M1-intro.txt

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Resonance is what describes what is relevant for two level
systems and also-- and we will touch
upon this-- resonances are the way how precision
measurements are made.
So what is a resonance?
Well, we can first look at a classic resonance.
Well, a resonance is something where we have some variable,
and it varies periodically.
So, in other words, there's a variable,
which can be anything.
It can be the population of quantum state.
It can be an electric field.
It can be the position of an atom.
It can be any thing you can think about and any thing you
can measure.
And if this variable varies periodically,
you have a resonance.
Of course, the periodic variation
usually requires that you drive the system.
So you first drive it and then the system oscillates.
And this means now that when you drive the system--
so this may be a free oscillation,
but now you drive this system-- with a variable frequency,
and what you then observe is-- you observe a peak.
So the phenomenon of resonance is
that you have something which can periodically vary,
and then you drive it, you see a peaked response
when driven with a variable frequency.

We are interested in atomic physics
in every single possible aspect of these resonance: the shape
of the curve, how we can modify it,
what happens when we drive it strongly,
when we drive it weakly.
I mean, this-- resonances is really the language
we talked to atoms with.
So, but here, I just want to give
a light-hearted introduction.
The first thing we want to add to the phenomenon
that there is a resonance at a certain frequency
is finite damping, that would mean
that after the system is driven, the oscillation does not last
for an infinite amount of time.
And that implies that when we drive the system,
and look at the response as a function of frequency,
there is a finite width delta f for the driven system,
and as we will see in many ways, the damping time and delta f
are related by Fourier transform.

And we usually characterize oscillators
by the sharpness of the resonance.
And the sharpness of the resonance
is the ratio of the width of the resonance and the frequency,
or the inverse of it.
So if you have an oscillator at kilohertz and resonance
is one Hertz wide, we say the resonance has
a Q, a quality factor of 1,000.
And it means you can observe a thousand oscillations
before the oscillation decays away.
So what is special about atomic physics here?
Why do I emphasize it in the introduction
of an atomic physics course?
Well, the system is that in atomic physics,
we often have exquisitely isolated system, an atom
in an ultrahigh vacuum chamber, or systems
which are prepared with all of the tools
and precision which we have developed
over decades in atomic physics.
And therefore, the result is, that in atomic physics,
our oscillators are characterized by an extremely
high quality factor Q.
And let me give you an example.
If we look at an optical excitation--

Maybe let me point out that something you all
should try when you take a class in atomic phsyics-- or even
more so, when you do research in atomic physics--
that you have a few numbers in your mind which match.
So, every single person in this room
should know what is the frequency of light.
How many Hertz is-- what is the frequency of a laser?
The number I usually use for those estimates
is 10 to the 15 Hertz.
Who knows what wavelengths this laser is?
10 to the 15 is--

Well, I've used some visible light, but the speed of light
is 3 times 10 to the 10.
So therefore, if I just use a power, 1o to the 15 Hertz,
it has to be 300 nanometer.
OK.
So, never forget that for the rest of your life.
300 nanometer is 10 to the 15 Hertz.
This means the most of us who are working
with rubidium, lithium, and sodium, which
is 600 nanometer or 800 nanometer,
the frequency is more.
5 times 10 to the 14 or 3 times 10 to the 14,
but just as a ballpark number, 10 to the 15 Hertz
is 300 nanometer.
OK.
So if you have an optical excitation,
and many atoms have that, what is the Q,
what is the quality factor of this resonance?
Well, when you stabilize your laser to a vapor cell
and you look at the resonance, then you
observe in the vapor cell, that you have room temperature
Doppler broadening.
We talk about Doppler broadening later
in this course-- that usually corresponds to a frequency
width on the order of a gigahertz.
And that means that your quality factor is on the order of 10
to the 6.
A million.
That's pretty good, a million oscillations.
It's a very pure oscillator.
But of course, you can do much better
if you do Doppler free spectroscopy,
either by having an atomic beam which
is intersected at a right angle, or even better,
put the atoms in an optical lattice.
And this is what people are now doing with the optical lattice
clocks, that they put an atom in an optical lattice
where the Doppler broadening is completely eliminated.
If you take a metastable level that the lifetime
of the excited state is maybe one second,
and strontium and other atoms have those levels-- have
those metastable levels, then you
can actually get a line width, which is one Hertz,
and the Q factor is on the order of 10 to the 15.
I will show you a graph and an example of such an experiment,
and one Hertz line width of an optical transition
for an optical atomic clock experiment in the next class
on Monday when I want to discuss other aspects of it,
but this is one of the world's best
oscillator you can imagine.
10 to the 15.
It's a mind-boggling number.

