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PROFESSOR: Back to
master equations

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00:00:24,000 --> 00:00:26,340
an optical Bloch equations.

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00:00:26,340 --> 00:00:28,940
I hope you remember
that on Monday we

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00:00:28,940 --> 00:00:32,070
derived, under very
general conditions,

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00:00:32,070 --> 00:00:35,950
how dissipation, relaxation
comes about in a quantum

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00:00:35,950 --> 00:00:38,880
system, namely because we
have a quantum system which

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interacts with a reservoir.

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00:00:41,070 --> 00:00:45,020
The total system evolves, as
quantum mechanics tells us,

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00:00:45,020 --> 00:00:46,275
with a unitary time evolution.

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00:00:50,820 --> 00:00:53,610
When we derive life what
happens with the small system,

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00:00:53,610 --> 00:00:55,810
due to the coupling
with a bigger system,

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00:00:55,810 --> 00:00:59,310
there are now suddenly-- the
time evolution of the density

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matrix has relaxation
terms, and we derived that.

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00:01:03,250 --> 00:01:05,360
And the special equation
we are interested

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00:01:05,360 --> 00:01:08,140
in for two level
system interacting

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00:01:08,140 --> 00:01:10,350
with the vacuum--
interacting with a vacuum

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00:01:10,350 --> 00:01:14,380
for spontaneous emission
are optical Bloch equations.

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00:01:14,380 --> 00:01:18,510
Today I want to discuss with you
some characteristic solutions

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00:01:18,510 --> 00:01:20,190
of the optical Bloch equations.

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00:01:20,190 --> 00:01:25,480
But before I do that, let
me first discuss with you

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00:01:25,480 --> 00:01:28,820
the assumptions we made to
derive a master equation.

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00:01:28,820 --> 00:01:32,960
And I hope you remember the Born
approximation and the Markov

30
00:01:32,960 --> 00:01:34,220
approximation.

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00:01:34,220 --> 00:01:38,001
The Born approximation says
that the reservoir will never

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00:01:38,001 --> 00:01:38,500
change.

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00:01:38,500 --> 00:01:42,100
The reservoir is always, for
instance, in the vacuum state.

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00:01:42,100 --> 00:01:44,230
And the Markov
approximation said

35
00:01:44,230 --> 00:01:46,520
that the correlations
in the reservoir

36
00:01:46,520 --> 00:01:48,510
are delta function related.

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00:01:48,510 --> 00:01:51,360
It can absorb photons
instantaneously.

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00:01:51,360 --> 00:01:53,320
The vacuum has no memory time.

39
00:01:53,320 --> 00:01:54,370
It's like a black hole.

40
00:01:54,370 --> 00:01:56,280
It sucks up everything.

41
00:01:56,280 --> 00:02:00,410
But it's always what it was
initially, namely the vacuum.

42
00:02:00,410 --> 00:02:05,440
Do you know examples where those
assumptions are not fulfilled?

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00:02:05,440 --> 00:02:06,701
I make it so natural.

44
00:02:06,701 --> 00:02:08,325
These are the most
natural assumptions,

45
00:02:08,325 --> 00:02:11,310
but what are systems which
cannot be described with this

46
00:02:11,310 --> 00:02:12,480
approach.

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00:02:12,480 --> 00:02:15,768
AUDIENCE: I was thinking maybe
like an atom in a cavity.

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00:02:15,768 --> 00:02:18,670
If you change the
vacuum somehow.

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00:02:18,670 --> 00:02:20,100
PROFESSOR: An atom in a cavity.

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00:02:20,100 --> 00:02:21,230
Very good.

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00:02:21,230 --> 00:02:23,320
If an atom emits--
now it depends.

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00:02:23,320 --> 00:02:27,560
We talk today about the master
equation for photons and atoms

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00:02:27,560 --> 00:02:29,470
in the cavity,
but we will assume

54
00:02:29,470 --> 00:02:31,980
that the cavity
is rapidly damped.

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00:02:31,980 --> 00:02:37,310
If it's a high Q cavity,
the atom emits a photon,

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00:02:37,310 --> 00:02:38,960
but the cavity has memory.

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00:02:38,960 --> 00:02:44,230
It stores the photon, and after
one period of the vacuum Rabi

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00:02:44,230 --> 00:02:47,030
oscillation, the photon
goes back to the atom.

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00:02:47,030 --> 00:02:49,970
So the vacuum has memory time.

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00:02:49,970 --> 00:02:52,580
The vacuum has now
a time constant,

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00:02:52,580 --> 00:02:55,680
which is the same as the
time constant for the atom,

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00:02:55,680 --> 00:02:58,280
namely the period of
the Rabi oscillation.

63
00:02:58,280 --> 00:03:01,420
Big violation of the
Markov approximation.

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00:03:01,420 --> 00:03:05,100
We do not have the very short
correlation time of the system.

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00:03:05,100 --> 00:03:09,700
We just integrate over in
the much longer time scale,

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00:03:09,700 --> 00:03:13,560
during which we are interested
in the dynamics of the system.

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00:03:13,560 --> 00:03:15,040
Maybe another example.

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00:03:15,040 --> 00:03:19,060
Actually, this example violates
both the Markov approximation,

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00:03:19,060 --> 00:03:22,020
also the Born approximation,
because the vacuum

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00:03:22,020 --> 00:03:24,536
has changed when
it has a photon.

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00:03:24,536 --> 00:03:26,160
Do you know an example
where maybe only

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00:03:26,160 --> 00:03:28,630
the Born approximation
is violated,

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00:03:28,630 --> 00:03:34,594
namely the reservoir is
changing due to interaction

74
00:03:34,594 --> 00:03:35,260
with the system?

75
00:03:48,840 --> 00:03:49,900
Well?

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00:03:49,900 --> 00:03:53,790
Before we talked about the
system in a big reservoir.

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00:03:53,790 --> 00:03:56,350
What happens when you make
the reservoir smaller?

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00:04:01,050 --> 00:04:02,960
Well then when it
absorbs energy.

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00:04:02,960 --> 00:04:04,140
It will heat up and such.

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00:04:04,140 --> 00:04:05,670
It will change.

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00:04:05,670 --> 00:04:08,450
The Born approximation
is actually,

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00:04:08,450 --> 00:04:11,210
when it would come to energy
transfer, the assumption

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00:04:11,210 --> 00:04:13,770
that the reservoir has an
infinite heat capacity.

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00:04:13,770 --> 00:04:16,010
It can just take
whatever the system wants

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00:04:16,010 --> 00:04:18,954
to deliver without changing,
for instance, it's temperature.

86
00:04:21,660 --> 00:04:23,470
OK, so at least
you know now there

87
00:04:23,470 --> 00:04:26,140
are systems for which
the treatment has

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00:04:26,140 --> 00:04:28,660
to be generalized, and you
understand maybe better

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00:04:28,660 --> 00:04:31,950
the nature of
approximations we have made.

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00:04:31,950 --> 00:04:32,855
[INAUDIBLE]

91
00:04:32,855 --> 00:04:33,938
AUDIENCE: Just a question.

92
00:04:33,938 --> 00:04:36,753
For the last case which is where
the Born approximation is not

93
00:04:36,753 --> 00:04:40,399
violated, is there
any example of that?

94
00:04:46,837 --> 00:04:48,920
PROFESSOR: Well for instance,
spontaneous emission

95
00:04:48,920 --> 00:04:51,170
into the vacuum, but
the speed of light

96
00:04:51,170 --> 00:04:54,810
is so slow that it takes forever
for the photon to escape.

97
00:04:54,810 --> 00:04:57,190
The Markov approximation
is more about the time

98
00:04:57,190 --> 00:05:01,460
scale for the
reservoir to react.

99
00:05:01,460 --> 00:05:06,200
And the Born approximation
is if the reservoir is really

100
00:05:06,200 --> 00:05:08,955
big enough to simply absorb
everything without changing.

101
00:05:13,739 --> 00:05:14,280
I don't know.

102
00:05:14,280 --> 00:05:17,190
You could maybe think about
you have some reservoir,

103
00:05:17,190 --> 00:05:19,630
but it has very bad heat
conduction or something.

104
00:05:19,630 --> 00:05:22,330
The transport is just very
slow, and therefore you

105
00:05:22,330 --> 00:05:24,430
do not have the hierarchy
of the two time scales.

106
00:05:28,200 --> 00:05:31,600
OK, any other questions
about the master equation?

107
00:05:31,600 --> 00:05:32,695
Optical Bloch equations?

108
00:05:35,540 --> 00:05:43,572
Well, then we can
learn more about

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00:05:43,572 --> 00:05:48,040
the characteristic features
of the optical Bloch equations

110
00:05:48,040 --> 00:05:49,275
by looking at solutions.

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00:05:57,500 --> 00:06:01,705
And I would like to discuss
three aspects of solution.

112
00:06:04,620 --> 00:06:15,400
I want to discuss the spectrum
and the intensity of light

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00:06:15,400 --> 00:06:16,525
emitted by the atoms.

114
00:06:21,050 --> 00:06:32,275
Secondly, we want to talk about
transient and steady state

115
00:06:32,275 --> 00:06:32,775
solutions.

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00:06:37,560 --> 00:06:40,970
And finally, actually I put it
in the chapter of optical Bloch

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00:06:40,970 --> 00:06:47,890
equations, but I
will use a cavity

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00:06:47,890 --> 00:06:49,650
for that example,
which will actually

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00:06:49,650 --> 00:06:51,067
go beyond optical
Bloch equations.

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00:06:51,067 --> 00:06:52,733
It's also nice for
you to see that there

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00:06:52,733 --> 00:06:54,530
is more than the
optical Bloch equation,

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00:06:54,530 --> 00:06:56,130
but it's the same formalism.

123
00:06:56,130 --> 00:07:00,440
And this is when we discuss the
damping of the damped vacuum

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00:07:00,440 --> 00:07:01,190
Rabi oscillations.

125
00:07:06,470 --> 00:07:08,040
I also picked this
one example sort

126
00:07:08,040 --> 00:07:10,980
of as an appetizer,
because I can introduce

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00:07:10,980 --> 00:07:13,560
two concepts for you
in this last example.

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00:07:13,560 --> 00:07:15,910
One is the quantum Zeno effect.

129
00:07:15,910 --> 00:07:18,370
And the other one is the
adiabatic elimination

130
00:07:18,370 --> 00:07:21,300
of coherences, which
plays a major role

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00:07:21,300 --> 00:07:24,100
in many, many
theoretical treatments.

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00:07:24,100 --> 00:07:31,480
But let's start with the first
part, namely the spectrum

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00:07:31,480 --> 00:07:34,800
and intensity of emitted light.

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00:07:34,800 --> 00:07:38,790
And I want to start off this
discussion with a clicker

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00:07:38,790 --> 00:07:39,290
question.

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00:07:39,290 --> 00:07:43,620
Do everybody take a
clicker out of the box?

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00:07:43,620 --> 00:07:48,850
So the situation is that
we have an atom which

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00:07:48,850 --> 00:07:55,020
is excited by laser light
of frequency omega L.

139
00:07:55,020 --> 00:07:57,920
The atom has frequency omega 0.

140
00:08:00,670 --> 00:08:03,370
I'm discussing the limit
of very low intensity.

141
00:08:07,570 --> 00:08:11,270
And the question
for you is what is

142
00:08:11,270 --> 00:08:16,260
the spectrum of the
emitted radiation?

143
00:08:22,570 --> 00:08:27,500
So in other words,
we have our atom.

144
00:08:27,500 --> 00:08:30,670
We have the laser
beam which comes here.

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00:08:30,670 --> 00:08:39,750
And now the atom emits
in all directions.

146
00:08:39,750 --> 00:08:44,190
And you collect and
frequency analyze the light.

147
00:08:44,190 --> 00:08:50,852
And I want to give you
four possibilities how

148
00:08:50,852 --> 00:08:56,330
the spectrum of the emitted
light may look like.

149
00:08:56,330 --> 00:08:58,850
Let's assume that on
the frequency axis

150
00:08:58,850 --> 00:09:01,990
this is the laser frequency.

151
00:09:01,990 --> 00:09:06,570
And this is the
frequency of the atom.

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00:09:06,570 --> 00:09:11,270
And the first possibility
is that we observed

153
00:09:11,270 --> 00:09:14,900
a spectrum which is centered
at the atomic frequency where

154
00:09:14,900 --> 00:09:17,270
the line meets gamma.

155
00:09:17,270 --> 00:09:21,810
The second option is that
we observe a delta function,

156
00:09:21,810 --> 00:09:26,290
a very sharp peak at
the frequency omega 0.

