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00:00:21,540 --> 00:00:24,270
PROFESSOR: Good afternoon.

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00:00:24,270 --> 00:00:25,570
We are on the finish line.

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00:00:25,570 --> 00:00:27,610
Two more weeks to go.

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00:00:27,610 --> 00:00:31,740
Our last chapter is coherence,
and I can promise you

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00:00:31,740 --> 00:00:35,390
this chapter of coherence
has some highlights,

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00:00:35,390 --> 00:00:40,030
so we are not going to
and more boring subject.

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00:00:40,030 --> 00:00:41,950
Actually, some of
the best subjects,

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00:00:41,950 --> 00:00:48,000
some of the most exciting
topics are still to come.

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00:00:48,000 --> 00:00:52,780
So today, we continue our
discussion of coherence.

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00:00:52,780 --> 00:00:57,240
As I pointed out
last week, we first

18
00:00:57,240 --> 00:00:59,480
talk about coherence
in single atoms

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00:00:59,480 --> 00:01:01,040
and then coherence
between atoms.

20
00:01:05,129 --> 00:01:09,150
In the first part
on coherence, I

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00:01:09,150 --> 00:01:12,460
want to come back to this
topic of spontaneous emission,

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00:01:12,460 --> 00:01:17,120
because many of us have
deep rooted misconceptions

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00:01:17,120 --> 00:01:19,040
about what spontaneous
emission is.

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00:01:21,970 --> 00:01:28,470
We discussed on Wednesday that
spontaneous emission is not

25
00:01:28,470 --> 00:01:33,830
so spontaneous as many
of us assume because it's

26
00:01:33,830 --> 00:01:37,040
a unitary time evolution
with an operator

27
00:01:37,040 --> 00:01:38,825
with a term in the Hamiltonian.

28
00:01:38,825 --> 00:01:41,680
It is exactly this
operator which

29
00:01:41,680 --> 00:01:45,460
takes the wave function of
the total system, the atoms

30
00:01:45,460 --> 00:01:48,940
and the light, to
whatever it is later on.

31
00:01:48,940 --> 00:01:52,520
There is no random phase,
there is no random variable

32
00:01:52,520 --> 00:01:57,680
in this time evolution,
exclamation mark.

33
00:01:57,680 --> 00:02:01,200
But there are certain
aspects associated

34
00:02:01,200 --> 00:02:05,300
with spontaneous emission,
and I want to address them.

35
00:02:05,300 --> 00:02:08,539
On the other hand, if you think
about spontaneous emission

36
00:02:08,539 --> 00:02:11,150
in the most fundamental way,
the first thing you should think

37
00:02:11,150 --> 00:02:13,470
about it, vacuum
Rabi oscillation.

38
00:02:13,470 --> 00:02:16,860
Here you see in the
simplest possible system

39
00:02:16,860 --> 00:02:18,870
what spontaneous
emission can do for you.

40
00:02:23,540 --> 00:02:28,710
The way we want to discuss
an important aspect

41
00:02:28,710 --> 00:02:32,550
of spontaneous emission, we want
to go beyond the vacuum Rabi

42
00:02:32,550 --> 00:02:34,930
oscillation, is the following.

43
00:02:34,930 --> 00:02:37,250
We start with an atom
in the ground state

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00:02:37,250 --> 00:02:40,740
and the cavity is
in the vacuum state,

45
00:02:40,740 --> 00:02:43,550
but now we take a
short pulse of a laser

46
00:02:43,550 --> 00:02:45,700
and we prepare the atom.

47
00:02:45,700 --> 00:02:50,440
And because the laser
outputs a coherent state,

48
00:02:50,440 --> 00:02:53,050
the coherent state has
a well defined phase,

49
00:02:53,050 --> 00:02:56,190
and this phase appears
in the superposition

50
00:02:56,190 --> 00:02:59,290
between ground and excited
state because this superposition

51
00:02:59,290 --> 00:03:01,400
is created with the
matrix element which

52
00:03:01,400 --> 00:03:04,670
has the electric
field of the laser.

53
00:03:04,670 --> 00:03:09,400
But then we allow spontaneous
emission to happen,

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00:03:09,400 --> 00:03:11,140
and spontaneous
emission to happen

55
00:03:11,140 --> 00:03:14,310
means we take our operator
which I just showed you,

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00:03:14,310 --> 00:03:17,510
we propagate forward
in time in such a way

57
00:03:17,510 --> 00:03:21,300
that we just go through half
a cycle of a vacuum Rabi

58
00:03:21,300 --> 00:03:24,170
oscillation, which
means everything which

59
00:03:24,170 --> 00:03:27,970
was in the excited state
is now in the ground state.

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00:03:27,970 --> 00:03:32,700
And by just exactly propagating
this system forward in time,

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00:03:32,700 --> 00:03:36,650
we obtain this state,
and that's something

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00:03:36,650 --> 00:03:40,080
I hope very, very
insightful which

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00:03:40,080 --> 00:03:43,180
we arrived at the end
of the last lecture,

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that the quantum
state of the atom

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00:03:47,230 --> 00:03:52,010
has been perfectly mapped
onto the photon field.

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00:03:52,010 --> 00:03:54,900
So all the information
which was in the atom

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00:03:54,900 --> 00:03:58,120
before spontaneous
emission is now

68
00:03:58,120 --> 00:04:01,360
available in the photon field.

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00:04:01,360 --> 00:04:05,880
So the next thing to
address is the phase phi.

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00:04:05,880 --> 00:04:09,810
What is the phase of the
spontaneously emitted photon,

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00:04:09,810 --> 00:04:11,643
and this is what we
want to understand now.

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00:04:23,330 --> 00:04:31,130
So how well can we
measure the phase phi?

73
00:04:31,130 --> 00:04:34,900
You should first assume
the phase phi is perfectly

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00:04:34,900 --> 00:04:37,960
determined with
extremely high accuracy

75
00:04:37,960 --> 00:04:39,720
if you use a laser
beam which has

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00:04:39,720 --> 00:04:41,490
a macroscopic electric field.

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00:04:41,490 --> 00:04:43,700
The phase phi is a
classical variable

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00:04:43,700 --> 00:04:46,600
and can be determined
with arbitrary precision.

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00:04:46,600 --> 00:04:50,530
And therefore the phase phi,
which we have imprinted first

80
00:04:50,530 --> 00:04:56,190
into the atomic wave function
and then in the photon field,

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00:04:56,190 --> 00:04:57,220
is an exact number.

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00:04:57,220 --> 00:04:59,250
It comes from the laser beam.

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00:05:05,320 --> 00:05:11,910
What I'm showing here
is phase space plots

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00:05:11,910 --> 00:05:16,190
for the photon field.

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00:05:16,190 --> 00:05:20,570
I know we talk about
photons and two

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00:05:20,570 --> 00:05:26,080
dimensional phase-space
distribution mainly in 8-422,

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00:05:26,080 --> 00:05:28,905
but I think the pictures
speak for themselves.

88
00:05:34,990 --> 00:05:37,510
A lot of you have seen the
harmonic oscillator and

89
00:05:37,510 --> 00:05:40,930
harmonic oscillator,
if you start

90
00:05:40,930 --> 00:05:45,320
to prepare the system here, this
is position, this is momentum,

91
00:05:45,320 --> 00:05:47,530
the system evolves in a circle.

92
00:05:47,530 --> 00:05:51,630
A lot of you have seen, if we
regard photon states as states

93
00:05:51,630 --> 00:05:54,030
of a harmonic oscillator,
which they are,

94
00:05:54,030 --> 00:05:57,690
that you have flux states,
which are just circles,

95
00:05:57,690 --> 00:06:03,170
or the vacuum state is just
a tiny circle at the center.

96
00:06:03,170 --> 00:06:05,040
And if you have
a coherent state,

97
00:06:05,040 --> 00:06:07,625
a coherent state is maybe
a little blob out there,

98
00:06:07,625 --> 00:06:11,070
and for the coherent state,
you can determine the phase

99
00:06:11,070 --> 00:06:14,610
because the angle of this little
blob relative to the origin

100
00:06:14,610 --> 00:06:16,760
is well determined.

101
00:06:16,760 --> 00:06:19,770
I think you all have
seen a version of that.

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00:06:19,770 --> 00:06:24,350
So anyway, what
is done here is I

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00:06:24,350 --> 00:06:29,300
show you this phase-space
plot for the photon field,

104
00:06:29,300 --> 00:06:33,550
and what happens is if
initially, the excited

105
00:06:33,550 --> 00:06:37,410
state was zero, this is
just the ground state

106
00:06:37,410 --> 00:06:39,100
of the harmonic oscillator.

107
00:06:39,100 --> 00:06:40,880
It's a circle.

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00:06:40,880 --> 00:06:43,600
If the excited
state was occupied

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00:06:43,600 --> 00:06:47,890
with unity probability, it's
a flux state with n equals 1,

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00:06:47,890 --> 00:06:51,720
and here you see the phase-space
plot of a flux state with n

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00:06:51,720 --> 00:06:52,770
equals 1.

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00:06:52,770 --> 00:06:56,550
And of course, you realize if
you have exactly one photon

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00:06:56,550 --> 00:06:59,070
or one atom in an excited state.

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00:06:59,070 --> 00:07:01,480
There is no phase
information left

115
00:07:01,480 --> 00:07:05,450
because the phase is
actually the relative phase

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00:07:05,450 --> 00:07:08,400
in the superposition between
ground and excited state.

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00:07:08,400 --> 00:07:10,150
If you have an
excited state, only

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00:07:10,150 --> 00:07:11,990
an excited state
with a phase factor,

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00:07:11,990 --> 00:07:15,490
you know if a phase factor
can simply be factored out

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00:07:15,490 --> 00:07:18,150
of the total wave function,
it's never measurable.

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00:07:18,150 --> 00:07:20,470
What is measurable
are phases which

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00:07:20,470 --> 00:07:26,960
are relative phases between two
amplitudes which are populated.

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00:07:26,960 --> 00:07:30,140
And of course, not
surprisingly, if we now

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00:07:30,140 --> 00:07:34,880
vary the excited state
fraction of the atom, that's

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00:07:34,880 --> 00:07:38,040
a probability to have a
photon in the photon field

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00:07:38,040 --> 00:07:42,330
from zero or one in
between, we sort of see

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00:07:42,330 --> 00:07:44,820
that this phase-space
distribution, it

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00:07:44,820 --> 00:07:50,580
points along the 45 degree axis
and we can measure the phase.

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00:07:50,580 --> 00:07:52,880
And the most accurate
phase measurement

130
00:07:52,880 --> 00:07:56,210
can be done if the superposition
between ground and excited

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00:07:56,210 --> 00:08:00,490
state is 50-50, or, talking
about the photon field,

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00:08:00,490 --> 00:08:03,795
we have a 50-50 superposition
state between no photon and one

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00:08:03,795 --> 00:08:05,260
photon.

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00:08:05,260 --> 00:08:09,650
But the phase here is
indetermined and the phase here

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00:08:09,650 --> 00:08:16,520
has quite a bit of
variance because if you

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00:08:16,520 --> 00:08:20,550
have a single photon,
there's only so much accuracy

137
00:08:20,550 --> 00:08:23,580
for the phase.

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00:08:23,580 --> 00:08:25,350
It would require
more discussion,

139
00:08:25,350 --> 00:08:27,845
but sometimes you talk even
about an uncertainty relation,

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00:08:27,845 --> 00:08:30,460
delta n delta phi equals 1.

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00:08:30,460 --> 00:08:32,150
So if you have one
photon, you can only

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00:08:32,150 --> 00:08:35,090
measure the phase with
precision on the order of unity.

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00:08:35,090 --> 00:08:37,070
If you had millions
of photons, then you

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00:08:37,070 --> 00:08:40,889
can do very accurate
phase measurements.

145
00:08:45,710 --> 00:08:50,340
So what we have is-- let me
just summarize the conclusion.

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00:08:50,340 --> 00:09:07,230
So the phase phi is best
defined in the atom,

147
00:09:07,230 --> 00:09:12,300
and therefore also in
the photon field, when

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00:09:12,300 --> 00:09:18,950
we have an equal superposition
of spin up and spin down,

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00:09:18,950 --> 00:09:20,760
of ground and excited state.

