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PROFESSOR: Are we ready?

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00:00:29,570 --> 00:00:33,560
So good afternoon.

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00:00:33,560 --> 00:00:37,260
Just a reminder, this week we
see each other three times--

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00:00:37,260 --> 00:00:41,080
today on Wednesday and
Friday in this other lecture

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00:00:41,080 --> 00:00:42,680
hall for our mid-term exam.

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00:00:45,710 --> 00:00:48,740
Today we will finish
the big chapter

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00:00:48,740 --> 00:00:50,939
on light-atom interaction.

15
00:00:50,939 --> 00:00:52,730
But we're not getting
rid of it, because we

16
00:00:52,730 --> 00:00:55,290
will be transitioning
to an important aspect

17
00:00:55,290 --> 00:00:56,800
of light-atom interaction.

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00:00:56,800 --> 00:00:59,490
And these are line shifts
and line broadening.

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00:00:59,490 --> 00:01:04,910
So today we start the next big
chapter-- line shifts and line

20
00:01:04,910 --> 00:01:05,880
broadening.

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00:01:05,880 --> 00:01:08,980
But before I do that, we have to
finish light-atom interaction.

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00:01:08,980 --> 00:01:12,750
And I want to come back
to the rotating wave

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00:01:12,750 --> 00:01:14,190
approximation revisited.

24
00:01:14,190 --> 00:01:16,490
So I'll revisit the revisit
of the rotating wave

25
00:01:16,490 --> 00:01:17,810
approximation.

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00:01:17,810 --> 00:01:19,650
And sometimes when
I have discussions

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00:01:19,650 --> 00:01:21,850
with students after
class, I realize

28
00:01:21,850 --> 00:01:25,220
that something which I
sort of casually mentioned

29
00:01:25,220 --> 00:01:27,811
is either confusing or
interesting for you.

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00:01:27,811 --> 00:01:29,310
And there are two
aspects I actually

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00:01:29,310 --> 00:01:31,040
want to come back here.

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00:01:31,040 --> 00:01:32,640
So several people
reacted to that,

33
00:01:32,640 --> 00:01:35,260
but some felt it was maybe a
little bit too complicated.

34
00:01:35,260 --> 00:01:37,440
Or others asked me
about some details.

35
00:01:37,440 --> 00:01:39,150
So let me come back
to two aspects.

36
00:01:39,150 --> 00:01:41,220
And I hope you find
them interesting.

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00:01:41,220 --> 00:01:51,300
One is when we sorted
out all those terms,

38
00:01:51,300 --> 00:01:54,200
those [? need two ?] angular
momentum selection rules.

39
00:01:54,200 --> 00:01:56,470
But I made sort of the
innocent comment-- well,

40
00:01:56,470 --> 00:02:00,560
if you have omega minus omega
in a time-dependent Hamiltonian,

41
00:02:00,560 --> 00:02:03,950
one term is responsible
for absorption,

42
00:02:03,950 --> 00:02:05,460
one is for emission.

43
00:02:05,460 --> 00:02:09,110
And when more than one
person asked me about it,

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00:02:09,110 --> 00:02:12,810
I think many more than
one person in class

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00:02:12,810 --> 00:02:15,220
would like to know
more about it.

46
00:02:15,220 --> 00:02:19,530
So therefore, let me spend the
first few minutes in explaining

47
00:02:19,530 --> 00:02:24,210
why is a time-dependent
term in the Hamiltonian

48
00:02:24,210 --> 00:02:27,660
with plus or minus
omega, why is one of them

49
00:02:27,660 --> 00:02:29,890
responsible for
absorption, and one

50
00:02:29,890 --> 00:02:33,680
is responsible for emission?

51
00:02:33,680 --> 00:02:36,170
Well, we have
Schrodinger's equation,

52
00:02:36,170 --> 00:02:43,810
which says that the change
of the amplitude in state one

53
00:02:43,810 --> 00:02:45,890
has a term.

54
00:02:45,890 --> 00:02:49,100
And if it started out with
population in state two--

55
00:02:49,100 --> 00:02:51,460
let's say perturbation theory,
we start in state two--

56
00:02:51,460 --> 00:02:55,270
then it is the only term where
the differential equation,

57
00:02:55,270 --> 00:02:57,280
through an off-diagonal
matrix element

58
00:02:57,280 --> 00:03:00,790
puts amplitude from
state two into state one.

59
00:03:00,790 --> 00:03:04,550
So what I'm writing down here
is just Schrodinger's equation.

60
00:03:04,550 --> 00:03:08,550
And the operator V
is the drive field

61
00:03:08,550 --> 00:03:11,790
connecting state
two to state one.

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00:03:11,790 --> 00:03:14,870
And so if I just
integrate this equation

63
00:03:14,870 --> 00:03:21,000
for a short time between
time t and t plus delta t,

64
00:03:21,000 --> 00:03:23,400
and I'm asking, did we
change the population

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00:03:23,400 --> 00:03:27,270
of state one, which is
now our final state?

66
00:03:27,270 --> 00:03:38,390
Well, then you integrate over
that for time interval delta t.

67
00:03:38,390 --> 00:03:43,930
But now comes the point
that the initial state has,

68
00:03:43,930 --> 00:03:46,570
in its time-dependent
wave function,

69
00:03:46,570 --> 00:03:50,600
a vector which is e to
the minus i omega 2 t.

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00:03:50,600 --> 00:03:53,140
The final state,
which I called one,

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00:03:53,140 --> 00:03:58,840
has-- because it's a complex
conjugate-- plus omega 1 t.

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00:03:58,840 --> 00:04:05,380
And let's just
assume we have here

73
00:04:05,380 --> 00:04:08,562
the proportionality
to e to the i omega t.

74
00:04:08,562 --> 00:04:11,020
And let me just say omega can
now be positive and negative.

75
00:04:11,020 --> 00:04:12,746
It will be part of
the answer whether it

76
00:04:12,746 --> 00:04:14,550
should be positive or negative.

77
00:04:14,550 --> 00:04:20,910
Well, this integral here becomes
an integral of e to the i,

78
00:04:20,910 --> 00:04:27,990
omega 1 minus omega 2 plus
omega t integrated with time.

