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ROBERT FIELD: Today's
lecture is one where--
00:00:25.410 --> 00:00:28.270
it's a lecture I've
never given before.
00:00:28.270 --> 00:00:33.210
And it's very much
related to the experiments
00:00:33.210 --> 00:00:36.580
we are doing right now
in my research group.
00:00:36.580 --> 00:00:45.730
And so basically, we
have a chirped pulse
00:00:45.730 --> 00:00:53.530
of microwave radiation, which
is propagating through a sample.
00:00:53.530 --> 00:00:59.770
It causes all of the
molecules in the sample
00:00:59.770 --> 00:01:03.280
to be prepared in some way.
00:01:03.280 --> 00:01:05.590
We call it polarized.
00:01:05.590 --> 00:01:09.400
And this polarization
relaxes by what
00:01:09.400 --> 00:01:11.950
we call free induction decay.
00:01:11.950 --> 00:01:14.350
And they produce
a signal which--
00:01:14.350 --> 00:01:16.881
so we have a pulse
of radiation that
00:01:16.881 --> 00:01:18.130
propagates through the sample.
00:01:21.130 --> 00:01:24.670
The two-level
systems in the sample
00:01:24.670 --> 00:01:28.570
all get polarized, which
we'll talk about today.
00:01:28.570 --> 00:01:31.640
And they radiate
that polarization.
00:01:31.640 --> 00:01:34.690
And we collect it
in a detector here.
00:01:34.690 --> 00:01:38.830
And so the two
important things are
00:01:38.830 --> 00:01:41.610
this is a time
independent experiment,
00:01:41.610 --> 00:01:46.120
and that we have a whole
bunch of molecules.
00:01:46.120 --> 00:01:48.460
And they're interacting
with the radiation
00:01:48.460 --> 00:01:51.790
in a way which is complicated.
00:01:51.790 --> 00:01:53.500
Because this is not--
00:01:53.500 --> 00:01:56.650
each one of them
has quantum states,
00:01:56.650 --> 00:01:59.860
but all of the
particles in this sample
00:01:59.860 --> 00:02:03.790
are somehow interacting
with the radiation field
00:02:03.790 --> 00:02:08.639
in a way which is uncorrelated.
00:02:08.639 --> 00:02:12.070
So we could say all
of these particles
00:02:12.070 --> 00:02:13.500
are either bosons or fermions.
00:02:13.500 --> 00:02:15.990
But we're not going
to symmeterize
00:02:15.990 --> 00:02:17.900
or anti-symmeterize.
00:02:17.900 --> 00:02:22.070
Each of these particles
is independent.
00:02:22.070 --> 00:02:25.370
And we need a way of describing
the quantum mechanics
00:02:25.370 --> 00:02:29.540
for an ensemble of
independent particles.
00:02:29.540 --> 00:02:35.040
So it's a big step towards
useful quantum mechanics.
00:02:35.040 --> 00:02:37.830
And I'm not going
to be able to finish
00:02:37.830 --> 00:02:40.620
the lecture as I planned it.
00:02:40.620 --> 00:02:43.620
So you should know
where I'm going.
00:02:43.620 --> 00:02:49.880
And I'm going to be introducing
a lot of interesting concepts.
00:02:49.880 --> 00:02:54.210
The first 2/3 of
lectures votes are typed.
00:02:54.210 --> 00:02:56.010
And you could have seen them.
00:02:56.010 --> 00:03:00.810
And the rest of them will
be typed later today.
00:03:00.810 --> 00:03:04.830
This is based on material
in Mike Fayer's book, which
00:03:04.830 --> 00:03:06.300
is referenced in your notes.
00:03:09.320 --> 00:03:13.590
This book is really accessible.
00:03:13.590 --> 00:03:15.390
It's not nearly
as elegant as some
00:03:15.390 --> 00:03:19.350
of the other treatments of
interaction of radiation
00:03:19.350 --> 00:03:22.650
with two-level systems.
00:03:22.650 --> 00:03:24.720
Now I talked about
interaction of radiation
00:03:24.720 --> 00:03:27.740
with two-level systems
in lecture number 19.
00:03:27.740 --> 00:03:30.120
And this is a completely
different topic
00:03:30.120 --> 00:03:35.400
from that, because
in that, we were
00:03:35.400 --> 00:03:43.380
interested in many transitions.
00:03:43.380 --> 00:03:49.990
Let me just say the
radiation field that interact
00:03:49.990 --> 00:03:53.390
with the molecule is weak.
00:03:53.390 --> 00:03:57.350
It interacts with
all the molecules,
00:03:57.350 --> 00:04:03.650
and the theory is
for a weak pulse--
00:04:06.790 --> 00:04:10.900
and the important point in
lecture 19 was resonance.
00:04:13.790 --> 00:04:18.860
And so we made the
dipole approximation.
00:04:18.860 --> 00:04:26.000
And each two-level system
is separately resonant
00:04:26.000 --> 00:04:30.740
and is weakly interacted
with, and does something
00:04:30.740 --> 00:04:32.830
to the radiation field.
00:04:32.830 --> 00:04:36.140
Now here, we're
going to be talking
00:04:36.140 --> 00:04:41.480
about a two-level
system, only two levels.
00:04:41.480 --> 00:04:44.180
And the radiation
field is really strong,
00:04:44.180 --> 00:04:45.890
or is as strong as you want.
00:04:45.890 --> 00:04:50.290
And it does something
to the two-level system
00:04:50.290 --> 00:04:54.870
which results in a signal.
00:04:54.870 --> 00:04:58.180
And because the radiation
field is strong,
00:04:58.180 --> 00:05:02.110
it's not just a matter of taking
two levels and mixing them.
00:05:02.110 --> 00:05:04.550
The mixing coefficients
are not small.
00:05:04.550 --> 00:05:07.350
It's not linear response.
00:05:07.350 --> 00:05:10.500
The mixing is sinusoidal.
00:05:10.500 --> 00:05:13.220
The stronger the
radiation field,
00:05:13.220 --> 00:05:16.680
the mixing changes, and all
sorts of interesting things
00:05:16.680 --> 00:05:18.000
happen.
00:05:18.000 --> 00:05:21.930
So this is a much harder
problem than what was
00:05:21.930 --> 00:05:26.200
discussed in lecture number 19.