157
00:09:26,290 --> 00:09:29,640
The third possibility is
the we observe a sharp peak

158
00:09:29,640 --> 00:09:32,210
at the laser frequency.

159
00:09:32,210 --> 00:09:38,550
And the fourth option is that
we observe radiation centered

160
00:09:38,550 --> 00:09:42,250
at the laser frequency,
but with gamma.

161
00:09:42,250 --> 00:09:45,110
So I would ask
you for your vote.

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00:09:45,110 --> 00:09:51,650
A, B, C, or D.

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00:09:51,650 --> 00:09:55,570
Was there any questions
about the choices?

164
00:09:55,570 --> 00:09:57,460
40 for in spectra.

165
00:09:57,460 --> 00:10:00,390
The low intensity limit
of light scattered

166
00:10:00,390 --> 00:10:04,800
emitted by an atom which is
excited with a laser beam.

167
00:10:04,800 --> 00:10:07,040
We can assume close
to resonance, but not

168
00:10:07,040 --> 00:10:09,910
on resonance, so we can
distinguish between the laser

169
00:10:09,910 --> 00:10:12,000
frequency and the
atomic frequency.

170
00:10:25,960 --> 00:10:28,216
Anybody had a chance?

171
00:10:28,216 --> 00:10:28,715
OK.

172
00:10:36,130 --> 00:10:37,730
OK, good.

173
00:10:37,730 --> 00:10:41,860
A, B, C, D.

174
00:10:41,860 --> 00:10:43,890
Yes, let's discuss it.

175
00:10:46,520 --> 00:10:48,950
I want to draw your attention
to one thing I said,

176
00:10:48,950 --> 00:10:49,940
namely low intensity.

177
00:10:56,520 --> 00:11:01,760
At low intensity, you can assume
that there's only one photon

178
00:11:01,760 --> 00:11:07,850
scattered at a time, because
the intensity-- we work

179
00:11:07,850 --> 00:11:10,860
in the limit of
infinitesimal intensity.

180
00:11:10,860 --> 00:11:15,080
So I want to as you
now, why don't you

181
00:11:15,080 --> 00:11:19,560
consider energy conservation and
think about the problem again?

182
00:11:33,340 --> 00:11:37,330
OK, maybe I'm indicating I'm
not happy with the answer.

183
00:11:45,951 --> 00:11:47,950
Maybe an obvious question
which sometimes people

184
00:11:47,950 --> 00:11:50,450
get confused about, what
about Doppler shifts?

185
00:11:50,450 --> 00:11:53,280
We assume the atom is infinitely
heavy and cannot move.

186
00:11:53,280 --> 00:11:55,450
Or we trap it in an ion trap.

187
00:11:55,450 --> 00:11:57,610
It's in the ground
state of the ion trap,

188
00:11:57,610 --> 00:12:00,960
and no kinetic
energy is exchanged.

189
00:12:00,960 --> 00:12:03,260
So don't get confused,
and I think none of you

190
00:12:03,260 --> 00:12:06,315
got confused about Doppler
shifts in kinetic energy here.

191
00:12:06,315 --> 00:12:09,225
We eliminate that.

192
00:12:09,225 --> 00:12:09,725
OK.

193
00:12:15,310 --> 00:12:20,055
Well it's moving in
the right direction.

194
00:12:23,260 --> 00:12:27,440
Now the two most frequent
answers are those two.

195
00:12:31,820 --> 00:12:35,680
OK, I've asked you to take
energy conservation really

196
00:12:35,680 --> 00:12:37,140
seriously.

197
00:12:37,140 --> 00:12:39,690
If you start with an
atom in the ground state,

198
00:12:39,690 --> 00:12:43,500
and the monochromatic photon at
the laser frequency, and you've

199
00:12:43,500 --> 00:12:45,920
done a wonderful job,
which many labs can do.

200
00:12:45,920 --> 00:12:49,720
You've stabilized your laser
with a precision of 1 hertz.

201
00:12:49,720 --> 00:12:53,930
And 1 Hertz, this is sort of
what I do as a delta function.

202
00:12:53,930 --> 00:12:56,910
So if you take energy
conservation absolutely

203
00:12:56,910 --> 00:13:00,780
seriously, the
photon, which can--

204
00:13:00,780 --> 00:13:03,470
the atom is in the ground state
after it has scattered light,

205
00:13:03,470 --> 00:13:07,660
so the energy of the
atom has not changed.

206
00:13:07,660 --> 00:13:10,590
And you can only fulfill
energy conservation

207
00:13:10,590 --> 00:13:14,360
if there is no change in
the spectrum of the light.

208
00:13:14,360 --> 00:13:18,150
And that means delta function
in, delta function out.

209
00:13:21,200 --> 00:13:26,560
Now I know I say it as if
it would be the simplest

210
00:13:26,560 --> 00:13:30,910
thing in the world, but
maybe you have questions.

211
00:13:30,910 --> 00:13:33,960
Does anybody want to
maybe argue or defend

212
00:13:33,960 --> 00:13:35,960
why the light
should be broadened?

213
00:13:42,250 --> 00:13:45,680
And actually discuss
very soon other examples.

214
00:13:45,680 --> 00:13:50,360
We go to higher intensity,
and at higher intensity

215
00:13:50,360 --> 00:13:52,600
other things end and
other things are possible.

216
00:13:52,600 --> 00:13:55,210
But at the low
intensity limit, do

217
00:13:55,210 --> 00:13:56,670
you have any
suggestion what could

218
00:13:56,670 --> 00:13:58,120
cause the broadening of light?

219
00:14:04,180 --> 00:14:07,035
AUDIENCE: I guess this is more
if [INAUDIBLE] but then look

220
00:14:07,035 --> 00:14:10,790
at it as pinned down,
but you're going

221
00:14:10,790 --> 00:14:14,360
to see a Doppler shift from the
recoil as the light comes out.

222
00:14:14,360 --> 00:14:20,370
So it accepts the light,
and then it moves something,

223
00:14:20,370 --> 00:14:23,180
so then it has some velocity.

224
00:14:23,180 --> 00:14:25,432
And then when it
emits, it's going

225
00:14:25,432 --> 00:14:27,210
to be emitting in
a random direction.

226
00:14:27,210 --> 00:14:28,626
And so its final
velocity is going

227
00:14:28,626 --> 00:14:32,540
to be in some basically
arbitrary direction,

228
00:14:32,540 --> 00:14:34,180
because the two
[INAUDIBLE] boosts

229
00:14:34,180 --> 00:14:37,460
that you get from
accepting and emitting.

230
00:14:37,460 --> 00:14:40,670
PROFESSOR: OK now first of
all, I wanted for this problem

231
00:14:40,670 --> 00:14:45,740
to assume that the mass
of the atom is infinite.

232
00:14:45,740 --> 00:14:48,040
And at that point,
the recoil energy

233
00:14:48,040 --> 00:14:50,300
has 1 over the mass term.

234
00:14:50,300 --> 00:14:52,840
The recoil energy is 0
and we can neglect it.

235
00:14:52,840 --> 00:14:55,080
So the limit I'm
talking about is sound.

236
00:14:55,080 --> 00:14:55,850
But you're right.

237
00:14:55,850 --> 00:14:58,440
If you want to bring in the
recoil energy, what would then

238
00:14:58,440 --> 00:15:01,260
happen is if an atom
absorbs a photon

239
00:15:01,260 --> 00:15:03,730
and emits it pretty much
in the same direction,

240
00:15:03,730 --> 00:15:05,710
there is no momentum exchange.

241
00:15:05,710 --> 00:15:08,730
Then it will be close
to the laser frequency.

242
00:15:08,730 --> 00:15:13,910
But then you will find,
depending on the angle now,

243
00:15:13,910 --> 00:15:18,320
there is a shift between
0 and 2 recoil energies.

244
00:15:18,320 --> 00:15:21,550
If you would, however, measure
the light at a specific angle,

245
00:15:21,550 --> 00:15:23,180
you would still find
a delta function,

246
00:15:23,180 --> 00:15:25,060
but you would find it shifted.

247
00:15:25,060 --> 00:15:27,100
But this is really
a complication

248
00:15:27,100 --> 00:15:29,535
which involves now the
mechanical effect of light

249
00:15:29,535 --> 00:15:30,975
and the recoil of a photon.

250
00:15:34,100 --> 00:15:36,070
It doesn't change the result.

251
00:15:36,070 --> 00:15:39,890
Energy conservation requires
the spectrum to be, so to speak,

252
00:15:39,890 --> 00:15:42,280
a delta function
in total energy.

253
00:15:42,280 --> 00:15:44,420
And what we have
just discussed is

254
00:15:44,420 --> 00:15:46,440
that, depending on the
scattering angle, maybe

255
00:15:46,440 --> 00:15:49,085
there's a kinetic
energy to consider.

256
00:16:06,350 --> 00:16:09,330
So what else can
broaden the light?

257
00:16:09,330 --> 00:16:10,104
Yes?

258
00:16:10,104 --> 00:16:12,080
AUDIENCE: Finite pulse time?

259
00:16:12,080 --> 00:16:13,080
PROFESSOR: Yes.

260
00:16:13,080 --> 00:16:16,460
So things are very different
if I use a pulse laser.

261
00:16:16,460 --> 00:16:20,640
But here I said the incident
light is monochromatic,

262
00:16:20,640 --> 00:16:22,220
just one frequency.

263
00:16:22,220 --> 00:16:24,760
And therefore you don't
have any time resolution

264
00:16:24,760 --> 00:16:25,720
in the experiment.

265
00:16:25,720 --> 00:16:27,710
The moment you have
time resolution,

266
00:16:27,710 --> 00:16:30,780
you would introduce into
the whole discussion

267
00:16:30,780 --> 00:16:34,190
Fourier [INAUDIBLE] and such.

268
00:16:34,190 --> 00:16:41,440
But frankly, I ask this question
every time I teach the course.

269
00:16:41,440 --> 00:16:44,502
And every time the vast
majority of the class

270
00:16:44,502 --> 00:16:45,460
gives the wrong answer.

271
00:16:47,972 --> 00:16:49,430
Let me ask you
something else which

272
00:16:49,430 --> 00:16:52,830
is maybe a little bit simpler,
and that's the following.

273
00:16:52,830 --> 00:16:55,540
You have a harmonic
oscillator which

274
00:16:55,540 --> 00:16:58,840
has an eigen frequency omega 0.

275
00:16:58,840 --> 00:17:00,610
You drive the
harmonic oscillator

276
00:17:00,610 --> 00:17:04,869
at frequency omega L.
What is the oscillation

277
00:17:04,869 --> 00:17:05,990
of the oscillator?

278
00:17:05,990 --> 00:17:09,470
Is it omega 0, the harmonic
oscillator frequency?

279
00:17:09,470 --> 00:17:13,662
Or omega L, the drive frequency?

280
00:17:13,662 --> 00:17:15,520
AUDIENCE: [INAUDIBLE].

281
00:17:15,520 --> 00:17:17,859
PROFESSOR: Of course,
the drive frequency.

282
00:17:17,859 --> 00:17:21,109
There may be a short
transient, but if we do not

283
00:17:21,109 --> 00:17:24,920
switch on the drive suddenly,
we avoid the transient.

284
00:17:24,920 --> 00:17:28,140
The steady state solution is
the harmonic oscillator always

285
00:17:28,140 --> 00:17:32,150
oscillates at the
drive frequency.

286
00:17:32,150 --> 00:17:34,992
Now, isn't the atom a
harmonic oscillator?

287
00:17:38,040 --> 00:17:41,300
In 8421 we discuss the
oscillator strings.

288
00:17:41,300 --> 00:17:43,740
We discuss that
the polarizability

289
00:17:43,740 --> 00:17:46,880
of the atom, the dipole
moment, the AC stock shift

290
00:17:46,880 --> 00:17:50,320
can really be regarded as an
oscillating dipole moment.

291
00:17:50,320 --> 00:17:53,170
So the atom for that
purpose is nothing else

292
00:17:53,170 --> 00:17:55,940
than a driven
harmonic oscillator.

293
00:17:55,940 --> 00:18:00,050
And when you drive the
atom at a laser frequency,

294
00:18:00,050 --> 00:18:05,620
the dipole moment oscillates
at which frequency?

295
00:18:05,620 --> 00:18:07,350
The laser frequency.

296
00:18:07,350 --> 00:18:09,380
Not the resonant frequency.

297
00:18:09,380 --> 00:18:12,310
And therefore, the light
which is scattered--

298
00:18:12,310 --> 00:18:14,170
think about it semi classically.