79
00:04:31,460 --> 00:04:34,770
And this is an
oscillating function,

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00:04:34,770 --> 00:04:37,360
where if you integrated
with over time,

81
00:04:37,360 --> 00:04:48,320
it will average to
zero unless omega

82
00:04:48,320 --> 00:04:52,350
is equal or at least
close to the frequency

83
00:04:52,350 --> 00:04:58,560
difference between
initial and excited state.

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00:04:58,560 --> 00:05:00,540
So actually, what
you encounter here

85
00:05:00,540 --> 00:05:05,520
is-- well, what I've derived
for you here is actually,

86
00:05:05,520 --> 00:05:07,380
you can say energy conservation.

87
00:05:07,380 --> 00:05:09,180
I didn't assume it.

88
00:05:09,180 --> 00:05:12,480
It is built into the time
evolution of the Schrodinger

89
00:05:12,480 --> 00:05:16,890
equation, that you can
only go from state one

90
00:05:16,890 --> 00:05:21,290
to state two or state two to one
if the drive term has a Fourier

91
00:05:21,290 --> 00:05:25,110
component omega, which
makes up for the difference.

92
00:05:25,110 --> 00:05:28,030
Or I'm using different
language now.

93
00:05:28,030 --> 00:05:31,210
If, through the drive
term, you provide photons,

94
00:05:31,210 --> 00:05:36,260
you provide quanta of
energy, where omega fulfills

95
00:05:36,260 --> 00:05:38,760
the equation for
energy conservation.

96
00:05:38,760 --> 00:05:42,630
And you also see from this
result, when omega 1 is higher

97
00:05:42,630 --> 00:05:46,330
than omega 2, omega
has to be negative.

98
00:05:46,330 --> 00:05:49,360
When the reverse is true,
omega has to be positive.

99
00:05:49,360 --> 00:05:51,530
So that's why I said,
the e to the plus

100
00:05:51,530 --> 00:05:54,350
i omega t term is
responsible for absorption.

101
00:05:54,350 --> 00:05:56,175
The e to the minus
i omega t term

102
00:05:56,175 --> 00:06:03,610
is responsible for
stimulated emission.

103
00:06:03,610 --> 00:06:06,380
You also see, of
course-- but I stop here,

104
00:06:06,380 --> 00:06:08,400
because I think you've
heard it often enough.

105
00:06:08,400 --> 00:06:12,220
If you integrate over
short time delta t,

106
00:06:12,220 --> 00:06:14,420
this equation has
to be fulfilled

107
00:06:14,420 --> 00:06:17,490
only to within 1 over delta t.

108
00:06:17,490 --> 00:06:22,120
This is sort of the
energy-time uncertainty.

109
00:06:22,120 --> 00:06:26,120
For short times,
the photon energy

110
00:06:26,120 --> 00:06:30,720
does not need to match
exactly the energy difference.

111
00:06:30,720 --> 00:06:33,480
And you also realize
when we think

112
00:06:33,480 --> 00:06:35,950
about omega is
close to resonance,

113
00:06:35,950 --> 00:06:38,840
then e to the i omega
t does absorption.

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00:06:38,840 --> 00:06:40,510
But if you're in
the ground state,

115
00:06:40,510 --> 00:06:43,947
e to the minus i
omega t, leads now

116
00:06:43,947 --> 00:06:46,440
to a very rapid
oscillation here,

117
00:06:46,440 --> 00:06:49,930
which is close to [? 2 ?]
omega oscillation.

118
00:06:49,930 --> 00:06:53,240
And we've discussed that in the
context of the AC stark shift,

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00:06:53,240 --> 00:06:55,650
that this gives rise to
the [? proxy ?] shift.

120
00:06:55,650 --> 00:07:01,690
We've also discussed that this
term is rapidly oscillating

121
00:07:01,690 --> 00:07:05,442
and it's nothing else than the
counter-rotating term which

122
00:07:05,442 --> 00:07:07,400
we usually neglect when
we do the rotating wave

123
00:07:07,400 --> 00:07:08,740
approximation.

124
00:07:08,740 --> 00:07:11,140
So everything we've
discussed in this context--

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00:07:11,140 --> 00:07:14,130
counter-rotating term, energy
conservation, Heisenberg's

126
00:07:14,130 --> 00:07:18,210
uncertainty,
time-energy uncertainty

127
00:07:18,210 --> 00:07:22,520
actually comes from
this kind of formalism.

128
00:07:22,520 --> 00:07:23,115
Any question?

129
00:07:27,698 --> 00:07:33,010
Of course, if you quantize
the electromagnetic field,

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00:07:33,010 --> 00:07:36,890
then you don't have a drive
term with e to the i omega t.

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00:07:36,890 --> 00:07:40,050
You just have a and a
[? degas ?] for the photons.

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00:07:40,050 --> 00:07:42,675
And the question, which
term absorbs a photon

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00:07:42,675 --> 00:07:45,640
or creates a photon,
does not exist.

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00:07:45,640 --> 00:07:48,239
Because you know it whether
it's a or a [? dega. ?] But

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00:07:48,239 --> 00:07:49,780
you have the two
choices, whether you

136
00:07:49,780 --> 00:07:53,120
want to use a fully quantized
field with photon operators

137
00:07:53,120 --> 00:07:56,890
or whether you want to use
the time-dependent formalism,

138
00:07:56,890 --> 00:07:59,119
using a semi-classical
or classical field

139
00:07:59,119 --> 00:08:00,285
in the Schrodinger equation.

140
00:08:04,100 --> 00:08:10,160
The second comment
I wanted to do

141
00:08:10,160 --> 00:08:14,840
is using the
semi-classical picture,

142
00:08:14,840 --> 00:08:18,750
I was sort of going with you
through some examples when

143
00:08:18,750 --> 00:08:22,120
the rotating wave approximation
is necessary, when not.

144
00:08:22,120 --> 00:08:24,410
When do you have
counter-rotating terms.