00:05:26.200 --> 00:05:28.670
And in order to
discuss it, I'm going
00:05:28.670 --> 00:05:36.320
to use some important tricks
and refer to something
00:05:36.320 --> 00:05:40.190
called the density matrix.
00:05:40.190 --> 00:05:44.840
The first trick is we
have this equation which
00:05:44.840 --> 00:05:48.080
can easily be derived.
00:05:48.080 --> 00:05:49.820
And most of this
lecture, I'm going
00:05:49.820 --> 00:05:53.360
to be skipping derivations.
00:05:53.360 --> 00:05:56.000
Some of the derivations are
going to be in the notes.
00:05:56.000 --> 00:05:58.550
So we have some operator.
00:05:58.550 --> 00:06:01.640
And we want to know the time
dependence of the expectation
00:06:01.640 --> 00:06:02.810
value of that operator.
00:06:02.810 --> 00:06:17.140
And it's possible to
show that the expectation
00:06:17.140 --> 00:06:22.570
value of the operator a is
given by the expectation
00:06:22.570 --> 00:06:28.050
value of the computator
of a with the Hamiltonian
00:06:28.050 --> 00:06:33.870
plus the expectation value
of the partial derivative
00:06:33.870 --> 00:06:36.410
of the operator a.
00:06:36.410 --> 00:06:42.980
So this is a general and useful
equation for the time dependent
00:06:42.980 --> 00:06:45.260
of anything.
00:06:45.260 --> 00:06:50.510
And it's derived
simply by taking the--
00:06:50.510 --> 00:06:55.970
applying the chain rule
to this sort of thing.
00:07:03.040 --> 00:07:06.900
So we have three terms.
00:07:06.900 --> 00:07:12.070
And when you do that, you
end up getting this equation.
00:07:12.070 --> 00:07:16.090
So this is just this
ordinary equation.
00:07:16.090 --> 00:07:19.540
And anyway, so this
is what happens.
00:07:29.150 --> 00:07:33.340
So we're going to have
some notation here.
00:07:33.340 --> 00:07:34.340
We have a wave function.
00:07:37.550 --> 00:07:39.390
And this is a
capital psi, so this
00:07:39.390 --> 00:07:42.300
is a wave function,
a time dependent wave
00:07:42.300 --> 00:07:45.260
function that satisfies the
time dependent shorter equation.
00:07:45.260 --> 00:07:48.420
And we're going to replace
that by just something called
00:07:48.420 --> 00:07:51.500
little t.
00:07:51.500 --> 00:07:59.660
And we can write this thing psi
of x and t as the sum over n
00:07:59.660 --> 00:08:07.340
Cn psi n of x.
00:08:07.340 --> 00:08:18.520
And this becomes in a bracket
notation, becomes C Cn n.
00:08:21.240 --> 00:08:27.270
So we have a complete ortho
normal set of functions.
00:08:29.820 --> 00:08:34.250
And this thing is
normalized to one.
00:08:42.610 --> 00:08:45.220
And now I'm going to introduce
this thing called the density
00:08:45.220 --> 00:08:45.720
matrix.
00:08:54.540 --> 00:09:02.030
This is a very useful quantum
mechanical quantity which
00:09:02.030 --> 00:09:06.160
replaces the wave function.
00:09:06.160 --> 00:09:10.540
It repackages everything we know
from the time dependent shorter
00:09:10.540 --> 00:09:15.290
equation and the short in your
picture of a wave function.
00:09:15.290 --> 00:09:16.330
It's equivalent.
00:09:16.330 --> 00:09:19.940
It's just arranging
it in a different way.
00:09:19.940 --> 00:09:25.360
And this different way
is extremely powerful,
00:09:25.360 --> 00:09:31.000
because what it does is it gets
rid of a lot of complexity.
00:09:31.000 --> 00:09:33.790
I mean, when you have the
time dependent wave functions,
00:09:33.790 --> 00:09:39.460
you have this e to the minus i
E t over h-bar always kicking
00:09:39.460 --> 00:09:40.780
around.
00:09:40.780 --> 00:09:47.950
And we get rid of that
for most everything.
00:09:47.950 --> 00:09:52.820
And it also enables us to
do really, really beautiful,
00:09:52.820 --> 00:09:56.290
simple calculations of the
time dependence of expectation
00:09:56.290 --> 00:09:57.560
values.
00:09:57.560 --> 00:10:00.280
It's also a quantity
where, if you
00:10:00.280 --> 00:10:05.890
have a whole bunch of different
molecules in the system,
00:10:05.890 --> 00:10:08.395
each one of them has
a density matrix.
00:10:11.110 --> 00:10:13.700
And those density matrices add.
00:10:13.700 --> 00:10:17.860
And you have the density
matrix for an ensemble.
00:10:17.860 --> 00:10:22.320
And so if the populations of
different levels are different,
00:10:22.320 --> 00:10:29.680
the weights for each of the
levels, or each of the systems,
00:10:29.680 --> 00:10:30.550
is taken care of.
00:10:30.550 --> 00:10:35.320
But we don't worry about
coherences between particles
00:10:35.320 --> 00:10:39.160
unless we create coherence
between particles.
00:10:39.160 --> 00:10:41.620
So this is a really
powerful thing.
00:10:41.620 --> 00:10:46.170
And it's unlike
the wave function,
00:10:46.170 --> 00:10:53.320
it's observable, because
the diagonal elements
00:10:53.320 --> 00:10:58.480
of this matrix are populations.
00:10:58.480 --> 00:11:04.300
And the off diagonals elements,
which we call coherences,
00:11:04.300 --> 00:11:06.430
are also observable.
00:11:06.430 --> 00:11:10.810
And if you look at the Fourier
transform of the emission
00:11:10.810 --> 00:11:15.890
from this system, it will
consist of several frequencies.
00:11:15.890 --> 00:11:20.380
And those frequencies are
the off diagonal elements
00:11:20.380 --> 00:11:24.320
with the amplitude, the relative
weights of those frequencies.
00:11:24.320 --> 00:11:27.850
And so one can determine
everything in the density
00:11:27.850 --> 00:11:29.860
matrix experimentally.
00:11:29.860 --> 00:11:32.950
Now it's really--
it's still indirect
00:11:32.950 --> 00:11:35.160
because you're making
experimental measurements.
