- [Instructor] Thank you.
Okay, so this is an important
slide because it shows us
the connection between the
oscillator and the spin,
and how we realize the
spin by limiting ourselves
to a two qubit subspace
because of the infermanisity
contained in this term alpha,
which is set by the scale
of the zero point quantum fluctuations,
which I think you've seen me
talk about a million times now
and I emphasize highly fundamental.
Great.
I think in the interest of time,
I will only gloss over
the following section,
which helps bridge the
world of the classical
to the quantum.
I noticed that there were
many questions about,
well, how do you really go
from the classical world,
let's say, if I'm standing
here and hop over the barrier
into the world of fuzziness,
where things are a bit uncertain
and we have shown here cat states
and things aren't exactly in
one position at all times?
And we've done it a bit
formally with Lagrangians
and Hamiltonians and directs
quantization and so forth.
But you can actually in practice,
skip all of that and really
leapfrog if you want,
from the classical world
over into the quantum world
very quickly using the idea of the energy,
so-called energy participation ratio.
In other words, asking
the very simple question,
where is the energy
distributed in my circuit?
And if you ask that question repeatedly,
you can do the quantization very directly,
and you can forget about
all the canonical steps
of Hamiltonians and Lagrangians
and transformations.
And you can even apply
this to distributed circus
to circuits that are extended, right?
Not just to lumped elements,
circus that are ELLs and Cs,
but to circuits that extend over space,
to circuits that have many junctions,
to circuits that have
many modes and so forth.
And if you're interested
in the details of that,
and you're a practitioner in the field,
I'll refer you to the papers down here
and just highlight the
idea that a crucial link
that allows you to bridge
the world of quantum,
where you talk about Planck's constant
and the zero point quantum fluctuations
of say a Josephson
junction, a J in a mode M.
Right now we've only talked
about one oscillator mode
and one Joseph's injunction,
but when you make more
complicated circuits,
you can have M modes and J junctions.
They're linked to the
classical frequencies
in Joseph junction
energies, which are known
by the question, what
fraction of the energy
of a mode is stored in the junction?
And this is the question that
we can answer very easily,
but it serves as a stepping
stone, as a bridge to leapfrog
from the classical to quantum.
And again, if you're
interested in the details,
those are kind of detailed
in these two papers,
especially in the upcoming one here,
which I'll just flash for
the more interested readers.
And so that leads me into
the transplant qubit,
how do we control it?
So I'll briefly review
some of what you did
in the lab yesterday, and
maybe try to give more of
an intuition and a
picture rather than going
into too much math for
the rest of this talk.
This is what the physical
transmission picture looks like.
Again, all of this occurs
in real physical devices,
but at the same time, it
allows us to implement things
in the abstract, right?
To say, well, here, as you
saw in the first few lectures,
I have some qubits and I
can apply an X poly-gate,
a white poly-gate, a Z poly-gate.
The kinds of gates you read
about and Nielsen and Schwann.
A Hadamard, an S gate, a T
gate, or really more generally
any SU2 unitary rotation on a single qubit
or in many qubits in parallel.
So how exactly has
everything we said so far
linked between the gates in the abstract
and the real physical world over here.
And that's through qubit control.
And missing piece in terms of
what we haven't quite yet said
is that we've analyzed
everything about the qubit,
but there was a question,
how do you measure it?
How do you control it?
Well, here is a control line.
This is a transmission line,
just like the coaxial cable at the back of
your old school TV, which can
support electromagnetic waves.
Now, if I send an electromagnetic wave.
which can run from the
right over to the left,
once it impinges upon the oscillator here
through this weak coupling capacitor,
it can apply a force that steers
the electromagnetic fields
in such a way that tries to push the atom
from maybe the ground
state to the excited state
or vice versa.
And so when we draw these
arrows at certain frequencies,
we're really talking
about control clauses,
which have a carrier
frequency or a frequency here
that's related to the
wavelength, and is set typically
by the oscillation
frequency of this qubit,
which we call Omega Q, which
is the energy splitting
between these two.
Right.
And then the amplitude of this pulse
is given by the capital Omega.
And it can be shaped as a function of time
for various reasons.
The Hamiltonian from the
classical point of view,
corresponds to a coupling to charge
because we have a capacitor.
So a capacitor will couple
charge to charge here
between the transmission
line and the transmon qubit.
And if you remember the charge operator Q
was minus I, Q, ZPF
times a dagger minus A.