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00:09:20,760 --> 00:09:24,660
And you also get that from the
Bloch vector picture, if you

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00:09:24,660 --> 00:09:26,750
have a Bloch vector which
is pointing like that,

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00:09:26,750 --> 00:09:29,000
it doesn't have a phase,
it's just pointing up.

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00:09:29,000 --> 00:09:31,420
If it's pointing down,
there is no phase.

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00:09:31,420 --> 00:09:34,260
But if it's a 50%
superposition state,

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00:09:34,260 --> 00:09:38,040
it points in the xy plane and
you have the best definition

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00:09:38,040 --> 00:09:40,905
of the relative phase of
the amplitude between ground

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00:09:40,905 --> 00:09:41,930
and excited state.

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00:10:06,964 --> 00:10:09,130
So I mentioned this Heisenberg
uncertainty relation.

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00:10:11,870 --> 00:10:15,070
The fact is just looking
at these phase-space plots,

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00:10:15,070 --> 00:10:24,980
you realize the angle
which we can determine here

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00:10:24,980 --> 00:10:28,890
for the photon distribution
will have quite a variance,

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00:10:28,890 --> 00:10:30,540
but now I want to
discuss with you

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00:10:30,540 --> 00:10:32,180
how would we
actually go about it,

164
00:10:32,180 --> 00:10:34,910
how would we measure the
phase of the photon field?

165
00:10:34,910 --> 00:10:40,150
And this requires a homodyne
experiment, a beat experiment

166
00:10:40,150 --> 00:10:44,310
where we interfere the emitted
photon with a local oscillator,

167
00:10:44,310 --> 00:10:46,860
which is the laser beam which
was used in the first place

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00:10:46,860 --> 00:10:50,610
to excite the atom.

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00:10:50,610 --> 00:10:53,730
And what we will find out,
and it's clear that we cannot

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00:10:53,730 --> 00:10:59,800
obtain a sharp
value of the phase,

171
00:10:59,800 --> 00:11:04,960
but these fluctuations in the
phase do not come from any

172
00:11:04,960 --> 00:11:09,569
partial trace, do not
come from any fluctuations

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00:11:09,569 --> 00:11:10,360
in the Hamiltonian.

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00:11:13,430 --> 00:11:15,280
Just to address that,
when we write down

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00:11:15,280 --> 00:11:20,030
the term in the Hamiltonian,
the e dot d term,

176
00:11:20,030 --> 00:11:24,930
yes, depending on the
basis set, depending

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00:11:24,930 --> 00:11:27,270
how you define
spin up, spin down,

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00:11:27,270 --> 00:11:29,920
and what phase factors you
put into your basis set,

179
00:11:29,920 --> 00:11:33,300
you may have a phase popping
up in the Hamiltonian,

180
00:11:33,300 --> 00:11:35,750
but this phase is
purely definitional.

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00:11:35,750 --> 00:11:37,950
The phase I'm talking
about is really

182
00:11:37,950 --> 00:11:40,470
a relative phase
between two amplitudes,

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00:11:40,470 --> 00:11:43,590
and it is independent
of a phase which

184
00:11:43,590 --> 00:11:47,997
may be your choice by choosing
the basis set in which you

185
00:11:47,997 --> 00:11:49,080
formulate the Hamiltonian.

186
00:11:52,550 --> 00:11:59,780
Therefore, when we
measure the phase,

187
00:11:59,780 --> 00:12:08,530
and we find that there
are fluctuations,

188
00:12:08,530 --> 00:12:14,560
they actually come
form the quantum nature

189
00:12:14,560 --> 00:12:15,576
of the states involved.

190
00:12:23,280 --> 00:12:28,380
Let's talk about the
measurement and let

191
00:12:28,380 --> 00:12:30,610
me set it up genetically.

192
00:12:30,610 --> 00:12:36,610
Here is our atom, here
is the laser beam,

193
00:12:36,610 --> 00:12:44,250
and we want to create a
Mach-Zehnder interferometer.

194
00:12:47,660 --> 00:12:49,730
Let me just use another
color for the laser beam.

195
00:12:57,080 --> 00:12:59,060
Why don't we take
sodium today, which

196
00:12:59,060 --> 00:13:03,360
has emission in the orange?

197
00:13:07,090 --> 00:13:08,900
The idea is the following.

198
00:13:08,900 --> 00:13:13,670
We have a laser beam which
is used to excite the atom,

199
00:13:13,670 --> 00:13:16,940
and here we have a switch.

200
00:13:16,940 --> 00:13:22,682
And what we let through
is only a certain pulse.

201
00:13:22,682 --> 00:13:24,890
Let's say if we want to have
a coherent superposition

202
00:13:24,890 --> 00:13:26,420
between ground
and excited state,

203
00:13:26,420 --> 00:13:29,510
it would be pi over 2 pulse.

204
00:13:29,510 --> 00:13:32,910
Then, after the atom
has absorbed the pulse,

205
00:13:32,910 --> 00:13:37,090
we switch off the light pass.

206
00:13:37,090 --> 00:13:45,230
So then in the second
stage, the atom can emit,

207
00:13:45,230 --> 00:13:50,530
and the emitted light interferes
with the local oscillator,

208
00:13:50,530 --> 00:13:55,760
which is the laser beam, and
we can measure the beat node

209
00:13:55,760 --> 00:13:58,700
on the detector.

210
00:13:58,700 --> 00:14:04,045
This is the scheme how we
do a homodyne measurement.

211
00:14:10,439 --> 00:14:12,480
And so we assume we have
a very short pulse which

212
00:14:12,480 --> 00:14:13,490
excites the atom.

213
00:14:13,490 --> 00:14:17,980
Then we switch off the laser
in the upper pass and the light

214
00:14:17,980 --> 00:14:19,860
which reaches the
detector for homodyne

215
00:14:19,860 --> 00:14:22,270
is only the light
which has been emitted

216
00:14:22,270 --> 00:14:25,130
by the atom maybe
a nanosecond later.

217
00:14:25,130 --> 00:14:33,940
So we do a homodyne measurement
of the phase of the wave

218
00:14:33,940 --> 00:14:38,635
or the wave train
emitted by the atom.

219
00:14:45,280 --> 00:14:50,190
And the distribution of
measurements for the phase,

220
00:14:50,190 --> 00:14:52,326
I don't want to give you
mathematical expressions,

221
00:14:52,326 --> 00:14:53,700
but it's pretty
much what you can

222
00:14:53,700 --> 00:14:58,200
read from the drawing
I've shown you.

223
00:14:58,200 --> 00:15:13,024
So for a pi over 2 pulse,
we retrieve the phase phi,

224
00:15:13,024 --> 00:15:13,940
but with fluctuations.

225
00:15:18,270 --> 00:15:21,590
Let's now come to
the interesting case

226
00:15:21,590 --> 00:15:22,790
that we have a pi pulse.

227
00:15:25,310 --> 00:15:31,530
The pi pulse prepares the
atom in an excited state,

228
00:15:31,530 --> 00:15:36,170
and at t equals 0
after the excitation,

229
00:15:36,170 --> 00:15:38,370
there's absolutely no coherence.

230
00:15:38,370 --> 00:15:41,600
The density operator
for the atom

231
00:15:41,600 --> 00:15:45,930
has just one in the column
and row for the excited state.

232
00:15:45,930 --> 00:15:48,060
There's no off diagonal
matrix element.

233
00:15:48,060 --> 00:15:50,010
There is no phase information.

234
00:15:50,010 --> 00:15:55,230
So at t equals 0, no
coherence, no phase.

235
00:15:58,480 --> 00:16:02,200
So now we have excited
the atom with a pi pulse,

236
00:16:02,200 --> 00:16:05,190
but there is no
phase information

237
00:16:05,190 --> 00:16:08,470
in the atomic system,
and that would also

238
00:16:08,470 --> 00:16:12,500
mean that when we now
start mapping the quantum

239
00:16:12,500 --> 00:16:15,560
state of the atom onto the
quantum state of the light,

240
00:16:15,560 --> 00:16:19,900
there won't be any specific
phase for the light.

241
00:16:19,900 --> 00:16:23,810
We could say after the
spontaneous emission is over

242
00:16:23,810 --> 00:16:25,860
before we do any
measurement process,

243
00:16:25,860 --> 00:16:29,980
we have mapped a flux state
of the atom onto a flux state,

244
00:16:29,980 --> 00:16:32,450
n equals 1, of the
photon field, and there

245
00:16:32,450 --> 00:16:37,470
is no phase associated
with a number state.

246
00:16:37,470 --> 00:16:40,090
But let's be a little
bit more specific here.

247
00:16:40,090 --> 00:16:43,820
Let's assume we can have
an ensemble of atoms,

248
00:16:43,820 --> 00:16:46,330
we can repeat the
measurement many times,

249
00:16:46,330 --> 00:16:50,880
and let's ask the question, what
happens after the atom which

250
00:16:50,880 --> 00:16:55,840
was originally in the excited
state has decayed to 50%?

251
00:16:55,840 --> 00:17:01,510
Well, then we have a
wave function which

252
00:17:01,510 --> 00:17:09,550
is a superposition of
ground and excited state,

253
00:17:09,550 --> 00:17:18,760
and there is a phase phi
now, but this phase phi

254
00:17:18,760 --> 00:17:19,845
is completely random.

255
00:17:24,450 --> 00:17:27,609
So for those of you who are
concerned that I call it a wave

256
00:17:27,609 --> 00:17:30,690
function, you can
be more specific

257
00:17:30,690 --> 00:17:32,710
in the sense of
quantum Monte Carlo,

258
00:17:32,710 --> 00:17:35,400
that at any quantum Monte Carlo
wave function, at any given

259
00:17:35,400 --> 00:17:43,250
moment you have a wave function,
but the ensemble of your atom

260
00:17:43,250 --> 00:17:45,320
is now an ensemble
of all those wave

261
00:17:45,320 --> 00:17:47,650
functions with a
random phase phi.

262
00:17:47,650 --> 00:17:50,360
This is a way how
you can decompose

263
00:17:50,360 --> 00:17:53,120
the statistical
operator of the system,

264
00:17:53,120 --> 00:17:56,460
but the result is
the phase is random.

265
00:17:56,460 --> 00:18:00,520
If the phase is random,
that means no coherence.

266
00:18:00,520 --> 00:18:02,070
The statistical
operator does not

267
00:18:02,070 --> 00:18:04,510
have an off diagonal
matrix element.

268
00:18:04,510 --> 00:18:08,610
It also means that,
if you would ask

269
00:18:08,610 --> 00:18:14,640
what is the ensemble average
of the dipole moment,

270
00:18:14,640 --> 00:18:17,870
the dipole moment is
given by the Bloch vector.

271
00:18:17,870 --> 00:18:23,920
Well, if all phases are equally
populated in your ensemble,

272
00:18:23,920 --> 00:18:28,960
the dipole moment
average is 2-0.

273
00:18:28,960 --> 00:18:33,190
But, of course, you
have a d squared value,

274
00:18:33,190 --> 00:18:36,890
a value of the dipole
moment, which is not zero.

275
00:18:40,690 --> 00:18:45,530
So here we have now a situation
where the photon field has

276
00:18:45,530 --> 00:18:50,990
a random phase because
we lost the phase

277
00:18:50,990 --> 00:18:52,690
information of the
laser beam when

278
00:18:52,690 --> 00:18:56,080
we put the atom into
an excited state,

279
00:18:56,080 --> 00:19:03,907
and you may now ask, what
is the origin of this phase

280
00:19:03,907 --> 00:19:04,406
uncertainty?

281
00:19:09,380 --> 00:19:11,490
And at least the
qualitative answer

282
00:19:11,490 --> 00:19:12,896
is it's vacuum fluctuations.

283
00:19:19,790 --> 00:19:22,540
You can take the concept
of vacuum fluctuations

284
00:19:22,540 --> 00:19:23,680
a little bit further.

285
00:19:23,680 --> 00:19:28,240
I'm just mentioning it,
but I will not work it out.

286
00:19:28,240 --> 00:19:33,020
The fact that the phase of
this photon which is random

287
00:19:33,020 --> 00:19:37,750
is somewhat associated
with vacuum fluctuations,

288
00:19:37,750 --> 00:19:42,165
you can address this question
when you talk about two atoms.

289
00:19:44,920 --> 00:19:47,060
So we have two atoms.