145
00:08:24,410 --> 00:08:27,780
And yes, everything I
told you is, I think,

146
00:08:27,780 --> 00:08:31,211
is the best possible way
how you can-- I assume,

147
00:08:31,211 --> 00:08:33,710
because I haven't found a better
one-- the best possible way

148
00:08:33,710 --> 00:08:36,360
to present it and
explain it to using

149
00:08:36,360 --> 00:08:40,030
the time-dependent
electromagnetic field.

150
00:08:40,030 --> 00:08:42,190
But I realized
after class that it

151
00:08:42,190 --> 00:08:44,930
may be useful to
quickly state what

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00:08:44,930 --> 00:08:52,990
I have said in using
the photon picture.

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00:08:52,990 --> 00:09:00,360
If we have circular
polarization,

154
00:09:00,360 --> 00:09:05,200
we have, for given frequency,
annihilation and creation

155
00:09:05,200 --> 00:09:06,300
operator.

156
00:09:06,300 --> 00:09:09,970
But let's assume that the
mode we are considering

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00:09:09,970 --> 00:09:13,620
is right-handed
circularly polarized,

158
00:09:13,620 --> 00:09:17,350
so the operator creates a
photon at frequency omega

159
00:09:17,350 --> 00:09:21,910
with this right-handed
circular polarization.

160
00:09:27,665 --> 00:09:34,880
So that means now if
we start from a level m

161
00:09:34,880 --> 00:09:38,900
and we have now
light-atom interaction,

162
00:09:38,900 --> 00:09:46,620
the operator which
annihilates a photon

163
00:09:46,620 --> 00:09:51,080
with circular polarization
because of angular momentum

164
00:09:51,080 --> 00:09:55,410
conservation can only
take us to a level

165
00:09:55,410 --> 00:09:57,750
where the magnetic quantum
number is n plus 1.

166
00:10:03,110 --> 00:10:07,700
Well, the operator A
[? dega ?] creates a photon

167
00:10:07,700 --> 00:10:10,760
through stimulated emission.

168
00:10:10,760 --> 00:10:12,515
And so this is now
our two-level system.

169
00:10:19,930 --> 00:10:23,050
And now we should ask the
question, in terms of rotating

170
00:10:23,050 --> 00:10:25,480
wave approximation,
is necessary or not,

171
00:10:25,480 --> 00:10:28,410
are there
counter-rotating terms?

172
00:10:28,410 --> 00:10:32,300
Well, the counter-rotating terms
are the non-intuitive terms,

173
00:10:32,300 --> 00:10:37,970
where you start out
in the lowest state.

174
00:10:37,970 --> 00:10:42,060
But now instead of absorbing
a photon, you emit a photon.

175
00:10:42,060 --> 00:10:47,410
And the operator for
emission is this one.

176
00:10:47,410 --> 00:10:51,465
So I can now ask, is
there another term?

177
00:10:54,450 --> 00:10:59,230
Why don't we stick to the
blue color for the photons?

178
00:10:59,230 --> 00:11:03,810
That there is a term which
is driven by the operator

179
00:11:03,810 --> 00:11:06,800
A [? dega ?]
circularly polarized.

180
00:11:06,800 --> 00:11:11,570
Well, the answer is,
there may be such a term,

181
00:11:11,570 --> 00:11:16,000
but the state we need has
now a magnetic quantum

182
00:11:16,000 --> 00:11:20,920
number of m minus 1 because
of angular momentum selection

183
00:11:20,920 --> 00:11:22,780
rules.

184
00:11:22,780 --> 00:11:29,220
So this here is the
counter-rotating term,

185
00:11:29,220 --> 00:11:32,440
which you may or may not
neglect, depending whether you

186
00:11:32,440 --> 00:11:35,710
want to make the rotating
wave approximation or not.

187
00:11:35,710 --> 00:11:38,270
So therefore, if you got
a little bit confused

188
00:11:38,270 --> 00:11:40,330
about the different
cases I considered

189
00:11:40,330 --> 00:11:46,440
at the end of the
last lecture, then

190
00:11:46,440 --> 00:11:49,520
you may just summarize the
many examples I gave you

191
00:11:49,520 --> 00:11:51,770
in the lecture just
as a note which

192
00:11:51,770 --> 00:11:53,740
you should keep in
the back of your head,

193
00:11:53,740 --> 00:11:56,820
namely-- let me first
phrase it in words

194
00:11:56,820 --> 00:11:57,910
and then write it down.

195
00:11:57,910 --> 00:12:00,580
If you have circular
polarization and angular

196
00:12:00,580 --> 00:12:05,260
momentum selection rules,
then the counter-rotating term

197
00:12:05,260 --> 00:12:10,320
may require a third level and is
not part of two-level physics.

198
00:12:10,320 --> 00:12:13,560
So if you have a situation where
the third level does not exist,

199
00:12:13,560 --> 00:12:17,030
you do not have a
counter-rotating term.

200
00:12:17,030 --> 00:12:20,480
However, in all situations
I've encountered in the lab,

201
00:12:20,480 --> 00:12:22,015
this third level does exist.

202
00:12:25,681 --> 00:12:26,180
OK.

203
00:12:26,180 --> 00:12:30,360
So let me just write that
down-- counter-rotating term

204
00:12:30,360 --> 00:12:49,530
for circularly
polarized radiation

205
00:12:49,530 --> 00:12:58,900
requires a third level
which may not exist.

206
00:12:58,900 --> 00:13:00,420
And then you don't
have this term.

207
00:13:03,300 --> 00:13:07,550
But it does exist in most cases.

208
00:13:13,070 --> 00:13:16,510
Anyway, just an additional
clarification for the topics

209
00:13:16,510 --> 00:13:17,980
we had on Wednesday.

210
00:13:17,980 --> 00:13:20,411
Any questions about that?

211
00:13:20,411 --> 00:13:20,910
Yes.

212
00:13:20,910 --> 00:13:24,210
STUDENT: Are there cases other
than a spin-1/2 system where it

213
00:13:24,210 --> 00:13:26,360
doesn't exist?