00:11:35.160 --> 00:11:37.570
But we think about this
thing in a way we don't
00:11:37.570 --> 00:11:39.800
think about the wave function.
00:11:39.800 --> 00:11:41.520
It's really important.
00:11:41.520 --> 00:11:44.270
And this is the
gateway to almost all
00:11:44.270 --> 00:11:49.050
of modern quantum mechanics
and statistical mechanics--
00:11:49.050 --> 00:11:50.780
quantum statistical mechanics.
00:11:50.780 --> 00:11:53.060
And so this is a really
important concept.
00:11:53.060 --> 00:11:56.600
And we've protected
you from it until now.
00:11:56.600 --> 00:12:00.170
And since this is the last
lecture both, in this course
00:12:00.170 --> 00:12:03.260
and in my teaching of
this course forever,
00:12:03.260 --> 00:12:07.940
I want to talk about
this gateway phenomena.
00:12:10.880 --> 00:12:11.855
So what is this?
00:12:15.950 --> 00:12:20.210
Well, we denote it by this,
this strange notation.
00:12:20.210 --> 00:12:24.860
I mean, you're used to
this kind of notation where
00:12:24.860 --> 00:12:28.890
we have the overlap
of bra with a ket
00:12:28.890 --> 00:12:32.250
and or abroad with itself.
00:12:32.250 --> 00:12:34.590
But this is different.
00:12:34.590 --> 00:12:38.420
You know, this is a number
and this is a matrix.
00:12:46.650 --> 00:12:54.290
And if we have a
two-level system,
00:12:54.290 --> 00:13:07.800
then we can say that t is
equal to C1 of t plus C2 of t.
00:13:07.800 --> 00:13:12.530
So state 1, state 2, and
we have time dependence.
00:13:12.530 --> 00:13:14.640
Now those could be--
00:13:14.640 --> 00:13:16.800
there's lots of stuff
that could be in here.
00:13:16.800 --> 00:13:18.745
And this is going to be
a solution of the time
00:13:18.745 --> 00:13:19.870
dependent shorter equation.
00:13:22.730 --> 00:13:27.060
So since it's an
unfamiliar topic,
00:13:27.060 --> 00:13:30.840
I'm going to spend more time
talking about the mechanics
00:13:30.840 --> 00:13:35.140
than how you use that
to solve this problem.
00:13:35.140 --> 00:13:37.560
But let's just look at it.
00:13:37.560 --> 00:13:40.380
So we have for a
two-level level system,
00:13:40.380 --> 00:13:43.530
we have-- it's a matrix
of 1, 1; a 1, 2; a 2, 1;
00:13:43.530 --> 00:13:45.480
and a 2, 2 element.
00:13:45.480 --> 00:13:52.520
And so we want the rho
1, 1 matrix on there.
00:13:52.520 --> 00:13:54.890
And so it's going
to be a 1 here,
00:13:54.890 --> 00:13:57.290
then we're going
to have a 1 here.
00:13:57.290 --> 00:14:12.190
And then we have the C1, 1.
00:14:16.900 --> 00:14:22.990
Plus C2, 2.
00:14:28.700 --> 00:14:43.780
And then we have C1
star 1 plus C2 star 2.
00:14:52.290 --> 00:14:55.490
And so have I got-- am
I doing it right now?
00:14:55.490 --> 00:15:04.410
So the first thing we do is
we look at this inside part.
00:15:04.410 --> 00:15:10.250
And we have C1, C1 star.
00:15:10.250 --> 00:15:14.710
And we have 1, 1.
00:15:14.710 --> 00:15:20.225
And we have C2, C2 star.
00:15:24.154 --> 00:15:31.330
Now I'm getting in trouble,
because I want this to come out
00:15:31.330 --> 00:15:35.900
to be only C1, C1 star.
00:15:35.900 --> 00:15:38.191
So what am I doing wrong?
00:15:38.191 --> 00:15:40.860
AUDIENCE: [INAUDIBLE]
both have C2 halves
00:15:40.860 --> 00:15:44.008
the left hand and the right
hand half are both [INAUDIBLE]..
00:15:50.924 --> 00:15:54.500
See, on the left side, you
have 1 on 1 [INAUDIBLE]..
00:15:56.842 --> 00:15:57.800
ROBERT FIELD: So here--
00:16:12.820 --> 00:16:25.120
And that's C2 C1
star, but that's 0.
00:16:25.120 --> 00:16:29.730
And so anyway, I'm
not going to say more.
00:16:29.730 --> 00:16:37.340
But this combination is
1, this combination is 0,
00:16:37.340 --> 00:16:41.790
this combination is 0,
this combination is 1.
00:16:41.790 --> 00:16:44.120
And we end up getting--
00:16:44.120 --> 00:16:46.300
and then we end up
just getting this.
00:16:55.380 --> 00:17:02.756
Rho 1, 2 is equal to C1 C2 star.
00:17:02.756 --> 00:17:08.609
Rho 2, 1 is equal to C2 C1 star.
00:17:08.609 --> 00:17:15.119
And rho 2, 2 is
equal to C2, C2 star.
00:17:15.119 --> 00:17:18.760
So we have the elements
of this matrix.
00:17:18.760 --> 00:17:24.730
And they are expressed in terms
of these mixing coefficients
00:17:24.730 --> 00:17:25.810
for the states 1 and 2.
00:17:32.850 --> 00:17:56.470
Now, if we look at this, we can
see that rho 1, 1 plus rho 2, 2
00:17:56.470 --> 00:18:04.277
is equal to C1, C1
star plus C2, C2 star.
00:18:04.277 --> 00:18:05.860
And that's the
normalization integral.
00:18:05.860 --> 00:18:06.360
That's 1.
00:18:08.950 --> 00:18:16.650
And we have 1, 1; 1, 2
is equal to 2, 1 star.
00:18:16.650 --> 00:18:25.650
And so each rho is formation.
00:18:28.870 --> 00:18:34.330
So the density matrix
is normalized to 1.
00:18:34.330 --> 00:18:36.100
And it's Hermitian matrix.
00:18:36.100 --> 00:18:41.080
And we can use all sorts of
tricks for Hermitian matrices.
00:18:41.080 --> 00:18:45.170
Now we're interested in
the time dependence of rho.