290
00:19:47,060 --> 00:19:52,190
We excite them both with our
pi pulse into an excited state,

291
00:19:52,190 --> 00:19:57,300
and then, as time
goes by, we will

292
00:19:57,300 --> 00:20:03,320
have atoms which create photons.

293
00:20:06,970 --> 00:20:08,510
And at least as
long as the atoms

294
00:20:08,510 --> 00:20:13,320
are well localized within
optical wavelengths,

295
00:20:13,320 --> 00:20:16,180
you could play with the
idea that if they're

296
00:20:16,180 --> 00:20:19,290
vacuum fluctuations,
maybe the two atoms

297
00:20:19,290 --> 00:20:21,600
will see the same
vacuum fluctuations.

298
00:20:27,820 --> 00:20:35,144
And therefore, indeed, you will
actually observe correlations

299
00:20:35,144 --> 00:20:36,060
in the relative phase.

300
00:20:39,980 --> 00:20:43,940
So if you measure the phase of
the light emitted spontaneously

301
00:20:43,940 --> 00:20:48,150
by the two atoms, you
will find a correlation

302
00:20:48,150 --> 00:20:52,220
which is due to the fact that--
I'm waving my hands here,

303
00:20:52,220 --> 00:20:54,750
but that the spontaneous
emission was triggered

304
00:20:54,750 --> 00:20:57,370
by the same random
vacuum fluctuations.

305
00:20:57,370 --> 00:21:00,670
So the absolute phase will be
completely random from time

306
00:21:00,670 --> 00:21:06,444
to time, but the relative
phase will be correlated.

307
00:21:06,444 --> 00:21:07,860
But what we are
talking about here

308
00:21:07,860 --> 00:21:09,670
is correlations
between two atoms.

309
00:21:12,650 --> 00:21:20,430
We will talk later
about superradiance,

310
00:21:20,430 --> 00:21:24,400
and maybe this will
make it much clearer

311
00:21:24,400 --> 00:21:28,140
what it means if several atoms
emit spontaneously together.

312
00:21:31,730 --> 00:21:34,520
Any questions?

313
00:21:34,520 --> 00:21:35,494
Yes, Colin?

314
00:21:35,494 --> 00:21:36,994
AUDIENCE: Are there
any requirements

315
00:21:36,994 --> 00:21:40,396
on these two atoms being located
within an optical wavelength

316
00:21:40,396 --> 00:21:42,840
of each other?

317
00:21:42,840 --> 00:21:44,910
PROFESSOR: Yes and no.

318
00:21:44,910 --> 00:21:47,610
In the simplest example
of superradiance,

319
00:21:47,610 --> 00:21:50,900
we want to put them to within
one optical wavelength,

320
00:21:50,900 --> 00:21:53,330
and then we do not
have any phase vectors,

321
00:21:53,330 --> 00:21:56,420
but we will talk about it
next week, that we also

322
00:21:56,420 --> 00:21:59,890
have superradiance
in extended samples,

323
00:21:59,890 --> 00:22:03,210
and then we only get the
superradiance, the coherence

324
00:22:03,210 --> 00:22:06,010
between atoms, into
a smaller solid angle

325
00:22:06,010 --> 00:22:14,010
where the are different
phases are very well defined.

326
00:22:14,010 --> 00:22:16,180
If you would now average
the spontaneous emission

327
00:22:16,180 --> 00:22:19,160
over different directions, you
would get propagation phases

328
00:22:19,160 --> 00:22:24,370
and the atoms would only be
coherent in one solid angle

329
00:22:24,370 --> 00:22:28,339
but not be coherent in
another solid angle.

330
00:22:28,339 --> 00:22:29,005
Other questions?

331
00:22:32,790 --> 00:22:34,320
That's, to the best
off my knowledge

332
00:22:34,320 --> 00:22:38,320
at the most fundamental limit,
what spontaneous emission is,

333
00:22:38,320 --> 00:22:44,390
how accurately a spontaneously
emitted photon carries forward

334
00:22:44,390 --> 00:22:48,550
the phase of the laser beam
which excited the atom,

335
00:22:48,550 --> 00:22:52,050
and then eventually when we
have completely lost the phase

336
00:22:52,050 --> 00:22:55,990
because we excited the
atom to an excited state.

337
00:22:55,990 --> 00:22:59,310
Everything that we discussed
will be actually carried

338
00:22:59,310 --> 00:23:02,440
to the next level when
we discuss superradiance

339
00:23:02,440 --> 00:23:07,650
because then we have n atoms--
n can be a big number-- which

340
00:23:07,650 --> 00:23:11,180
excite together, and
if they emit photons,

341
00:23:11,180 --> 00:23:15,500
the phase of this n photon field
can be very precisely measured.

342
00:23:15,500 --> 00:23:17,880
So some of the
uncertainties we have here

343
00:23:17,880 --> 00:23:20,327
simply come from the fact
that, if you have only one

344
00:23:20,327 --> 00:23:22,660
photon or one atom,
there are naturally

345
00:23:22,660 --> 00:23:25,710
quantum fluctuations of
any phase measurement.

346
00:23:25,710 --> 00:23:30,745
But that part will go away when
we go to ensembles of atoms

347
00:23:30,745 --> 00:23:34,890
where we have many atoms,
and superradiance is then

348
00:23:34,890 --> 00:23:38,230
the way how we can
revisit the subject,

349
00:23:38,230 --> 00:23:41,520
how well can you retrieve
the phase of the laser field

350
00:23:41,520 --> 00:23:43,170
from the spontaneously
emitted photons.

351
00:23:46,350 --> 00:23:47,130
Other questions?

352
00:23:47,130 --> 00:23:47,711
Nancy?

353
00:23:47,711 --> 00:23:49,184
AUDIENCE: [INAUDIBLE]
single atom?

354
00:23:49,184 --> 00:23:53,603
So pi over 2, I can see that
we do a homodyne measurement

355
00:23:53,603 --> 00:23:55,567
and get the phase out.

356
00:23:55,567 --> 00:23:59,249
Do we need dipole moments
for pi [INAUDIBLE], or this

357
00:23:59,249 --> 00:24:01,131
is just a science
that we're going

358
00:24:01,131 --> 00:24:03,029
to use for many other things?

359
00:24:03,029 --> 00:24:04,320
PROFESSOR: What's the question?

360
00:24:04,320 --> 00:24:05,471
We have pi over--

361
00:24:05,471 --> 00:24:06,846
AUDIENCE: What
measurements do we

362
00:24:06,846 --> 00:24:08,694
need if we have just one atom?

363
00:24:08,694 --> 00:24:12,600
Do we make any measurements,
or no phase information?

364
00:24:12,600 --> 00:24:15,470
PROFESSOR: I think the
measurement is, in a way,

365
00:24:15,470 --> 00:24:17,860
what I indicated here.

366
00:24:17,860 --> 00:24:21,580
We excite the atom, then
we switch off this pulse,

367
00:24:21,580 --> 00:24:25,760
and then we take this
short pulse of light.

368
00:24:25,760 --> 00:24:28,440
It's a wave train
which has a duration

369
00:24:28,440 --> 00:24:31,660
on the order of the natural
lifetime of the atom,

370
00:24:31,660 --> 00:24:34,770
and this wave
train is interfered

371
00:24:34,770 --> 00:24:38,500
with a local oscillator,
and the interference term

372
00:24:38,500 --> 00:24:40,720
allows us to retrieve the phase.

373
00:24:40,720 --> 00:24:44,700
And if you use a strong
local oscillator,

374
00:24:44,700 --> 00:24:48,820
then we pretty much
retrieve the quantum limit

375
00:24:48,820 --> 00:24:50,690
of the measurement,
and the quantum limit

376
00:24:50,690 --> 00:24:53,060
of the measurement
is what I showed you

377
00:24:53,060 --> 00:24:57,083
in these cartoon drawings of
the phase-space distribution.

378
00:24:57,083 --> 00:24:58,958
AUDIENCE: So essentially,
we can [? read a ?]

379
00:24:58,958 --> 00:24:59,833
flux state like this?

380
00:25:11,730 --> 00:25:14,090
PROFESSOR: If you
have a flux state

381
00:25:14,090 --> 00:25:16,165
and you repeat the
measurement many times,

382
00:25:16,165 --> 00:25:17,900
we will measure random phase.

383
00:25:20,930 --> 00:25:24,820
So what happens here is--
let me put it this way.

384
00:25:24,820 --> 00:25:27,540
The homodyne
detection is a way how

385
00:25:27,540 --> 00:25:29,940
we want to measure the
phase, and whenever

386
00:25:29,940 --> 00:25:31,650
you want to measure
the phase, you

387
00:25:31,650 --> 00:25:33,370
get a phase because
the number you

388
00:25:33,370 --> 00:25:35,480
get from a phase
measurement is a phase.

389
00:25:35,480 --> 00:25:37,440
But if you have a
flux state which

390
00:25:37,440 --> 00:25:40,740
has not a specific phase
but an equal probability

391
00:25:40,740 --> 00:25:43,850
for all phases
between zero and 2 pi,

392
00:25:43,850 --> 00:25:46,570
then, if you repeat a
phase measurement many,

393
00:25:46,570 --> 00:25:50,300
many times, you will get a
random result for the phase.

394
00:25:50,300 --> 00:25:53,049
AUDIENCE: I think that's what
my question originally was.

395
00:25:53,049 --> 00:25:54,465
What measurement
would you perform

396
00:25:54,465 --> 00:25:57,260
for this [? pi phase? ?]
Would you still do a phase

397
00:25:57,260 --> 00:25:58,784
measurement?

398
00:25:58,784 --> 00:25:59,950
PROFESSOR: It's your choice.

399
00:25:59,950 --> 00:26:01,825
If you want to do a
phase measurement, that's

400
00:26:01,825 --> 00:26:03,730
a way to do it, and
then for flux state,

401
00:26:03,730 --> 00:26:06,400
you will get a random phase.

402
00:26:06,400 --> 00:26:08,720
But maybe for the flux
state, of course, you

403
00:26:08,720 --> 00:26:11,690
can say in hindsight, the flux
state doesn't have a phase

404
00:26:11,690 --> 00:26:13,960
so maybe you shouldn't
bother measuring the phase.

405
00:26:13,960 --> 00:26:15,580
The special thing
about the flux state

406
00:26:15,580 --> 00:26:18,400
that it has exactly one
photon, and so maybe you

407
00:26:18,400 --> 00:26:23,240
want to have a measurement
which is measuring

408
00:26:23,240 --> 00:26:25,190
the special character
of the flux state,

409
00:26:25,190 --> 00:26:28,300
namely that you have
sub-Poissonian distribution

410
00:26:28,300 --> 00:26:30,100
of the photons.

411
00:26:30,100 --> 00:26:33,570
Of course, this aspect of
just having one photon gets

412
00:26:33,570 --> 00:26:36,220
completely lost when
you have a beam splitter

413
00:26:36,220 --> 00:26:38,425
and you have zillions of
photons in your laser beam

414
00:26:38,425 --> 00:26:39,925
with all the
Poissonian fluctuations

415
00:26:39,925 --> 00:26:42,600
in the coherent state
and you superimpose it.

416
00:26:42,600 --> 00:26:44,937
But this is nothing else
than complementarity.

417
00:26:44,937 --> 00:26:47,520
You can either measure the phase
or you can measure the photon

418
00:26:47,520 --> 00:26:50,050
number, and the question is,
what are you interested in?

419
00:26:57,160 --> 00:27:06,410
This is one aspect of coherence
in a two level system,

420
00:27:06,410 --> 00:27:12,280
namely that we have a phase
in the two level system

421
00:27:12,280 --> 00:27:15,796
and the question is,
how can we measure it?

422
00:27:15,796 --> 00:27:17,920
And the answer is we can
map it on the photon field

423
00:27:17,920 --> 00:27:20,730
and then perform a quantum
measurement on the photon

424
00:27:20,730 --> 00:27:21,230
field.

425
00:27:24,880 --> 00:27:28,340
I want to continue with some
other aspect of coherence

426
00:27:28,340 --> 00:27:29,150
in single atom.

427
00:27:31,780 --> 00:27:37,790
Let me just point out
one important aspect

428
00:27:37,790 --> 00:27:44,860
about coherence in
a two level system,

429
00:27:44,860 --> 00:27:55,720
and this is related to something
very mundane, the precession

430
00:27:55,720 --> 00:27:59,470
of spin-- when it's
a two level system,

431
00:27:59,470 --> 00:28:01,730
it's spin 1/2-- in
a magnetic field.