214
00:13:26,360 --> 00:13:31,020
PROFESSOR: Well, I mentioned
the example last class--

215
00:13:31,020 --> 00:13:34,700
if you do spectroscopy
of an S2P transition

216
00:13:34,700 --> 00:13:38,090
in the magnetic field
of a neutron star.

217
00:13:38,090 --> 00:13:40,250
Then the same one
splitting is so huge

218
00:13:40,250 --> 00:13:43,867
that, well, you can assume that
this has been shifted so far

219
00:13:43,867 --> 00:13:45,700
away that it has been
completely suppressed.

220
00:13:48,780 --> 00:13:52,580
Other than that, well, we
have the trivial situation

221
00:13:52,580 --> 00:13:54,190
which we discussed in NMR.

222
00:13:54,190 --> 00:13:56,040
If you simply have
spin one half,

223
00:13:56,040 --> 00:13:58,670
then the total number
of levels is only two,

224
00:13:58,670 --> 00:14:01,290
because we're talking about
spin up and spin down.

225
00:14:01,290 --> 00:14:04,320
Or I constructed,
in the last class,

226
00:14:04,320 --> 00:14:09,050
the forbidden transition, a
doublet S to doublet S state.

227
00:14:09,050 --> 00:14:11,790
So that's two pairs
of S equals one half.

228
00:14:11,790 --> 00:14:15,530
And then we are
missing this state

229
00:14:15,530 --> 00:14:19,880
to couple any counter-rotating
term into the system.

230
00:14:23,636 --> 00:14:24,135
OK.

231
00:14:41,490 --> 00:14:45,380
The next subject is saturation.

232
00:14:45,380 --> 00:14:48,290
Now, in this chapter,
I want to talk

233
00:14:48,290 --> 00:14:51,080
about saturation in general.

234
00:14:51,080 --> 00:14:55,740
I want to discuss monochromatic
light but also broadband light.

235
00:14:55,740 --> 00:15:00,810
And I want to introduce
concepts of saturation intensity

236
00:15:00,810 --> 00:15:04,080
of absorption cross section,
certain things which

237
00:15:04,080 --> 00:15:08,432
I find extremely useful if I
want to understand what happens

238
00:15:08,432 --> 00:15:09,890
when light interacts
with a system.

239
00:15:12,640 --> 00:15:15,420
Just sort of to
whet your appetite,

240
00:15:15,420 --> 00:15:17,890
I will sort of show you
that the absorption cross

241
00:15:17,890 --> 00:15:20,270
section of a two-level
system is independent

242
00:15:20,270 --> 00:15:22,920
whether you have a strong
or weak transition.

243
00:15:22,920 --> 00:15:26,170
Some people think the
cross section should be--

244
00:15:26,170 --> 00:15:27,670
but there is a
difference which is

245
00:15:27,670 --> 00:15:31,240
important between monochromatic
and broadband light.

246
00:15:31,240 --> 00:15:34,340
But in the end, the
concepts are very simple.

247
00:15:34,340 --> 00:15:36,300
I should say, sometimes
I feel it's almost too

248
00:15:36,300 --> 00:15:39,762
simple to present it in class.

249
00:15:39,762 --> 00:15:41,470
On the other hand, if
I don't present it,

250
00:15:41,470 --> 00:15:43,770
I can't make a few comments
and guide you through.

251
00:15:43,770 --> 00:15:45,710
So my conclusion
now is, at least

252
00:15:45,710 --> 00:15:50,050
for now, I show you
some prepared slides,

253
00:15:50,050 --> 00:15:53,240
and I sort of step you through,
and make a few annotations,

254
00:15:53,240 --> 00:15:55,470
and point out certain things.

255
00:15:55,470 --> 00:15:57,690
We have already
partially transitioned

256
00:15:57,690 --> 00:16:01,730
to teach you this material
through homework assignment.

257
00:16:01,730 --> 00:16:04,570
This week's homework assignment,
which is due on Wednesday,

258
00:16:04,570 --> 00:16:08,400
is almost completely
on saturation.

259
00:16:08,400 --> 00:16:11,930
And I will make a few comments
where what I present you today

260
00:16:11,930 --> 00:16:14,999
is an extension or different
from what you're learning

261
00:16:14,999 --> 00:16:16,290
in the homework and vice versa.

262
00:16:22,720 --> 00:16:25,370
Yes.

263
00:16:25,370 --> 00:16:29,100
If I wanted to present
you saturation,

264
00:16:29,100 --> 00:16:32,560
power broadening, and all
that in the purest form,

265
00:16:32,560 --> 00:16:36,020
I would just preset you with
the Optical Bloch equations.

266
00:16:36,020 --> 00:16:37,180
We can solve them.

267
00:16:37,180 --> 00:16:40,650
And then we have everything
we want-- a result which

268
00:16:40,650 --> 00:16:43,450
explains saturation
and a result which

269
00:16:43,450 --> 00:16:46,740
explains power broadening.

270
00:16:46,740 --> 00:16:48,760
And you do some of
it in your homework.

271
00:16:48,760 --> 00:16:50,410
However, what I
want to show here

272
00:16:50,410 --> 00:16:54,430
is that saturation is
actually a general feature

273
00:16:54,430 --> 00:16:57,920
of a two-level system if you
have sort of three rates, which

274
00:16:57,920 --> 00:17:00,480
I will explain to you in
a moment-- very similar

275
00:17:00,480 --> 00:17:02,350
to Einstein's A
and B coefficient--

276
00:17:02,350 --> 00:17:05,030
that all such systems
have saturation.

277
00:17:05,030 --> 00:17:08,339
And then you may immediately
solve the Optical Bloch

278
00:17:08,339 --> 00:17:10,500
creation for
monochromatic radiation.

279
00:17:10,500 --> 00:17:13,430
But for broadband
radiation, we usually

280
00:17:13,430 --> 00:17:15,900
don't use the Optical
Bloch equations,

281
00:17:15,900 --> 00:17:19,369
because for infinitely
broad light,

282
00:17:19,369 --> 00:17:20,920
there is no coherence
for which we

283
00:17:20,920 --> 00:17:24,589
need Optical Bloch equations.

