sum has to be zero.
In other words, the current
flowing from the capacitor
has to equal the current
going into the inductor.
The plus sign here has to do
with the reference direction
since we've decided to
denote current going
in the same direction.
So the capacitive current is pointing up
and you see that when it
goes into the inductor
it has to flip sign.
And so that's why there's
this plus sign here.
Now, the second law that
comes from Maxwell's equations
from Faraday's law of
induction specifically
is Kirchhoff's voltage law.
And it just tells us that
the sum of the voltages
across a loop
or along a loop.
Remember that this is nothing
but just the voltage at
branch B has to be equal zero.
And again, there's an orientation
which is indicated by the plus-minus sign.
And so if I look at loop one
and I trace
how the green arrow here
aligns with the brown arrow.
I can write down that for loop one,
I have a plus sign for
the capacitive flux.
And now because the brown
arrow is anti-aligned
with the purple arrow,
I have a negative sign
for the flux of the inductor.
And this is kind of
the formal construction
which you can exploit to
generalize to arbitrary circuits.
But it says an incredibly
simple thing at this stage
which is that the voltage
across the capacitor is equal
to the voltage across the inductor, right.
Moving this phi over
to the right hand side.
Very surprising.(laughs)
Of course this is expected
from electrostatics
because the voltage is independent
of the path taken from
one point to the other.
Now we can use that
relationship, we can use the fact
that the capacitive flux is
equal to the magnetic flux.
And we can use the
constitute of relationship
that if you remember Q is equal to CV.
In other words
that Q dot at the capacitor
which is the current at the capacitor
is equal to C phi
double dot at the capacitor.
But we also know that the
magnetic flux of the capacitor
is just equal to the magnetic
flux of the inductor.
And by the way, since they're equal
we might as well drop the indices
and just call them magnetic flux phi.
And so this is the term that leads
here if upon substitution
of the charge equation
on the left.
And on the right hand side,
we can again use that the fact that
we know that the magnetic
flux is linked to current
of an inductor by the constitute
of relationship phi is LI.
And we also know that I is equal to
the rate of change of the charge
across the inductor.
And so using the second relationship
we can then obtain the condition here.
Alright.
And now what you might
already begin to recognize
is a very simple equation
of simple harmonic motion.
Here we come to our first oscillator.
Because we see that a change in
the second time derivative
of phi is related
to the position of phi.
So what does that look like?
Okay, here's the same equation
again bringing it back.
We can rewrite this in the suggestive
usual simple harmonic
oscillator motion way.
Where we have something
that looks like F equals ma
essentially with the harmonic
frequency omega naught
which is equal to one over the square root
of the inductance times the capacitance.
That will give us the resonance frequency
since this flux phi
can now oscillate
at something like phi
naught e to the negative
say I Omega t.
To make an analogy
between the electromagnetic circuits
which tend to be a bit more abstract
and the very familiar
harmonic oscillators.
We can look at a spring
which is suspended from...
At a mass suspended through spring
from a ceiling.
The equilibrium position of the spring
will denote as phi zero or x zero.
X for position or flux
for phi in our case.
Phi t are the deviations from equilibrium
of that magnetic flux or
of course for a spring
it's just the deviations of position.
And the velocity is the rate
of change of magnetic flux
or usually we'll also write
that as v of t
for a mechanical oscillator.
Okay.
So to illustrate what
the equation on the left
here is doing for the oscillation
of the electromagnetic flux.
We really have just the
very standard familiar
mass spring system that
oscillates back and forth.
Except this time what's oscillating,
isn't the position of a mechanical spring.
It's the amount of
magnetic flux
rather than the amount of deviation
of the spring and the mass.
And I think Brian, maybe I should pause
if there are any questions.
- [Brian] Sure thing I'm
gonna pull up just a few here.
Just for everyone to know,
I did go through and kinda
combed through the questions.
If you were asking a
question about something
outside of today's lecture,
I did either mark those as
irrelevant and then deleted them.
Sorry, not to say they're irrelevant.
They are not irrelevant
but just mark them as not right for today.
So we wanna ask questions
that are good for today.
And with that in mind, this question says,
"Am I understanding
correctly that Q sorry,
CQED allows the measurement of a qubit
without destroying the superposition?"
- [Zlatko] Yeah, thank you.