432
00:28:07,617 --> 00:28:09,200
In other words, I
just want to quickly

433
00:28:09,200 --> 00:28:15,100
remind you in a few minutes
that for any two level system,

434
00:28:15,100 --> 00:28:17,340
we can always map
it on spin 1/2.

435
00:28:17,340 --> 00:28:21,100
I was really
emphasizing this message

436
00:28:21,100 --> 00:28:22,780
throughout the whole course.

437
00:28:22,780 --> 00:28:25,220
But for spin 1/2, if you
think of spin up or down

438
00:28:25,220 --> 00:28:28,752
in a magnetic field, there
is a very clear visualization

439
00:28:28,752 --> 00:28:29,460
of the coherence.

440
00:28:29,460 --> 00:28:32,000
If you have a coherence of
position of spin up and down,

441
00:28:32,000 --> 00:28:35,210
the phase of the
superposition decides

442
00:28:35,210 --> 00:28:38,030
whether the spin
points in x or y.

443
00:28:38,030 --> 00:28:42,510
So the precession of a spin
in the transverse xy plane

444
00:28:42,510 --> 00:28:45,690
is actually the
manifestation of coherence,

445
00:28:45,690 --> 00:28:47,880
and it's not just
the special coherence

446
00:28:47,880 --> 00:28:50,810
of spin 1/2 in a magnetic
field because all two level

447
00:28:50,810 --> 00:28:52,670
systems are isomorphic to that.

448
00:28:52,670 --> 00:28:55,785
You can always use it as
an intuitive visualization

449
00:28:55,785 --> 00:28:59,250
of what coherences are.

450
00:28:59,250 --> 00:29:02,080
So what I just want to
point out is the relation

451
00:29:02,080 --> 00:29:06,130
to the quantum mechanical
or classical precession

452
00:29:06,130 --> 00:29:13,150
of spin in a magnetic
field, that it is simply

453
00:29:13,150 --> 00:29:19,480
an effect of coherence
within one atom, coherence

454
00:29:19,480 --> 00:29:20,790
between two levels in an atom.

455
00:29:24,080 --> 00:29:31,760
So if spin points
in the x direction,

456
00:29:31,760 --> 00:29:36,280
it is a coherent superposition
of plus z and minus z,

457
00:29:36,280 --> 00:29:41,810
spin up and spin down, but
this is a situation at time t

458
00:29:41,810 --> 00:29:43,660
equals 0.

459
00:29:43,660 --> 00:29:49,320
If we let time evolve,
spin up and spin down

460
00:29:49,320 --> 00:29:52,230
evolve with the Larmor
frequency, actually

461
00:29:52,230 --> 00:29:56,730
effect of 1/2, but
with opposite phases

462
00:29:56,730 --> 00:30:00,730
because one has plus the Larmor,
h bar omega Larmor over 2.

463
00:30:00,730 --> 00:30:04,595
The other has minus h bar
omega Larmor over 2 as energy.

464
00:30:09,667 --> 00:30:11,750
In other words, if you
look at the relative phase,

465
00:30:11,750 --> 00:30:13,890
it's a beat node with
the same unsplitting

466
00:30:13,890 --> 00:30:16,290
between spin up and spin down.

467
00:30:16,290 --> 00:30:20,430
But that means now, due
to this coherent time

468
00:30:20,430 --> 00:30:26,630
evolution of the two
amplitudes, that the spin

469
00:30:26,630 --> 00:30:31,110
precesses in the xy plane.

470
00:30:36,090 --> 00:30:38,990
The statistical operator
for the spin 1/2,

471
00:30:38,990 --> 00:30:45,850
we have 50-50 population
in spin up and spin down,

472
00:30:45,850 --> 00:30:57,050
but now, the phase
here precesses

473
00:30:57,050 --> 00:31:00,520
as e to the minus
and e to the plus

474
00:31:00,520 --> 00:31:09,630
i omega lt, which means, if you
use the statistical operator

475
00:31:09,630 --> 00:31:15,160
and find the expectation
value for the x spin.

476
00:31:15,160 --> 00:31:19,710
It means we take the
statistical operator describing

477
00:31:19,710 --> 00:31:22,120
the pure state of
a two level system,

478
00:31:22,120 --> 00:31:25,580
we multiply with a
Pauli matrix in x,

479
00:31:25,580 --> 00:31:28,840
and this is a prescription
to get the expectation

480
00:31:28,840 --> 00:31:34,930
value for sigma x, and we
find it's cosine omega lt.

481
00:31:34,930 --> 00:31:41,930
So the spin is precessing
the x component changes

482
00:31:41,930 --> 00:31:42,555
cosinusoidally.

483
00:31:42,555 --> 00:31:47,570
That would mean the y
component changes sinusoidally.

484
00:31:47,570 --> 00:31:53,280
Let me just contrast it to
the case of no coherence,

485
00:31:53,280 --> 00:32:07,660
and this would mean off
diagonal matrix elements are 0.

486
00:32:07,660 --> 00:32:11,085
then if you have a
statistical operator where

487
00:32:11,085 --> 00:32:15,490
the off diagonal matrix elements
are zero, in one minute,

488
00:32:15,490 --> 00:32:19,890
you can show that
then any expectation

489
00:32:19,890 --> 00:32:24,700
value for the x or y component
of the spin vanishes.

490
00:32:30,620 --> 00:32:34,560
In that case, if you have a
statistical mixer between spin

491
00:32:34,560 --> 00:32:37,940
up and down, of course there
is no phase determined,

492
00:32:37,940 --> 00:32:40,600
and it is the phase of
the superposition state

493
00:32:40,600 --> 00:32:44,030
which tells you where
between 0 and 2 pi

494
00:32:44,030 --> 00:32:48,010
the spin is pointing
in the xy plane.

495
00:32:48,010 --> 00:32:49,720
I want to come
back later on when

496
00:32:49,720 --> 00:32:56,680
I discuss an example of
coherent spectroscopy,

497
00:32:56,680 --> 00:33:00,540
that if you excite coherently
a superposition of spin

498
00:33:00,540 --> 00:33:02,890
up and spin down,
you can perform

499
00:33:02,890 --> 00:33:05,530
some form of coherent
spectroscopy, which I want

500
00:33:05,530 --> 00:33:08,230
to explain first in
general and then come back

501
00:33:08,230 --> 00:33:09,460
to the spin as an example.

502
00:33:29,440 --> 00:33:33,680
When we talk about
coherent spectroscopy,

503
00:33:33,680 --> 00:33:36,470
I want to just in 10
minutes or 15 minutes

504
00:33:36,470 --> 00:33:40,930
show you some spectroscopic
techniques which

505
00:33:40,930 --> 00:33:45,695
exploit the coherence between
several quantum states.

506
00:33:48,490 --> 00:33:50,340
I do it for a number of reasons.

507
00:33:50,340 --> 00:33:54,690
One is coherent
spectroscopy actually

508
00:33:54,690 --> 00:34:01,550
allows us to obtain information
about the level structure

509
00:34:01,550 --> 00:34:06,120
even if this level structure is
much narrower than the Doppler

510
00:34:06,120 --> 00:34:07,010
width.

511
00:34:07,010 --> 00:34:11,989
So it is a sub-Doppler
technique to exploit coherence.

512
00:34:11,989 --> 00:34:15,610
And before people had lasers,
before people invented

513
00:34:15,610 --> 00:34:18,449
sub-Doppler laser
spectroscopy, often,

514
00:34:18,449 --> 00:34:21,070
coherent spectroscopy
was the only way

515
00:34:21,070 --> 00:34:25,150
how you could obtain detailed
structure of the atom.

516
00:34:25,150 --> 00:34:27,730
The reason why I explain
coherent spectroscopy

517
00:34:27,730 --> 00:34:31,600
is to just give you a little bit
idea about that you appreciate

518
00:34:31,600 --> 00:34:34,590
how smart people were before
lasers were developed,

519
00:34:34,590 --> 00:34:40,100
but also, it illustrates
what coherence can do for us.

520
00:34:40,100 --> 00:34:45,090
It's a nice example for
the concept of coherence.

521
00:34:45,090 --> 00:34:47,780
When I was a graduate
student, textbooks

522
00:34:47,780 --> 00:34:52,949
had dozens of pages, 50 pages
on coherent spectroscopy,

523
00:34:52,949 --> 00:34:55,650
the Hanle effect, quantum
beat measurements.

524
00:34:55,650 --> 00:35:00,130
It's all old fashioned because
with a laser, and especially

525
00:35:00,130 --> 00:35:03,210
cold atoms and the laser,
we have such wonderful tools

526
00:35:03,210 --> 00:35:05,815
to go to the ultimate
fundamental precision

527
00:35:05,815 --> 00:35:07,330
of quantum measurements.

528
00:35:07,330 --> 00:35:10,976
But still, coherence
is important.

529
00:35:14,500 --> 00:35:21,520
Let me talk about
one method, which

530
00:35:21,520 --> 00:35:23,790
is called quantum
beat spectroscopy.

531
00:35:31,670 --> 00:35:34,110
The selling point
about quantum beat

532
00:35:34,110 --> 00:35:44,400
spectroscopy is that it allows
the measurement of narrow level

533
00:35:44,400 --> 00:35:49,110
spacings-- just think
about Zeeman splitting

534
00:35:49,110 --> 00:36:01,541
in a magnetic field-- without
any form of narrow band

535
00:36:01,541 --> 00:36:02,040
excitation.

536
00:36:07,880 --> 00:36:09,380
You can also put it like this.

537
00:36:09,380 --> 00:36:14,241
If you don't have any way to
selectively excite levels,

538
00:36:14,241 --> 00:36:16,240
but you're interested
what is the level spacing,

539
00:36:16,240 --> 00:36:18,910
but you cannot have
a narrow band laser,

540
00:36:18,910 --> 00:36:23,500
have atoms which stand still and
scan and get peak, peak, peak,

541
00:36:23,500 --> 00:36:27,130
what you can still do is you can
just excite all of the levels

542
00:36:27,130 --> 00:36:28,450
at once.

543
00:36:28,450 --> 00:36:31,210
In other words, you hit
the atom with a board laser

544
00:36:31,210 --> 00:36:34,440
like with a sledgehammer,
and then you see a beat node,

545
00:36:34,440 --> 00:36:37,790
you see some blinking,
a quantum beat

546
00:36:37,790 --> 00:36:40,250
between the excitation
of the levels.

547
00:36:40,250 --> 00:36:40,980
That's the idea.

548
00:36:46,100 --> 00:36:48,110
We assume we have a
ground state and then we

549
00:36:48,110 --> 00:36:52,030
have an excited state manifold,
and in this excited state

550
00:36:52,030 --> 00:36:57,930
manifold, we have several
levels distributed

551
00:36:57,930 --> 00:36:59,180
over an energy interval delta.

552
00:37:02,550 --> 00:37:04,680
Yes, we don't have a
narrow band source.

553
00:37:04,680 --> 00:37:08,430
We may just have a
classic light source,

554
00:37:08,430 --> 00:37:17,620
but if we use a short pulse
that the pulse duration is

555
00:37:17,620 --> 00:37:21,530
much smaller than the splitting
between energy levels,

556
00:37:21,530 --> 00:37:31,990
then we create a coherent
superposition of those levels.

557
00:37:34,680 --> 00:37:38,720
So therefore, what
we create at time t

558
00:37:38,720 --> 00:37:44,620
equals 0 is a
coherent superposition

559
00:37:44,620 --> 00:37:47,530
of energy eigenlevels.

560
00:37:47,530 --> 00:37:50,930
And the important thing is that
this is at time t equals 0,

561
00:37:50,930 --> 00:37:57,790
but now, when time goes
on, each amplitude,

562
00:37:57,790 --> 00:38:02,650
each part of the wave function,
evolves with its frequency

563
00:38:02,650 --> 00:38:13,820
omega i, and if we would
then look at, let's say,

564
00:38:13,820 --> 00:38:19,576
the emission spectrum
as a function of time,

565
00:38:19,576 --> 00:38:25,570
we will find that-- I will give
you a little bit more details

566
00:38:25,570 --> 00:38:31,560
later-- that yes, there
is a decay approximately

567
00:38:31,560 --> 00:38:35,530
with the natural
spontaneous emission

568
00:38:35,530 --> 00:38:40,210
time with the inverse of
the natural line widths.