284
00:17:24,589 --> 00:17:26,859
If you only have the
Optical Bloch equations,

285
00:17:26,859 --> 00:17:30,216
you have solved for saturation
in one limiting case,

286
00:17:30,216 --> 00:17:32,215
and you don't see that
the concept of saturation

287
00:17:32,215 --> 00:17:34,400
is much broader.

288
00:17:34,400 --> 00:17:40,270
So therefore, let us assume
that we have a two-level system.

289
00:17:40,270 --> 00:17:46,140
And we have the
couple two levels

290
00:17:46,140 --> 00:17:48,040
with a rate--
which you can think

291
00:17:48,040 --> 00:17:51,060
of the rate of absorption, the
rate of stimulated emission.

292
00:17:51,060 --> 00:17:55,310
And I call the rate
the unsaturated rate.

293
00:17:55,310 --> 00:17:58,080
In addition, there
is some dissipation,

294
00:17:58,080 --> 00:18:01,710
some spontaneous decay
described by gamma.

295
00:18:05,190 --> 00:18:09,000
So Ru is the unsaturated
rate for absorption

296
00:18:09,000 --> 00:18:12,000
and for stimulated emission.

297
00:18:18,760 --> 00:18:24,110
Of course, you know even before
you solve those equations

298
00:18:24,110 --> 00:18:27,830
that there must be some
saturation built in.

299
00:18:27,830 --> 00:18:30,970
If you would look at
the fraction of atoms

300
00:18:30,970 --> 00:18:36,440
in the excited state and
you change the laser power,

301
00:18:36,440 --> 00:18:43,240
which means changing
the unsaturated rate,

302
00:18:43,240 --> 00:18:47,140
things cannot shoot up forever,
because you cannot put more

303
00:18:47,140 --> 00:18:50,980
than 100% of the population
into the excited state.

304
00:18:50,980 --> 00:18:54,810
However, the effect that when
we increase the laser power,

305
00:18:54,810 --> 00:18:57,810
we do upwards absorption
and downward stimulation--

306
00:18:57,810 --> 00:19:00,710
means you won't even get 100%.

307
00:19:00,710 --> 00:19:02,470
The maximum you can get is 50%.

308
00:19:05,200 --> 00:19:07,140
And what I'm just
drawing for you is

309
00:19:07,140 --> 00:19:08,916
this is a phenomenon
of saturation,

310
00:19:08,916 --> 00:19:10,665
and now we want to
understand the details.

311
00:19:18,520 --> 00:19:25,720
So using this rate
equation, we are

312
00:19:25,720 --> 00:19:29,930
defining-- this is now a
definition-- the saturated rate

313
00:19:29,930 --> 00:19:36,410
is the net transfer from A to B.

314
00:19:36,410 --> 00:19:39,090
Because we have an absorption
and stimulated emission,

315
00:19:39,090 --> 00:19:41,070
the net transfer is the
[? unsaturated ?] rate

316
00:19:41,070 --> 00:19:43,080
times the population difference.

317
00:19:43,080 --> 00:19:45,420
And this is our saturated rate.

318
00:19:45,420 --> 00:19:48,200
But of course, we normalize
everything for atom.

319
00:19:48,200 --> 00:19:52,590
So therefore, our saturated
rate has a rate coefficient S

320
00:19:52,590 --> 00:19:55,830
times the total number of
atoms, or the total population

321
00:19:55,830 --> 00:19:56,710
in both states.

322
00:20:02,590 --> 00:20:05,460
Eventually, we are
interested in steady state.

323
00:20:05,460 --> 00:20:07,890
We can immediately
solve the rate equation

324
00:20:07,890 --> 00:20:12,460
in terms of steady state,
which is done there.

325
00:20:12,460 --> 00:20:16,450
And we find that for
those, we can now

326
00:20:16,450 --> 00:20:20,040
eliminate one of the
states from the equation,

327
00:20:20,040 --> 00:20:22,320
because we have the
steady state ratio.

328
00:20:22,320 --> 00:20:25,190
And then we find that
the saturated rate

329
00:20:25,190 --> 00:20:28,070
is gamma over 2 times
an expression which

330
00:20:28,070 --> 00:20:30,390
involves a saturation parameter.

331
00:20:30,390 --> 00:20:33,270
So in other words, it's just
almost trivial solutions

332
00:20:33,270 --> 00:20:36,780
of very simple equations,
which describe the saturation

333
00:20:36,780 --> 00:20:42,005
phenomenon I outlined
for you at the beginning.

334
00:20:46,810 --> 00:20:49,830
This solution has the
two limiting cases

335
00:20:49,830 --> 00:20:55,080
which we want to see-- that at
a very low unsaturated rate,

336
00:20:55,080 --> 00:20:57,580
the saturated rate is the
unsaturated rate because there

337
00:20:57,580 --> 00:20:58,990
is no saturation.

338
00:20:58,990 --> 00:21:02,390
And secondly, if we would
go to infinite power,

339
00:21:02,390 --> 00:21:05,780
the saturated rate
becomes gamma over 2,

340
00:21:05,780 --> 00:21:08,620
because we have equalized
the population between ground

341
00:21:08,620 --> 00:21:09,850
and excited state.

342
00:21:09,850 --> 00:21:12,820
One half of the atoms
are in the excited state,

343
00:21:12,820 --> 00:21:15,340
and they dissipate or scatter
light with the rate gamma.

344
00:21:21,560 --> 00:21:22,150
All right.

345
00:21:30,312 --> 00:21:30,895
Any questions?

346
00:21:34,380 --> 00:21:38,530
We now want to specialize it
to a situation which we often

347
00:21:38,530 --> 00:21:42,360
encounter, namely
monochromatic radiation.

348
00:21:42,360 --> 00:21:47,120
And for monochromatic
radiation, the unsaturated rate

349
00:21:47,120 --> 00:21:51,510
follows-- well, I factored
out something here.

350
00:21:51,510 --> 00:21:59,334
But it follows the
normalized line shape,

351
00:21:59,334 --> 00:22:00,250
which is a Lorentzian.