00:18:45.170 --> 00:18:48.160
And so we're going to use
this wonderful equation up
00:18:48.160 --> 00:18:50.800
here in order to get the
time dependence of rho
00:18:50.800 --> 00:18:54.410
because rho like a,
is Hermitian operator.
00:18:54.410 --> 00:18:56.330
And so we could do that.
00:18:56.330 --> 00:19:01.360
And so the time
dependence of rho
00:19:01.360 --> 00:19:16.272
is going to be equal to the
time dependence of t, t.
00:19:16.272 --> 00:19:18.680
Where we operating first here.
00:19:18.680 --> 00:19:22.470
And then t time dependent.
00:19:35.220 --> 00:19:38.810
And when we do this,
what we end up getting--
00:19:38.810 --> 00:19:44.260
well, so we have a time
dependence of a wave function.
00:19:44.260 --> 00:19:46.770
So we use the time
dependence shorter equation.
00:19:46.770 --> 00:19:48.300
And we insert that.
00:19:48.300 --> 00:19:51.940
And using the time dependence
shorter equation we have things
00:19:51.940 --> 00:19:52.440
like--
00:20:15.390 --> 00:20:18.790
so every time we take
the wave function--
00:20:18.790 --> 00:20:25.420
the derivative of a function,
we get a Hamiltonian and so on.
00:20:25.420 --> 00:20:31.290
And so what we can express this
time dependence of the density
00:20:31.290 --> 00:20:36.410
matrix by just using the time--
inserting the time dependent
00:20:36.410 --> 00:20:38.310
shorter equation repeatedly.
00:20:38.310 --> 00:20:43.860
This is why I say this is
repackaging the Schrodinger
00:20:43.860 --> 00:20:46.740
picture, repackaging
the wave function
00:20:46.740 --> 00:20:54.120
and writing everything in
terms of these matrices.
00:21:02.010 --> 00:21:04.390
So that's the first--
00:21:04.390 --> 00:21:05.620
that's what happens here.
00:21:09.241 --> 00:21:09.740
Sorry.
00:21:09.740 --> 00:21:12.020
That's what happens here.
00:21:12.020 --> 00:21:18.470
And then we write this
one, and we get plus 1
00:21:18.470 --> 00:21:29.330
over minus i h-bar t, t H of t.
00:21:33.360 --> 00:21:42.210
And we recognize that that is
just one over i h-bar times H
00:21:42.210 --> 00:21:42.710
rho.
00:21:46.170 --> 00:21:48.900
So the time dependence
of the density matrix
00:21:48.900 --> 00:21:50.595
is given by this computator.
00:21:53.210 --> 00:21:56.150
And the computators are
kind of neat because usually
00:21:56.150 --> 00:22:00.050
what happens is these
two-level things have
00:22:00.050 --> 00:22:02.390
very different
structures, and you
00:22:02.390 --> 00:22:06.470
get rid of something you don't
want to deal with anymore.
00:22:06.470 --> 00:22:10.790
And so now we actually
evaluate these things.
00:22:10.790 --> 00:22:19.430
And we do a lot of algebra.
00:22:19.430 --> 00:22:23.690
And we get these equations
of motion for the elements
00:22:23.690 --> 00:22:25.740
of the density matrix.
00:22:25.740 --> 00:22:30.260
And so we find the
time dependence
00:22:30.260 --> 00:22:34.070
of the diagonal
element for state 1
00:22:34.070 --> 00:22:36.110
is opposite that for state 2.
00:22:36.110 --> 00:22:40.720
In other words,
population from state 1
00:22:40.720 --> 00:22:42.757
is being transferred
into state 2.
00:22:45.440 --> 00:22:50.900
And that is equal to
minus i over h-bar times
00:22:50.900 --> 00:23:01.250
H 1, 2 rho 2, 1 minus
h 2, 1 rho 1, 2.
00:23:05.930 --> 00:23:11.590
And we have rho 1,
2 time dependence
00:23:11.590 --> 00:23:19.380
is equal to rho 2, 1
time dependent star.
00:23:19.380 --> 00:23:26.530
And that comes out to be minus
i h-bar minus i over h-bar,
00:23:26.530 --> 00:23:40.800
H 1, 1 minus H 2, 2 rho 1,
2 rho 2, 2 minus rho 1, 1
00:23:40.800 --> 00:23:42.030
times H 1, 2.
00:23:44.760 --> 00:23:46.680
This is very
interesting, but now we
00:23:46.680 --> 00:23:48.930
have a couple
differential equations
00:23:48.930 --> 00:23:50.940
and we can solve them.
00:23:50.940 --> 00:23:54.700
But we want to do a trick
where we write the Hamiltonian
00:23:54.700 --> 00:23:57.885
as a sum of two terms.
00:24:00.660 --> 00:24:03.480
This is the time--
the independent part,
00:24:03.480 --> 00:24:05.460
and this is the
time dependent part.
00:24:05.460 --> 00:24:07.240
This is the part that
gives us trouble.
00:24:07.240 --> 00:24:09.930
This is the part that
takes us into territory
00:24:09.930 --> 00:24:13.490
that I haven't talked
about in time independent.
00:24:13.490 --> 00:24:15.720
But it's still-- it's
perturbation theory.
00:24:15.720 --> 00:24:20.130
This is supposed to
be something that
00:24:20.130 --> 00:24:23.880
is different from and
usually smaller than H 0.
00:24:23.880 --> 00:24:26.130
And so we do this.
00:24:26.130 --> 00:24:39.960
So H 0, operating on any
function gives En times n.
00:24:39.960 --> 00:24:42.030
And so we could
call these E zeroes,
00:24:42.030 --> 00:24:44.154
but we don't need
to do that anymore.
00:24:49.850 --> 00:24:52.565
And now we do a lot of algebra.
00:25:01.400 --> 00:25:04.700
We discover that the time
dependence of the density
00:25:04.700 --> 00:25:13.160
matrix is given by
minus i over h-bar times
00:25:13.160 --> 00:25:20.490
H 1 of t times the
density matrix.
00:25:20.490 --> 00:25:24.190
So this is very much like what
we did before, but now we have
00:25:24.190 --> 00:25:27.080
that the time dependence
is entirely due to the time
00:25:27.080 --> 00:25:28.370
independent Hamiltonian.