569
00:38:40,210 --> 00:38:45,040
But we observe some
oscillations which

570
00:38:45,040 --> 00:38:48,340
is the interference term of
the different terms in the wave

571
00:38:48,340 --> 00:38:48,840
function.

572
00:38:51,350 --> 00:38:56,180
So therefore, if we
would take this spectrum

573
00:38:56,180 --> 00:38:59,590
and perform a
Fourier transform, we

574
00:38:59,590 --> 00:39:08,810
will actually observe
different peaks.

575
00:39:11,720 --> 00:39:15,280
This is frequency, and
the frequency peaks

576
00:39:15,280 --> 00:39:19,550
are at discrete frequencies
corresponding to frequency

577
00:39:19,550 --> 00:39:22,750
differences between
the excited state.

578
00:39:22,750 --> 00:39:25,320
And ideally, the
widths of these is

579
00:39:25,320 --> 00:39:26,930
determined by the
natural line widths.

580
00:39:32,310 --> 00:39:38,620
So in other words, what
we have actually done

581
00:39:38,620 --> 00:39:43,950
is we have done a version of
the double slit experiment.

582
00:39:43,950 --> 00:39:49,630
We have ground state, we
had our excited states,

583
00:39:49,630 --> 00:39:58,860
e sub i, and our
broad band source

584
00:39:58,860 --> 00:40:02,370
was creating a
coherent excitation,

585
00:40:02,370 --> 00:40:07,300
and then we were observing
the light which came out.

586
00:40:07,300 --> 00:40:10,560
We were performing a
multi-slit experiment.

587
00:40:10,560 --> 00:40:13,890
We had a laser pulse and then
we see photons coming out,

588
00:40:13,890 --> 00:40:17,790
but it is fundamentally
not observable

589
00:40:17,790 --> 00:40:21,410
which intermediate state was
responsible for the scattering.

590
00:40:21,410 --> 00:40:25,530
So therefore, we have
in the Feynman sense

591
00:40:25,530 --> 00:40:34,730
several indistinguishable
paths going

592
00:40:34,730 --> 00:40:37,090
through different
internal states.

593
00:40:37,090 --> 00:40:41,254
And therefore, we get
an interference effect.

594
00:40:41,254 --> 00:40:42,920
Some of what I'm
saying we will retrieve

595
00:40:42,920 --> 00:40:45,580
later on when we talk
about three level systems.

596
00:40:45,580 --> 00:40:48,215
We will also have
situations that sometimes we

597
00:40:48,215 --> 00:40:51,710
go through two possibilities
for the intermediate state,

598
00:40:51,710 --> 00:40:54,860
and if we have no way,
even in principle,

599
00:40:54,860 --> 00:40:57,890
to figure out which
intermediate state was involved,

600
00:40:57,890 --> 00:41:00,672
we have to sum up
the amplitudes,

601
00:41:00,672 --> 00:41:02,130
and that's when we
get a beat node.

602
00:41:13,560 --> 00:41:20,030
This technique is a
Doppler free technique

603
00:41:20,030 --> 00:41:24,730
because, even if you
take a single pulse

604
00:41:24,730 --> 00:41:28,070
from a light source, you have
a Doppler broadening, which

605
00:41:28,070 --> 00:41:30,960
is k dot v, v, the
thermal velocity,

606
00:41:30,960 --> 00:41:33,340
and this can be
much, much broader.

607
00:41:33,340 --> 00:41:38,300
You will still see
the quantum beats,

608
00:41:38,300 --> 00:41:41,610
maybe I should say in principle,
because the beat happens

609
00:41:41,610 --> 00:41:43,395
at the much smaller
frequency delta.

610
00:41:47,320 --> 00:41:55,200
Or maybe I should say that
the Doppler shift is reduced

611
00:41:55,200 --> 00:41:59,930
by the splitting of the excited
states over the frequency

612
00:41:59,930 --> 00:42:01,550
of the exciting laser.

613
00:42:01,550 --> 00:42:04,550
Of course, if you have your
different atoms emitting

614
00:42:04,550 --> 00:42:07,120
at different frequencies,
you have a Doppler shift,

615
00:42:07,120 --> 00:42:09,490
but since you measure
the difference frequency,

616
00:42:09,490 --> 00:42:12,090
you only get the Doppler shift
associated with the difference

617
00:42:12,090 --> 00:42:12,589
frequency.

618
00:42:18,590 --> 00:42:21,170
Now let me come back
to the previous example

619
00:42:21,170 --> 00:42:23,920
I had about the spin 1/2 system.

620
00:42:23,920 --> 00:42:27,890
If you assume you have a spin
1/2 system, spin up and spin

621
00:42:27,890 --> 00:42:31,310
down, which is excited
with a laser, which

622
00:42:31,310 --> 00:42:33,790
is linear polarization,
you would then

623
00:42:33,790 --> 00:42:38,560
create a superposition of up
and down which, let's say,

624
00:42:38,560 --> 00:42:42,270
is now a dipole moment which
points in the x direction.

625
00:42:42,270 --> 00:42:45,020
A dipole moment which
points in the x direction

626
00:42:45,020 --> 00:42:48,810
will not emit light along
the axis of the dipole moment

627
00:42:48,810 --> 00:42:53,230
because of the dipolar
emission pattern.

628
00:42:53,230 --> 00:42:55,290
It will only emit to the side.

629
00:42:55,290 --> 00:43:02,320
But I mentioned to you that
the dipole moment or the spin,

630
00:43:02,320 --> 00:43:04,700
which is originally
in x, will now

631
00:43:04,700 --> 00:43:08,470
oscillate with a Larmor
frequency in the xy plane.

632
00:43:08,470 --> 00:43:10,240
So the picture you
can actually have

633
00:43:10,240 --> 00:43:13,260
of such a quantum beat
and quantum superposition

634
00:43:13,260 --> 00:43:15,970
is like the lighthouse.

635
00:43:15,970 --> 00:43:18,450
You have a searchlight
at the lighthouse,

636
00:43:18,450 --> 00:43:21,360
and the searchlight
is just rotating

637
00:43:21,360 --> 00:43:23,040
at the Larmor frequency.

638
00:43:23,040 --> 00:43:25,030
For instance, you wouldn't
see light right now,

639
00:43:25,030 --> 00:43:27,470
now you don't see light,
now you don't see light.

640
00:43:27,470 --> 00:43:30,520
It's really like a
classical lighthouse

641
00:43:30,520 --> 00:43:34,000
which is emitting light
at the Larmor frequency.

642
00:43:34,000 --> 00:43:35,980
So if you have a
fluorescence detector which

643
00:43:35,980 --> 00:43:38,290
looks at the atoms from
a certain direction,

644
00:43:38,290 --> 00:43:41,720
you will pretty much see
the lighthouse effect

645
00:43:41,720 --> 00:43:45,470
that the fluorescence of this
coherent superposition of atoms

646
00:43:45,470 --> 00:43:48,420
goes on and off, on
and off, on and off.

647
00:43:48,420 --> 00:43:51,420
This is sort of a very
nice visualization how

648
00:43:51,420 --> 00:43:57,090
you obtain what I
showed here, a beat

649
00:43:57,090 --> 00:43:58,820
node in your detected signal.

650
00:44:11,640 --> 00:44:14,820
Let me talk about another
aspect of coherence.

651
00:44:14,820 --> 00:44:17,830
And of course, they
are all related.

652
00:44:17,830 --> 00:44:20,520
Coherence is always related
to the phase, to beat nodes,

653
00:44:20,520 --> 00:44:22,780
to superposition.

654
00:44:22,780 --> 00:44:25,880
Let me now talk about
one aspect which

655
00:44:25,880 --> 00:44:27,430
is related to delayed detection.

656
00:44:36,522 --> 00:44:37,980
When I was a graduate
student and I

657
00:44:37,980 --> 00:44:41,060
learned about spectroscopic
techniques, somehow

658
00:44:41,060 --> 00:44:44,120
I was so fascinated
by techniques

659
00:44:44,120 --> 00:44:48,840
which could measure
spectroscopically transitions

660
00:44:48,840 --> 00:44:51,019
better than the
natural line widths.

661
00:44:51,019 --> 00:44:51,560
I don't know.

662
00:44:51,560 --> 00:44:54,190
Maybe from what I
had read before,

663
00:44:54,190 --> 00:44:55,980
what I learned as
an undergraduate,

664
00:44:55,980 --> 00:44:58,640
the natural line
widths appeared to be

665
00:44:58,640 --> 00:45:01,430
the natural fundamental limit.

666
00:45:01,430 --> 00:45:04,260
So the topic I'm
teaching right now

667
00:45:04,260 --> 00:45:07,360
has always had a certain
fascination for me,

668
00:45:07,360 --> 00:45:10,090
but you will, of course,
also realize that in the end,

669
00:45:10,090 --> 00:45:12,629
the answer is rather simple.

670
00:45:12,629 --> 00:45:14,920
Once you know the answer,
most answers are very simple.

671
00:45:22,140 --> 00:45:26,070
So we want to talk
about delayed detection.

672
00:45:26,070 --> 00:45:30,485
Let's say we excite the system.

673
00:45:35,150 --> 00:45:37,370
You can think about a
quantum beat experiment.

674
00:45:37,370 --> 00:45:41,192
You have a short pulse and
then your quantum beats happen.

675
00:45:55,960 --> 00:46:00,280
And now the question
is, normally,

676
00:46:00,280 --> 00:46:03,450
when you do a measurement
on a decaying system,

677
00:46:03,450 --> 00:46:06,880
you're always limited by
the natural line widths,

678
00:46:06,880 --> 00:46:10,220
by the inverse of the lifetime.

679
00:46:10,220 --> 00:46:13,550
But now, maybe you
want to be smart

680
00:46:13,550 --> 00:46:20,980
and you say, well, I start the
detection, I only detect atoms

681
00:46:20,980 --> 00:46:27,400
after a time t0, which
is much, much larger

682
00:46:27,400 --> 00:46:30,270
than the natural line widths.

683
00:46:30,270 --> 00:46:33,965
And the question
I have is, can you

684
00:46:33,965 --> 00:46:45,470
obtain, with such a measurement,
a spectral resolution which

685
00:46:45,470 --> 00:46:54,200
is narrower than the
natural line widths?

686
00:46:54,200 --> 00:46:57,850
Well, we can give
two possible answers.

687
00:46:57,850 --> 00:47:01,170
One is yes, because
you're looking

688
00:47:01,170 --> 00:47:04,850
at atoms which have survived
for a long time, so to speak.

689
00:47:04,850 --> 00:47:07,346
These atoms are longer lived.

690
00:47:07,346 --> 00:47:08,720
We have just
selected atoms which

691
00:47:08,720 --> 00:47:10,920
happened to survive
for several lifetimes.

692
00:47:22,070 --> 00:47:25,510
But then there should
some lingering doubts.

693
00:47:25,510 --> 00:47:29,080
If you have a sample which
undergoes radioactive decay,

694
00:47:29,080 --> 00:47:32,720
and you would go to
your favorite supplier

695
00:47:32,720 --> 00:47:35,620
and buy uranium,
which has already

696
00:47:35,620 --> 00:47:37,850
decayed for a
billion years, it's

697
00:47:37,850 --> 00:47:42,060
the same uranium which
existed a billion years ago.

698
00:47:42,060 --> 00:47:44,990
You will not be able to
perform any measurement

699
00:47:44,990 --> 00:47:48,990
on your well aged uranium,
which has a higher

700
00:47:48,990 --> 00:47:52,420
resolution than if you had
lived a billion years ago

701
00:47:52,420 --> 00:47:56,690
and had done your measurement
with younger uranium.

702
00:47:56,690 --> 00:47:59,540
So in other words,
the exponential decay

703
00:47:59,540 --> 00:48:00,260
is self similar.

704
00:48:00,260 --> 00:48:04,390
It starts at any moment
and it looks exponential

705
00:48:04,390 --> 00:48:05,540
no matter where you start.

706
00:48:19,230 --> 00:48:21,770
I didn't bring clickers
today, but with which

707
00:48:21,770 --> 00:48:23,350
answer would you side?

708
00:48:23,350 --> 00:48:26,770
Is it possible or
it's not possible?