And so that's exactly what we
see here in the Hamiltonian
of the drive, is that we have
the amplitude of the drive
multiplying essentially the Q quadrature.
Within the qubit space manifold,
that reduces to say a rotation
Y like this potentially.
Now, let's look a little bit more quickly
at what happens when we
reduce this intra Hamiltonian
from the point of view of a spin.
So the harmonic oscillator
creation in animation operator
is reduced to spin operators,
Sigma and Sigma dagger.
And I see that I've written
way too many daggers
in this picture.
So that's a small typo, apologies,
where the lowering operator Sigma
will take the state one
into the state zero.
And so it looks like this.
Now, typically this
frequency here Omega will,
or this amplitude here of the
drive will have an amplitude,
Omega naught at a carrier frequency Omega,
and might have some phase five.
Now this phase is actually very
crucial because it allows us
to tune the access around
which the rotation will happen.
Just one minute.
There we go.
Hopefully you can see me
or hear me, thank you.
We can expand this sign, of course,
in terms of a complex part,
which has two exponentials,
one that rotates clockwise, one
to rotates counterclockwise,
and we can again apply the
ideas of the rotating wave
approximation by treating the
rotation of these A operators.
Remember it was A, E to the
minus, I Omega naught T.
So you're starting to
see the same ingredients
that we labored for the
first four or five hours
of this lectures come over and over again,
and just get applied
in new and unique ways.
And now if you multiply
out this Omega from here
into that product of
also oscillating terms
and apply the rotating wave approximation,
which I think was done in the
lab in one way, in a version,
you can reduce the
Hamiltonian into two pieces.
In this interaction picture
a piece that depends
on the tuning and sets the
amplitude of your rotation
of the Hamiltonian and the axis
of the energy or the Z axis,
which is the drive frequency
minus the natural frequency
of resonance.
Plus a term that goes
like Sigma, Sigma dagger,
and depends on the phase
of the drive applied.
Now, depending on the phase
that you choose to set
in your drive, you can
choose to set the effect
of Hamiltonian to have either a rotation
around the X quadrature
of the block sphere,
which is depicted here in the figure.
Or you can choose to have a
rotation around the Y quadrature
of the bluff vector.
And so, for instance, if the
spin of the qubits starts
in a state where the block
vector points down like this,
then using this type of
Hamiltonian where say Delta is zero
and Omega is equal to sum
number that will give us
the rate of rotation, it's the Rabi rate.
Then you'll see that the state will rotate
in this plane around the Y axis here.
And I think this was
to some extent covered
in your lab yesterday.
And very neatly, you can
use the right hand rule
to understand how a
state such as row here,
or this is the state side if you prefer,
will rotate around the axis
defined by this Hamiltonian.
And so by choosing the
phase of the drive theta,
and by choosing for how
long this pulse is applied
at a given rate, we can
implement the X poly,
the white poly, the Z poly, the H, the T,
the S gate is done by typically
a change in the phase five.
It's in practice, a software rotation,
or it's a delay in the
phases of future pulses.
And that allows you to
then create in general,
any unitary SU2 operation,
meaning the general unitary
rotation on the qubit state.
And so I think in the lab yesterday,
you have applied exactly
this kind of rotation
in a real experiment on the
qubits, or you will soon,
which allows you to
calibrate the qubit pulses
in the cubic gates for single qubits.
Thus by steering the atom
with this drive Omega T,
you can very time or the
amplitude in this case
of the drive pulse Omega.
And as a function of that,
you'll see that you're going
to rotate for a fixed time
the qubit starting typically
in the ground state
and at a given rotation angle,
you can go up to the excited state,
and that allows you to
calibrate a pipe boss,
which is say an X gate
or Y gate and so forth.
And it's very important to also realize
that this control post that we apply
can be supplied either by us,
but it can also be supplied
by almost anyone else
that's coupled to that transmission lines,
such as an evil environment.
And that can create as
well as wanted rotation,
unwanted rotation.
Now imagine that instead of
us sending a nice coherent
and beautiful pulse, we
rather have a noisy pulse
that's controlled by
say, maybe a thermal bath
or noisy electronics, or
some other incoherent source.
This kind of noise, which
by the way, could also be,
again, quantum zero point fluctuations.
This kind of noise can then
drive the qubit stochastically,
and that can lead to uncontrolled
random bit and face flips.
And this is a fundamental manifestation
of a major topic in
condensed matter physics,
which we'll only touch on here called
the fluctuation-dissipation theorem,
which says that anytime you
have control of a system,
you're also equally opening
it up to a proportional amount
of dissipation and noise
because noise can come in
from here and drive the system.