299
00:18:14,170 --> 00:18:15,670
This is like an antenna.

300
00:18:15,670 --> 00:18:16,630
It's an electron.

301
00:18:16,630 --> 00:18:17,390
It's a charge.

302
00:18:17,390 --> 00:18:19,140
It's a dipole moment
which oscillates

303
00:18:19,140 --> 00:18:20,490
at the drive frequency.

304
00:18:20,490 --> 00:18:23,407
And the light, which is emitted
by this accelerated charge

305
00:18:23,407 --> 00:18:24,490
is at the drive frequency.

306
00:18:27,634 --> 00:18:31,330
The atom is, for that
purpose, nothing else

307
00:18:31,330 --> 00:18:33,610
than a driven
harmonic oscillator.

308
00:18:33,610 --> 00:18:36,960
And I'm saying that
probably several times

309
00:18:36,960 --> 00:18:40,480
in that course, when you
have a semi classical picture

310
00:18:40,480 --> 00:18:45,490
and a fully quantum picture,
and they do not fully agree.

311
00:18:45,490 --> 00:18:48,300
I've never gone wrong with
the semi classical picture.

312
00:18:48,300 --> 00:18:51,100
But myself and
many of my students

313
00:18:51,100 --> 00:18:56,730
have come to very wrong answers
by somehow holding on just

314
00:18:56,730 --> 00:18:58,580
to the quantum
picture, and now seeing

315
00:18:58,580 --> 00:19:00,700
what the semi
classical limit is.

316
00:19:00,700 --> 00:19:03,650
You may think too
much here in the atom

317
00:19:03,650 --> 00:19:06,340
not as a harmonic oscillator,
but a two level system.

318
00:19:06,340 --> 00:19:09,120
The two level
system gets excited,

319
00:19:09,120 --> 00:19:13,820
and then after it is excited, it
emits at the natural frequency.

320
00:19:13,820 --> 00:19:16,830
But you've taken something
to seriously here.

321
00:19:16,830 --> 00:19:23,480
It's a two level system, but you
cannot read the system that you

322
00:19:23,480 --> 00:19:26,000
first populate a level,
and then you emit,

323
00:19:26,000 --> 00:19:29,860
because the energy of the
laser is not exactly the energy

324
00:19:29,860 --> 00:19:31,810
needed to populate the level.

325
00:19:31,810 --> 00:19:34,800
So therefore you shouldn't do
first order perturbation theory

326
00:19:34,800 --> 00:19:37,040
where you populate a level
and then you wait and wait

327
00:19:37,040 --> 00:19:38,340
for the emitted photon.

328
00:19:38,340 --> 00:19:39,760
This is light scattering.

329
00:19:39,760 --> 00:19:42,790
It's a second order problem.

330
00:19:42,790 --> 00:19:44,540
One photon in, one photon out.

331
00:19:44,540 --> 00:19:48,420
And you have to conserve energy.

332
00:19:48,420 --> 00:19:50,070
Sorry for hammering
on it, but it's

333
00:19:50,070 --> 00:19:52,640
really an example
where you can maybe

334
00:19:52,640 --> 00:19:55,570
realize certain
misconceptions, and I really

335
00:19:55,570 --> 00:19:57,659
want to make sure that
you fully understand

336
00:19:57,659 --> 00:19:59,075
what's going on
in this situation.

337
00:20:01,740 --> 00:20:02,240
Questions?

338
00:20:06,090 --> 00:20:08,450
Well, then let's make
it more interesting.

339
00:20:08,450 --> 00:20:12,420
Let's introduce
higher intensity.

340
00:20:12,420 --> 00:20:14,170
So let me first maybe
show you-- we've

341
00:20:14,170 --> 00:20:17,630
talked a lot about
diagrams-- what

342
00:20:17,630 --> 00:20:20,910
the diagram for this process is.

343
00:20:20,910 --> 00:20:24,720
So we have the laser
frequency, and then we

344
00:20:24,720 --> 00:20:25,940
have the photon emitted.

345
00:20:32,540 --> 00:20:36,580
And of course, the length of
this vector, this frequency

346
00:20:36,580 --> 00:20:38,445
is the same as the
laser frequency.

347
00:21:14,010 --> 00:21:18,086
So let's move on now and
talk about higher intensity.

348
00:21:28,300 --> 00:21:30,940
The high intensity
case we discuss

349
00:21:30,940 --> 00:21:35,890
in great detail in 8421,
but you pretty much I think

350
00:21:35,890 --> 00:21:37,780
know also from basic
quantum physics

351
00:21:37,780 --> 00:21:39,840
what happens if you
take a two level system

352
00:21:39,840 --> 00:21:40,800
and drive it strongly.

353
00:21:49,010 --> 00:21:50,920
If you drive it
strongly, stronger

354
00:21:50,920 --> 00:21:53,800
than spontaneous
emission, that means

355
00:21:53,800 --> 00:21:55,280
there is a limit
where you can just

356
00:21:55,280 --> 00:21:57,820
forget about spontaneous
emission in leading order,

357
00:21:57,820 --> 00:22:02,520
and the system is nothing else
than a two level system coupled

358
00:22:02,520 --> 00:22:06,680
by a strong monochromatic drive.

359
00:22:06,680 --> 00:22:10,300
So I assume you've all
heard about this solution?

360
00:22:10,300 --> 00:22:14,690
Get the Rabi oscillation between
ground and excited state.

361
00:22:14,690 --> 00:22:25,600
So the high intensity
limit are Rabi oscillations

362
00:22:25,600 --> 00:22:28,950
at the Rabi frequency.

363
00:22:28,950 --> 00:22:35,830
So what happens now to the
spectrum of the emitted light

364
00:22:35,830 --> 00:22:38,640
if you are in the
limit that you have

365
00:22:38,640 --> 00:22:40,580
Rabi oscillation of
the atomic system?

366
00:22:56,060 --> 00:22:58,440
And I want you to think
classically or semi

367
00:22:58,440 --> 00:22:58,940
classically.

368
00:23:01,675 --> 00:23:03,670
AUDIENCE: Three peaks.

369
00:23:03,670 --> 00:23:06,160
PROFESSOR: Three peaks
you generate side bends.

370
00:23:06,160 --> 00:23:10,250
Classically, you have an
emitter, an oscillating charge.

371
00:23:10,250 --> 00:23:13,310
But now you want to throw
in that the atom goes

372
00:23:13,310 --> 00:23:15,320
from the ground to
the excited state,

373
00:23:15,320 --> 00:23:17,435
from the excited to
the ground state.

374
00:23:17,435 --> 00:23:19,970
So, so to speak, when
it's in the ground state,

375
00:23:19,970 --> 00:23:20,800
it cannot emit.

376
00:23:20,800 --> 00:23:23,600
When it's in the excited
state, it can emit and such.

377
00:23:23,600 --> 00:23:27,430
So you should think about
an oscillating dipole

378
00:23:27,430 --> 00:23:30,580
which is now
intensity modulated.

379
00:23:30,580 --> 00:23:33,030
And you know when you take
a classic light source which

380
00:23:33,030 --> 00:23:38,700
is monochromatic, but put on top
of it an intensity modulation

381
00:23:38,700 --> 00:23:42,570
at frequency omega Rabi,
the solution of that

382
00:23:42,570 --> 00:23:44,370
are three peaks.

383
00:23:44,370 --> 00:23:48,560
The carrier, which is the
laser frequency, plus two side

384
00:23:48,560 --> 00:23:52,170
bends at omega Rabi.

385
00:23:52,170 --> 00:23:57,360
So therefore, what we now
expect for the spectrum is we

386
00:23:57,360 --> 00:24:00,730
have the laser
frequency, and this

387
00:24:00,730 --> 00:24:04,430
is sort of our result
for low intensity.

388
00:24:04,430 --> 00:24:14,182
However, when we-- in the
limit of strong drive,

389
00:24:14,182 --> 00:24:15,890
and you will see in
a moment where strong

390
00:24:15,890 --> 00:24:17,400
drive clearly comes in.

391
00:24:17,400 --> 00:24:21,450
So if you have this modulation
at the Rabi frequency,

392
00:24:21,450 --> 00:24:25,940
you obtain side bends
at the Rabi frequency.

393
00:24:28,970 --> 00:24:34,900
And the fact that there is
a structure with three peaks

394
00:24:34,900 --> 00:24:38,630
is actually the
famous Mollow triplet.

395
00:24:38,630 --> 00:24:40,170
I know when I was
a graduate student

396
00:24:40,170 --> 00:24:42,170
there was big excitement
because people measured

397
00:24:42,170 --> 00:24:43,880
for the first time
the Mollow triplet.

398
00:24:43,880 --> 00:24:46,020
You need high resolution
lasers and such,

399
00:24:46,020 --> 00:24:48,530
and people were just
ready to measure that.

400
00:24:48,530 --> 00:24:54,570
Now it seems something which we
just mention in basic courses

401
00:24:54,570 --> 00:24:57,800
because it's such
a basic phenomenon.

402
00:24:57,800 --> 00:25:01,680
OK, but let's work a
little bit more on that,

403
00:25:01,680 --> 00:25:09,190
namely the Rabi frequency is
the Rabi frequency of the drive,

404
00:25:09,190 --> 00:25:12,230
or you can say the
coupling matrix element.

405
00:25:12,230 --> 00:25:15,980
But when we have a
detuning-- and this

406
00:25:15,980 --> 00:25:17,710
is what you want
to discuss now--

407
00:25:17,710 --> 00:25:20,640
you have to add the
detuning in quadrature.

408
00:25:20,640 --> 00:25:24,010
So this is a frequency at
which the atomic population

409
00:25:24,010 --> 00:25:27,510
oscillates, and it is
just semi classically

410
00:25:27,510 --> 00:25:30,320
the modulation of the
atomic population which

411
00:25:30,320 --> 00:25:32,560
creates the side bends.

412
00:25:32,560 --> 00:25:38,030
Now if you take that to the
limit of large detuning,

413
00:25:38,030 --> 00:25:44,850
or small drive, small Rabi
frequency, this becomes delta.

414
00:25:47,840 --> 00:25:55,830
So that actually means if you
go to our stick diagram here,

415
00:25:55,830 --> 00:25:59,430
the laser frequency is, of
course, detuned by delta

416
00:25:59,430 --> 00:26:01,110
from the atomic frequency.

417
00:26:01,110 --> 00:26:04,590
But the Rabi frequency is delta,
and this is the Rabi frequency.

418
00:26:10,860 --> 00:26:15,520
Maybe some people who pressed
A and B feel now vindicated,

419
00:26:15,520 --> 00:26:23,410
because now you have a component
of the emission spectrum, which

420
00:26:23,410 --> 00:26:26,770
is at the atomic
resonance frequency.

421
00:26:26,770 --> 00:26:29,760
And you have a
second peak, which

422
00:26:29,760 --> 00:26:33,670
is omega 0 minus 2
delta or 2 omega Rabi.

423
00:26:42,500 --> 00:26:44,110
Question.

424
00:26:44,110 --> 00:26:45,760
How would you give
a simple answer

425
00:26:45,760 --> 00:26:50,190
if I would ask you but what
about energy conservation?

426
00:26:50,190 --> 00:26:52,530
I was hammering so much
on energy conservation.

427
00:26:52,530 --> 00:26:56,490
How can we conserve energy
by have a laser photon

428
00:26:56,490 --> 00:26:58,615
and emitting one at a
different frequency?

429
00:27:01,400 --> 00:27:03,890
AUDIENCE: [INAUDIBLE].

430
00:27:03,890 --> 00:27:06,630
PROFESSOR: It's
compensated with that.

431
00:27:06,630 --> 00:27:08,770
And now let me ask
you another question.

432
00:27:08,770 --> 00:27:12,470
If you would scatter n photons
and you do an experiment,

433
00:27:12,470 --> 00:27:17,360
you would expect that, due to
Poisson fluctuation and such--

434
00:27:17,360 --> 00:27:22,990
we've talked about fluctuations
of intensity and squeezed light

435
00:27:22,990 --> 00:27:23,680
and all that.

436
00:27:23,680 --> 00:27:25,340
But unless you do
something special,

437
00:27:25,340 --> 00:27:29,600
you would expect if you excite
an atom it emits n photons.

438
00:27:29,600 --> 00:27:30,430
You experiment.

439
00:27:30,430 --> 00:27:32,017
You measure plus
minus square root

440
00:27:32,017 --> 00:27:33,600
and you measure
Poisson in statistics.