709
00:48:26,770 --> 00:48:29,230
Maybe just hints
of who thinks it

710
00:48:29,230 --> 00:48:34,890
is possible by taking advantage
of the longtime survivors?

711
00:48:34,890 --> 00:48:36,070
OK.

712
00:48:36,070 --> 00:48:38,890
Who thinks it's impossible?

713
00:48:38,890 --> 00:48:39,630
A few.

714
00:48:39,630 --> 00:48:41,050
Good.

715
00:48:41,050 --> 00:48:43,120
The answer is actually depends.

716
00:48:43,120 --> 00:48:47,910
If you would go and just look
at the longtime survivors,

717
00:48:47,910 --> 00:48:50,470
you would not be able to do
a more precise measurement.

718
00:48:50,470 --> 00:48:52,630
You need a little
bit of information

719
00:48:52,630 --> 00:48:55,060
from the earlier time.

720
00:48:55,060 --> 00:49:00,840
So in that sense, the question
is a little bit deliberately

721
00:49:00,840 --> 00:49:04,280
confusing, and I want to show
you how the mathematics work.

722
00:49:04,280 --> 00:49:08,100
It's just five minutes to show
the mathematics of a Fourier

723
00:49:08,100 --> 00:49:11,400
transform, and
the result will be

724
00:49:11,400 --> 00:49:16,270
if you have information about
something at t equals 0,

725
00:49:16,270 --> 00:49:20,150
and then you look at the long
time survivors, you in essence

726
00:49:20,150 --> 00:49:23,500
have a longer integration
period for your measurement,

727
00:49:23,500 --> 00:49:25,750
and then the Fourier
transform of that measurement

728
00:49:25,750 --> 00:49:27,440
can be very narrow.

729
00:49:27,440 --> 00:49:30,430
But if you do the dumb thing,
you just go to the store

730
00:49:30,430 --> 00:49:34,405
and buy very well aged uranium
or very well aged atoms,

731
00:49:34,405 --> 00:49:35,905
and you then start
your measurement,

732
00:49:35,905 --> 00:49:37,070
you have no chance.

733
00:49:37,070 --> 00:49:39,880
You are always back to
the spontaneous decay

734
00:49:39,880 --> 00:49:41,040
to the natural line widths.

735
00:49:50,340 --> 00:49:53,350
All I have to do is
actually just a few lines

736
00:49:53,350 --> 00:49:58,380
of mathematics and
Fourier transform.

737
00:49:58,380 --> 00:50:02,950
Let us assume we have the
situation we discussed earlier.

738
00:50:02,950 --> 00:50:10,810
We have a quantum beat where we
have a beat frequency omega 0.

739
00:50:10,810 --> 00:50:13,620
Just think about the
searchlight, the atoms which

740
00:50:13,620 --> 00:50:15,750
oscillate with a
Larmor frequency,

741
00:50:15,750 --> 00:50:18,870
and you have some
cosine omega Larmor t

742
00:50:18,870 --> 00:50:23,440
factor in the intensity
of the light you observe.

743
00:50:23,440 --> 00:50:27,830
But now, because the atom in
the excited state is decaying,

744
00:50:27,830 --> 00:50:33,970
everything will decay with
the natural line width.

745
00:50:33,970 --> 00:50:37,300
This is sort of what we
observe, and the question

746
00:50:37,300 --> 00:50:39,980
is, if you observe
that in real time,

747
00:50:39,980 --> 00:50:43,980
can we then retrieve
spectral information from it

748
00:50:43,980 --> 00:50:46,130
which is more
accurate than gamma?

749
00:50:48,910 --> 00:50:51,000
All I want to do is I want
to discuss the Fourier

750
00:50:51,000 --> 00:50:54,670
transform of this
function, s of t.

751
00:50:54,670 --> 00:51:00,480
Let me use
dimensionless variables.

752
00:51:00,480 --> 00:51:04,910
We measure frequencies
in units of gamma.

753
00:51:04,910 --> 00:51:08,795
We use a Lorentzian, which
is just 1 plus x squared.

754
00:51:18,020 --> 00:51:23,440
We just said we want to start
the measurement at time t0.

755
00:51:23,440 --> 00:51:27,370
So the question is, if you
start later and later and later,

756
00:51:27,370 --> 00:51:29,314
do we get higher
accuracy because we're

757
00:51:29,314 --> 00:51:30,355
talking to the survivors?

758
00:51:37,190 --> 00:51:41,660
So let's perform the
Fourier transform,

759
00:51:41,660 --> 00:51:44,710
and let's use complex
notation, e to the i omega t.

760
00:51:47,340 --> 00:51:51,620
I will measure times in units
of the inverse line widths.

761
00:51:55,940 --> 00:52:01,120
So we performed the
Fourier transform,

762
00:52:01,120 --> 00:52:04,110
and by doing e to the
i omega t, I actually

763
00:52:04,110 --> 00:52:10,400
performed the Fourier transform
for the cosine and for the sine

764
00:52:10,400 --> 00:52:12,970
by using the real part
and the imaginary part

765
00:52:12,970 --> 00:52:14,750
of the complex number.

766
00:52:14,750 --> 00:52:17,090
So we will actually
be able to look

767
00:52:17,090 --> 00:52:18,960
at the real and imaginary part.

768
00:52:22,450 --> 00:52:30,960
The Fourier transform has
a real and imaginary part,

769
00:52:30,960 --> 00:52:36,220
so let me call the real part
F of x and the imaginary part

770
00:52:36,220 --> 00:52:37,560
G of x.

771
00:52:48,247 --> 00:52:49,080
You can do the math.

772
00:52:49,080 --> 00:52:53,040
It's a straightforward integral.

773
00:52:53,040 --> 00:52:57,450
In both cases, will we find
that if you do our measurement,

774
00:52:57,450 --> 00:53:01,680
well, the longer we wait,
the more signal we lose.

775
00:53:01,680 --> 00:53:04,970
This is common to all
delayed measurements.

776
00:53:04,970 --> 00:53:07,710
You're really now talking
to an exponentially smaller

777
00:53:07,710 --> 00:53:09,340
and smaller signal.

778
00:53:09,340 --> 00:53:13,190
Also, because of the
exponential decay,

779
00:53:13,190 --> 00:53:15,750
we get an envelope
which is a Lorentzian.

780
00:53:20,300 --> 00:53:23,250
But then, and this is
the interesting part,

781
00:53:23,250 --> 00:53:36,570
we have cosine xT
minus x sine xT.

782
00:53:40,830 --> 00:53:55,110
So now we have factors
which depend on capital T,

783
00:53:55,110 --> 00:53:59,260
and t is larger
the longer we wait.

784
00:53:59,260 --> 00:54:05,140
It is actually those
parts with sine and cosine

785
00:54:05,140 --> 00:54:21,040
which determine whether
we can get resolution

786
00:54:21,040 --> 00:54:22,830
below the natural line widths.

787
00:54:26,970 --> 00:54:29,530
What is important
is-- and this should

788
00:54:29,530 --> 00:54:31,710
be sort of an eye
opener for you--

789
00:54:31,710 --> 00:54:36,980
if you simply measure intensity,
if you look at the power

790
00:54:36,980 --> 00:54:40,900
spectrum, you take the real part
plus imaginary part and just

791
00:54:40,900 --> 00:54:46,850
look at the absolute
value, then,

792
00:54:46,850 --> 00:54:51,020
because of cosine squared
plus sine squared equals 1,

793
00:54:51,020 --> 00:54:56,430
all the cosine and sine part,
the last part of the expression

794
00:54:56,430 --> 00:55:00,680
above cancels out
and you find that you

795
00:55:00,680 --> 00:55:03,570
have an exponential
loss of signal

796
00:55:03,570 --> 00:55:07,690
but your spectral distribution
is always a Lorentzian.

797
00:55:07,690 --> 00:55:14,900
So you always have a Lorentzian
line shape completely

798
00:55:14,900 --> 00:55:20,420
independent of the
delay time, capital T.

799
00:55:20,420 --> 00:55:22,420
And this is what some
of you maybe thought.

800
00:55:22,420 --> 00:55:24,680
If I start the
measurement later and all

801
00:55:24,680 --> 00:55:27,760
I can do is look at the
power of the emitted light,

802
00:55:27,760 --> 00:55:28,750
I have no advantage.

803
00:55:28,750 --> 00:55:29,830
I cannot go sub-natural.

804
00:55:32,730 --> 00:55:42,735
However, if you
look at the function

805
00:55:42,735 --> 00:55:52,275
F of x for large values of
x, you find oscillations.

806
00:55:55,040 --> 00:55:58,750
So if you look at the sine
or cosine Fourier transform,

807
00:55:58,750 --> 00:56:01,660
the real or imaginary
part separately,

808
00:56:01,660 --> 00:56:06,830
you find oscillations,
and those oscillations,

809
00:56:06,830 --> 00:56:11,420
similar to Ramsey fringes,
have a central peak.

810
00:56:11,420 --> 00:56:16,410
The central peak is
narrow and the width

811
00:56:16,410 --> 00:56:20,540
is now given by not the
inverse of the natural lifetime

812
00:56:20,540 --> 00:56:22,711
but the inverse
of the delay time

813
00:56:22,711 --> 00:56:23,960
you wait for your measurement.

814
00:56:28,320 --> 00:56:37,700
So the fact is now we
had a signal s of t which

815
00:56:37,700 --> 00:56:45,030
we assumed was a quantum beat
with a well defined phase,

816
00:56:45,030 --> 00:56:48,980
and then it was
exponentially decaying.

817
00:56:48,980 --> 00:56:52,210
If we would now perform
a Fourier transform where

818
00:56:52,210 --> 00:56:57,290
we do a Fourier transform
with cosine omega t plus phi,

819
00:56:57,290 --> 00:57:01,410
we can get a narrow signal,
but we have no idea what phi is

820
00:57:01,410 --> 00:57:04,500
or if, in repetitions
of the experiment,

821
00:57:04,500 --> 00:57:09,050
phi would be random, then this
is sort of what the math does.

822
00:57:09,050 --> 00:57:12,770
If phi is random, it is the
same as if we simply measure

823
00:57:12,770 --> 00:57:15,610
the power spectrum because
we cannot distinguish between

824
00:57:15,610 --> 00:57:17,840
the cosine and sine
Fourier transform.

825
00:57:22,020 --> 00:57:25,872
So if the phase
phi is random, that

826
00:57:25,872 --> 00:57:40,920
means you only measure
what was F plus IG before.

827
00:57:48,970 --> 00:57:53,460
In other words, the situation
is extremely in the end

828
00:57:53,460 --> 00:57:54,170
very simple.

829
00:57:54,170 --> 00:57:55,730
If you have quantum
beats which start

830
00:57:55,730 --> 00:58:00,140
with a well defined phase,
and you know the phase was,

831
00:58:00,140 --> 00:58:04,230
let's say, zero here, and now
you have the decaying function,

832
00:58:04,230 --> 00:58:08,070
and now you look at the
quantum beat over there, well,

833
00:58:08,070 --> 00:58:13,810
in a way, you had
n beats between t

834
00:58:13,810 --> 00:58:16,370
equals 0 and your
measurement, and then

835
00:58:16,370 --> 00:58:18,950
your resolution
goes with 1 over n,

836
00:58:18,950 --> 00:58:22,370
but you have to know what
the phase was at t equals 0,

837
00:58:22,370 --> 00:58:25,420
and then, by looking
at delayed detection,

838
00:58:25,420 --> 00:58:31,440
you can do spectroscopy below
the natural line widths.

839
00:58:31,440 --> 00:58:34,250
So therefore, what
is crucial here

840
00:58:34,250 --> 00:58:53,470
is knowledge, or at
least reproducibility,

841
00:58:53,470 --> 00:58:59,490
of the phase phi, and then
you can get narrow lines.

842
00:59:03,040 --> 00:59:06,870
Examples for techniques where
you excite the system at t

843
00:59:06,870 --> 00:59:09,850
equals 0 and then you
can do delayed detection

844
00:59:09,850 --> 00:59:12,440
of quantum beats.

845
00:59:12,440 --> 00:59:14,400
I mentioned earlier
in the course

846
00:59:14,400 --> 00:59:22,080
the Ramsey spectroscopy where
you have one Ramsey zone where

847
00:59:22,080 --> 00:59:23,520
you prepare your Bloch vector.