And likewise, energy from
the system can leak out
into the environment.
But I only want to expose
you to the ideas here.
And these are a little
bit maybe more advanced
as indicated by the dangerous bend symbol,
but can nonetheless
provide a nice intuition
for a lot of what you
see in the experiments.
And these in effect can
be one way to understand
the energy relaxation time of the qubits,
why qubits aren't infinitely long lived.
So they have a relaxation
and energy of T one,
as well as why they have a coherence,
defacing time, potentially T two,
which is the rate of loss of the coherence
between the ground and excited state.
There was a question earlier
in the discussion about,
how have these coherence times
been effected by developments
in materials, by developments
in the community?
And especially since the
early days of finding out
that the degree of phase difference
between two superconducting
islands connect coherently,
even though it's a very
macroscopic phenomenon.
And that looks like this.
Here is a, somewhat
famous plot at this time,
where it starts all the way back
with the first superconducting
qubits in the late nineties,
which had lifetimes T1s of about
a nanosecond or even below.
That is, once you put
energy in the excited state
of the qubit, it would relax
in about one nanosecond.
A very short amount of time
down to the ground state.
And through a series of really
incredible developments,
we have seen an almost exponential growth,
both in the coherence and
the energy relaxation time,
the T one and T two, of both
qubits end cavities over
almost eight orders of magnitude
in the span of 20 years.
And more recently, we have seen
a continuation of this trend
where individual qubits, again,
not qubits in large scale chips,
but individual qubits lead the way.
Again, these points represent
best reported individual
qubit results for different varieties
and exploratory devices.
Of course, multi qubit
systems are more complicated
because you have many
more devices happening.
And often when you give up
again in isolation of the qubit,
you also give up in control.
And at the end of the day,
the thing that matters
is how much control can
you exert in a given amount
of time that the qubit stays coherent
and can contain its energy.
But I personally find this
a very optimistic graph.
Now there's some advanced topics here,
which I'll only flash through and leave it
for further questions at the end of this.
And that will lead me into a
primer on quantum measurement.
And for the rest of this talk,
or really, of this lecture,
I'll focus mostly on the
ideas and try to convey to you
a few key points rather than going
into a lot of the details.
Perhaps the best way to
introduce measurements
is with a cartoon.
And I'll let you read
this cartoon very briefly.
Now, as you can observe, the
main thing here is observation.
Can I observe it?
That's the tricky question.
And measurements is really
where I think a lot of
the beauty, subtlety and really uniqueness
of quantum physics comes in.
All the weirdness that
we tend to associate
with quantum physics, and in my view,
most of it stems from
really the measurement part.
And here's a classical measurement
example to also show you
that not all the weirdness
is truly quantum.
So you have to be careful
in distinguishing.
Imagine you want to ask yourself,
how do I measure whether
this barrel has oil
or it doesn't have oil in it?
Our one way to find that out is to say,
well, if there are
fumes in the oil barrel,
then I can find that out
in a very explosive way.
And by the way, I'd like to
acknowledge Howard Wiseman
for this example.
This is a example from his great
book " Wiseman and Milburn"
on measurements.
And it will illustrate two
basic classes of measurements.
The first is destructive.
Suppose I take a match to the barrel.
If I want to know, are there
fumes in the barrel or not?
I can take that match,
drop it in the barrel.
If there are fumes in the barrel,
obviously there'll be a very
loud recognizable signal.
If there are no fumes in the barrel,
we'll also see that a very
dull situation happens.
Now, this type of measurement,
classical measurement,
right now is a demolition measurement.
It changes quite fundamentally
the state of the system.
It destroys the state of the system,
or even destroys the entire
system upon performance.
And this is an example
of photon absorption.
On the other hand,
imagine that we did something
slightly more sophisticated,
which is we took the barrel of oil fumes.
And now look that the reflection of say
a laser above the barrel.
If there are fumes in the oil barrel,
they can escape through
a hole and their escape
can then modulate, say the
amplitude or scattered light
from a laser.
So we can detect that with a camera.
And this is a very much
more, a much more rather,
undestructive or nondestructive,
non demolition type of measurement.
The measurement that measures
the state of the system
without destroying it in
the more indirect manner.
And so this is a kind of example,
a classical example of say
a dispersive measurement
of readout cavity, which
is what we'll talk about.