As we'll see
it depends very much on the type
of measurement you do.
If you do a measurement of energy,
the measurement will place you.
Well, if it's also a
standard projective type
of von Neumann measurement,
which is exactly the kind of measurement
that we actually do in practice.
The effect of the measurement
due to the necessary
and unavoidable measurement back action,
is to disturb the state of the system
so as to randomly land if
you start in a super position
into either the ground
state or the excited state,
meaning the zero logical
or the one logical.
So if you started in a super
position of zero plus one,
a projective measurement
of the Z quadrature,
which is basically the energy of sigma Z,
will project you into
either the ground state
or the excited state for zero or the one.
So it will destroy in that
sense the superposition.
And of course if you
start in zero plus one
and you measure in an basis
of an operator which measures
are you in zero plus one or
are you in zero minus one,
then that will not
destroy the superposition.
So it depends on the measurement
but typically the measurement
that we do does destroy it.
- [Brian] Okay.
And then one last quick question.
Some people were asking
for just a quick reminder
about the dot notation that
you're using in your equations.
- [Zlatko] Yes, thank you.
This is a physics shorthand
for time derivative.
And you notice that everywhere
here I've omitted to write
so you can think of phi is
really equal to phi of t.
This is my shorthand for writing these.
I don't know why it's a little pixelated
on my screen here when I write.
So I apologize for that.
And of course phi dot in this notation
is defined to just be the time derivative
or the rate of change in time
of the variable that's underneath it.
Right.
And if you have two dots
then that will indicate
that you have
there you go, two dots.
That will indicate that you have
basically done this operation twice.
And so you have a second
time derivative like this.
Okay.
Thank you.
Okay, so then this takes us
into a bit of a dangerous bend.
You see I've indicated
here that we're going to
temporarily assume some
knowledge of classical mechanics
just on our way to a Hamiltonian,
a quantum Hamiltonian.
But I'll basically describe
all the necessary steps
and keep it at a kind of
mixed level of idea and rigor.
We saw that from very
simple considerations
of Kirchoff's law We can
derive an equation of motion
that relates how the
magnetic flux oscillates.
You notice that we completely
eliminated the charge
and we only operate with one variable phi.
So we can think of this variable phi
as the position of the spring, right.
And then phi dot as the velocity.
Now this starts to sound a
lot like mechanics in a way.
And we can use the regular
tools of classical mechanics
to describe this type of dynamical system.
At the level of equations I
could just swap all the symbols
from phis to Xs for position.
Now, the standard description
begins with the so-called Lagrangian.
The Lagrangian is just the difference
between the kinetic and
the potential energy.
All right so the Lagrangian
which is a function of a phi and phi dot
or basically the position
and the velocity,
is the capacitive energy
which is a function of just velocity.
That's what it means to be kinetic.
And the energy function
of the potential energy,
which is just a function of the position.
That's what it means
to be potential energy.
Alright.
If I substitute in what
capacitive energy is
in terms of magnetic flux.
And what inductive energy
is in terms of magnetic flux
which we derived I think two slides ago.
I have one half C voltage squared
minus phi squared over two L.
And this is a very, very
standard common equation
you'll see.
Now, the next step in going
from a circuit to a Hamiltonian
is to identify the so-called momentum
in the language of mechanics
or in the language of
circuits, the charge.
The charge here will
play the role of momentum
because I've chosen the
magnetic flux as my position.
In the language of Lagrangian mechanics,
this is called a canonically
conjugate variable.
But I can obtain the charge Q
by taking a partial
derivative on the Lagrangian
with respect to phi dot.
And if I do that you notice that
d phi squared,
d phi dot is equal to zero
because we treat position and velocity
as two independent variables.
So that's zero.
And then the partial
derivative with respect
to the position here squared, or sorry,
the velocity squared
with respect to d phi dot
that's just equal to the usual
two phi dot.
It's just like doing dx squared dx
oops is equal to two x.
And of course, all this says by the way,
from the Lagrangian point of view,
you notice this says Q equals CV, right.
Or charge is equal to
capacitance times voltage,
nothing new.
We've already seen this except this case
there's a little slight of hand in that
who's charge and who's voltage?
Well, we've already
built in the constraints
about relating the fact that
the flux of the inductor
is equal to the flux of the capacitor
and how their charges are linked.