441
00:27:36,815 --> 00:27:39,920
But now let's
expect you observe n

442
00:27:39,920 --> 00:27:42,224
plus photons on the
upper side bend.

443
00:27:45,560 --> 00:27:49,280
And you measure and you observe
n minus photons on the lower

444
00:27:49,280 --> 00:27:51,730
side bend.

445
00:27:51,730 --> 00:27:56,110
Would you expect that
n plus and n minus both

446
00:27:56,110 --> 00:27:58,430
have now Poisson
fluctuations, or would you

447
00:27:58,430 --> 00:27:59,680
expect something else?

448
00:28:07,520 --> 00:28:10,930
In other words, let
me be very specific,

449
00:28:10,930 --> 00:28:17,250
if you count n plus-- we can
make it a clicker question.

450
00:28:20,160 --> 00:28:22,540
So you have a
counting experiment,

451
00:28:22,540 --> 00:28:31,320
and you have n plus photons
in the upper side bend and n

452
00:28:31,320 --> 00:28:34,797
minus in the lower side bend.

453
00:28:37,930 --> 00:28:44,910
And if you simply look
at the variance, which

454
00:28:44,910 --> 00:28:48,200
is the square of the standard
deviation from n plus,

455
00:28:48,200 --> 00:28:50,610
you find that it is
Poisson distributed.

456
00:28:54,100 --> 00:28:56,440
So the question
which I have now is

457
00:28:56,440 --> 00:29:02,090
what is the variance between
n plus minus n minus.

458
00:29:02,090 --> 00:29:08,505
Is it simply the variance
of n plus plus the variance

459
00:29:08,505 --> 00:29:09,355
of n minus?

460
00:29:14,270 --> 00:29:19,520
The second answer I
want to give you is 0.

461
00:29:19,520 --> 00:29:22,340
And the third answer
is something else.

462
00:29:27,510 --> 00:30:04,760
So this is A. This
is B. And this is C.

463
00:30:04,760 --> 00:30:05,888
Yes.

464
00:30:05,888 --> 00:30:08,030
Energy conservation.

465
00:30:08,030 --> 00:30:10,590
So there shouldn't
be any fluctuation.

466
00:30:10,590 --> 00:30:16,044
The emission of the upper side
bend and the lower side bend

467
00:30:16,044 --> 00:30:16,960
have to be correlated.

468
00:30:19,510 --> 00:30:24,420
And you would immediately
verify that if I

469
00:30:24,420 --> 00:30:27,990
show you what is the diagram.

470
00:30:27,990 --> 00:30:30,950
We have ground
and excited state.

471
00:30:30,950 --> 00:30:35,180
We excite with a laser
which has detuning.

472
00:30:35,180 --> 00:30:39,000
But then we emit a photon
on the lower side bend.

473
00:30:39,000 --> 00:30:42,270
The second laser photon
can now resonantly reach

474
00:30:42,270 --> 00:30:45,530
the excited state.

475
00:30:45,530 --> 00:30:49,840
And so you see we have absorbed
two photons from the laser

476
00:30:49,840 --> 00:30:50,640
beam.

477
00:30:50,640 --> 00:30:53,050
And what we have
emitted in this diagram

478
00:30:53,050 --> 00:30:55,540
is one lower and
one upper side bend.

479
00:30:55,540 --> 00:31:01,020
This is what happens
in second order-- well,

480
00:31:01,020 --> 00:31:03,300
second order in the laser
beam, but fourth order

481
00:31:03,300 --> 00:31:03,980
in the diagram.

482
00:31:03,980 --> 00:31:05,770
We emit four photons.

483
00:31:05,770 --> 00:31:10,390
And each time the
system does it,

484
00:31:10,390 --> 00:31:14,560
goes through that,
it fulfills energy.

485
00:31:14,560 --> 00:31:19,430
Or if I call the upper side
bend click and the lower side

486
00:31:19,430 --> 00:31:20,720
bend clack.

487
00:31:20,720 --> 00:31:24,160
The atom only makes click
clack click clack clack click

488
00:31:24,160 --> 00:31:25,950
clack clack click clack.

489
00:31:25,950 --> 00:31:27,876
It never does click
click clack clack.

490
00:31:27,876 --> 00:31:32,410
It cannot do two clicks because
it has to fulfill this diagram,

491
00:31:32,410 --> 00:31:34,330
and they always come in pair.

492
00:31:34,330 --> 00:31:36,322
The two photons always
come correlated.

493
00:31:42,300 --> 00:31:51,750
OK, so we understand
now the spectrum.

494
00:31:51,750 --> 00:31:54,650
We understand when the
resonant photons emerge.

495
00:31:54,650 --> 00:31:56,760
They emerge in
the high intensity

496
00:31:56,760 --> 00:31:58,435
limit due to Rabi oscillations.

497
00:32:01,060 --> 00:32:08,890
The question is now what is
the widths of those peaks?

498
00:32:12,240 --> 00:32:18,870
And I'm want to short you
that too, because no you

499
00:32:18,870 --> 00:32:23,640
realize some things are easy,
but other things are harder.

500
00:32:23,640 --> 00:32:26,740
In order to get that, you have
to solve the optical Bloch

501
00:32:26,740 --> 00:32:30,470
equations, but I want to show
you in the next half hour

502
00:32:30,470 --> 00:32:33,456
how we can at least get the
salient features of this.

503
00:32:33,456 --> 00:32:34,580
Colin, you have a question?

504
00:32:34,580 --> 00:32:39,760
AUDIENCE: The picture you just
offered about the [INAUDIBLE].

505
00:32:39,760 --> 00:32:42,245
That's assuming that
we're not depleting

506
00:32:42,245 --> 00:32:45,094
the carrier [INAUDIBLE].

507
00:32:45,094 --> 00:32:46,718
Because if we're
depleting the carrier,

508
00:32:46,718 --> 00:32:49,700
wouldn't we expect
to see sort of

509
00:32:49,700 --> 00:32:55,745
like a high order, almost
like a Bessel function.

510
00:32:55,745 --> 00:32:56,620
PROFESSOR: All right.

511
00:32:56,620 --> 00:32:57,810
OK, yes.

512
00:32:57,810 --> 00:33:00,470
AUDIENCE: First order on
to the second [INAUDIBLE].

513
00:33:00,470 --> 00:33:03,100
PROFESSOR: Yes, thanks for
bringing that to our attention.

514
00:33:03,100 --> 00:33:04,480
There is an assumption.

515
00:33:04,480 --> 00:33:06,810
When I said we
have a laser beam,

516
00:33:06,810 --> 00:33:08,860
I assume we have
a laser beam which

517
00:33:08,860 --> 00:33:14,670
delivers zillions of
photons in such a way

518
00:33:14,670 --> 00:33:16,820
that depletion
doesn't play a role.

519
00:33:16,820 --> 00:33:20,420
What that means technically
we replace a laser beam by a c

520
00:33:20,420 --> 00:33:22,700
number, and a c
number never changes,

521
00:33:22,700 --> 00:33:24,580
therefore cannot be depleted.

522
00:33:24,580 --> 00:33:29,630
But if you would use a
very weak laser beam,

523
00:33:29,630 --> 00:33:31,350
we have to modify the answer.

524
00:33:31,350 --> 00:33:37,100
The extreme case would be if
we use single photon sources,

525
00:33:37,100 --> 00:33:39,360
then of course we can
never scatter two photons

526
00:33:39,360 --> 00:33:43,310
because there's only one
photon in the source.

527
00:33:43,310 --> 00:33:46,465
But already if you would
have hundreds or photons

528
00:33:46,465 --> 00:33:49,570
in your cavity and you
put in a single atom,

529
00:33:49,570 --> 00:33:51,870
the scheduling of
single photons would not

530
00:33:51,870 --> 00:33:54,290
cause major depletion effects.

531
00:33:54,290 --> 00:33:55,191
[INAUDIBLE]?

532
00:33:55,191 --> 00:33:58,137
AUDIENCE: In regards to the semi
classical [? fabrication ?],

533
00:33:58,137 --> 00:34:00,101
we said we have
a Rabi frequency,

534
00:34:00,101 --> 00:34:02,065
and when it's excited
it's likely to emit,

535
00:34:02,065 --> 00:34:04,892
and when it's in the ground
state it's not likely to emit.

536
00:34:04,892 --> 00:34:06,975
So whatever frequency you
emit should be modulated

537
00:34:06,975 --> 00:34:11,148
by your Rabi frequency,
but aren't you

538
00:34:11,148 --> 00:34:13,849
most likely to emit when
you have a dipole moment?

539
00:34:13,849 --> 00:34:16,873
Which would be in the
intermediate states between e

540
00:34:16,873 --> 00:34:20,006
and g, in which case those
occur at twice the Rabi

541
00:34:20,006 --> 00:34:21,693
frequencies I think.

542
00:34:27,489 --> 00:34:29,579
PROFESSOR: OK, now
you go to subtleties

543
00:34:29,579 --> 00:34:31,969
of a semi classical picture.

544
00:34:31,969 --> 00:34:34,409
Number one is-- I just want
to say we're a little bit

545
00:34:34,409 --> 00:34:36,630
dangerous, on
slippery slope here.

546
00:34:36,630 --> 00:34:39,400
One is I really discuss
the spectrum here.

547
00:34:39,400 --> 00:34:41,909
When I discuss the
spectrum, I don't

548
00:34:41,909 --> 00:34:43,909
know when the photon is emitted.

549
00:34:43,909 --> 00:34:47,780
So when I said we have
a Rabi oscillation,

550
00:34:47,780 --> 00:34:52,340
I told you that the
system is modulated,

551
00:34:52,340 --> 00:34:54,400
and this gives
rise to side bends.

552
00:34:54,400 --> 00:35:03,010
But I can only spectrally
solve the side bends

553
00:35:03,010 --> 00:35:06,910
if I fundamentally do not have
the time resolution to measure

554
00:35:06,910 --> 00:35:09,910
when in the Rabi cycle
is the photon emitted.

555
00:35:09,910 --> 00:35:12,880
The moment I would localize
when the photon is emitted,

556
00:35:12,880 --> 00:35:15,500
whether the atom is in the
ground or in the excited state,

557
00:35:15,500 --> 00:35:17,670
if I have a time resolution
better than the Rabi

558
00:35:17,670 --> 00:35:20,380
oscillation, I do not have
the spectral resolution

559
00:35:20,380 --> 00:35:22,310
to resolve the side bends.

560
00:35:22,310 --> 00:35:28,050
So therefore the question
when the photon is emitted

561
00:35:28,050 --> 00:35:32,095
is not compatible with
observing the spectrum.

562
00:35:35,630 --> 00:35:46,420
The second question, I
think, can be addressed.

563
00:35:50,620 --> 00:35:51,790
And that's the following.

564
00:35:51,790 --> 00:35:56,110
I will show you in
the next 45 minutes

565
00:35:56,110 --> 00:35:58,770
during this class
that the steady states

566
00:35:58,770 --> 00:36:02,710
solution of the
optical Bloch equations

567
00:36:02,710 --> 00:36:05,830
give us a rate of
photon scattering

568
00:36:05,830 --> 00:36:09,580
which is simply gamma times
the excited state fraction.

569
00:36:09,580 --> 00:36:13,940
So I think from that I would
say the more you have atoms

570
00:36:13,940 --> 00:36:17,560
in the excited state, the
larger is the scattering rate.

571
00:36:17,560 --> 00:36:20,420
So therefore, the
semi classical picture

572
00:36:20,420 --> 00:36:25,240
that you have only a dipole
moment when you are halfway

573
00:36:25,240 --> 00:36:29,380
between ground and excited
state is overly simplistic here.

574
00:36:33,550 --> 00:36:37,520
Also, I just want to give
you one word of warning.

575
00:36:37,520 --> 00:36:39,810
It's really an
important comment.

576
00:36:39,810 --> 00:36:42,710
And this is the following.

577
00:36:42,710 --> 00:36:46,780
The oscillating dipole
moment is a picture

578
00:36:46,780 --> 00:36:50,035
which uses sort of the
analogy with the atom

579
00:36:50,035 --> 00:36:51,696
and the harmonic oscillator.

580
00:36:51,696 --> 00:36:53,070
And I mean yeah,
I really sort of

581
00:36:53,070 --> 00:36:54,903
was a bit provocative
a few minutes ago when

582
00:36:54,903 --> 00:36:58,160
I said just regard the atom
as a harmonic oscillator.

583
00:36:58,160 --> 00:36:59,720
But you have to be careful.