And it traditionally in quantum physics,
most measurements were done
using demolition measurements
of photons, photon counters.
But the measurements we do here
in the superconducting qubit
community and the measurements
you do on the quantum bits,
and the measurements we like
to do for quantum computers
are of this latter kind
of non demolition kind,
the kind that will project
the system into a state
and not destroy it.
So if we measure that the barrel has fumes
after we've done the measurement,
the barrel still has fumes.
It hasn't exploded on us.
Now this is a classical discussion,
but we can extend it to say
a few basic characteristics
of quantum measurements.
Now, quantum measurement,
unlike classical measurement
must necessarily disturb the
system through back action.
And this is represented in
the non commuting behaviors
of position and momentum,
which also As representative
leads to Heisenberg's
uncertainty relationship.
And there's a cartoon I'll invite you
to take a look at here, which summarizes
the uncertainty relationship.
Of course, that uncertainty
relationship leads to a number
of key quantum limits,
such as fundamental limits
to precision, the standard
quantum limit, et cetera.
That also leads to other problems such as,
there is no joint probability distribution
of the ability to measure
both X and P at the same time,
because measuring one
starts to disturb the other,
measuring the other disturbs the one.
So we can't really write
a joint distribution
in the same way we can classically.
However, we can still get some
intuition of what's happening
by writing these more
generalized so-called,
quasi probability, such as
the Q function or Wigner,
which I depicted here.
And again, this is a
bit of a dangerous bend
of going into slightly
more advanced ideas,
but I'd like to expose
you to the broader context
of quantum measurements here.
And measurements always increase
the entropy in the system.
That's a very fundamental
property of these measurements.
Now, that's unless the
system is in an eigen state
of the measurement and they
lead to some deeper threads,
such as contextuality.
In other words, in quantum physics,
you don't really see what you get,
you rather get what you see.
And that's a statement about the character
of the disturbance due to the measurement.
The measurement in a way
is providing the action
or the force that steers
the atom into the ground
or excited state upon the measurement.
This is a very near and dear to my heart.
I think many of you on Twitter commented
on my quantum jumps
paper, published nature,
and that's essentially what
it discusses and describes
in great detail is how it is
actually the measurement force
that comes if you want
from that transmission line
or through the cavity that
is able to evolve the atom
in a non unitary manner
and create the projection,
create the collapse of the
wave function into either say,
the ground or the excited state.
Now, having done this very
general overview of measurements,
let's specialize a bit more to circuits.
There are two options
at the very basic level.
One is that we look at a qubit
through direct monitoring,
meaning that we directly
hook up the qubit over here,
through a coupling capacitor,
up to some sort of
input-output transmission line,
some sort of cable that
can guide control signals
and waves from the experimentalists.
This is the experimentalists.
And receive data back and forth
between the measurement apparatus,
which will do the projection
and the control apparatus,
which will send the signals.
Now, this is not what we
actually do, for several reasons.
One it's a demolition measurement,
and we prefer to keep things there
even after we've measured them.
And second, it tends to,
by this fluctuation-dissipation theorem,
which I alluded to earlier,
it creates the ability for
the information and the qubit
to leak out far too easily
through this coupling capacitor
into the line and never come back.
And that leads to things
such as T one and T two.
And of course, any noise
from the system here,
a noisy signal here can just come in
and perform arbitrary Rabi
rotations in the qubit,
again, disturbing information.
And we don't like that.
We'd like to rather isolate the qubit,
by at least one degree
of freedom through say,
a cavity readout mode.
And this is the standard
circuit quantum electrodynamics
dispersive readout
measurement, which allows you
to couple the qubit to a cavity mode.
The cavity allows you to
isolate the noise coming
from the environment
and hitting the cavity
from the qubit directly.
It's only a second order effect.
And likewise, the qubit cannot
leak out information directly
into the environment, it has
to go through the cavity.
Now, the regime will operate in,
is that the cavity or
the oscillator again,
will be far the tune from the qubit.
So they want one to
exchange information easily.
And very importantly,
this type of measurement
allows you to do a dispersive,
a non demolition measurement,
just like with the oil barrel,
when you can shine a
laser over the oil barrel
to look at the reflection of the fumes.
So it inhibits the spontaneous emission,
and it allows you to perform
a nondestructive measurement
of the energy of this system.
Or essentially of what will
be the Z hat quadrature,
the qubit Z.
And how is this actually
implemented in practice?
Well, as you've already gathered,
the bread and butter are resonators.