And so we've eliminated actually
all of those other variables.
Now this equation looks just
like classical mechanics.
So we have momentum,
we have position
and we can write the F
equals ma (laughs)of circuits
or also called the
Euler-Lagrange equations.
The sort of mathematically
rigorous way to do that
is to write down this expression.
We have a full time derivative
on the rate of change
of the Lagrangian with
respect to two phi dot
or the voltage is equal to dL dphi.
If you plug in these numbers,
basically just use again the
expressions I wrote up here.
And just substitute
those into this equation,
you will find again
that you have recovered
the equation of motion.
And the way we actually
constructed the Lagrangian
officially is by starting
the equation of motion
and making sure the
Euler-Lagrange equation
express exactly this condition.
Now, if you remember how
we defined the current
across the capacitor and the
current across the inductor,
you'll see that again
this equation is nothing
but Kirchhoff's current law.
So Kirchhoff's current
law here is F equals ma
or is the Euler-Lagrange equation, right.
And then the two currents of the charge
and the current of the inductor
which are up up here serve to describe
the actual evolution of the system.
Now, having done all of this
nice mathematical trickery
we can use
the momentum that we've
defined which is charge.
And the position which is phi
and the two energy functionals
to define the Hamiltonian.
And the Hamiltonian now is
a functional of the position
which is the magnetic flux phi
and the momentum which is the charge Q.
And we arrive at the
all important equation
that the Hamiltonian of an
electromagnetic LC circuit
is phi squared over two L
plus Q squared over two C.
Where we have simply rewritten
the energies in terms
of the momentum and the position
or formerly done a Legendre transform.
Let me maybe pause Brian
since this is a slightly
more advanced slide.
Are there any questions on this?
- [Brian] I will ask actually in the chat
if there are any questions
on this directly.
Normally we see there's
about a 10 to 15 second delay
between me asking this out loud
and people actually
acknowledging it in the chat.
So I'm just gonna stall
for about 15 seconds here.
But yeah, we definitely
wanna see if there's any
very topical questions about this.
So I will pause for a
second and watch the chat.
And maybe after someone
responds I'll let you know.
- [Zlatko] Okay, good.
- [Brian] So start with I see here
based on a calibrated
pulse since the qubit,
you determine what the
absorb frequency is.
And that will determine the
energy level that's changed?
- [Zlatko] Oh yeah.
I think the question is
maybe about the calibration
in the frequency.
And how it we know it.
Well, you see that the
frequency of the qubit
will essentially come from this equation.
There'll be a non-linearity we add
which will slightly renormalize it
but the frequency of the
qubit will be set by exactly
these two parameters L and C.
And I think on the previous slide here
I had this equation where we defined oops
the frequency omega
naught of the resonance.
Now, once we have the non-linearity,
the frequency of the qubit
will be slightly renormalized
due to the non-linearity and
will change as a fact of that.
But will be still very close
to the linear frequency here
of the resonance.
And so the microwave pulses
we shine will be at this
dressed renormalized
frequency of oscillation.
- [Brian] Okay.
And we had a few people
asking if you could please
just go over the Hamiltonian one more time
and just explain in detail.
A few people are confused
about how that connects
with the Lagrangian.
- [Zlatko] Yes.
And I'll give a nice physical
picture in the next few slides
to explain it in more detail.
Well you notice that the two
are very interestingly related.
The Lagrangian is a particular
type of useful oops function,
which Lagrange worked on.
which defines the kinetic
minus the potential energy.
Now that's not all, it's not
just kinetic minus potential.
It's kinetic minus potential
in terms of the position
which is phi.
And the velocity which is phi dot.
The Hamiltonian on the other hand
is basically the kinetic
plus the potential
so there's a sign difference.
But that's not really the
most essential difference.
Even more important
difference between the two
is that we have swapped out the second
part of the function here.
It's no longer in terms of velocity
it's in terms of momentum, right.
So when you do mechanics you see that
there's a number of advantages
in reasons that why you
should use momentum as
opposed to velocity.
And in this case, we exactly
have the same picture
that we need to work in
terms of flux in charge.
We can't use current and voltage.
Basically current and voltage don't serve
the purpose in these equations.
Motion is the generalized
position in momentum.
We have to move one level up
to flux in charge.