584
00:36:59,720 --> 00:37:02,770
There is one fundamental
limit between-- one

585
00:37:02,770 --> 00:37:07,330
fundamental difference between
the harmonic oscillator

586
00:37:07,330 --> 00:37:09,230
and a two level atom.

587
00:37:09,230 --> 00:37:10,960
And the fact is the following.

588
00:37:10,960 --> 00:37:17,610
This is a two level atom, and
this is a harmonic oscillator.

589
00:37:17,610 --> 00:37:19,570
So the difference
is the following.

590
00:37:19,570 --> 00:37:22,110
An harmonic oscillator
can be excited.

591
00:37:22,110 --> 00:37:24,160
You can pump more and
more energy in it.

592
00:37:24,160 --> 00:37:27,230
You can go to larger
amplitude coherent states,

593
00:37:27,230 --> 00:37:29,860
and there is no non linearity.

594
00:37:29,860 --> 00:37:31,820
The atom saturates.

595
00:37:31,820 --> 00:37:33,050
And this is the difference.

596
00:37:33,050 --> 00:37:34,730
Saturation.

597
00:37:34,730 --> 00:37:37,690
When saturation comes, when you
put more than a few percent,

598
00:37:37,690 --> 00:37:40,330
let's say 50%, into
the excited state,

599
00:37:40,330 --> 00:37:43,100
you have to be very,
very, very careful

600
00:37:43,100 --> 00:37:45,380
with analogies with the
harmonic oscillator.

601
00:37:45,380 --> 00:37:49,890
So if you have most of the
population in the ground

602
00:37:49,890 --> 00:37:53,400
state, a little bit here,
and almost nothing here,

603
00:37:53,400 --> 00:37:57,320
then we can describe the atom
as an harmonic oscillator.

604
00:37:57,320 --> 00:38:01,810
And everything you glean
from the model of an electron

605
00:38:01,810 --> 00:38:06,740
tethered with a spring is
not only qualitatively,

606
00:38:06,740 --> 00:38:08,330
it's quantitatively
correct if you

607
00:38:08,330 --> 00:38:10,600
throw in the oscillator
strength of the atom.

608
00:38:10,600 --> 00:38:14,080
But the moment you pile
up more population here,

609
00:38:14,080 --> 00:38:18,000
and you just ask
about 50-50, at least

610
00:38:18,000 --> 00:38:20,010
I would say
immediately be careful.

611
00:38:20,010 --> 00:38:23,560
Do not take the harmonic
oscillator fully seriously

612
00:38:23,560 --> 00:38:26,200
at this point, because there's
a fundamental difference.

613
00:38:26,200 --> 00:38:28,870
In the harmonic oscillator,
when you're 50% here,

614
00:38:28,870 --> 00:38:30,650
you would have
things over there.

615
00:38:30,650 --> 00:38:33,400
The harmonic oscillator
stays linear.

616
00:38:33,400 --> 00:38:35,340
But the two level system
starts to saturate.

617
00:38:41,250 --> 00:38:41,750
Good.

618
00:38:48,200 --> 00:38:53,230
So the question is now we
understand the stick diagram

619
00:38:53,230 --> 00:38:56,510
of the Mollow
triplet that we have

620
00:38:56,510 --> 00:38:57,990
always a line at the carrier.

621
00:39:01,662 --> 00:39:03,440
Let me just put marks here.

622
00:39:03,440 --> 00:39:08,230
We have the carrier and
the resonance frequency,

623
00:39:08,230 --> 00:39:12,810
but the question is now what
is the width of the spectrum.

624
00:39:15,930 --> 00:39:19,340
The general answer
is rather difficult,

625
00:39:19,340 --> 00:39:21,030
but let me give it to you.

626
00:39:21,030 --> 00:39:23,950
And I will derive it from
the optical Bloch equation.

627
00:39:23,950 --> 00:39:28,900
Let me discuss that one
limit where the detuning is

628
00:39:28,900 --> 00:39:30,800
larger than anything else.

629
00:39:30,800 --> 00:39:34,070
And the other case
is the resonant case.

630
00:39:38,980 --> 00:39:46,220
In general we
started out by saying

631
00:39:46,220 --> 00:39:50,950
we have a delta function
at low intensity.

632
00:39:50,950 --> 00:39:53,810
If you now go to higher
and higher intensity,

633
00:39:53,810 --> 00:39:56,650
you still sort of have the delta
function from the lowest order

634
00:39:56,650 --> 00:40:01,550
diagram, but the higher order
diagram become more important.

635
00:40:01,550 --> 00:40:09,340
And the spectrum is in
general a superposition

636
00:40:09,340 --> 00:40:14,060
of three broadened peaks
and the delta function.

637
00:40:14,060 --> 00:40:16,320
At low intensity, you only
have the delta function.

638
00:40:16,320 --> 00:40:19,800
At high intensity, the
weight of the elastic scatter

639
00:40:19,800 --> 00:40:23,050
the delta function goes to 0.

640
00:40:23,050 --> 00:40:25,250
And now let's
discuss the widths.

641
00:40:25,250 --> 00:40:28,790
And that shows you that even
light scattering by a two level

642
00:40:28,790 --> 00:40:33,550
system, those very, very
simple Jaynes Cummings model

643
00:40:33,550 --> 00:40:38,530
has interesting and maybe
non intuitive aspect.

644
00:40:38,530 --> 00:40:41,620
The side bends have
a width of gamma.

645
00:40:41,620 --> 00:40:46,060
The carrier has a
width of 2 gamma.

646
00:40:46,060 --> 00:40:51,680
But when you are on resonance,
the situation changes.

647
00:40:55,300 --> 00:41:00,540
The carrier has now a width
of gamma and side bends

648
00:41:00,540 --> 00:41:02,631
have a width of 3/2 gamma.

649
00:41:06,240 --> 00:41:15,060
But if you look at it, the
sum of the three widths

650
00:41:15,060 --> 00:41:17,266
is always 4 gamma.

651
00:41:17,266 --> 00:41:19,580
And you will understand
that in a few minutes.

652
00:41:30,830 --> 00:41:37,740
OK, so this was more
qualitative discussion.

653
00:41:37,740 --> 00:41:48,330
Let's now work
more quantitatively

654
00:41:48,330 --> 00:41:52,780
and describe the Mollow triplet.

655
00:41:57,840 --> 00:42:07,960
The Hamiltonian for our system
is the atomic Hamiltonian

656
00:42:07,960 --> 00:42:13,840
times the the z Pauli matrix.

657
00:42:13,840 --> 00:42:18,390
The z Pauli matrix
is excited excited

658
00:42:18,390 --> 00:42:22,940
minus ground ground state.

659
00:42:22,940 --> 00:42:31,220
And because of the
conservation of probability,

660
00:42:31,220 --> 00:42:32,310
it's 2 times the excited.

661
00:42:32,310 --> 00:42:36,300
Also just mathematically
it's just that.

662
00:42:36,300 --> 00:42:39,310
We have the three
Hamiltonian of the radiation

663
00:42:39,310 --> 00:42:42,920
field, which is a dagger a.

664
00:42:42,920 --> 00:42:47,620
And then we have the coupling
of a two level system, which

665
00:42:47,620 --> 00:42:55,810
is the product of the
atomic dipole moment written

666
00:42:55,810 --> 00:42:59,110
as sigma plus plus sigma minus
raising and lowering operator.

667
00:43:03,390 --> 00:43:07,960
Just to remind us,
sigma plus is e g.

668
00:43:07,960 --> 00:43:12,950
And sigma minus is g e.

669
00:43:12,950 --> 00:43:19,040
And then we have to multiply
with a plus a dagger.

670
00:43:19,040 --> 00:43:22,930
If you want, the atomic
part is the dipole moment,

671
00:43:22,930 --> 00:43:26,660
and this is the electric field
both written as operators.

672
00:43:31,680 --> 00:43:35,030
So this is the electric field.

673
00:43:35,030 --> 00:43:38,950
And if you look at
the coupling term--

674
00:43:38,950 --> 00:43:41,200
we've discussed it before.

675
00:43:41,200 --> 00:43:44,110
You have four terms.

676
00:43:44,110 --> 00:43:45,905
But in the rotating
wave approximation,

677
00:43:45,905 --> 00:43:49,980
which we want to use
here, we keep only

678
00:43:49,980 --> 00:43:53,160
the two resonant
terms, which is when

679
00:43:53,160 --> 00:43:57,630
we raise an excitation
in the atom,

680
00:43:57,630 --> 00:44:00,810
we lose an excitation
in the light field.

681
00:44:00,810 --> 00:44:04,150
Or if you go from the
excited to the ground state,

682
00:44:04,150 --> 00:44:05,890
we emit a photon.

683
00:44:05,890 --> 00:44:08,800
So we have two terms which
form the rotating wave

684
00:44:08,800 --> 00:44:09,910
approximation.

685
00:44:09,910 --> 00:44:12,880
The other two terms
are highly of resonant

686
00:44:12,880 --> 00:44:14,370
and can often be neglected.

687
00:44:20,600 --> 00:44:23,590
So this rotating
wave approximation

688
00:44:23,590 --> 00:44:28,050
becomes of course the
better, the more we

689
00:44:28,050 --> 00:44:30,710
go in resonant, because in
one term is fully resonant

690
00:44:30,710 --> 00:44:34,140
and the other one is
very off resonant.

691
00:44:34,140 --> 00:44:43,790
And in this case, we can
simplify the Hamiltonian

692
00:44:43,790 --> 00:44:48,770
because the number of
excitations is conserved.

693
00:44:48,770 --> 00:44:51,680
The number of
excitations is the number

694
00:44:51,680 --> 00:44:54,980
of excitations in
the photon field

695
00:44:54,980 --> 00:45:02,840
plus the number
of excited atoms.

696
00:45:02,840 --> 00:45:04,583
So this is the
operator which measures

697
00:45:04,583 --> 00:45:05,666
the number of excitations.

698
00:45:09,660 --> 00:45:12,390
If in a [INAUDIBLE]
counter rotating term,

699
00:45:12,390 --> 00:45:14,995
this operator is conserved.

700
00:45:20,220 --> 00:45:33,910
So if I introduce as
usual the detuning--

701
00:45:33,910 --> 00:45:35,420
this is what my notes say.

702
00:45:35,420 --> 00:45:38,060
It seems to have
the opposite sign.

703
00:45:38,060 --> 00:45:41,090
I'm not sure if I made a
sign error here or later,

704
00:45:41,090 --> 00:45:42,550
or if I changed the definition.

705
00:45:42,550 --> 00:45:45,720
Let's just move on and
see how it works out.

706
00:45:45,720 --> 00:45:49,020
So if you now do the
rotating wave approximation,

707
00:45:49,020 --> 00:45:54,730
our Hamiltonian has now-- by
introducing the operator N,

708
00:45:54,730 --> 00:45:58,220
the number of [INAUDIBLE]
times h bar omega.

709
00:45:58,220 --> 00:46:02,320
However, we have
multiplied the N operator

710
00:46:02,320 --> 00:46:04,010
with the photon energy.

711
00:46:04,010 --> 00:46:06,970
If the energy is in
the atom, we have

712
00:46:06,970 --> 00:46:12,500
made a mistake, which is
delta, and we correct that

713
00:46:12,500 --> 00:46:16,090
by using the z Pauli matrix.

714
00:46:16,090 --> 00:46:22,200
And then our interaction
term is a dagger times sigma

715
00:46:22,200 --> 00:46:26,240
minus plus a times sigma plus.

716
00:46:46,320 --> 00:46:52,590
Let's now discuss
what are the eigen

717
00:46:52,590 --> 00:46:58,620
states of this Hamiltonian
In the simple case

718
00:46:58,620 --> 00:47:00,030
when the detuning is 0.

719
00:47:14,870 --> 00:47:18,510
And we have sort of two
levels of eigen state.

720
00:47:18,510 --> 00:47:23,620
One is the ground
state with n photons.

721
00:47:23,620 --> 00:47:27,870
And one is the excited
state with n photons.

722
00:47:27,870 --> 00:47:32,400
0, 1, 2.

723
00:47:32,400 --> 00:47:37,220
So we can label this
by the photon number.

724
00:47:37,220 --> 00:47:39,690
And the excited state
of course starts

725
00:47:39,690 --> 00:47:41,690
with a higher
energy for 0 photon.

726
00:47:45,300 --> 00:47:47,470
This is the energy
with one photon.

727
00:47:47,470 --> 00:47:56,100
And in the case of 0 detuning,
the two letters are degenerate.

728
00:47:56,100 --> 00:48:09,510
And if we introduce the
interaction, if g is now non 0,

729
00:48:09,510 --> 00:48:11,110
the levels split.