So here's the familiar now transmon qubit
and this squiggly line over here
is a co-planner wave guide resonator,
which is essentially a
wire that has a resonance
that oscillates, and it can
be represented yet again
by another LC circuit.
And this will be the readout resonator.
The input-output ports here
can be transmission line.
If you want, can come over here and couple
to this squiggly through
some weak coupling capacitor
like this.
And this is where we can feed
in the control system signals
and also take out information
from the quantum chip,
turn it into a classical signal.
Now what's amazing is
that the oscillator here
is not just the bread and
butter of the readout,
but also in the exact same treatment,
almost the same way we've done it,
we can treat two qubits
and treat two cubit gates.
I won't have the time to
get into it very much,
but I'd only like to draw the distinction
or rather the similarity
between the analysis
in this approach.
I'll present between a
qubit coupled to a resonator
and a qubit coupled to qubit.
And so let's take back
the familiar picture
of an oscillator, which has
a capacitor and an inductor,
and a qubit, which has
a Joseph's injunction
with some Joseph energy
EGA and a capacitance C.
The picture that we can
use to represent these two
is the following that
we have a qubit coupled
through a coupling
capacitor to an oscillator.
That oscillator in turn
is coupled through another
coupling capacitor to the
input-output transmission line.
So you can see this
Matryoshka doll nested chain
of coupling.
And as we send waves,
let's try this again.
As we send a wave,
that wave will impinge
upon the oscillator.
It will reflect carrying
back some information
about the response of the oscillator.
And the essential point here will be that
because this oscillator is
coupled to a nonlinear system,
or in other words, the qubit here,
the way that the reflection
of this wave occurs
will depend on the state of the qubit.
Whether the qubit is in the
ground state or the one state
will lead to a different manner
in which the squiggle reflects.
Now very similarly between
the picture of the qubit
coupled to an oscillator, is
the picture of a qubit coupled
to yet another qubit as we'll see soon.
But first, let me just show
the Hamiltonian of the qubit,
which we saw as Q squared over to C,
minus EGA co-sign of the flux.
On the right hand side,
we have the Hamiltonian,
which goes like Q squared
over two C plus five squared
over 2L.
I will note that the
only difference between
our readout resonator on the right
and another transmon qubit is the swapping
between the inductor with
the nonlinear inductor.
And so the Hamiltonian down
here will simply look like this.
Two copies of the same thing.
So in both cases,
whether you have two
qubits or two oscillators,
you have two Hamiltonians
that look like that.
I only point out that
the charge is modified
from the uncoupled case, by
the presence of this capacitor.
So it's not as simple as just saying, oh,
this Hamiltonian can just
get added to this Hamiltonian
by some coupling.
There is a certain dressing
and renormalization that occurs
due to the loading effect
of this coupling capacitor.
Now that's a detail that we
can almost ignore completely
when we use the energy participation,
which we'll automatically
take that into account.
In both cases, we can always
apply the same recipe,
which is to say that we
linearize the system,
whether we have two
junctions or one junction.
So in the case of a readout resonator,
coupled to a qubit or in
the case of a qubit, oops,
coupled to another qubit.
And the idea as before,
is going to be that we
linearize the system,
we break up the pieces into
a linear and nonlinear piece.
Then we find the normal
modes of the system,
which have resonant frequencies.
And here, I only will give
you the big picture overview
and won't go into the details.
We expand just as before
into a linear piece
and the nonlinear piece,
at the same way that we did
for the case of the single transmon.
However, there is one major difference,
and that is that the magnetic
flux across the junction,
which is a five, one ,is no
longer a simple thing as before.
It doesn't just depend on one mode,
which was described by A and A dagger.
It depends on two modes.
Because now there are two
oscillators in the system
and they're coupled.
So the voltage or the
flux across this inductor,
which is Joseph's injunction will depend
on how much energy is in both modes.
Because both modes involved
the junction to some degree.
And in other words, we can
rewrite that the magnetic flux
across the junction, which
here I've labeled as five, one,
and I can put a hat on
it for the quantum case,
is no longer just the
zero point fluctuations
of the junction times
eight dagger per say,
the but rather has added to it,
the zero point fluctuations
of B plus B dagger.
Now what is B and B dagger?
Well, B and B dagger
represent the oscillations
of the second mode in
the hybridized limit,
and A represents the first mode.
And so following the same recipe,
but now with the extra added complication,
we can write the linear
part of the Hamiltonian,
not as one H bar Omega A dagger A,
but as two terms in H
bar Omega A dagger A,
which involve the cavity where
the readout resonator here
and the qubit.