00:25:33.250 --> 00:25:35.650
So everything
associated with H 0
00:25:35.650 --> 00:25:38.050
is gone from this
equation of motion.
00:25:43.030 --> 00:25:47.320
So now let's just be specific.
00:25:47.320 --> 00:25:49.210
So here is a two-level
level system.
00:25:49.210 --> 00:25:51.330
This is state 1.
00:25:51.330 --> 00:25:53.750
This is state 2.
00:25:53.750 --> 00:25:59.280
This difference is delta E.
00:25:59.280 --> 00:26:03.950
And we're going to call
that h-bar omega 0.
00:26:03.950 --> 00:26:08.690
So this is the
frequency difference
00:26:08.690 --> 00:26:10.420
between levels 1 and 2.
00:26:13.300 --> 00:26:25.478
H 0 is equal to minus h h-omega
over 2 h-bar omega over 2 0, 0.
00:26:25.478 --> 00:26:26.702
We like that, right?
00:26:26.702 --> 00:26:27.285
It's diagonal.
00:26:31.940 --> 00:26:36.000
h1 is where all
the trouble comes.
00:26:36.000 --> 00:26:50.370
And we're going to call that
h-bar times e x 1, 2 E0 cosine
00:26:50.370 --> 00:26:52.620
omega t.
00:26:52.620 --> 00:26:55.430
This is not an energy.
00:26:55.430 --> 00:26:58.440
This is an electric field.
00:26:58.440 --> 00:27:01.940
So this is the strength
of the perturbation.
00:27:01.940 --> 00:27:08.000
And this is the dipole matrix
element between levels 1 and 2.
00:27:08.000 --> 00:27:14.610
So we have a dipole moment times
an electric field multiplied
00:27:14.610 --> 00:27:16.410
by h-bar.
00:27:16.410 --> 00:27:24.780
So this quantity here, has
units of angular frequency.
00:27:24.780 --> 00:27:30.580
And we call it omega 1,
which is the Rabi frequency.
00:27:30.580 --> 00:27:33.545
It gets a special name
because Rabi was special.
00:27:36.290 --> 00:27:38.860
And so we're going to be--
00:27:38.860 --> 00:27:42.530
and this is-- expresses the
strength of the interaction.
00:27:42.530 --> 00:27:46.230
So we have a molecular
antenna mu 1, 2.
00:27:46.230 --> 00:27:47.830
And we have the external field.
00:27:47.830 --> 00:27:50.180
And they're interacting
with each other.
00:27:50.180 --> 00:27:53.200
And so this is the
strength of the badness,
00:27:53.200 --> 00:27:54.220
except its goodness.
00:27:54.220 --> 00:27:55.750
Because we want to
see transitions.
00:28:00.710 --> 00:28:04.340
So now we do a little bit
of playing with notation
00:28:04.340 --> 00:28:07.815
because there's just a lot
of stuff that's going on.
00:28:07.815 --> 00:28:10.130
and we have to understand it.
00:28:10.130 --> 00:28:13.760
So we're going to
call the state--
00:28:13.760 --> 00:28:18.140
we're going to separate
the time dependent--
00:28:18.140 --> 00:28:21.290
the time independent part of
the wave functions from the time
00:28:21.290 --> 00:28:22.520
dependent.
00:28:22.520 --> 00:28:25.610
And so state 1--
00:28:25.610 --> 00:28:28.970
this is the full time
dependent wave function.
00:28:28.970 --> 00:28:41.561
And it's going to be
minus i omega 0 t over 2.
00:28:41.561 --> 00:28:48.160
in Other words, we should
have had zeros here--
00:28:52.160 --> 00:28:58.780
times 1, right.
00:28:58.780 --> 00:29:00.710
So this is the time
independent part,
00:29:00.710 --> 00:29:03.300
and this is the
time dependent part.
00:29:03.300 --> 00:29:15.420
And 2 is e to the minus i
omega 0 t over 2, 2 prime.
00:29:15.420 --> 00:29:19.680
Notice these two guys
have the same sign.
00:29:19.680 --> 00:29:23.190
This bothers me a lot.
00:29:23.190 --> 00:29:28.930
But it's true, because we
have opposite signs here,
00:29:28.930 --> 00:29:31.990
and we have a bra and a ket.
00:29:31.990 --> 00:29:34.480
And they end up
having the same signs.
00:29:37.870 --> 00:29:44.700
So that means that h
1 looks like this, 0,
00:29:44.700 --> 00:30:00.810
0 omega 1 cosine omega t e
to the minus i omega 0 t.
00:30:00.810 --> 00:30:11.770
And here we have omega 1 cosine
omega t e to the pi omega 0 t.
00:30:11.770 --> 00:30:13.640
So this is a 2 by 2 matrix.
00:30:13.640 --> 00:30:16.390
Diagonal elements are 0.
00:30:16.390 --> 00:30:19.960
Off diagonal elements
are this omega.
00:30:19.960 --> 00:30:23.170
The strength of the
interaction times the frequency
00:30:23.170 --> 00:30:29.650
of the applied radiation
times the oscillating factor.
00:30:39.960 --> 00:30:42.600
So now we go back
and we calculate
00:30:42.600 --> 00:30:48.600
the equation of motion,
bringing in this h 1 term.
00:30:48.600 --> 00:30:59.330
And so we have minus
i over h-bar h 1 rho.
00:30:59.330 --> 00:31:01.850
And we get some complicated
equations of motions.
00:31:05.430 --> 00:31:09.540
And I don't really
want to write them out,
00:31:09.540 --> 00:31:16.370
because it takes a while,
and they're in your notes.
00:31:16.370 --> 00:31:22.080
And I'm going to make the
crucial approximation,
00:31:22.080 --> 00:31:26.350
the rotating wave approximation.
00:31:26.350 --> 00:31:31.860
Notice we have a cosine omega t.
00:31:31.860 --> 00:31:35.620
We can write that as e
to the i omega t plus e
00:31:35.620 --> 00:31:37.560
to the minus i omega t.
00:31:37.560 --> 00:31:40.320
And so basically
what we're doing
00:31:40.320 --> 00:31:43.574
is we're going to do a trick.
00:31:43.574 --> 00:31:44.990
We have the
Hamiltonian, and we're
00:31:44.990 --> 00:31:48.040
going to go to a rotating
coordinate system.