352
00:22:03,760 --> 00:22:07,170
And therefore, our
unsaturated rate

353
00:22:07,170 --> 00:22:09,530
is proportion to
the laser power.

354
00:22:09,530 --> 00:22:12,410
But I usually like to express
the laser power through a Rabi

355
00:22:12,410 --> 00:22:14,660
frequency or the Rabi
frequency squared.

356
00:22:14,660 --> 00:22:18,685
So our unsaturated rate
follows this Lorentzian.

357
00:22:21,270 --> 00:22:28,980
And on resonance,
this part is one.

358
00:22:28,980 --> 00:22:32,100
Our rate is omega Rabi
squared over comma.

359
00:22:32,100 --> 00:22:35,300
And the definition for the
saturation parameter of one,

360
00:22:35,300 --> 00:22:39,440
or for the saturation intensity
is that the unsaturated rate

361
00:22:39,440 --> 00:22:42,110
has to be gamma over 2.

362
00:22:42,110 --> 00:22:44,550
So therefore, by
omega Rabi squared

363
00:22:44,550 --> 00:22:46,830
over gamma is the
unsaturated rate,

364
00:22:46,830 --> 00:22:49,720
it should be gamma
over 2 for saturation,

365
00:22:49,720 --> 00:22:51,830
for saturation parameter of one.

366
00:22:51,830 --> 00:22:55,040
So therefore, our saturation
pentameter resonance

367
00:22:55,040 --> 00:22:58,850
is given by this expression.

368
00:22:58,850 --> 00:23:06,880
And if you use the
previous result

369
00:23:06,880 --> 00:23:10,450
and apply it to this
unsaturated rate,

370
00:23:10,450 --> 00:23:16,270
we find a saturated
rate which shows now

371
00:23:16,270 --> 00:23:20,200
the new phenomenon
of power broadening.

372
00:23:20,200 --> 00:23:26,530
Let me illustrate
it in two ways.

373
00:23:26,530 --> 00:23:30,000
The saturated rate involves
a saturation parameter,

374
00:23:30,000 --> 00:23:33,030
and the unsaturated
rate is a Lorentzian.

375
00:23:33,030 --> 00:23:36,230
But this Lorentzian appears
now in the numerator

376
00:23:36,230 --> 00:23:38,350
and the denominator.

377
00:23:38,350 --> 00:23:40,370
So it appears twice.

378
00:23:40,370 --> 00:23:42,900
But with a one-step
manipulation,

379
00:23:42,900 --> 00:23:45,740
you can transform it
into a single Lorentzian.

380
00:23:45,740 --> 00:23:49,160
But this single Lorentzian
is now power-broadened.

381
00:23:52,620 --> 00:23:55,735
It no longer has
[? width ?] of gamma

382
00:23:55,735 --> 00:23:58,160
over-- of the natural
line with gamma.

383
00:23:58,160 --> 00:24:02,028
It has an additional term,
and this is power broadening.

384
00:24:02,028 --> 00:24:02,980
STUDENT: [INAUDIBLE]?

385
00:24:05,727 --> 00:24:06,310
PROFESSOR: No.

386
00:24:06,310 --> 00:24:10,620
It's still-- the resonance
is at [? 0 ?] [INAUDIBLE].

387
00:24:14,940 --> 00:24:16,350
The equations are trivial.

388
00:24:16,350 --> 00:24:17,750
It's really just
substituting one

389
00:24:17,750 --> 00:24:19,970
and getting from an
expression, simplifying it

390
00:24:19,970 --> 00:24:23,650
to simple Lorentzian.

391
00:24:23,650 --> 00:24:25,260
I just want to
emphasize the result.

392
00:24:28,000 --> 00:24:29,640
If you drive a
transition, we have

393
00:24:29,640 --> 00:24:44,290
now-- the [? width ?] of the
Lorentzian is now gamma over 2

394
00:24:44,290 --> 00:24:45,940
if we have no saturation.

395
00:24:45,940 --> 00:24:48,610
But then if we crank up
the saturation parameter,

396
00:24:48,610 --> 00:24:51,246
the [? width ?] increases with
a square root of the power.

397
00:24:51,246 --> 00:24:52,370
That's an important result.

398
00:24:52,370 --> 00:24:56,420
The square root of the
power leads to broadening.

399
00:24:56,420 --> 00:25:01,170
Now let me give you a
pictorial description

400
00:25:01,170 --> 00:25:03,490
of what we have done here.

401
00:25:03,490 --> 00:25:09,410
If we start with the Lorentzian
and we increase the power,

402
00:25:09,410 --> 00:25:12,470
you sort of want to drive
the system with the stronger

403
00:25:12,470 --> 00:25:13,950
Lorentzian.

404
00:25:13,950 --> 00:25:17,460
But we know we have a
ceiling, which is saturation.

405
00:25:17,460 --> 00:25:20,080
And of course, when
you drive it stronger,

406
00:25:20,080 --> 00:25:23,640
you reach the
ceiling on resonance

407
00:25:23,640 --> 00:25:27,720
earlier than you reach the
ceiling when you transfer it

408
00:25:27,720 --> 00:25:29,630
away from resonance.

409
00:25:29,630 --> 00:25:31,790
So therefore, if you
start with the red curve,

410
00:25:31,790 --> 00:25:34,732
crank up the power,
you will get more

411
00:25:34,732 --> 00:25:37,830
of a factor, more of
a result in the wings

412
00:25:37,830 --> 00:25:40,420
because you are not
yet saturated there.

413
00:25:40,420 --> 00:25:43,540
And this graphical construction,
which I have just sort of

414
00:25:43,540 --> 00:25:49,160
indicated to you, lead now to
a curve which is broadened,

415
00:25:49,160 --> 00:25:51,980
broader than the
original Lorentzian.

416
00:25:51,980 --> 00:25:54,190
And this is the reason
behind power broadening.

417
00:25:59,670 --> 00:26:02,130
I want to mention
one thing here.

