M1L2b.txt

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We talked so much about harmonic oscillators, the precision
in which we can determine the frequency
of the harmonic oscillator.
And I'm going to give you two outstanding examples
for atomic clocks.
The two extreme examples are well,
the two best atomic clocks of the build.
One is, the cesium atom with the fountain clock.
Well, we'll talk about it later, but some of you
know that the cesium atom has hyperfine structure in one
state.
The electron and nucleus being a parallel in the other state,
the are anti-parallel.
And the transition frequency is 10 gigahertz.
9.0 something but for the matter of this discussion
it's 10 gigahertz.
And the definition of the second--
the definition of time , the definition of one second is
in terms of so and so many cycles of this transition.
So this transition has this frequency.
And for decades, this frequency was determined
in an atomic beam with an atomic cesium beam
and you interrogated it with microwave fields.
But now with cold atoms, we can achieve much, much longer
interrogation times by completely--
by almost completely eliminating the atomic motion.
And the current experiment is that you
have a cloud laser cooled to microkelvin temperature.
You launch it into a fountain.
The cloud goes up and goes down.
And you interrogate it twice.
And the interrogation time is one second.
And this interrogation time is no longer as in the conventional
atomic clocks, limited by the thermal velocity of the atom.
It's limited by gravity.
If you want to increase the time to 10 seconds,
you need a 100 meter tower.
And nobody wants to build that.
It would really be a big atomic clock.
So therefore, you usually deal with an interrogation time
on the order of one second.
So we know that based on Fourier's theorem,
there's a factor of 2 pi,
but the line widths, if you were to record now the spectrum,
the line width delta omega would be, there's a factor of two,
but it would be one over one second.
And therefore, the fractional accuracy is 10 to the 11
is on the order of 10 to the minus 11.
However, the accuracy of the best cesium fountains
is now 10 to the minus 16.
So people are able to split the line to one part in 100,000,
which requires exquisite knowledge, exquisite knowledge
of the line shape and all systematic effects.
But well, an atomic clock is a piece of art.

I just want to emphasize I sometimes
feel when I talk about the uncertainty of which you
can measure a classical oscillator
or such is almost trivial.
You should know all about it.
But I can just say without looking at anybody,
that I just had a lunch discussion with some
of my graduate students, and we talked about laser stabilization.
And one graduate student asked, but if the natural
line width of a transition is 10 megahertz,
can be stabilize a laser to better than a megahertz.
Of course we can.
I mean, here we have a natural line width of 10
to the minus 11.
But we can get an accuracy of a signal, which
is 10 to the minus 16.
So a microwave oscillator, which is the microwave equivalent
of a laser, can now be locked to the cesium transition
with the precision of 10 to the minus 16,
100,000 times better than the line widths.
The other example I want to give you
is the strontium optical clock.
And there was a nice, really nice paper
in "Nature." just a few weeks ago.

Here is a level diagram of atomic strontium.
And what happens is, there is a very fast transition,
is to p transition for laser cooling and trapping
and all that.
but then there is a very, very slow transition
forbidden to a triplet state, which is metastable.
And this transition, this-- those states
have a very long lifetime.
And therefore, this transition is extremely narrow.
So what is state of the art for this strontium clock?

Well, in the experiment, they use an interrogation time.
So they observe the atom for delta t,
which is n the order of 160 milliseconds.
And putting in the two pis in the right place,
that could mean the frequency resolution
is one hertz.
And you see that on the left hand side,
when we record the resonance, the blue one is about one hertz
and there's something broader.
This is when they are not actively feeding back
the magnetic field.
So they have to do I'm for this fantastic position
you control everything.
But the blue line is sort of, what we call,
a sub clock transition.
It's about one Hertz.
Now compare to a cesium clock, the big advantage
is, that the strontium clock operates in the optical domain.
And this is a frequency at 5 times 10 to the 14 hertz.
So assume that if the frequency is much, much faster, even
if you have a shorter interrogation time,
your relative accuracy is better.
And here the Q value, the data, the nu over delta nu,
is on the order of 10 times 10 to the minus 15.
Fantastic.
15 orders.
15 orders of magnitude.
And they are splitting the line.
Not as extreme as in the cesium atomic clock
by effect of 100,000, by a factor of which
is on the order of 300.
And with that, they have an accuracy, which is now really
the record for the best performance
of any atomic clock, which is 6 times 10 to the minus 18.

Now, this is now in the optical domain.
You may wonder about the laser.
How stable is the laser?
The laser is stabilized to an optical cavity.
But the laser, because of thermal fluctuations,
not technical fluctuations, thermal
fluctuations in the mirror.
Because of thermal noise, it's limited in the short term
to 10 to the minus 16.
So the short term stability between one and 1000
second is 10 to the minus 16.
So they use a laser, which has a-- they
use a laser, which is the stability of 10
to the minus 16.
Every 1.3 seconds they take a data point.
Each data point, or the spectral width
is 2 times 10 to the-- sort of the laser is 10 to the 16.
The line they record is 10 to the 15.
But since the thermal noise is completely random
and by averaging it, they can determine the line center
to better than 10 to the 17.

So I think this just illustrates the precision at which you
can observe the harmonic oscillator.
And what I like about this example,
it shows you that both the transition you're recording
and the laser itself, 10 to the 15, 10 to the 16,
are worse then the final precision
of the measurement, which is better than 10 to the 17.

Makes sense, but yes.
That's what it is.
Questions about that?
Yes, Colin.
[Colin] What was the accuracy of the most recent Ytterbium result?
[Colin] Is it better than 6 times 10 to the minus 18?
No.
The 6 times 10 to the minus 18 is really the world record.
[Colin] Is that the JILA clock?

There was an aluminum ion clock which
had I think 10 to the minus 17 precision,
but 6 times 10 to the minus 18 is close to that.
So they are close,
or they are better now than the aluminum clock.
But the aluminum clock is a single ion.
You have to average for much, much longer time
to get this precision.
So there's a big advantage for the strontium
clock, which is maybe many atoms in an optical lattice.
And they say that they have improved on the best previous,
latest clock by a factor of 20.
But I think the improvement of factor of 20--
you always have to distinguish between sensitivity
and absolute precision.
In the absolute precision, you also
have to control all systematic effects.
And the big step here, which was really
boosting the absolute precision was they completely controlled
the black-body environment.
So you have to really know with the high precision what
is the effective temperature of the black body radiation,
because at the 10 to the minus 18 level,
it causes a shift of the atomic resonance.

We talk later about it, but it's the AC Stark effect
of the black body radiation, which
becomes an important systematic at that level of precision.
I don't really know exactly the number on the previous atomic clocks,
but now this is the gold standard
of frequency metrology.
Yes.
What you have here is, you have sort
of a classical harmonic oscillator.
But the classical harmonic oscillator
is based on doing measurements on single strontium
atoms and single cesium atoms.
So you see that the accuracy, and we had a discussion on it
last class, the accuracy of which you can observe a quantum
mechanical oscillator and a classical oscillator,
it's pretty much the same.
It is the same.
It's the same kind of-- if you have a good signal to noise,
you can improve your precision for the line center substantially.