Well, it's clear why clocks have gone atomic.
Mechanical systems are actually not bad, but of course,
not nearly as good.
If you take a quartz oscillator well,
you can build pretty good clocks out of quartz oscillators.
You have quality factors which vary between a few thousands
and a million.
The best values are reached at low temperature.
Even in the advent of atomic clocks,
quarts oscillators or sapphire oscillators
still play a role, because you need flywheels.
An atomic clock, you may interrogate
only every-- the Ramsey spectroscopy-- every tens
of seconds, you get a signal, and in between, you
need a flywheel.
And then, clocks, which have a very high signal to noise
ratio, but not accuracy, help you to interpolate
between measurements.
We see actually a renaissance of mechanical systems, in the form
of micro-mechanical oscillators.
It was only achieved in the last two or three years,
that micro mechanical oscillators could
be cooled to the absolute ground state,
and there's a lot of interest of coupling
the motion of a mechanical oscillator
to an atomic oscillator, because they have different properties
and for quantum computation, and other
explorations of Hilbert space,
you want to have different oscillators
and take-- combine the best of the properties.
So therefore, there is a real renaissance
in mechanical oscillators.
And those micro-mechanical oscillators
have often quality factors of 10 to the 5.
This is a micro-fabricated device.
It looks like a little mushroom.
And what happens is, this mushroom-type structure
can confine light which travels around the perimeter.
it's the so-called whispering gallery mode.
It's similar to an acoustic mode which can travel
in the dome of a big cathedral.
That's how it was discovered.
It's an amazing effect.
I wish somebody would demonstrate to me.
But if you go to one of the ancient cathedrals
and you're in a dome, somebody can talk in one direction.
The sound can travel around and you can hear it.
There is a guided, special mode which can travel around
the perimeter of a dome.
And here, in the microscopic domain,
it's light which is confined in such a resonator.
So this is a resonator for whispering
gallery mode and that can have a Q on the order of a billion.

So the idea here, is that you have
either one of those mushrooms, or a glass sphere and the light
can sort of travel around.
And this is the characteristics of this mode.
Well, you can go from a tiny glass sphere
to astronomical dimensions.

And you also find oscillators and the Q of those oscillators
is not really bad.
How good is the Q of the rotation of the Earth?

It fulfills all of our requirements
for resonance and oscillator.
It's a periodic phenomenon.
The Earth goes-- the Earth rotates around the sun
once a year.
And the question is, how stable is it.
Well, the number is 10 to the 7.
It has a Q of 10 to the 7.
So the precision of the rotation of the Earth
is better than one part in a million.
You can also look at the rotation of neutron star.
If those neutron star emit flashes of x-rays-- these
are pulsars-- and you can measure
the rotation of neutron stars, those neutron stars
have a quality factor of 10 to the 10.
And of course, if-- as the name says--
if a resonance has a high quality factor,
it can be used for quality research.
The narrower the line is, the more sensitive
you are to tiny, little changes.
And you probably know this pulsar with Q of 10 to the 10,
well, has been used for the first-- also
indirect-- observation of gravitational waves.
The pulsar rotates with a very precise frequency
and you can measure it with one part of 10 to the 10.
And people have seen that over the years,
the frequency of rotation became smaller.
And you can figure out that it becomes
smaller by just one part in 10 to the 10
because you have this precision.
And what happens is, when the pulsar rotates,
when the neutron star rotates, it emits gravitational waves,
and the gravitational wave is energy
which is taken away from the kinetic energy of rotation,
and therefore the pulsar slows down.
So having an oscillator with such a high Q
has allowed researchers to find a small effect
in the damping of this oscillator,
which in this case, the gravitational waves.
Of course, the story I will tell you, is about very small
changes of atomic oscillators, which
led to the discovery of the Lamb shift
and to quantum electrodynamics.
But the story is the same.
A high quality oscillator is a tool for discovery.
OK.
So, we've talked about resonances.