418
00:26:02,130 --> 00:26:04,570
For the classroom
discussion, I have

419
00:26:04,570 --> 00:26:09,100
assumed that the light-atom
interaction can be described

420
00:26:09,100 --> 00:26:13,090
by Fermi's golden rule, which
we know is a limitation.

421
00:26:17,300 --> 00:26:20,080
When the system is,
in effect, incoherent

422
00:26:20,080 --> 00:26:22,750
or no longer coherent,
we had a long discussion

423
00:26:22,750 --> 00:26:26,820
about Rabi oscillation,
Fermi's golden rule

424
00:26:26,820 --> 00:26:28,420
in the last two weeks.

425
00:26:31,430 --> 00:26:34,590
But what I'm doing is
mathematically correct.

426
00:26:34,590 --> 00:26:36,800
The Optical Bloch
equation, which

427
00:26:36,800 --> 00:26:39,160
you'll use in your
homework assignment,

428
00:26:39,160 --> 00:26:42,450
will include the transition
from Rabi oscillation

429
00:26:42,450 --> 00:26:43,822
towards Fermi's golden rule.

430
00:26:43,822 --> 00:26:46,030
And I'm just considering
this [? fundamental ?] case.

431
00:27:03,060 --> 00:27:04,740
OK.

432
00:27:04,740 --> 00:27:07,750
I've talked about
saturation of a transition.

433
00:27:07,750 --> 00:27:10,880
I've mentioned that we
have defined the saturation

434
00:27:10,880 --> 00:27:14,440
parameter such that when we have
saturation parameter of one,

435
00:27:14,440 --> 00:27:16,440
we sort of get into
the non-linear regime

436
00:27:16,440 --> 00:27:18,110
where saturation happens.

437
00:27:18,110 --> 00:27:20,230
And of course, for
an experimentalist,

438
00:27:20,230 --> 00:27:24,930
the next question is, at what
intensity does that happen?

439
00:27:24,930 --> 00:27:27,420
This is summarized
in those equations.

440
00:27:27,420 --> 00:27:29,250
It's as simple as
possible algebra.

441
00:27:29,250 --> 00:27:31,080
You just combine two equations.

442
00:27:31,080 --> 00:27:32,940
I don't want to do it here.

443
00:27:32,940 --> 00:27:36,150
And we have a result for the
saturation intensity, which

444
00:27:36,150 --> 00:27:38,870
has two features, which
I want to point out.

445
00:27:38,870 --> 00:27:43,620
One is [? its case ?]
with omega cube.

446
00:27:43,620 --> 00:27:47,470
So the higher the frequency
of your transition is,

447
00:27:47,470 --> 00:27:51,030
the harder it is to saturate.

448
00:27:51,030 --> 00:27:55,040
Of course, it has something
to do with that in saturation,

449
00:27:55,040 --> 00:27:57,130
you have an
unsaturated rate, which

450
00:27:57,130 --> 00:27:59,420
is one half of the
spontaneous emission rate.

451
00:27:59,420 --> 00:28:01,640
And you remember that the
spontaneous emission rate

452
00:28:01,640 --> 00:28:03,080
was proportional to omega cube.

453
00:28:03,080 --> 00:28:06,230
So that's why we have,
again, the omega cube factor.

454
00:28:06,230 --> 00:28:25,571
And in addition, the larger--
actually, it depends.

455
00:28:25,571 --> 00:28:26,070
Sorry.

456
00:28:26,070 --> 00:28:26,778
I made a mistake.

457
00:28:36,020 --> 00:28:41,120
Well, you can write the
results in several ways.

458
00:28:41,120 --> 00:28:43,420
If you have an intensity
and you go back to photons,

459
00:28:43,420 --> 00:28:44,970
you get factors of omega.

460
00:28:44,970 --> 00:28:50,660
So when I said omega cubed comes
from the natural line widths,

461
00:28:50,660 --> 00:28:53,590
yes, it does, but it's
not the only omega factor.

462
00:28:53,590 --> 00:28:55,810
You can write the
result actually

463
00:28:55,810 --> 00:28:57,690
that you have a gamma
squared dependence,

464
00:28:57,690 --> 00:29:00,370
because one gamma comes from
the matrix element squared

465
00:29:00,370 --> 00:29:03,240
and one comes because you need
to compete in your excitation

466
00:29:03,240 --> 00:29:04,540
with spontaneous emission.

467
00:29:07,480 --> 00:29:10,760
So anyway, this is
sort of the result.

468
00:29:10,760 --> 00:29:13,820
And you can calculate it
for your favorite atom.

469
00:29:13,820 --> 00:29:16,400
And for alkaline
atoms, we usually find

470
00:29:16,400 --> 00:29:18,620
that the saturation
intensity is a few milliwatt

471
00:29:18,620 --> 00:29:37,250
per square centimeter

472
00:29:37,250 --> 00:29:42,410
Well, we can now repeat
some of this exercise

473
00:29:42,410 --> 00:29:44,690
for the broadband case.

474
00:29:44,690 --> 00:29:48,910
In the broadband case,
the unsaturated rate,

475
00:29:48,910 --> 00:29:51,980
which is the rate for absorption
in stimulated emission,

476
00:29:51,980 --> 00:29:56,070
following Einstein's treatment
of the AB coefficient

477
00:29:56,070 --> 00:29:59,060
is used by using
Einstein's B coefficient

478
00:29:59,060 --> 00:30:03,200
times the spectral intensity.

479
00:30:03,200 --> 00:30:06,630
And now we want the same
situation as before.

480
00:30:06,630 --> 00:30:08,440
We want to reach saturation.

481
00:30:08,440 --> 00:30:11,960
And saturation happens when
this is comparable with gamma.

482
00:30:11,960 --> 00:30:13,600
And it's purely a
definition that we

483
00:30:13,600 --> 00:30:15,160
say it should be gamma over 2.

484
00:30:15,160 --> 00:30:18,620
But we are consistent
with what we did before.

485
00:30:18,620 --> 00:30:21,870
And if you just
take this equation

486
00:30:21,870 --> 00:30:25,550
and calculate what the
saturation intensity is,

487
00:30:25,550 --> 00:30:29,770
well, gamma is nothing else
than the Einstein A coefficient.