Good.
- [Brian] Good.
Yeah, so for now let's jump
back into your lecture.
Thank you very much for answering those.
- [Zlatko] Right.
And again I'll stress the analogy
that position that is summarized here
between mass and spring
constant and so forth.
So let's explore what
the Hamiltonian means
and what it does.
And two key parameters
we saw are the product
of the capacitance and inductance
which give the resonance
frequency, omega naught.
And there's a little typo here.
You can also take the other product
which is the ratio of the
inductance and capacitance.
And I'll call that the
impedance for reasons
that will become apparent later.
But it's essentially a way to transform
the variables mathematically
but physically
their product
or inverse gives you
the resonance frequency.
And then the impedance will have to do
with the susceptibility or how easy it is
maybe to couple to this element.
So let's look at what
the Hamiltonian says.
I think this will answer the question.
Well, the Hamiltonian
consists of two parts.
It has the potential energy function
which is phi squared over two L.
Which is a quadratic in energy.
So here I have the magnetic
flux which serves this position
on the bottom on the horizontal
and on the vertical y-axis we have energy.
And a classical particle of course,
you can think of as a little
red dot that is trapped
on this parabola.
And can move up and down
this parabola, right?
And it's sloshes its energy between,
for a given energy let's say E
it sloshes its energy
between kinetic and potential
or inducted and capacitive in this case.
And the velocity is given
by the capacitive energy,
one half C phi dot squared.
And I love this quote by Henri Poincare.
"It is by logic that we prove,
but by intuition that we discover."
And so let's build a little
bit further intuition
on the Hamiltonian.
Let's suppose just for simplicity,
we set the inductance and
capacitance to equal one for now.
Then I can say, "Well,
let me pick an energy."
Suppose I want to put an amount of
this much in the resonator
which we'll call h bar omega
naught n plus one half.
As very suggestive form you've noticed
I've introduced a quantum action,
action of quantum
and quantum of action, excuse
me, in quantum physics,
Planck's constant, h bar.
Omega naught is the resonance
frequency of the oscillator.
n is just some number which I can choose.
And one half.
This is of course, the energy of
a quantum harmonic oscillator,
even though we haven't
quantized yet this oscillator.
But I can choose to set the energy
of the classical oscillator
equal to that of a quantum one.
Why not?
That's totally allowed.
And that equals one half
Q squared plus phi squared
in terms of the Hamiltonian, right.
The Hamiltonian is supposed
to describe the total energy
of the circuit.
Now what does that remind you of?
Well if you fix the energy
and you say that you have
X squared plus P squared
or Q squared plus phi
squared equal to fix energy.
In phase space, in the space
of position and momentum
or phi and Q.
The equal potentials of energy
give circles for the trajectory.
So the very first circle here
corresponds to n equals zero
or as we'll see in quantum
that will be the ground state
n equals one, two and so forth.
In other words, each of
these rings corresponds
to one more excitation of energy
in the circuit.
Now there's no reason
that in classical physics
I have to constrain myself
to be at one of these rings.
I could be anywhere in between
but in quantum we'll see
effect of the (indistinct).
You can't do that.
Now, suppose that we want to describe
how the particle behaves, the
classical oscillator behaves.
The oscillator I can represent
its state by this red dot.
The red dot indicates
the position and momentum
of the particle.
From the Hamiltonian equations of motion
which are just another way of
rewriting Kirchhoff's laws.
We can figure out that the rate of change
of the magnetic flux,
if I'm at a point here will
depend on my position Q
and vice versa.
Q will depend on phi.
And so if I start here I can
denote with a little vector
which way I will go if I'm
the particle oscillating.
And so you can do this for a
number of points on the ring.
And you see that the trajectory
that the particle will follow
is in this case circular.
And it will oscillate
between position and momentum
or between flux and charge, right.
This is just another way to represent
that the electric charges on the capacitor
are flowing through the inductor
to the other plate and back and forth.
And that's what we see
that charge goes positive
then it turns into flux which corresponds
to current flowing inductor
then goes to negative charge.
So we flip the polarity of the
capacitor and back and forth.
Now we can introduce
a very helpful variable
by saying that we can
recognize that this particle
is just moving in a ring, in a circle.
Of course that should remind
you of complex numbers.