This is the linear piece.
The nonlinear piece now is
again as before a fourth power
of A plus A dagger, but in this time
there's also a B plus B dagger,
which comes from the previous slide.
Now you might be wondering at this point,
we've done a lot of steps
and skipped a lot of detail.
How do I know what the value
of these zero point fluctuations is?
Because we saw in all
the current discussion
that the amplitude of
zero point fluctuations
is absolutely key to understand.
And to that, I provide
you with a shortcut.
And the shortcut is to say
that the zero point fluctuation
amplitude of junction J
in mode M is equal to,
or is given by the frequency of that mode,
the linear frequency of that mode,
divided by the energy of that junction,
times a simple number, a participation.
How much of the energy of the mode
is stored in this junction?
So for instance, if these two oscillators
were perfectly hybridized on resonance,
the participation would be one half.
On the other hand, if
one of the oscillators
was completely uncoupled
from the other one,
if we removed the coupling capacitor here,
then the participation of
the Joseph's injunction
in the mode of the qubit would be one
since there's only one
inductor in that mode.
And the participation of
the Joseph's injunction
in the mode of the resonator is zero.
And the participation is
a number that's bounded
between zero and one.
And it is that simple,
easily to calculate classically number
bounded between zero and one
that completely determines
the amount of quantum
zero point fluctuations
that the magnetic flux will experience.
This participation is entirely determined
by the circuit parameters here.
And as before we can apply the
rotating wave approximation,
which allows us to arrive at a Hamiltonian
that looks like this
of the coupled system.
And in fact, I'm going to
actually let us jump ahead
into a simpler version of it.
So after we've written
up all of those steps
and followed the same procedures,
as in the earlier part
of this lecture, we can
arrive at an expression
that's somewhat simple.
And it's the following.
That we'll explain how
the readout operates.
The effect of Hamiltonian having
done all the approximations
we labeled out in the first
part is given by this expression
where we have the number of
photons in the readout cavity,
multiplied by the number
of photons in the qubit,
which for a qubit is either zero or one.
As these are the states
admissible for the qubit.
And of course,
this is all happening at
the frequency of the cavity.
In other words, if I look at
now the transition spectrum,
we previously saw that for a simple qubit,
we can write down the
frequency of the transition
between the zero and one
levels as Omega qubit,
and then the frequency of
transition between the first
and second level as
Omega qubit minus alpha.
These are split by the end chronicity.
Now we can almost entirely
ignore and forget everything else
where the qubit happening in
this part of frequency region,
because we restrict ourselves
to working with a qubit
rather than the full nonlinear oscillator.
And so we bound ourselves
to this narrow window
and frequencies where
we can control the qubit
and apply an arbitrary SU2
operation on the qubit.
The cavity is now a new
addition to the spectrum.
At the frequency Omega cavity, of course,
the cavity or the resonator here,
which I can identify with
this piece in the diagram
has not one resonance as a
classical harmonic oscillator,
but in effect it has two resonances.
And that's because of
this nonlinear coupling
between the first mode
and the second mode.
In other words, imagine
that I have a state side,
which is equal to now the
product of two things.
Let's say the product of
the qubit ground state zero
and the cavity state one, right?
If I applied the
Hamiltonian on this state,
then the energy of the state we'll see
is H bar Omega cavity.
'Cause I have one photon in the cavity.
And that's it.
The A dagger A times zeroes over here.
The term A dagger A times
zero will give me zero.
However, if we now consider
the effect on the state,
not zero, zero, but on the
state, say zero zero one.
Let me maybe rewrite that since
we're running out of space.
Let's say that we now
apply the Hamiltonian
on the state side equals one, one,
where one indicates that there
is one photon in the qubit.
and the second one indicates
that there is one photon
in the cavity.
The energy of the state H on side
is equal to H bar Omega C times,
or rather minus, excuse
me, minus H bar Chi,
which I can write as H bar Omega C,
which is the cavity frequency minus Chi.
The cavity and resonator
I'm using interchangeably.
So I apologize if that's a bit confusing.
Right.
And this occurs because again,
A dagger A on one Q is equal to one.
Is equal to the number one
applied on the state one Q.
That's because A dagger A is
the photon number operator.
And these are icon States of
the Fulton number operator.
And B dagger B applied
on the cavity state,
one cavity is again, one
times the cavity state.
Okay.