00:31:48.040 --> 00:31:52.950
And if we choose the rotational
coordinate the rotation
00:31:52.950 --> 00:32:01.440
frequency right, we can almost
exactly cancel omega 0 terms.
00:32:01.440 --> 00:32:05.430
And so we have two terms,
one rotating like this,
00:32:05.430 --> 00:32:09.720
which is canceling or
trying to cancel omega 0,
00:32:09.720 --> 00:32:15.870
and one rotating like this,
which is adding to omega 0.
00:32:15.870 --> 00:32:19.100
And so what we end up getting
is a slowly oscillating term,
00:32:19.100 --> 00:32:22.640
which we like, and a
rapidly oscillating term,
00:32:22.640 --> 00:32:25.040
which we can throw away.
00:32:25.040 --> 00:32:27.620
That's the approximation.
00:32:27.620 --> 00:32:30.170
And this is commonly used.
00:32:30.170 --> 00:32:33.830
And I can write this in
terms of transformations.
00:32:33.830 --> 00:32:37.550
And although we think about
going to a rotating coordinate
00:32:37.550 --> 00:32:45.060
system, for each
two-level system,
00:32:45.060 --> 00:32:47.580
we can rotate at a
different frequency
00:32:47.580 --> 00:32:52.380
to cancel or make nearly
canceling the off diagonal
00:32:52.380 --> 00:32:53.100
elements.
00:32:53.100 --> 00:32:57.390
So although the
molecule doesn't rotate
00:32:57.390 --> 00:33:00.180
at different frequencies,
our transformation
00:33:00.180 --> 00:33:03.900
attacks the coupling
between states individually.
00:33:03.900 --> 00:33:06.870
And you can imply as
many rotating wave core
00:33:06.870 --> 00:33:08.550
transformations as you want.
00:33:08.550 --> 00:33:09.997
But we have a two-level system.
00:33:09.997 --> 00:33:10.830
So we only have one.
00:33:17.790 --> 00:33:28.320
And so we do this.
00:33:28.320 --> 00:33:35.940
And we skip a lot of steps,
because it's complicated
00:33:35.940 --> 00:33:39.600
and because we don't
have a lot of time.
00:33:39.600 --> 00:33:47.100
We now have the time
dependence of the 1, 1 element.
00:33:47.100 --> 00:33:48.930
And it's expressed as omega 1.
00:33:48.930 --> 00:33:50.760
I've skipped a lot of steps.
00:33:50.760 --> 00:33:53.010
But you can do those steps.
00:33:53.010 --> 00:33:55.500
The important thing is what
we're going to see here.
00:33:55.500 --> 00:34:06.760
We have e to the i omega
0 minus omega t rho 1, 2.
00:34:06.760 --> 00:34:12.000
And we have a minus e
to the minus i omega
00:34:12.000 --> 00:34:17.679
0 minus omega t times rho 2, 1.
00:34:20.659 --> 00:34:26.610
And we have 2, 2 dot is
equal to minus rho 1, 1 dot.
00:34:26.610 --> 00:34:30.090
And we have rho 1, 2 dot--
00:34:30.090 --> 00:34:32.380
this is the important guy--
00:34:32.380 --> 00:34:39.520
is equal to i omega 1 over
2 e to the minus i omega
00:34:39.520 --> 00:34:46.872
0 minus omega t rho
1, 1 minus rho 2, 2.
00:34:46.872 --> 00:34:49.330
So we have a whole bunch of
coupled differential equations,
00:34:49.330 --> 00:34:53.230
but each of them have
these factors here where
00:34:53.230 --> 00:34:57.220
you have omega 0 minus omega.
00:34:57.220 --> 00:34:59.830
I've thrown away the
omega 0 plus omega terms.
00:35:02.610 --> 00:35:04.940
And now it really
starts to look good,
00:35:04.940 --> 00:35:08.300
because we can make these--
00:35:08.300 --> 00:35:14.960
so when we make omega equal to
omega 0, well, this is just 1.
00:35:14.960 --> 00:35:16.290
Everything is simple.
00:35:16.290 --> 00:35:17.990
We're on resonance.
00:35:17.990 --> 00:35:25.650
And so what we do is we create
another symbol, delta omega,
00:35:25.650 --> 00:35:28.670
which is omega 0 minus omega.
00:35:28.670 --> 00:35:30.900
So this is the oscillating
frequency applied.
00:35:30.900 --> 00:35:33.772
This is the intrinsic level
spacing in the molecule.
00:35:37.840 --> 00:35:42.640
And so we can now
write the solution
00:35:42.640 --> 00:35:47.447
to this differential
equation for each
00:35:47.447 --> 00:35:49.030
of the elements of
the density matrix.
00:35:55.460 --> 00:35:57.680
And we're going to actually
define another symbol.
00:36:00.350 --> 00:36:04.750
We going to have the
symbol omega sub e.
00:36:04.750 --> 00:36:07.360
This is not the
vibrational frequency.
00:36:07.360 --> 00:36:10.690
This is just a symbol that
is used a lot in literature,
00:36:10.690 --> 00:36:17.710
and that it comes out to
be delta omega squared
00:36:17.710 --> 00:36:20.830
plus omega 1 squared.
00:36:24.120 --> 00:36:27.950
So in solving the
density matrix equation,
00:36:27.950 --> 00:36:34.490
it turns out we care about
this extra frequency.
00:36:34.490 --> 00:36:37.850
If delta omega is 0, well, then
there's nothing surprising.
00:36:37.850 --> 00:36:39.680
Well, maybe e is just omega 1.
00:36:42.320 --> 00:36:48.500
But this allows for there to
be an effect of the detuning.
00:36:48.500 --> 00:36:52.790
So basically what you're doing
is when you go to the rotating
00:36:52.790 --> 00:37:02.420
coordinate system, you have an
intrinsic frequency separation.
00:37:02.420 --> 00:37:07.320
And so in the rotating
coordinate system,
00:37:07.320 --> 00:37:09.320
you have two levels
that are different.
00:37:09.320 --> 00:37:12.251
And there's a stark
effect between them.
00:37:12.251 --> 00:37:14.150
And you diagonalize
this stark effect
00:37:14.150 --> 00:37:17.600
using second order
perturbation theory or just
00:37:17.600 --> 00:37:20.800
the diagonizing the matrix.