And if you have a complex
number and you exponentiated
each of the I such as this
e to the minus I omega naught t.
That's a compact way to simply represent
a point in a 2D plane.
In other words, I can
succinctly write down
where the particle is
in this plane by saying,
"Well, if the particle is over here,
I can write that as a vector of phi and Q
or it may be notation advantageous
to use a complex number."
And I've picked here a very specific
scaling factor in
representation of how to define
this classical complex variable alpha,
which will exactly describe
the point in phase space
of the oscillator.
In such a way that it's
the most analogous variable
to the quantum operator,
the latter operators,
the creation and annihilation
operators that come up.
But it essentially acts
very much like them.
And so if we rewrite the
energy of the Hamiltonian
using this new variable
which mostly at this point
is just the mathematical trick.
We can see that the Hamiltonian
now becomes a function
of alpha and it's complex
conjugate one half h bar omega.
And I've just symmetrized
here the product of these two
complex numbers.
And the reason I have
symmetrized is because
when we do quantum quantization on this,
we have to worry about
the ordering of products.
Now at the level of
algebra all I've done is a
mathematical transformation.
At the level of pictures
we have done a very nice,
let's see if we can play
this animation again here.
We've done a simple transformation
where we've expressed circular
motion using complex numbers.
But this alpha will be quite
key in the quantum case
will be the basis of the description
of most of the transmon qubit.
And it's very important
to know that its motion
really proceeds in this
simple harmonic motion
as e to the I omega naught t.
And that will lead us into things
like the rotating wave approximation
and simplifying the dynamics quite a bit.
Okay.
And that brings us into
unveiling the quantum world.
And maybe Brian I should pause
here and take any questions.
- [Brian] Yep.
I think answering one question here
before you dive into that.
And then after you dive
in we'll have a lot ready
after the next break so
we can plan for that.
But right here, a question we have
now that we're pretty
deep into the lecture.
You mentioned physicists
can all view physics
as all spins or all oscillators.
And I know that's kind of a joke
and kind of not a joke.
But would you be able to
arrive to the same results
of today's lecture by following
the spin view of physics
instead of the oscillator one?
- [Zlatko] Yeah, that's a good question.
You can in the case of treating the qubit
directly from spins if you
start at the level of the atom.
You cannot use really spins
to describe the electromagnetic circuits
because they are really oscillators.
You can mathematically of
course, at a very formal level,
always map a spin to an oscillator,
an oscillator to a spin.
This is related to the so-called
Holstein-Primakoff transformation.
We don't need to get into that.
But there is a mapping essentially
which allows you to say,
"Well, if I have a spin
with n going to infinity
number of levels I can make
it look like an oscillator."
Obviously we've seen how
to take an oscillator
and make an n harmonic.
And make it look like a spin,
which was the first part
of the lecture.
So at a very formal
level you could do that
but I think in practice
especially for the language
of circuit quantum electrodynamics,
we can really think of
everything as oscillators,
nonlinear oscillators
and modified oscillators.
And that allows a lot of simplicity
because oscillators are simple.
And they form a simple basis.
But as soon as you add a
few extra ingredients on top
you can create very
complicated and difficult
and complex and rich dynamics in behavior,
which makes fun.(both laugh)
- [Brian] And then one more
thing just really quickly
people are asking about
what we just had on screen
in the last slide, the
equation that we had there.
Does alpha describe the path?
- [Zlatko] Oh, wonderful.
Yes, thank you.
That's a great way to phrase it.
Alpha describes the path or the trajectory
of the motion in phase space.
Exactly.
- [Brian] Great.
Thank you for that
- [Zlatko] Yeah.
So alpha itself describes
the position or the state.
It summarizes both the position phi.
It summarizes both the position phi
and the momentum Q or the flux in charge.
It fully specifies the
state of the system.
Basically anything and
everything you would want to know
about the system can be
calculated if you know phi and Q
or equivalent if you just know alpha.
Alpha is just the compact
way of writing the two.
But it's time evolution is
also really nice and simple.
I can write that alpha of
t is equal to alpha zero.
Alpha zero is just, well,
maybe if I started my system
not at phi equals, you know,
if I started my system over
here, then alpha of zero
would be the point over here.
And an alpha of t describes
the path or the trajectory
that the particle takes along this ring.