So with that little algebra,
what does this actually
look like in a spectrum of frequency?
Well, when you add in the
fact that this oscillator
isn't infinitely long lived,
because remember we said that
this oscillator is coupled
through a weak coupling
capacitor to the output line,
which means that energy from
this oscillator can leak out
into the transmission
line and lose itself.
That means that it doesn't have
just one definite frequency
because of dissipation.
So if we try to excite this oscillator
using this red excitation pulse here,
what we can see is one
of two possible cases,
looking at the reflected signal.
When we send a microwave
tone at a frequency
that is farther tuned, such as say here.
From the resonance of the cavity,
which we'll call Omega
cavity at this axis,
then the cavity response
is essentially zero.
And that's because when
you drive an oscillator
far away from residence, it
doesn't admit a lot of energy.
It isn't able to store a lot of energy
for a given excitation.
However,
as you get for that same
amplitude closer in frequency
towards the resonance frequency,
you have the resonance phenomenon,
which allows energy to build
up inside of the oscillator.
And so on resonance,
when you drive the readout cavity here
with its cavity frequency,
you notice that there
you can get in a lot of
energy into the oscillator.
And so the height here of
the curve, the response
is indicative of how much
energy you can excite
inside of the oscillator for driving it
at a particular frequency.
Now, there are two cases
still because the qubit
has two different positions.
And as we saw, the energy of the cavity
actually depends on
the state of the qubit.
And it's given by this
nonlinear four wave mixing
interaction up here.
And in particular, it's
given by this term,
by the cross curve.
And so the cavity frequency
will actually take either
of two values.
Either it will be at Omega cavity,
which is the case when the
qubit is in the ground state,
as we saw on the previous slide.
Or it will take the value
of Omega cavity minus Chi.
Chi is some numbers, some constant,
we call it a dispersive shift.
Typically, this value of Chi here
is of the order of a few
megahertz in energy scale.
So it's a very tiny little
number compared to Omega cavity,
which is on the order of 10 gigahertz.
But it's a large enough energy shift
that it can be easily
resolved in practice.
And what you're seeing now
is a conditional dynamic.
This body, the cavity has one
of two possible frequencies.
It's either at Omega cavity
or at Omega cavity minus Chi,
conditioned on whether the
qubit is in the ground state
or the qubit is in the excited state.
And so by shining a microwave tone,
typically at the middle
between the two peaks,
which provides in a sense
of optimal resolution,
we can distinguish the state
of the qubit indirectly
by looking at whether
the cavity is excited
in the green peak or in the red peak.
In other words, if the space
shift is either positive
or negative.
So in a way it's suffice us
to simplify the discussion
we've had and reduce it
onto an interaction picture
where we only really talk about
two things, the qubit term,
which is proportional to Z,
that's the operator of the
poly operator, Sigma Z,
or the qubit Z operator,
the energy in the qubit,
which relates to the transmon qubit.
And the cavity, here denoted in blue.
Oops!
Excuse me, I lost the annotation there.
So this is the qubit.
And this is the cavity.
And you notice that this
is a dispersive interaction
in the sense that this interaction, again,
it doesn't couple different States.
If we look at a perturbation
of the interaction
between say the fox state and N prime.
Remember that we have now two
modes, a qubit and a cavity.
So the first index here
refers to the qubit.
The second index refers
to how many photons
are in the cavity mode, times.
the interaction Hamiltonian
is only non-zero in the case
where we have M equal to N
and M equal to N and
the case where M prime
is equal to N prime.
And I see that I'm running
into the edge here.
In other words, only the energy changes,
but not the eigen states.
And that's a really nice interaction.
This is just like the pristine
case of the barrel, right?
We have a laser shining over
the oil fumes of the barrel
and not destroying anything.
You see, the eigen states don't change.
There's no exchange of energy.
We haven't lit the match to the qubit.
Now for the slightly
more advanced of view,
again, a dangerous bend symbol,
the way that you can in
practice resolve the state
of the qubit being in the
ground or in the excited
is to consider what the
integrated wave reflected back
from the readout cavity looks like.
And this is just another way
to present these two peaks,
these Lorenzen peaks,
except that now we consider
both the phase response
and the amplitude response.
So again, we're in the complex quadrature.
And again, we see these kinds
of complex action angled type
of variable, alpha, which
can take the oscillator
in one of two possible steady states.
So at its end, the oscillator
is a displaced vacuum state,
a coherence state, which
either sits here when the qubit
is in the ground state
and the cavity is driven.