730
00:48:14,630 --> 00:48:25,660
And the splitting is given by g.

731
00:48:25,660 --> 00:48:28,420
There's a factor of two.

732
00:48:28,420 --> 00:48:32,970
But now if you have
photons in the field,

733
00:48:32,970 --> 00:48:37,400
a dagger acting on the photon
field gives n plus one.

734
00:48:37,400 --> 00:48:40,960
So therefore for one photon
we have square root 2.

735
00:48:40,960 --> 00:48:44,530
In general, we get an expression
which is square root n plus 1.

736
00:48:48,230 --> 00:49:00,680
Or to write it
mathematically, if we

737
00:49:00,680 --> 00:49:05,150
have strong mixing
between an excited

738
00:49:05,150 --> 00:49:09,570
state with n atoms--
with n photons.

739
00:49:09,570 --> 00:49:14,450
And a ground state with n plus
1 photons, for the case of delta

740
00:49:14,450 --> 00:49:16,690
equals 0, it's just plus minus.

741
00:49:16,690 --> 00:49:19,870
You metric as a metric
contribution times 1

742
00:49:19,870 --> 00:49:21,790
over square root 2.

743
00:49:21,790 --> 00:49:25,180
And these are now with the
states we label plus minus.

744
00:49:25,180 --> 00:49:29,810
The upper and lower
state of each manifold.

745
00:49:29,810 --> 00:49:33,980
And n is the photon number
in the excited state.

746
00:49:33,980 --> 00:49:37,840
n plus 1 is the photon
number in the ground state.

747
00:49:37,840 --> 00:49:50,990
And the energy of those states
is the number of [INAUDIBLE] n

748
00:49:50,990 --> 00:49:53,330
plus 1.

749
00:49:53,330 --> 00:49:57,770
But then we have the
resonant splitting g.

750
00:49:57,770 --> 00:50:02,220
And because of the matrix
element of a and a dagger,

751
00:50:02,220 --> 00:50:03,750
we get square root n plus 1.

752
00:50:09,200 --> 00:50:16,130
Now if you use
very strong drive--

753
00:50:16,130 --> 00:50:24,050
and this addresses Colin's
question about depletion--

754
00:50:24,050 --> 00:50:29,210
we use a laser beam, which is
described by a Rabi frequency.

755
00:50:29,210 --> 00:50:32,460
This is proportional
to the electric field.

756
00:50:32,460 --> 00:50:35,530
So be introduce
the Rabi frequency,

757
00:50:35,530 --> 00:50:38,680
which is given by that.

758
00:50:38,680 --> 00:50:40,510
And the laser beam
has so many photons

759
00:50:40,510 --> 00:50:44,140
that we don't care about n plus
1 or scattering a few photons,

760
00:50:44,140 --> 00:50:49,530
so we simply replace n n plus 1
n minus 1, whatever contains n,

761
00:50:49,530 --> 00:50:53,689
by the z number, which is the
Rabi frequency of the laser

762
00:50:53,689 --> 00:50:54,189
beam.

763
00:50:58,980 --> 00:51:13,871
So this eventually means
that we have too many folds.

764
00:51:13,871 --> 00:51:15,070
Plus, minus.

765
00:51:15,070 --> 00:51:17,010
Plus, minus.

766
00:51:17,010 --> 00:51:19,800
So we have two
states plus minus,

767
00:51:19,800 --> 00:51:23,900
and they are periodic when we
add one photon to the field.

768
00:51:23,900 --> 00:51:30,037
Periodic in n.

769
00:51:34,710 --> 00:51:42,170
The splitting is the Rabi
frequency defined above.

770
00:51:42,170 --> 00:51:47,930
And now we realize that when
we have spontaneous emission

771
00:51:47,930 --> 00:51:58,980
between the manifold for n
minus 1 and the manifold n,

772
00:51:58,980 --> 00:52:03,800
we have two possibilities.

773
00:52:07,260 --> 00:52:12,720
To emit light on resonance.

774
00:52:12,720 --> 00:52:18,190
And we have then one possibility
to emit an upper side bend.

775
00:52:18,190 --> 00:52:20,770
And one possibility to
emit the lower side bend.

776
00:52:27,490 --> 00:52:32,480
So what is called
the dressed atom

777
00:52:32,480 --> 00:52:38,420
picture is nothing else than
the solution of an atom driven

778
00:52:38,420 --> 00:52:40,520
by a strong monochromatic field.

779
00:52:40,520 --> 00:52:43,410
The atom is dressed
with photons,

780
00:52:43,410 --> 00:52:46,330
and the eigen states are
no longer just atomic eigen

781
00:52:46,330 --> 00:52:47,040
states.

782
00:52:47,040 --> 00:52:52,550
These are dressed eigenstates
of the combined atomic system

783
00:52:52,550 --> 00:52:54,870
plus laser field.

784
00:52:54,870 --> 00:52:59,440
So this dressed atom picture
explains the Mollow triplet.

785
00:53:11,860 --> 00:53:13,590
And you can immediately
generalize it

786
00:53:13,590 --> 00:53:15,860
if you want to
arbitrarily tuning

787
00:53:15,860 --> 00:53:18,660
when the superposition
states plus minus do not

788
00:53:18,660 --> 00:53:20,450
have equal weight.

789
00:53:20,450 --> 00:53:23,030
But it's what you've
seen a million times,

790
00:53:23,030 --> 00:53:25,880
the diagonalization of
a two by two matrix.

791
00:53:25,880 --> 00:53:26,616
Colin?

792
00:53:26,616 --> 00:53:29,192
AUDIENCE: How does this
explain the [INAUDIBLE]?

793
00:53:29,192 --> 00:53:30,150
PROFESSOR: It does not.

794
00:53:30,150 --> 00:53:32,070
This is what I
just wanted to say.

795
00:53:32,070 --> 00:53:38,220
It explains the Mollow triplet,
but not the line widths.

796
00:53:41,670 --> 00:53:47,360
The line widths cannot be
obtained perturbatively.

797
00:53:47,360 --> 00:53:55,550
So for that we have to discuss
now the Bloch vector, which

798
00:53:55,550 --> 00:53:58,720
is a solution of the
optical Bloch equation.

799
00:53:58,720 --> 00:54:02,260
So now I'm going to explain
you not all glorious details,

800
00:54:02,260 --> 00:54:04,250
but the salient feature
of the line bits.

801
00:54:09,390 --> 00:54:18,030
So the density matrix
for two level atom

802
00:54:18,030 --> 00:54:24,190
can be written as-- let
me back up a moment.

803
00:54:24,190 --> 00:54:27,830
The density matrix has
four matrix elements.

804
00:54:27,830 --> 00:54:30,470
But if the sum of the
diagonal matrix element

805
00:54:30,470 --> 00:54:32,550
is one conservation
of probability,

806
00:54:32,550 --> 00:54:36,360
we have actually three
independent matrix elements.

807
00:54:36,360 --> 00:54:39,440
And those three
independent matrix elements

808
00:54:39,440 --> 00:54:44,400
can be parametrized by what
is called the Bloch vector.

809
00:54:44,400 --> 00:54:45,890
A lot of you have seen it.

810
00:54:45,890 --> 00:54:49,760
We also discuss it in
great detail in 8421.

811
00:54:49,760 --> 00:54:53,500
But I'm giving you
the definition.

812
00:54:53,500 --> 00:54:58,020
And sigma is the vector
of Pauli matrices.

813
00:54:58,020 --> 00:55:02,390
To be specific, the three
components of the Bloch vector

814
00:55:02,390 --> 00:55:04,370
are as follows.

815
00:55:04,370 --> 00:55:07,680
The z comportment
measures, you can say,

816
00:55:07,680 --> 00:55:09,920
the population inversion.

817
00:55:09,920 --> 00:55:15,430
The difference between ground
and excited state population.

818
00:55:15,430 --> 00:55:21,800
And the x and y component
measure the coherencies.

819
00:55:21,800 --> 00:55:27,090
Either the sum of
the coherencies

820
00:55:27,090 --> 00:55:29,880
or the difference
of the coherencies.

821
00:55:32,920 --> 00:55:35,370
And here is minus
imaginary unit.

822
00:55:44,960 --> 00:55:48,600
If you replace the density
matrix or the matrix elements

823
00:55:48,600 --> 00:55:54,310
sigma ee, or ee,
or gg by definition

824
00:55:54,310 --> 00:55:58,290
the optical Bloch equation
turns into differential

825
00:55:58,290 --> 00:55:59,230
equation for r.

826
00:56:04,320 --> 00:56:09,050
It's now a differential
equation for r.

827
00:56:16,330 --> 00:56:23,950
And let me just write it down.

828
00:56:23,950 --> 00:56:27,780
Let me just write down
the Hamiltonian part.

829
00:56:40,130 --> 00:56:44,670
This is the equation of
motion for the density matrix.

830
00:56:44,670 --> 00:56:48,819
I will add on in a minute
the relaxation part

831
00:56:48,819 --> 00:56:50,360
which comes from
the master equation.

832
00:56:50,360 --> 00:56:52,970
This is just the
Hamiltonian part.

833
00:56:52,970 --> 00:56:58,230
And this would result
into an equation

834
00:56:58,230 --> 00:57:02,410
of motion for the
Bloch vector, which

835
00:57:02,410 --> 00:57:11,388
has delta here minus
delta here minus g.

836
00:57:11,388 --> 00:57:17,070
g and the rest of the
matrix elements is 0.

837
00:57:17,070 --> 00:57:22,990
The Hamiltonian, using
rotation matrices,

838
00:57:22,990 --> 00:57:26,890
can be written by
a z rotation matrix

839
00:57:26,890 --> 00:57:31,992
plus g times the x rotation.

840
00:57:35,770 --> 00:57:56,120
And if you actually look at the
solution-- it's not a finite.

841
00:57:59,050 --> 00:58:00,970
It's an infinitesimal rotation.

842
00:58:00,970 --> 00:58:21,810
This matrix tells you that the
detuning does a z rotation,

843
00:58:21,810 --> 00:58:27,246
and the drive is responsible
for an x rotation.

844
00:58:36,650 --> 00:58:39,890
Let me tell you what happens.

845
00:58:39,890 --> 00:58:43,670
You have seen the Bloch
vector many cases.

846
00:58:43,670 --> 00:58:46,835
If you don't drive the
system, the Bloch vector

847
00:58:46,835 --> 00:58:48,880
in the ground state is down.

848
00:58:48,880 --> 00:58:51,220
In the upper state it's up.

849
00:58:51,220 --> 00:58:56,500
If it's in between-- that's
where Timo's dipole moment

850
00:58:56,500 --> 00:58:58,000
comes in-- we have
a superposition

851
00:58:58,000 --> 00:59:00,190
of ground and excited state.

852
00:59:00,190 --> 00:59:04,940
And it rotates at the
atomic frequency omega 0.

853
00:59:04,940 --> 00:59:08,550
But we are in the rotating
frame of the laser at omega,

854
00:59:08,550 --> 00:59:12,940
so in the rotating frame, when
the frame rotates with omega.

855
00:59:12,940 --> 00:59:17,270
And omega 0 rotation becomes
the rotation with delta.

856
00:59:17,270 --> 00:59:20,120
So therefore, the free
evolution of the atom

857
00:59:20,120 --> 00:59:24,760
is based on this Hamiltonian
that the Bloch vector rotates

858
00:59:24,760 --> 00:59:28,190
with delta in the rotating
frame of the laser.

859
00:59:28,190 --> 00:59:31,790
And it's a rotation
around the z-axis.

860
00:59:31,790 --> 00:59:35,860
If you drive the system now,
we take the Bloch vector

861
00:59:35,860 --> 00:59:39,910
from ground to excited,
there are Rabi oscillations.

862
00:59:39,910 --> 00:59:43,290
And this is now a rotation
around the x-axis.

863
00:59:43,290 --> 00:59:46,540
Well, whether it's x or y
is a convention of notation,

864
00:59:46,540 --> 00:59:49,235
and I've defined it such
that the driven system is

865
00:59:49,235 --> 00:59:50,640
an x rotation.

866
00:59:50,640 --> 00:59:53,300
The free evolution
is a z rotation.

867
00:59:53,300 --> 00:59:54,780
And that's all you have to know.

868
00:59:54,780 --> 00:59:56,490
This is the most
general solution

869
00:59:56,490 --> 01:00:00,070
of two level system
without dissipation

870
01:00:00,070 --> 01:00:02,300
that we have two
rotation angles.

871
01:00:02,300 --> 01:00:05,830
One is the free rotation,
which is the detuning.