Of course, there are resonances which are useful
and there are others which are less useful.
By useful, we mean they are reproducible.
You can really make a measurement
and trust it and repeat it and do it again.
And, that's not enough for being useful.
You also want to learn something about it.
So, we usually regard resonances as useful
when they are connected to by theory, to something
we are interested in.
It can either be fundamental constants
or-- well, let me see-- other parameters of interest.
If you want to measure the magnetic field with very
high precision and you look at atomic resonance,
it's only used when you have a theory which tells you
how the shift or the broadening of the resonance
is related to magnetic fields.
And this is again a specialty of AMO physics.
We have plenty of resonances which
are useful by those standards.

And if you compare to astrophysical
oscillators or quartz oscillators or fabricated
oscillators.
In atomic physics, we have the great advantage
that atoms are identical.
We know when we measure the transition of atomic hydrogen
in Japan, in Europe, in the United States,
the value has to be the same.
For other oscillators, you often don't know that.

And, the showcase of atomic physics
is the Rydberg constant, which is
the best known, the most accurately known constant
in all of physics.
And the reason is, because it can be directly measured
by performing spectroscopy in hydrogen with highly
stabilized lasers.
OK.
Of course, the question is, who's
interested in all those digits?
Why do you want to spend all of your Ph.D. or half of your life
in measuring the Rydberg constant to maybe 10 times more
precision?
Well, it depends.
It's maybe not something for everybody,
but there are some connoisseurs who
think that every digit has provided
new insight into nature.
And let me just give you one example.
If you measure the Rydberg constant very precisely,
you can know, and this has become a frontier of our field,
ask the question-- Is there a change with time
of fundamental constants?

So when you measure the Rydberg constant today with 10
to the minus 15 precision and measure it again
in a year, who is guaranteeing to you that you
will measure the same value?
So with the precision which I've just
given to you in this measurement of the Rydberg constant,
people are now able to see whether the Rydberg
constant has changed, 10 to the minus 15 per year.
Of course, you know the age of the universe
is 14 billion years.
That's about 10 to the 10 years, so even the worst case
is, if you would go back to the beginning of the universe
and the Rydberg constant would change at 10 to the minus 15
per year, it would have changed by 10 to the minus 5
over the age of the universe.
But this would be dramatic, because you can actually
show that life would not have developed.
The whole organic chemistry would've been different
if some fundamental constants of nature
had been different by one part in 10 to the minus 6, 7, or 8.
So you have extremely stringent limits
on how much fundamental constant could have changed
during the evolution of life, because life would not
have been the same if fundamental concept is changed.
The question, of course, is should
those fundamental constants change?
The answer is, we don't know.
But there is a whole research area,
in string theory, where they say that our universe is sort
of just one of many possible minima
in a multi-dimensional space.
And it's actually a dynamic minimum.
It changes the function of time.
So that people who wouldn't be surprised
if the world is not the same in the future
as it is right now, because the universe
or whatever defines fundamental constants,
is changing as a function of time.
So the question is, will it be during your lifetime,
or will it even be during, maybe your Ph.D.,
that one researcher says, we now have an error bar
and we find out, that yes, we measure
the fundamental constant using the most accurate atomic clock,
and a year later, we've measured something which is a tiny bit,
but significantly different.
The second aspect, why you should always measure things
as accurately as possible-- and this
is sort of the traditional of our field--
if you can measure something very accurately,
do it, because yes, there may be surprises.

And, for instance, when people looked at the Zeeman shift
with higher precision, they found--
we will talk about when we talk what
atoms in the magnetic field-- When people looked
at the anomalous Zeeman effect, the discovery of that,
is what nobody expected.
That particles electron has a spin.
Or, when people saw a tiny shift in the spectrum
of atomic hydrogen, it was a thousand megahertz splitting,
it was the Lamb shift.
This was the discovery of quantum electrodynamics.
And, we know that precision always
becomes a tool, a tool to control atomic systems,
control quantum mechanics with more precision.
For instance, if you can completely
hyperfine structure-- you can prepare atoms
in a certain hyperfine state.
If you don't have the resolution, you can't do that.
OK.
So we now want to talk-- go back to the resonance.
When we look at typical resonance,
we have a frequency omega, the resonance frequency omega
naught.

And we measure line with delta omega.
In many cases-- and we will discuss them in great detail--
the line shape is the Lorentzian.
And the Lorentzian is the imaginary part
of a one over x function.
Omega naught minus omega.
And then there is this parameter gamma.

Gamma, which appears in the Lorentzian,
is identical to the full widths at half maximum,
and the Q factor of a Lorentzian is omega_0 over gamma.