00:37:20.800 --> 00:37:24.200
And so that gives rise
to this extra term here,
00:37:24.200 --> 00:37:32.330
because you have the oscillation
frequency and the Rabi
00:37:32.330 --> 00:37:33.500
frequency.
00:37:33.500 --> 00:37:36.390
And anyway, when you
do the transformation,
00:37:36.390 --> 00:37:38.100
you get these terms.
00:37:38.100 --> 00:37:41.640
And so here is now the
solution in the rotating wave
00:37:41.640 --> 00:37:43.262
approximation.
00:37:43.262 --> 00:37:51.380
Rho 1, 1 is equal to 1 minus
omega 1 squared over omega
00:37:51.380 --> 00:37:58.760
e squared sine squared
omega 0 t over 2.
00:38:02.240 --> 00:38:08.520
We have rho 2, 2
is equal to just
00:38:08.520 --> 00:38:13.140
omega 1 squared over
the e squared sine
00:38:13.140 --> 00:38:20.050
squared omega e t over 2.
00:38:20.050 --> 00:38:22.320
We have omega 1, 2--
00:38:22.320 --> 00:38:26.030
rho 1, 2, which is equal to
something more complicated
00:38:26.030 --> 00:38:27.140
looking.
00:38:27.140 --> 00:38:37.260
Omega 1 over omega e
squared times i omega 0--
00:38:37.260 --> 00:38:50.530
omega e, sorry, or 2 sine omega
e t minus delta omega sine
00:38:50.530 --> 00:38:56.950
squared omega e t
over 2 times now
00:38:56.950 --> 00:39:03.790
e to the minus i delta omega t.
00:39:03.790 --> 00:39:05.260
It looks complicated.
00:39:05.260 --> 00:39:08.890
And we get a similar
term for who 2, 1.
00:39:08.890 --> 00:39:13.740
It's just equal to rho
1, 2 complex conjugate.
00:39:13.740 --> 00:39:20.310
And so now what we see
is these populations
00:39:20.310 --> 00:39:24.170
are oscillating not at--
00:39:27.180 --> 00:39:29.130
It's a e, not a 0.
00:39:32.680 --> 00:39:35.140
They're oscillating at a
slightly shifted frequency.
00:39:37.840 --> 00:39:40.710
But they're oscillating
sinusoidally.
00:39:40.710 --> 00:39:42.930
And we have an
amplitude term, which
00:39:42.930 --> 00:39:47.250
is omega 1 over omega
e quantity squared.
00:39:47.250 --> 00:39:50.850
Omega e is a little
bigger than omega 1.
00:39:50.850 --> 00:39:54.450
So this is less than 1.
00:39:54.450 --> 00:39:59.600
So it's just like the
situation in the absence
00:39:59.600 --> 00:40:03.290
of this oscillating
field, that you just
00:40:03.290 --> 00:40:07.820
get a slightly, slightly
shifted oscillation
00:40:07.820 --> 00:40:11.510
frequency, and a slightly
reduced co-factor.
00:40:14.690 --> 00:40:18.320
The coherence terms-- so these
are populations, populations
00:40:18.320 --> 00:40:20.460
going back and forth
between 1 and 2
00:40:20.460 --> 00:40:22.830
at a slightly shifted frequency.
00:40:22.830 --> 00:40:26.570
And then we have this,
which looks horrible.
00:40:29.510 --> 00:40:33.585
And now, for some more insights.
00:40:37.870 --> 00:40:45.970
If we make omega 1 much
larger than delta omega--
00:40:45.970 --> 00:40:48.340
in other words, the Rabi
frequency much larger
00:40:48.340 --> 00:40:52.330
than the tuning, it might
as well not be detuned.
00:40:52.330 --> 00:40:55.690
We get back the simple picture.
00:40:55.690 --> 00:41:03.670
We get rho 1 is equal to
cosine squared omega 1 of t
00:41:03.670 --> 00:41:05.430
over 2, et cetera.
00:41:12.350 --> 00:41:18.977
So we have what we
call free precession.
00:41:25.950 --> 00:41:29.270
Each of the elements
of the density--
00:41:29.270 --> 00:41:31.370
the density matrix
is telling you
00:41:31.370 --> 00:41:35.600
that the system is going
back and forth sinusoidally
00:41:35.600 --> 00:41:39.030
or co-sinusoidally
cosine squared.
00:41:39.030 --> 00:41:45.080
And what happens to
level 1 is the opposite
00:41:45.080 --> 00:41:46.790
of what happens at level 2.
00:41:46.790 --> 00:41:50.540
And everything is simple and
the system just oscillates.
00:42:01.860 --> 00:42:17.130
Suppose we apply radiation
or delta t a short time.
00:42:17.130 --> 00:42:20.970
And so what we're interested
in-- here is t equals 0.
00:42:20.970 --> 00:42:22.940
This is time.
00:42:22.940 --> 00:42:25.080
And this is t equals 0.
00:42:25.080 --> 00:42:27.690
And before t equals
0, we do something.
00:42:27.690 --> 00:42:28.830
We apply the radiation.
00:42:31.960 --> 00:42:37.340
And we apply the radiation
for a time, which gives rise
00:42:37.340 --> 00:42:39.015
to a certain flipping.
00:42:48.960 --> 00:42:51.600
And so what we choose.
00:42:51.600 --> 00:42:59.560
We have delta t is equal
to theta over omega 1,
00:42:59.560 --> 00:43:06.860
or theta is equal to delta t
omega 1, the Rabi frequency.
00:43:06.860 --> 00:43:11.280
And so if we choose a
flip angle, which we call
00:43:11.280 --> 00:43:17.200
say a pi pulse, theta is a pi.
00:43:17.200 --> 00:43:22.360
And what ends up happening is
that we transfer population
00:43:22.360 --> 00:43:25.390
entirely from
level 1 to level 2.
00:43:36.250 --> 00:43:42.780
When we do that, we get
no off diagonal elements
00:43:42.780 --> 00:43:43.970
of the density matrix.
00:43:43.970 --> 00:43:45.480
They are zero.
00:43:45.480 --> 00:43:55.330
So if at t equals 0, we
have everything in level 1,
00:43:55.330 --> 00:44:02.560
and we have applied this
0 pulse or a pi pulse,
00:44:02.560 --> 00:44:03.760
we have no coherence.