848
00:59:23,520 --> 00:59:25,850
Then the Bloch
vector oscillates,

849
00:59:25,850 --> 00:59:28,715
and if you simply look at the
phase angle of the Bloch vector

850
00:59:28,715 --> 00:59:32,970
after a very long time, you
have very high precision

851
00:59:32,970 --> 00:59:35,540
but you're dealing with an
exponentially small signal.

852
00:59:41,190 --> 00:59:48,970
Another example are heterodyne
or homodyne techniques,

853
00:59:48,970 --> 00:59:51,770
but you need something which
is phase sensitive in order

854
00:59:51,770 --> 00:59:58,320
to obtain sub-natural
line widths.

855
00:59:58,320 --> 01:00:04,490
A final comment is if you
want to get higher resolution

856
01:00:04,490 --> 01:00:08,260
with delayed detection,
yes, you can get it,

857
01:00:08,260 --> 01:00:11,420
but you was
exponentially in signal.

858
01:00:11,420 --> 01:00:14,220
And what does it
mean in practice?

859
01:00:14,220 --> 01:00:18,310
Well, if you know
your line shape,

860
01:00:18,310 --> 01:00:22,120
you know it's well
described by a Lorentzian,

861
01:00:22,120 --> 01:00:25,150
it is better to take
your full signal

862
01:00:25,150 --> 01:00:27,880
and use then an
excellent signal to noise

863
01:00:27,880 --> 01:00:31,060
to find the line center
and split the line.

864
01:00:31,060 --> 01:00:34,030
However, if there's
any ambiguity,

865
01:00:34,030 --> 01:00:38,170
there may be different,
not fully resourced lines

866
01:00:38,170 --> 01:00:40,260
under the Lorentzian
and you don't

867
01:00:40,260 --> 01:00:43,310
know how to split the
line, then it may be better

868
01:00:43,310 --> 01:00:45,600
to do delayed
detection and clearly

869
01:00:45,600 --> 01:00:48,395
see the structure of the lines
with sub-natural resolution.

870
01:00:52,610 --> 01:00:53,345
Any questions?

871
01:01:02,380 --> 01:01:08,210
This was coherence with two
levels, coherent excitation,

872
01:01:08,210 --> 01:01:10,925
coherent observation, some
spectroscopic techniques.

873
01:01:13,430 --> 01:01:19,530
Now we are ready to do
the next step, namely,

874
01:01:19,530 --> 01:01:23,840
to talk about coherence
in three level systems.

875
01:01:42,700 --> 01:01:46,410
If we have three levels,
we could think about it,

876
01:01:46,410 --> 01:01:49,910
we have terms which
connect level one to two,

877
01:01:49,910 --> 01:01:53,270
level two to three,
and level three to one

878
01:01:53,270 --> 01:01:55,890
in all possible
ways, but that's not

879
01:01:55,890 --> 01:01:59,180
what we want to assume here.

880
01:01:59,180 --> 01:02:03,460
The situation where we can
discuss some fundamentally new

881
01:02:03,460 --> 01:02:08,530
effects is when we have
two states connected

882
01:02:08,530 --> 01:02:11,600
through a third state.

883
01:02:21,310 --> 01:02:28,000
In other words, if
we have two levels,

884
01:02:28,000 --> 01:02:31,470
we are not allowing any
transition matrix element

885
01:02:31,470 --> 01:02:33,040
connecting the two.

886
01:02:33,040 --> 01:02:39,060
They are only connected
through a third state.

887
01:02:39,060 --> 01:02:43,560
This is for obvious reasons
called the lambda type system.

888
01:02:43,560 --> 01:02:49,000
You can turn it upside down
and you have the V type system

889
01:02:49,000 --> 01:02:54,920
or, if the intermediate
state is between the first

890
01:02:54,920 --> 01:03:02,590
and the second state, you
have a ladder type system.

891
01:03:02,590 --> 01:03:06,860
But once you start driving
it, it may not really matter.

892
01:03:06,860 --> 01:03:10,160
There may be a dressed
atom description

893
01:03:10,160 --> 01:03:15,060
where, if you drive two states
coherently in the dressed atom

894
01:03:15,060 --> 01:03:18,900
picture, you have degeneracy
between this level

895
01:03:18,900 --> 01:03:21,389
and one more photon
and this level.

896
01:03:21,389 --> 01:03:22,930
And then in the
dressed atom picture,

897
01:03:22,930 --> 01:03:24,890
which includes a
number of photons,

898
01:03:24,890 --> 01:03:27,070
the two levels have
become degenerate.

899
01:03:27,070 --> 01:03:29,910
So therefore, it's
very important

900
01:03:29,910 --> 01:03:32,490
for practical applications
or how to implement it

901
01:03:32,490 --> 01:03:35,640
in an atom what kind
of system you have,

902
01:03:35,640 --> 01:03:39,210
but for the description
of those systems,

903
01:03:39,210 --> 01:03:40,960
some of the
differences may simply

904
01:03:40,960 --> 01:03:44,930
disappear if you formulate
it in the dressed atom basis.

905
01:03:44,930 --> 01:03:47,770
Of course, there's an
important practical reason.

906
01:03:47,770 --> 01:03:50,120
Usually the lower states
are ground states,

907
01:03:50,120 --> 01:03:52,120
the upper states
are excited states.

908
01:03:52,120 --> 01:03:54,560
And here you have the
opportunity, and that's

909
01:03:54,560 --> 01:03:57,220
why the lambda type system
is the most important one,

910
01:03:57,220 --> 01:04:00,380
to have some coherent
superposition mediated

911
01:04:00,380 --> 01:04:03,080
by the third state, and
the coherent superposition

912
01:04:03,080 --> 01:04:05,385
is stable because it's
a coherent superposition

913
01:04:05,385 --> 01:04:07,180
of ground states.

914
01:04:07,180 --> 01:04:09,770
If any form of
coherent superposition

915
01:04:09,770 --> 01:04:14,340
involves an excited state, then
you have short lived states,

916
01:04:14,340 --> 01:04:19,164
and they are often not so
useful for certain phenomena.

917
01:04:22,800 --> 01:04:29,760
So if you think you know already
everything about how atoms

918
01:04:29,760 --> 01:04:34,356
interact with light
from two level atoms,

919
01:04:34,356 --> 01:04:39,670
I have to tell you that's not
the case because a three level

920
01:04:39,670 --> 01:04:42,270
system has many new effects.

921
01:04:51,500 --> 01:04:56,900
One, of course, is
that atoms can now

922
01:04:56,900 --> 01:05:06,160
interact with two
electromagnetic fields,

923
01:05:06,160 --> 01:05:20,470
and those two electromagnetic
fields can affect each other,

924
01:05:20,470 --> 01:05:26,480
and this can happen through
coherent or incoherent

925
01:05:26,480 --> 01:05:26,980
mechanisms.

926
01:05:39,120 --> 01:05:41,100
In other words, you
can say it simply.

927
01:05:41,100 --> 01:05:43,350
If you hit an atom with light
and you have a two level

928
01:05:43,350 --> 01:05:45,860
system, there is no way
how the atom can hide.

929
01:05:45,860 --> 01:05:48,770
It's always excited
by the laser.

930
01:05:48,770 --> 01:05:51,110
But if you have a
three level system,

931
01:05:51,110 --> 01:05:53,330
you may have a
situation where you

932
01:05:53,330 --> 01:05:56,490
have destructive interference
between what the two lasers can

933
01:05:56,490 --> 01:05:58,770
do to the atom,
and suddenly, there

934
01:05:58,770 --> 01:06:02,640
may be a state where the atom is
in the dark where the atom can

935
01:06:02,640 --> 01:06:04,160
hide from the laser beam.

936
01:06:04,160 --> 01:06:08,100
This is something which
is fundamentally new

937
01:06:08,100 --> 01:06:10,470
and has no counterpart
in a two level system.

938
01:06:15,210 --> 01:06:18,220
I've already pointed out
that the lambda system

939
01:06:18,220 --> 01:06:29,080
is the most important one
because it has two ground

940
01:06:29,080 --> 01:06:35,410
states which can be in a long
lived superposition state.

941
01:06:47,250 --> 01:06:56,500
What we want to discuss as
possible consequences is that

942
01:06:56,500 --> 01:07:08,960
in a three level system, you
can realize a lasing operation

943
01:07:08,960 --> 01:07:12,135
without having inversion
of the population

944
01:07:12,135 --> 01:07:14,140
of the ground and excited state.

945
01:07:14,140 --> 01:07:17,190
So if you always thought,
if I want to build a laser,

946
01:07:17,190 --> 01:07:18,990
the first thing I
have to do is make

947
01:07:18,990 --> 01:07:21,360
sure I have more atoms in the
excited than in the ground

948
01:07:21,360 --> 01:07:25,090
state, yes, this is valid
for two level system,

949
01:07:25,090 --> 01:07:28,300
but it is no longer valid
for a three level system.

950
01:07:31,100 --> 01:07:34,670
The reason why you want to
invert a two level system

951
01:07:34,670 --> 01:07:38,540
is you want to have stimulated
emission from the excited

952
01:07:38,540 --> 01:07:40,830
state which is stronger
than absorption

953
01:07:40,830 --> 01:07:42,620
from the ground state.

954
01:07:42,620 --> 01:07:46,340
But if you take advantage
of quantum coherence,

955
01:07:46,340 --> 01:07:49,760
you may have a situation
where two possibilities

956
01:07:49,760 --> 01:07:52,870
for stimulated emission
add up coherently,

957
01:07:52,870 --> 01:07:55,090
but the two possibilities
for absorption

958
01:07:55,090 --> 01:08:02,350
add up destructively.

959
01:08:02,350 --> 01:08:05,380
And therefore, you
can avoid destruction.

960
01:08:05,380 --> 01:08:07,240
You have only
stimulated emission,

961
01:08:07,240 --> 01:08:11,000
but you have not achieved
that through inversion.

962
01:08:11,000 --> 01:08:13,230
You have achieved that
through quantum coherence,

963
01:08:13,230 --> 01:08:16,189
a fundamentally new effect.

964
01:08:16,189 --> 01:08:19,160
So we have lasing
without inversion,

965
01:08:19,160 --> 01:08:21,609
we have the phenomenon
I mentioned already

966
01:08:21,609 --> 01:08:25,740
that atoms can hide in the
dark if the two laser beams

967
01:08:25,740 --> 01:08:29,910
in the excitation mechanism
destructively interfere.

968
01:08:29,910 --> 01:08:31,990
This is called
electromagnetically induced

969
01:08:31,990 --> 01:08:34,370
transparency.

970
01:08:34,370 --> 01:08:37,960
Systems which have sharp
resonances in three level

971
01:08:37,960 --> 01:08:45,540
systems are used for reducing
the group velocity of light,

972
01:08:45,540 --> 01:08:49,430
which goes under the
name slowing light,

973
01:08:49,430 --> 01:08:54,660
or even bringing light to a
standstill, stopping light.

974
01:08:54,660 --> 01:08:58,210
And three level
systems are also used

975
01:08:58,210 --> 01:09:01,410
for quantum mechanical memories
for quantum computation.

976
01:09:11,689 --> 01:09:12,580
Any questions?

977
01:09:17,050 --> 01:09:18,075
This is an introduction.

978
01:09:30,899 --> 01:09:37,420
Let me connect special effects
in a three level system

979
01:09:37,420 --> 01:09:43,458
to something which is very
basic and you've heard about it,

980
01:09:43,458 --> 01:09:44,624
and this is optical pumping.

981
01:09:49,069 --> 01:09:57,740
If we set up a system which
has two ground states, g and f,

982
01:09:57,740 --> 01:09:59,890
you may just think about
two hyperfine states

983
01:09:59,890 --> 01:10:03,330
in your favorite atom,
and then they are only

984
01:10:03,330 --> 01:10:06,890
coupled through
an excited state,

985
01:10:06,890 --> 01:10:14,360
you can now drive the
system with laser fields

986
01:10:14,360 --> 01:10:16,210
omega 1 and omega 2.

987
01:10:18,950 --> 01:10:21,810
Let me also use that
example of optical pumping

988
01:10:21,810 --> 01:10:28,060
to introduce some notation
which I will need to describe

989
01:10:28,060 --> 01:10:32,260
the system with a few equations.