872
01:00:05,830 --> 01:00:07,300
It is the z-axis.

873
01:00:07,300 --> 01:00:09,550
And the driven system
rotates around the x-axis.

874
01:00:15,300 --> 01:00:18,660
But now this is well known.

875
01:00:18,660 --> 01:00:20,810
This is boring.

876
01:00:20,810 --> 01:00:25,690
But now we want to add what
the master equation gives us.

877
01:00:25,690 --> 01:00:34,150
And the master equation uses
as its [? limb ?] platform

878
01:00:34,150 --> 01:00:37,960
where we have the sigma
plus jump operator.

879
01:00:37,960 --> 01:00:48,490
And I told you that we have
to use the following form.

880
01:00:48,490 --> 01:00:52,040
We'll hear more about
it, probably not today,

881
01:00:52,040 --> 01:00:52,910
but on Friday.

882
01:00:55,880 --> 01:01:00,530
You know that this gives
simply the optical--

883
01:01:00,530 --> 01:01:04,630
this gives us simply the
optical Bloch equations.

884
01:01:04,630 --> 01:01:10,330
We discussed it on Monday that
we have now a damping gamma

885
01:01:10,330 --> 01:01:12,470
term for the population.

886
01:01:12,470 --> 01:01:16,490
An excited state population
decays with a rate of gamma.

887
01:01:16,490 --> 01:01:20,510
And the coherences, the off
diagonal matrix elements,

888
01:01:20,510 --> 01:01:22,730
decay with a rate
of gamma over 2,

889
01:01:22,730 --> 01:01:26,380
and we spent a long time
discussing this vector of 2.

890
01:01:26,380 --> 01:01:31,850
So this would mean now that the
equation for the Bloch vector

891
01:01:31,850 --> 01:01:40,500
has gamma 2 terms, which
comes from the dampening

892
01:01:40,500 --> 01:01:43,280
of the coherences.

893
01:01:43,280 --> 01:01:48,115
And it has gamma, which is
the damping of the population.

894
01:01:54,000 --> 01:02:02,240
Well if that would be all,
everything is damped to 0,

895
01:02:02,240 --> 01:02:07,000
but everything is in the end
leads to the ground state

896
01:02:07,000 --> 01:02:08,700
population.

897
01:02:08,700 --> 01:02:11,990
So we have to add this term.

898
01:02:11,990 --> 01:02:14,940
It's just a different way of
the optical Bloch equations.

899
01:02:14,940 --> 01:02:17,470
We have discussed
in great lengths

900
01:02:17,470 --> 01:02:20,480
this form of the
optical Bloch equations.

901
01:02:20,480 --> 01:02:24,080
And the sigma plus sigma minus
is because these are just

902
01:02:24,080 --> 01:02:27,635
simple matrices with
one matrix element.

903
01:02:30,300 --> 01:02:32,900
You've seen the optical
Bloch equation, I think.

904
01:02:32,900 --> 01:02:35,200
We did also that
in 8421 21 that you

905
01:02:35,200 --> 01:02:38,620
have the derivative
of the density matrix

906
01:02:38,620 --> 01:02:41,990
includes now damping of the
off diagonal matrix elements

907
01:02:41,990 --> 01:02:44,646
by gamma over 2 damping by
the diagonal matrix elements

908
01:02:44,646 --> 01:02:45,145
by gamma.

909
01:02:50,470 --> 01:02:56,020
If you simply substitute the r
vector for the matrix element,

910
01:02:56,020 --> 01:03:00,190
you get this
equation in one step.

911
01:03:00,190 --> 01:03:06,880
These are now the
optical Bloch equations

912
01:03:06,880 --> 01:03:10,055
written as a differential
equation for the Bloch vector.

913
01:03:21,450 --> 01:03:25,910
OK, sorry if that was confusing,
but it's simple definition

914
01:03:25,910 --> 01:03:28,010
substitution no brainer.

915
01:03:28,010 --> 01:03:30,980
These are the optical
Bloch equations

916
01:03:30,980 --> 01:03:33,180
written in terms
of the Bloch vector

917
01:03:33,180 --> 01:03:36,020
and no longer in terms
of the density matrix.

918
01:03:36,020 --> 01:03:37,115
And questions about that?

919
01:03:42,370 --> 01:03:46,470
OK, because now I want
to draw conclusions.

920
01:03:46,470 --> 01:03:51,430
Our goal is to understand
the spectrum and the line

921
01:03:51,430 --> 01:03:53,820
widths of the Mollow triplet
of the emitted light.

922
01:03:56,450 --> 01:03:58,990
So I have to make
the connection.

923
01:03:58,990 --> 01:04:05,170
How do I make the connection
from the Bloch vector

924
01:04:05,170 --> 01:04:09,080
to emitted light?

925
01:04:09,080 --> 01:04:11,720
Well, it's done by
the dipole moment,

926
01:04:11,720 --> 01:04:14,630
because it is the
oscillating dipole which

927
01:04:14,630 --> 01:04:18,470
is responsible for emitting
and scattering light.

928
01:04:18,470 --> 01:04:20,880
The dipole moment
can of course be

929
01:04:20,880 --> 01:04:23,420
obtained from the solution
of the optical Bloch

930
01:04:23,420 --> 01:04:25,500
equation for the density matrix.

931
01:04:25,500 --> 01:04:28,020
It is the trace of
the density matrix

932
01:04:28,020 --> 01:04:31,870
times the operator
we are interested in.

933
01:04:31,870 --> 01:04:39,300
And this involves the
matrix element dge.

934
01:04:47,870 --> 01:04:50,500
The dipole moment has
only matrix elements

935
01:04:50,500 --> 01:04:53,600
between excited
and ground state.

936
01:04:53,600 --> 01:04:56,550
And ground and excited state.

937
01:05:02,580 --> 01:05:06,045
Trace rho other parentheses.

938
01:05:29,560 --> 01:05:33,780
So this is the matrix element.

939
01:05:33,780 --> 01:05:39,660
Let me now write down
the density matrix

940
01:05:39,660 --> 01:05:45,140
in its matrix element
rho ge plus rho eg.

941
01:05:45,140 --> 01:05:48,110
But now we want to
go from the rotating

942
01:05:48,110 --> 01:05:53,910
frame back to the lab frame.

943
01:05:53,910 --> 01:05:58,360
And the rotating frame
rotates at omega.

944
01:05:58,360 --> 01:06:03,540
So now I have to put back
e to the i omega t and e

945
01:06:03,540 --> 01:06:05,600
to the minus i omega t.

946
01:06:05,600 --> 01:06:08,890
This is sort of going back from
the rotating frame to the lab

947
01:06:08,890 --> 01:06:09,390
frame.

948
01:06:15,970 --> 01:06:18,080
And the picture
of we want to use

949
01:06:18,080 --> 01:06:23,780
is-- the intuitive picture
is to use the Bloch vector.

950
01:06:23,780 --> 01:06:27,340
So I'm expressing now
those matrix elements

951
01:06:27,340 --> 01:06:32,213
by the component x and
y of the Bloch vector.

952
01:06:35,920 --> 01:06:41,120
And since the x and y component
was the sum or difference

953
01:06:41,120 --> 01:06:45,750
of diagram matrix element of
the density the coherences.

954
01:06:45,750 --> 01:06:48,640
I get now cosine omega
t and sine omega t.

955
01:06:54,650 --> 01:07:03,510
So that tells us that one
component is in phase--

956
01:07:03,510 --> 01:07:11,970
well in phase means with respect
to the driving field we assume

957
01:07:11,970 --> 01:07:15,730
that the system is
driven by classical field

958
01:07:15,730 --> 01:07:18,130
e0 cosine omega t.

959
01:07:18,130 --> 01:07:22,190
And now we find that the
oscillating dipole moment,

960
01:07:22,190 --> 01:07:23,870
it's a driven
harmonic oscillator.

961
01:07:23,870 --> 01:07:26,700
And there is a part which is
driven in phase with the drive

962
01:07:26,700 --> 01:07:28,840
field, and one which
is in quadrature.

963
01:07:37,167 --> 01:07:38,500
So we have those two components.

964
01:07:41,440 --> 01:07:49,100
And the fact is
now the following.

965
01:07:49,100 --> 01:07:58,070
If you have an
oscillating dipole,

966
01:07:58,070 --> 01:08:04,730
this gives rise to immediate--
or more accurately,

967
01:08:04,730 --> 01:08:08,050
I should say scattered light.

968
01:08:08,050 --> 01:08:11,600
Remember, if you assume there
is photon first emitted and then

969
01:08:11,600 --> 01:08:14,780
a resonant photon absorbed,
this picture is wrong.

970
01:08:14,780 --> 01:08:17,569
You should rather think
of scattering light,

971
01:08:17,569 --> 01:08:20,460
but the light scattering is
induced by the oscillating

972
01:08:20,460 --> 01:08:22,630
dipole.

973
01:08:22,630 --> 01:08:26,439
And what happens
now is-- and this

974
01:08:26,439 --> 01:08:29,470
is why the Bloch vector
and the Bloch sphere

975
01:08:29,470 --> 01:08:32,590
is such a wonderful picture.

976
01:08:32,590 --> 01:08:38,470
The dipole moment are
the transverse component

977
01:08:38,470 --> 01:08:40,330
of the Bloch vector.

978
01:08:40,330 --> 01:08:47,630
So therefore, you can
say that if you're

979
01:08:47,630 --> 01:08:50,630
interested in the spectrum of
the emitted light, what you

980
01:08:50,630 --> 01:08:55,029
should find out is what
is the spectrum of r.

981
01:09:12,720 --> 01:09:15,189
Let me back up one step.

982
01:09:15,189 --> 01:09:21,040
If you want to get the
spectrum of the emitted light

983
01:09:21,040 --> 01:09:25,050
from first principles-- I'm
cheating a little bit here.

984
01:09:25,050 --> 01:09:27,350
I'm saying let's get it
from the dipole moment.

985
01:09:27,350 --> 01:09:29,800
You should actually get it
from the correlation function

986
01:09:29,800 --> 01:09:31,800
of the dipole moment.

987
01:09:31,800 --> 01:09:35,960
It's sort of the two time
correlation function.

988
01:09:35,960 --> 01:09:39,920
But you can show-- and this
takes another 10, 20, 30

989
01:09:39,920 --> 01:09:40,880
pages in API.

990
01:09:47,250 --> 01:09:51,950
The temporal correlation
function of the dipole moment

991
01:09:51,950 --> 01:09:56,850
fulfils the same differential
equation as the Bloch vector.

992
01:09:56,850 --> 01:09:59,660
So I've given you here
sort of the intuitive link

993
01:09:59,660 --> 01:10:02,390
that would oscillates is the
Bloch vector it emits it.

994
01:10:02,390 --> 01:10:04,720
But technically,
because you may not

995
01:10:04,720 --> 01:10:08,990
control the phase of each
atom, you should rather

996
01:10:08,990 --> 01:10:11,190
say the spectrum
is a correlation

997
01:10:11,190 --> 01:10:14,330
function of the
dipole, and we should

998
01:10:14,330 --> 01:10:16,660
look at the spectrum of
the correlation function.

999
01:10:16,660 --> 01:10:19,260
But instead, I'm looking
now at the spectrum

1000
01:10:19,260 --> 01:10:21,690
of the optical Bloch vector.

1001
01:10:21,690 --> 01:10:25,790
And you can show mathematically
exactly that they

1002
01:10:25,790 --> 01:10:27,530
obey the same
differential equation,

1003
01:10:27,530 --> 01:10:32,110
but it's a little bit
tedious to do that.

1004
01:10:32,110 --> 01:10:39,400
OK, so therefore if you take
this little inaccuracy--

1005
01:10:39,400 --> 01:10:42,400
forgive me this inaccuracy.

1006
01:10:42,400 --> 01:10:45,060
Once we know what
the optical Bloch

1007
01:10:45,060 --> 01:10:49,915
vector is doing spectrally,
we know the spectrum

1008
01:10:49,915 --> 01:10:52,620
of the emitted light.

1009
01:10:52,620 --> 01:11:01,980
And for that, for the
optical Bloch vector,

1010
01:11:01,980 --> 01:11:07,180
we simply look at
this matrix, and we

1011
01:11:07,180 --> 01:11:11,590
ask what are the
eigenvalues of this matrix?

1012
01:11:11,590 --> 01:11:20,420
This matrix has three by three
matrices, three eigenvalues.

1013
01:11:20,420 --> 01:11:22,890
The real part of
those eigenvalues

1014
01:11:22,890 --> 01:11:26,000
gives us the position
of the Mollow triplet.