00:44:10.130 --> 00:44:17.910
If we have a pi over
2 pulse, well then,
00:44:17.910 --> 00:44:22.890
we've equalized the
two-level populations,
00:44:22.890 --> 00:44:24.870
and we created a
maximum coherence.
00:44:27.990 --> 00:44:30.600
And this guy radiates.
00:44:30.600 --> 00:44:33.035
So now we have an
oscillating dipole.
00:44:33.035 --> 00:44:36.980
And it's broadcasting radiation.
00:44:36.980 --> 00:44:40.280
And so all of the
two-level systems,
00:44:40.280 --> 00:44:44.820
if you use a flip
angle of pi over 2,
00:44:44.820 --> 00:44:49.010
you get a maximum
polarization, they're
00:44:49.010 --> 00:44:52.680
radiating to my detector,
which is up there.
00:44:52.680 --> 00:44:53.850
And I'm happy.
00:44:53.850 --> 00:44:58.480
I detect their
resonance frequency.
00:44:58.480 --> 00:45:01.950
And so the experiments work.
00:45:01.950 --> 00:45:08.580
So we're pretty much done.
00:45:08.580 --> 00:45:11.900
So I mean, what we
are doing is we're
00:45:11.900 --> 00:45:18.020
creating a time
dependent dipole.
00:45:18.020 --> 00:45:21.380
And that dipole radiates
something which we call--
00:45:27.280 --> 00:45:31.810
if we have a sample
like this, that sample--
00:45:31.810 --> 00:45:34.000
all of the molecules
in the sample
00:45:34.000 --> 00:45:40.150
are contributing to the
radiation of this dipole.
00:45:40.150 --> 00:45:44.410
But they all have slightly
different frequencies,
00:45:44.410 --> 00:45:49.120
because the field that
polarized them wasn't uniform.
00:45:49.120 --> 00:45:52.230
In a perfect
experiment it would be.
00:45:52.230 --> 00:45:54.972
And so they have
different frequencies
00:45:54.972 --> 00:45:56.055
and they get out of phase.
00:45:59.530 --> 00:46:04.300
Or conservation of energy
as the two-level system
00:46:04.300 --> 00:46:06.550
radiates from the
situation where
00:46:06.550 --> 00:46:10.330
you have equal populations to
everybody in the lowest state,
00:46:10.330 --> 00:46:11.290
there is a decay.
00:46:11.290 --> 00:46:18.910
So there's decays that
causes the signal, which
00:46:18.910 --> 00:46:25.070
we call free induction
decay, to dephase or decay.
00:46:25.070 --> 00:46:27.917
But the important thing
is, you observe the signal
00:46:27.917 --> 00:46:29.500
and it tells you
what you want to know
00:46:29.500 --> 00:46:32.770
about the level system, the
two-level system, or the end
00:46:32.770 --> 00:46:35.000
level system.
00:46:35.000 --> 00:46:39.760
And it's a very powerful way of
understanding the interaction
00:46:39.760 --> 00:46:42.100
of radiation with
matter, because it
00:46:42.100 --> 00:46:45.670
focuses on near resonance.
00:46:45.670 --> 00:46:49.170
And near resonance for
one two-level system
00:46:49.170 --> 00:46:51.100
is not near resonance.
00:46:51.100 --> 00:46:54.430
For another-- and so
you're picking out one,
00:46:54.430 --> 00:46:56.650
and you get really good signals.
00:46:56.650 --> 00:46:58.540
And you can actually do--
00:46:58.540 --> 00:47:00.460
by chirping the pulse--
00:47:00.460 --> 00:47:03.909
you can have one two-level
system, and a little bit later,
00:47:03.909 --> 00:47:05.200
another two-level-level system.
00:47:05.200 --> 00:47:06.130
They all radiate.
00:47:06.130 --> 00:47:07.660
They all get polarized.
00:47:07.660 --> 00:47:10.270
They all radiate at
their own frequency.
00:47:10.270 --> 00:47:13.540
And you can detect the
signal in the time domain
00:47:13.540 --> 00:47:16.120
and get everything you want
in a simple experiment.
00:47:16.120 --> 00:47:19.630
This experiment has enabled
us to do spectroscopy
00:47:19.630 --> 00:47:22.860
a million times faster
than was possible before.
00:47:22.860 --> 00:47:25.690
A million is a big number.
00:47:25.690 --> 00:47:29.160
And so I think it's important.
00:47:29.160 --> 00:47:34.310
And I think that this
sort of theory is germane,
00:47:34.310 --> 00:47:38.480
not just for high
resolution frequency domain
00:47:38.480 --> 00:47:41.660
experiments, in
fact, it's basically
00:47:41.660 --> 00:47:42.830
a time domain experiment.
00:47:42.830 --> 00:47:45.320
You're detecting something
in the time domain,
00:47:45.320 --> 00:47:49.440
and Fourier transferring
back to the frequency domain.
00:47:49.440 --> 00:47:51.050
So there are ultra
fast experiments
00:47:51.050 --> 00:47:54.440
where you create
polarizations and they--
00:47:54.440 --> 00:47:57.170
it is what is modern
experimental physical
00:47:57.170 --> 00:47:59.430
chemistry.
00:47:59.430 --> 00:48:02.880
And the notes that
I will produce
00:48:02.880 --> 00:48:06.300
will be far clearer than
these lectures, this lecture.
00:48:06.300 --> 00:48:09.190
But it really is a gateway.
00:48:09.190 --> 00:48:12.530
And I hope that some of you
will walk through that gateway.
00:48:12.530 --> 00:48:16.170
And it's been a pleasure
for me lecturing to you
00:48:16.170 --> 00:48:18.840
for the last time in 5.61.
00:48:18.840 --> 00:48:22.020
I really enjoyed doing this.
00:48:22.020 --> 00:48:24.725
Thanks.
00:48:24.725 --> 00:48:27.620
[APPLAUSE]
00:48:27.620 --> 00:48:28.916
Thank them.
00:48:28.916 --> 00:48:32.740
[APPLAUSE]
00:48:32.740 --> 00:48:34.340
Well, I got to take
the hydrogen atom.
00:48:34.340 --> 00:48:38.240
[LAUGHTER]
00:48:38.240 --> 00:48:39.756
Thank you.