Or the cavity sits here
essentially in face space
when the atom is in the excited state.
Now these two different states,
they are distinguishable and measurable
by a measurement apparatus
on the transmission line.
These two states, we call pointer states
Of the measurement apparatus, the cavity,
and they correspond
directly to the qubit being
in the ground or the excited.
Now you notice that the Hamiltonian above
creates entanglement between
the qubit and the cavity.
So you might wonder where
did that entanglement go?
Well, the assumption here is
that the cavity is so leaky.
It's coupled rather so strongly
to this output transmission
line that any entanglement
created by the qubit and cavity
is quickly destroyed in the
sense of it leaves the system,
but it's then transmitted
down the measurement chain
and projected out into a classical signal.
And then that projection then
leads to a correlated state
rather than an entangled
state between the qubit state
between the qubit being either one.
Oops!
Being either a one or zero.
And again, on a slightly more advanced
and technical level of discussion,
the longer you integrate
the better you are able
to resolve the difference
as to whether the state
of the cavity is in the
ground or the excited,
and the higher resolution
you have on whether the qubit
is ground or excited.
Now, at some point,
these two peaks become
highly distinguishable,
and there's no point in
integrating any further
and that's when your
measurement has completed.
And that's when we say we've
done a projective measurement
of say the Z quadrature of the qubit.
And that's exactly how the
operations happen in T skid.
And that's the same way
that they're calibrated
is by tuning this time of
integration until there is
a sufficient distinguishability
between the pointer states
of the ground and the excited state.
And I won't get into the details
here of just how the signal
to noise ratio grows as a
function of the integration time
into distinguishability.
But I think this takes us,
in the interest of time,
to the road behind.
We reviewed with the harmonic
oscillator in great detail,
which led us straightforwardly
to the transmon qubit,
which we saw as an extension
that builds on top of the
basic ideas of zero point
quantum fluctuations fox states,
by adding nonlinearity.
We then dove into how you can
in a more advanced setting,
use external fields to steer
the atom among the states
and to create arbitrary gates.
And then using the same
ingredients of oscillators
and more oscillators
with some non-linearity
of the junction distributed throughout,
you can create much more
complicated and rich physics
and dynamics of coupled systems.
Systems that allow you to
perform non-destructive quantum,
and non demolition type of
measurement on the qubit
through a readout resonator.
Now unfortunately, I didn't
have a lot of time to go
into the details of that on
its own, that's its own course,
but we'd love to do that sometime.
But I hope I was able
to at least expose you
to some of these more advanced ideas
that really only build on
what we've already learned.
They take the same ingredients,
combine them liberally
and uniquely to create
new and unique features.
And that led us into the
final discussion of some more
of the advanced way that
you can shape measurements,
that you can distinguish pointer states,
and you can begin to resolve even how
the measurement process occurs.
You know, this sort of mysterious
type of Norman projection
while it doesn't occur instantaneously,
it doesn't occur all at once.
It occurs in a very special manner.
And that's what these measurements
and pointer state
description begins to unveil.
And this is of course, very
close and dear to my heart
and is what I did my dissertation on is,
taking that way even
further and being able
to actually resolve the
wave function collapse
and the way that you can even
anticipate the occurrence
of some of these measurements
that you can predict
what was once considered
fully unpredictable,
the leap of a quantum jump.
And the way that that
came about was with a lot
of experimentation, a lot of
fun and a lot of discussion.
And so I hope that you can
enjoy a similar benefit
from the labs that we'll
follow up today's chat,
today's lectures.
That I think Nick Braun
and company will deliver
to you in the rest of this day.
So with that I'll again,
invite you to please follow up
on the lectures by running
experiments in the real devices,
going to the labs,
checking out the references
that I've sprinkled
throughout the lecture series,
try some of the problems
given in the lectures.
And as you get more advanced,
go back to the slides with
the dangerous bend symbols
and see if you can maybe dig
a little deeper next time.
And most importantly,
what I've shown you
today is one set of rules
for how to do things.
But as you get to the next
stage of your careers,
try to break the rules.
And perhaps as final parting
thoughts I'd like to thank you.
And I'll just leave you with
this quote from Albert Einstein
that has, I think, summarized
some tidbits of wisdom
that I have found myself useful.
And so with that,
I'd like to thank you very
much for your attention.
Thank you for all the many questions.
And I hope we can get to
perhaps even some more
of those questions in the
final few minutes we have.