1015
01:11:26,000 --> 01:11:31,420
And the imaginary part gives
us the widths of those peaks.

1016
01:11:31,420 --> 01:11:33,730
So if you look at
this matrix-- and I'm

1017
01:11:33,730 --> 01:11:35,780
not doing the
complicated cases here.

1018
01:11:35,780 --> 01:11:42,830
If you look at this matrix for
delta equals 0 and g equals 0.

1019
01:11:42,830 --> 01:11:46,020
Well it is already
in diagonal form,

1020
01:11:46,020 --> 01:11:49,440
so it has three
imaginary eigenvalues

1021
01:11:49,440 --> 01:11:52,050
minus gamma over 2 minus
gamma 2 minus gamma.

1022
01:12:00,760 --> 01:12:12,170
So this matrix has three
complex eigenvalues.

1023
01:12:12,170 --> 01:12:14,900
And I want to discuss two cases.

1024
01:12:14,900 --> 01:12:26,810
For this case we have
minus gamma over 2

1025
01:12:26,810 --> 01:12:36,920
for the x and y component
of the optical Bloch vector.

1026
01:12:36,920 --> 01:12:41,050
And the z component
is minus gamma.

1027
01:12:41,050 --> 01:12:50,730
And this is the situation.

1028
01:12:57,390 --> 01:12:59,250
Apart from a factor
of two that we

1029
01:12:59,250 --> 01:13:01,420
have gamma, 2 gamma, and gamma.

1030
01:13:06,430 --> 01:13:08,280
The fact of 2 comes of course.

1031
01:13:08,280 --> 01:13:11,070
Are you asking what is the
spectrum of the electric field

1032
01:13:11,070 --> 01:13:12,740
or what is the
spectrum of the power?

1033
01:13:12,740 --> 01:13:15,390
If you go from e
to e squared, then

1034
01:13:15,390 --> 01:13:19,362
if you have an exponential
decay with gamma of e,

1035
01:13:19,362 --> 01:13:21,220
e squared decays with two gamma.

1036
01:13:21,220 --> 01:13:23,772
So that's where
factors of 2 come from.

1037
01:13:23,772 --> 01:13:28,191
AUDIENCE: Can you label the top
of the diagram with [INAUDIBLE]

1038
01:13:28,191 --> 01:13:29,664
just the delta equals 0.

1039
01:13:34,855 --> 01:13:35,480
PROFESSOR: Yes.

1040
01:13:44,984 --> 01:13:46,180
Give me a second.

1041
01:13:46,180 --> 01:13:51,080
AUDIENCE: [INAUDIBLE]
g equals 0, right?

1042
01:13:51,080 --> 01:13:52,870
PROFESSOR: The
fact is here we're

1043
01:13:52,870 --> 01:13:55,750
looking at limiting cases.

1044
01:13:55,750 --> 01:13:58,500
And the more important
thing is that this

1045
01:13:58,500 --> 01:14:00,490
is the case of weak drive.

1046
01:14:00,490 --> 01:14:05,790
So what is more important is
that the drive frequency is 0.

1047
01:14:05,790 --> 01:14:07,770
I'm not giving you
the full picture here.

1048
01:14:12,620 --> 01:14:16,180
The full glorious
derivation is in API.

1049
01:14:16,180 --> 01:14:18,110
And just tried to
sort of entertain

1050
01:14:18,110 --> 01:14:20,160
you by giving you
a few appetizers.

1051
01:14:20,160 --> 01:14:21,620
If you want, read it up there.

1052
01:14:21,620 --> 01:14:24,860
But what I want to show you
is they're two simple cases,

1053
01:14:24,860 --> 01:14:27,565
but it looks already
intriguing because here this

1054
01:14:27,565 --> 01:14:31,090
is wider than the side bends,
and here it's the opposite.

1055
01:14:31,090 --> 01:14:33,140
And I want to give
you a taste of it.

1056
01:14:33,140 --> 01:14:35,550
And I'm not doing it
rigorously, and you're right,

1057
01:14:35,550 --> 01:14:37,060
I have to think about it.

1058
01:14:37,060 --> 01:14:41,810
But the important part is
that the drive frequency is 0.

1059
01:14:41,810 --> 01:14:45,220
We are not driving, we
are not rotating in x.

1060
01:14:45,220 --> 01:14:48,980
The system is mainly
rotating around the z-axis.

1061
01:14:48,980 --> 01:14:57,690
And at that moment, the matrix
for the differential equation

1062
01:14:57,690 --> 01:15:01,420
for the optical Bloch vector
has three imaginary parts.

1063
01:15:01,420 --> 01:15:04,400
It's gamma, 2 gamma,
and gamma or half of it

1064
01:15:04,400 --> 01:15:05,920
if you look at the amplitude.

1065
01:15:05,920 --> 01:15:08,520
And now I want to just
show you in a second how

1066
01:15:08,520 --> 01:15:10,450
we can get the second part.

1067
01:15:10,450 --> 01:15:12,770
But yeah, I'm guilty as charged.

1068
01:15:12,770 --> 01:15:14,600
There are small
gaps in my argument.

1069
01:15:14,600 --> 01:15:17,130
But all I want to
show you is how

1070
01:15:17,130 --> 01:15:20,160
you find-- I want to
just sort of convince you

1071
01:15:20,160 --> 01:15:23,470
that the optical Bloch
equations with this matrix

1072
01:15:23,470 --> 01:15:25,470
have all the ingredients
to explain it.

1073
01:15:33,520 --> 01:15:36,980
OK, so what I've done
here is I've shown you

1074
01:15:36,980 --> 01:15:40,160
that we have those
three ingredients.

1075
01:15:40,160 --> 01:15:47,210
And actually the moment--
I should have said it

1076
01:15:47,210 --> 01:15:48,810
before answering your question.

1077
01:15:48,810 --> 01:15:53,620
When we now crank up delta,
what we're physically doing

1078
01:15:53,620 --> 01:16:09,620
is we're doing a rapid
rotation of the Bloch vector

1079
01:16:09,620 --> 01:16:11,540
around the z-axis.

1080
01:16:11,540 --> 01:16:13,860
So when we rapidly
around the z-axis,

1081
01:16:13,860 --> 01:16:16,090
we're not doing anything
to the z component,

1082
01:16:16,090 --> 01:16:18,990
but we're strongly mixing
the x and y component,

1083
01:16:18,990 --> 01:16:21,290
but since [? half ?]
the imaginary part--

1084
01:16:21,290 --> 01:16:22,600
I'm waving my hands now.

1085
01:16:22,600 --> 01:16:25,340
We're not changing the
imaginary part for x and y.

1086
01:16:25,340 --> 01:16:29,070
And we still have two imaginary
parts, which are gamma over 2.

1087
01:16:29,070 --> 01:16:31,450
One imaginary part
which is gamma.

1088
01:16:31,450 --> 01:16:33,200
I know I'm running out
of time, but I only

1089
01:16:33,200 --> 01:16:34,200
need a few more minutes.

1090
01:16:49,300 --> 01:16:53,940
So if you do a rapid
rotation around the z-axis,

1091
01:16:53,940 --> 01:16:57,850
we obtain three eigenvalues.

1092
01:17:01,640 --> 01:17:04,365
Since we rotate around the
z-axis, we have minus gamma.

1093
01:17:30,010 --> 01:17:32,566
Sorry, my notes are better
than what I remember.

1094
01:17:32,566 --> 01:17:33,690
I looked through the notes.

1095
01:17:33,690 --> 01:17:36,180
But what I wanted to
say was the following.

1096
01:17:36,180 --> 01:17:39,340
I first show you what
are the eigenvalues

1097
01:17:39,340 --> 01:17:43,750
of this matrix in one case,
which is just a warm up.

1098
01:17:43,750 --> 01:17:46,400
I'm telling you there
are those three values.

1099
01:17:46,400 --> 01:17:48,570
And now I'm saying
we are interested

1100
01:17:48,570 --> 01:17:51,520
in the physical situation
of rapid rotation.

1101
01:17:51,520 --> 01:18:00,030
The rapid rotation modifies
the three eigenvalues.

1102
01:18:00,030 --> 01:18:03,830
The three eigenvalues
are no longer real.

1103
01:18:03,830 --> 01:18:06,780
The rapid rotation
around the z-axis

1104
01:18:06,780 --> 01:18:11,210
adds an imaginary i delta
to the x and y-axis.

1105
01:18:11,210 --> 01:18:14,890
Because now we are rotating,
x and y are getting mixed.

1106
01:18:14,890 --> 01:18:19,110
And this means now that
we have three peaks.

1107
01:18:19,110 --> 01:18:22,560
They are located in the
rotating frame at 0.

1108
01:18:22,560 --> 01:18:27,750
This is the carrier
at plus minus delta.

1109
01:18:27,750 --> 01:18:29,450
These are the two side bends.

1110
01:18:29,450 --> 01:18:32,380
And this part are
the widths of it.

1111
01:18:32,380 --> 01:18:35,000
This is how you have to
interpret the result.

1112
01:18:35,000 --> 01:18:38,170
So we have to know three peaks.

1113
01:18:38,170 --> 01:18:40,730
And the full widths
of half maximum

1114
01:18:40,730 --> 01:18:44,970
for the intensity, which
is the amplitude squared

1115
01:18:44,970 --> 01:18:47,510
is gamma, 2 gamma, and gamma.

1116
01:18:53,870 --> 01:18:57,400
And we can now immediately
proceed to the next case.

1117
01:18:57,400 --> 01:19:03,270
If we are on resonance, and
we drive The system strongly,

1118
01:19:03,270 --> 01:19:08,850
now we are driving the
system strongly around,

1119
01:19:08,850 --> 01:19:12,860
not the z-axis,
around the x-axis.

1120
01:19:12,860 --> 01:19:17,850
So now if we add the strong
drive around the x-axis,

1121
01:19:17,850 --> 01:19:22,740
we have now eigenvalues
which is plus minus ig.

1122
01:19:22,740 --> 01:19:24,580
This is the rotation
we add, and this

1123
01:19:24,580 --> 01:19:27,120
gives e to the
i-- in rotation ge

1124
01:19:27,120 --> 01:19:29,675
to the i omega t gives an ig.

1125
01:19:32,890 --> 01:19:35,860
And what happens
is the following.

1126
01:19:35,860 --> 01:19:47,340
The rapid rotation
around the x-axis-- well,

1127
01:19:47,340 --> 01:19:51,110
if you rotate around the
x-axis, you're not touching x.

1128
01:19:51,110 --> 01:19:56,410
And without any rotation,
we had an eigenvalue

1129
01:19:56,410 --> 01:19:58,970
for x which was
minus gamma over 2.

1130
01:19:58,970 --> 01:20:01,230
And if you take the matrix
of whatever the system

1131
01:20:01,230 --> 01:20:10,356
and rotate it around the
x-axis, this is preserved.

1132
01:20:10,356 --> 01:20:11,210
AUDIENCE: Excuse me.

1133
01:20:11,210 --> 01:20:13,020
PROFESSOR: Yep, I
need one more minute.

1134
01:20:13,020 --> 01:20:15,310
Sorry.

1135
01:20:15,310 --> 01:20:20,910
But the rotation around
x strongly mixes y and z.

1136
01:20:20,910 --> 01:20:23,310
So therefore the two
other eigenvalues

1137
01:20:23,310 --> 01:20:26,320
appear at rotation ig.

1138
01:20:26,320 --> 01:20:30,820
And it is now the average
of gamma and gamma over 2,

1139
01:20:30,820 --> 01:20:33,300
which is 3/4 gamma.

1140
01:20:33,300 --> 01:20:38,260
And if you go to the
amplitude squared, it is 3/2.

1141
01:20:38,260 --> 01:20:40,670
So therefore we
have the situation

1142
01:20:40,670 --> 01:20:46,676
that we have those
three eigenvalues.

1143
01:20:49,800 --> 01:20:55,370
And this is responsible
for the three peaks

1144
01:20:55,370 --> 01:21:01,680
at plus minus g and 0.

1145
01:21:01,680 --> 01:21:05,860
And the widths of it
is I have to multiply

1146
01:21:05,860 --> 01:21:10,700
by a factor of 2
gamma and 3/2 gamma.

1147
01:21:10,700 --> 01:21:13,170
And this explains the
two limiting cases

1148
01:21:13,170 --> 01:21:15,620
I presented to you earlier.

1149
01:21:15,620 --> 01:21:18,840
No time for question, the
other people are waiting.

1150
01:21:18,840 --> 01:21:21,900
Reminder we have
class on Friday.