990
01:10:32,260 --> 01:10:34,840
We will use energy
level diagrams,

991
01:10:34,840 --> 01:10:38,030
and the energy is referred
to the lowest ground state.

992
01:10:38,030 --> 01:10:44,240
So here, we have an energy
splitting which is by omega gf,

993
01:10:44,240 --> 01:10:48,000
and the excited state
has a splitting of eg.

994
01:10:50,890 --> 01:10:55,930
We will call the photons in
one laser the photons created

995
01:10:55,930 --> 01:11:00,150
and annihilated with the
operator a and a dagger,

996
01:11:00,150 --> 01:11:04,360
and for the photons for
the other laser beam,

997
01:11:04,360 --> 01:11:06,215
we use c or c dagger.

998
01:11:09,930 --> 01:11:14,690
Now, there is a
very simple solution

999
01:11:14,690 --> 01:11:19,030
for this situation, a very
simple equilibrium situation,

1000
01:11:19,030 --> 01:11:22,650
if you have only one laser beam.

1001
01:11:22,650 --> 01:11:30,330
If you have only one laser
beam, omega 1 or omega 2,

1002
01:11:30,330 --> 01:11:32,200
it's clear what happens.

1003
01:11:32,200 --> 01:11:35,760
If you have only one laser
beam, let's say omega 1,

1004
01:11:35,760 --> 01:11:38,640
it doesn't talk at all to
the atoms in the state f.

1005
01:11:38,640 --> 01:11:40,390
They are left alone.

1006
01:11:40,390 --> 01:11:45,440
But the atoms in the state g are
excited to the excited state,

1007
01:11:45,440 --> 01:11:48,110
and then there may be a
certain branching ratio

1008
01:11:48,110 --> 01:11:53,915
for spontaneous emission,
but let's rather call it

1009
01:11:53,915 --> 01:11:55,760
fluorescence, two
photon scattering.

1010
01:11:55,760 --> 01:11:59,010
So there be a branching ratio
to go back to that state

1011
01:11:59,010 --> 01:12:02,020
or to go to the state f.

1012
01:12:02,020 --> 01:12:03,842
If the latter happens,
the atom doesn't

1013
01:12:03,842 --> 01:12:05,425
interact with the
laser light anymore.

1014
01:12:05,425 --> 01:12:07,720
If it goes back to
the original state,

1015
01:12:07,720 --> 01:12:10,749
the atom will try again and
again until after a while,

1016
01:12:10,749 --> 01:12:13,040
all the atoms have been
optically pumped into the state

1017
01:12:13,040 --> 01:12:14,300
f.

1018
01:12:14,300 --> 01:12:19,010
And the same would happen if
you have a laser, omega 2,

1019
01:12:19,010 --> 01:12:22,220
then you would pump all
the atoms into the state g.

1020
01:12:22,220 --> 01:12:25,790
So if you have only one laser
beam, omega 1 and omega 2,

1021
01:12:25,790 --> 01:12:32,090
then in equilibrium, the
equilibrium population

1022
01:12:32,090 --> 01:12:39,760
is 100% of the atoms are in
state f or g respectively,

1023
01:12:39,760 --> 01:12:42,990
and this is nothing
else than the phenomenon

1024
01:12:42,990 --> 01:12:44,408
of optical pumping.

1025
01:13:01,660 --> 01:13:03,010
We have a very simple solution.

1026
01:13:03,010 --> 01:13:05,470
We pump all the atoms
into one quantum state

1027
01:13:05,470 --> 01:13:07,970
if we have only one
of the laser beams.

1028
01:13:07,970 --> 01:13:12,410
But the question is now, can
we have a similar situation

1029
01:13:12,410 --> 01:13:13,950
when both laser beams are on?

1030
01:13:37,990 --> 01:13:40,740
And what I mean by
that is, is it possible

1031
01:13:40,740 --> 01:13:56,000
now to pump all the atoms
into a state which does not

1032
01:13:56,000 --> 01:13:59,940
scatter any light, which does
not react with the light, which

1033
01:13:59,940 --> 01:14:02,235
is never excited to
the excited state?

1034
01:14:14,930 --> 01:14:17,340
The answer is yes,
and this is what

1035
01:14:17,340 --> 01:14:19,130
we want to derive right now.

1036
01:14:23,960 --> 01:14:34,420
Before I go into any equation,
the result is pretty clear.

1037
01:14:34,420 --> 01:14:38,970
If you have, say, g and f
and they are both excited,

1038
01:14:38,970 --> 01:14:46,540
if the amplitude which you
put into the excited state

1039
01:14:46,540 --> 01:14:50,160
is the same but
has opposite sign,

1040
01:14:50,160 --> 01:14:53,560
the two amplitudes which
are added in a time delta t

1041
01:14:53,560 --> 01:14:56,110
destructively interfere
and you have not

1042
01:14:56,110 --> 01:15:00,250
put any amplitude into
the excited state.

1043
01:15:00,250 --> 01:15:03,050
And that means, if you have
this initial superposition

1044
01:15:03,050 --> 01:15:06,210
state where this complete
destructive interference

1045
01:15:06,210 --> 01:15:09,200
happened, this state will
be dark all the time.

1046
01:15:11,820 --> 01:15:14,770
But we want to assume--
also it doesn't matter--

1047
01:15:14,770 --> 01:15:17,430
that the two states,
the two lasers

1048
01:15:17,430 --> 01:15:19,350
have two different frequencies.

1049
01:15:19,350 --> 01:15:23,360
So you cannot say, this is just
the two lasers have a certain

1050
01:15:23,360 --> 01:15:26,660
phase and then the laser field
interferes and they reach

1051
01:15:26,660 --> 01:15:28,680
a space which is dark.

1052
01:15:28,680 --> 01:15:32,380
We assume that the atom
is sitting at the origin.

1053
01:15:32,380 --> 01:15:34,420
Again, we're not putting
in motion effects,

1054
01:15:34,420 --> 01:15:37,290
so it's an atom
with infinite mass.

1055
01:15:37,290 --> 01:15:39,490
Then we shine two
laser lights on it

1056
01:15:39,490 --> 01:15:41,645
and the atom is not in the dark.

1057
01:15:41,645 --> 01:15:45,310
It's not at a dark fringe of the
interference of the two laser

1058
01:15:45,310 --> 01:15:47,940
beams because if you have
two laser beams with two

1059
01:15:47,940 --> 01:15:50,940
different frequencies,
there will not

1060
01:15:50,940 --> 01:15:54,150
be any place in space
which is dark all the time.

1061
01:15:54,150 --> 01:15:55,920
You create maybe
interference fringes,

1062
01:15:55,920 --> 01:15:58,220
but the interference
fringes are rapidly

1063
01:15:58,220 --> 01:16:01,520
running with a beat node,
omega 1 minus omega 2.

1064
01:16:01,520 --> 01:16:05,570
So the atom is not in the
dark, but nevertheless, it

1065
01:16:05,570 --> 01:16:07,920
will not scatter light
if it is prepared

1066
01:16:07,920 --> 01:16:10,910
in a suitable coherent
superposition state.

1067
01:16:19,150 --> 01:16:33,070
We describe this situation
with a dipole Hamiltonian,

1068
01:16:33,070 --> 01:16:34,905
and we make the rotating
wave approximation.

1069
01:16:50,460 --> 01:16:53,588
The Hamiltonian has
three lines, three parts.

1070
01:16:57,950 --> 01:17:12,400
One is we describe each of the
laser fields as a single mode.

1071
01:17:12,400 --> 01:17:16,970
I call the frequency
now omega a and omega c,

1072
01:17:16,970 --> 01:17:19,600
just to connect it with
the operator c dagger, c.

1073
01:17:28,860 --> 01:17:37,120
For the atom, we use the
matrix, two by two matrices,

1074
01:17:37,120 --> 01:17:45,530
so this is the matrix if the
atom is in the ground state.

1075
01:17:45,530 --> 01:17:50,270
Coherences are
described by that.

1076
01:17:50,270 --> 01:17:55,930
And of course, without any
interaction with the laser

1077
01:17:55,930 --> 01:18:01,405
field, the atomic Hamiltonian is
atoms are in the ground state.

1078
01:18:14,370 --> 01:18:16,150
the state f has zero energy.

1079
01:18:16,150 --> 01:18:18,500
We use that as the
origin of the energy.

1080
01:18:18,500 --> 01:18:23,620
The state g has an
eigenenergy of omega gf,

1081
01:18:23,620 --> 01:18:31,590
and the excited state e has
an eigenenergy of omega ef.

1082
01:18:31,590 --> 01:18:38,120
But now the important
part is that we

1083
01:18:38,120 --> 01:18:40,100
want to have the coupling.

1084
01:18:40,100 --> 01:18:43,630
And actually, I realized I
was not saying it correctly.

1085
01:18:43,630 --> 01:18:50,220
Omega 1 and omega 2, these are
the Rabi frequencies of the two

1086
01:18:50,220 --> 01:18:54,960
fields, and the two fields are
at frequency omega a and omega

1087
01:18:54,960 --> 01:18:56,220
c.

1088
01:18:56,220 --> 01:19:03,470
So now we have the coupling
between the excited state

1089
01:19:03,470 --> 01:19:07,800
and the ground state via
photons a and a dagger.

1090
01:19:12,740 --> 01:19:18,940
And the coupling happens at
the Rabi frequency omega 1.

1091
01:19:18,940 --> 01:19:22,110
And then we have the
second laser field,

1092
01:19:22,110 --> 01:19:23,770
which is at Rabi
frequency omega 2.

1093
01:19:35,230 --> 01:19:41,930
We have the atomic raising
and lowering operator,

1094
01:19:41,930 --> 01:19:45,900
and we have the
photon c and c dagger.

1095
01:19:49,510 --> 01:19:50,860
That's a nice Hamiltonian.

1096
01:19:50,860 --> 01:19:52,105
It has three lines.

1097
01:19:58,020 --> 01:20:02,130
The important part here, which
we have explicitly assumed

1098
01:20:02,130 --> 01:20:08,610
is that each of the lasers, a
and a dagger, c and c dagger,

1099
01:20:08,610 --> 01:20:13,160
are only driving one transition.

1100
01:20:13,160 --> 01:20:17,490
One field is responsible
for connecting the state f

1101
01:20:17,490 --> 01:20:18,950
to the excited state.

1102
01:20:18,950 --> 01:20:21,440
The other field is responsible
for connecting the state

1103
01:20:21,440 --> 01:20:23,470
g to the excited state.

1104
01:20:23,470 --> 01:20:26,020
In practice, this
can be accomplished

1105
01:20:26,020 --> 01:20:28,340
by you have maybe polarization.

1106
01:20:28,340 --> 01:20:31,510
This is a plus one state,
this is a minus one state,

1107
01:20:31,510 --> 01:20:33,810
and the excited
state is m equals 0.

1108
01:20:33,810 --> 01:20:36,610
Then one laser
beam is sigma plus,

1109
01:20:36,610 --> 01:20:39,860
the other one is sigma
minus, so it can be polarized

1110
01:20:39,860 --> 01:20:42,040
and the two laser
beams can only talk

1111
01:20:42,040 --> 01:20:43,790
to one of the ground states.

1112
01:20:43,790 --> 01:20:48,050
Or you can have a situation
that you have a huge energy

1113
01:20:48,050 --> 01:20:49,030
separation.

1114
01:20:49,030 --> 01:20:51,140
Let's say you have a
large hyperfine splitting

1115
01:20:51,140 --> 01:20:53,215
and the two lasers are
separated by frequency.

1116
01:20:59,550 --> 01:21:01,970
I think I've set the stage,
but I think I should stop here

1117
01:21:01,970 --> 01:21:04,650
and on Wednesday, I'll show
you in the first few minutes

1118
01:21:04,650 --> 01:21:07,520
of the class that
this Hamiltonian has

1119
01:21:07,520 --> 01:21:10,610
a simple solution, which
is a dark state, which

1120
01:21:10,610 --> 01:21:13,540
is a superposition
state of g and f.

1121
01:21:13,540 --> 01:21:14,240
Any questions?

1122
01:21:16,820 --> 01:21:19,402
See you on Wednesday.

1123
01:21:19,402 --> 01:21:21,610
A few people haven't picked
up their midterm quizzes.

1124
01:21:21,610 --> 01:21:24,550
If you want them,
I have them here.