488
00:30:29,770 --> 00:30:32,420
Here we have the
Einstein B coefficient.

489
00:30:32,420 --> 00:30:35,300
And if we take the ratio
between the Einstein A and B

490
00:30:35,300 --> 00:30:39,240
coefficient, the matrix
element, everything

491
00:30:39,240 --> 00:30:42,810
which is specific to
the atom, cancels out.

492
00:30:42,810 --> 00:30:45,710
And the saturation intensity,
or the spectral density--

493
00:30:45,710 --> 00:30:47,870
it's the spectral density
now for broadband--

494
00:30:47,870 --> 00:30:52,430
only depends on speed of light
and the transition frequency

495
00:30:52,430 --> 00:30:52,980
cube.

496
00:30:52,980 --> 00:30:55,120
And it doesn't make a
difference whether you

497
00:30:55,120 --> 00:30:58,370
have a two-level system which
has a strong matrix element

498
00:30:58,370 --> 00:31:01,380
or weak matrix element.

499
00:31:01,380 --> 00:31:04,510
I could explain it to
you now at this point,

500
00:31:04,510 --> 00:31:08,530
but we want to hold
the idea that there

501
00:31:08,530 --> 00:31:11,330
is a difference
between single mode

502
00:31:11,330 --> 00:31:14,310
monochromatic and
broadband excitation

503
00:31:14,310 --> 00:31:16,545
until I have discussed
one more concept.

504
00:31:19,210 --> 00:31:21,980
And this is the cross section.

505
00:31:27,980 --> 00:31:29,660
Just to check, are
there any questions?

506
00:31:29,660 --> 00:31:30,707
Yes, Nancy.

507
00:31:30,707 --> 00:31:32,695
AUDIENCE: So in
the broadband case,

508
00:31:32,695 --> 00:31:34,683
the line shift
doesn't matter at all?

509
00:31:34,683 --> 00:31:36,174
Because in the
monochromatic case,

510
00:31:36,174 --> 00:31:38,162
we had a line shift [INAUDIBLE].

511
00:31:41,160 --> 00:31:45,110
Well, hold your question.

512
00:31:45,110 --> 00:31:46,160
The line shape matters.

513
00:31:46,160 --> 00:31:48,860
I will now discuss what is
the line shape of the atom.

514
00:31:48,860 --> 00:31:52,630
And the quick answer is, if
the atom has a line shape,

515
00:31:52,630 --> 00:31:54,670
we have to take the
atomic line shape

516
00:31:54,670 --> 00:31:59,230
and do a convolution with
the line of the radiation.

517
00:31:59,230 --> 00:32:01,620
And we have the two
situations where in one case,

518
00:32:01,620 --> 00:32:03,850
the monochromatic
light is narrower

519
00:32:03,850 --> 00:32:05,480
than the line
shape of the atoms.

520
00:32:05,480 --> 00:32:07,270
In the other case, it's broader.

521
00:32:07,270 --> 00:32:09,680
And this difference,
in the end, will

522
00:32:09,680 --> 00:32:12,430
be responsible for the
effect that the line widths

523
00:32:12,430 --> 00:32:16,600
of the atom, which is
the natural line widths,

524
00:32:16,600 --> 00:32:19,210
will cancel out in one
case and not in the other.

525
00:32:19,210 --> 00:32:23,204
But that's actually the result
of the next five minutes.

526
00:32:23,204 --> 00:32:23,870
Other questions?

527
00:32:27,510 --> 00:32:32,250
I know this topic
can get confusing,

528
00:32:32,250 --> 00:32:34,599
because we go from one
definition to the next.

529
00:32:34,599 --> 00:32:35,640
So let me just summarize.

530
00:32:35,640 --> 00:32:38,230
What I've said so far is
we [? derive ?] an atom.

531
00:32:38,230 --> 00:32:41,180
We have absorption, we
have stimulated emission.

532
00:32:41,180 --> 00:32:46,620
And we want to understand
the phenomenon of saturation.

533
00:32:46,620 --> 00:32:50,360
And based on the effect how
we define saturation, namely

534
00:32:50,360 --> 00:32:52,910
that the unsaturated
rate is gamma over 2,

535
00:32:52,910 --> 00:32:56,410
we got some nice results
for the saturation intensity

536
00:32:56,410 --> 00:32:59,410
and for power broadening
of a Lorentzian.

537
00:32:59,410 --> 00:33:04,250
So it's pretty much having a
definition and running with it.

538
00:33:04,250 --> 00:33:11,150
And now we want to
express the same physics

539
00:33:11,150 --> 00:33:14,470
by using the concept
of a cross section

540
00:33:14,470 --> 00:33:17,980
for the following reason.

541
00:33:17,980 --> 00:33:20,960
You can do physics, you
can do atomic physics

542
00:33:20,960 --> 00:33:24,049
without ever thinking
about a cross section.

543
00:33:24,049 --> 00:33:26,590
You can just say, I have a laser
beam of a certain intensity,

544
00:33:26,590 --> 00:33:29,900
and I scatter light.

545
00:33:29,900 --> 00:33:32,970
But often, when we
scatter something--

546
00:33:32,970 --> 00:33:35,910
and you may be familiar,
from atomic collisions--

547
00:33:35,910 --> 00:33:38,700
you often want to write
the scattering rate

548
00:33:38,700 --> 00:33:42,800
as a density times cross
section times relative velocity.

549
00:33:42,800 --> 00:33:46,150
And this sort of has
this intuitive feeling.

550
00:33:46,150 --> 00:33:48,400
If you have a stream of
particles in your accelerator

551
00:33:48,400 --> 00:33:52,060
or a stream of photons
in your laser beam,

552
00:33:52,060 --> 00:33:54,400
you can now hold
onto the picture

553
00:33:54,400 --> 00:33:58,620
that each atom in your
target is a little disk.

554
00:33:58,620 --> 00:34:01,970
If the particle of photons hits
the disk, something happens.

555
00:34:01,970 --> 00:34:04,790
If it misses the
disk, nothing happens.

556
00:34:04,790 --> 00:34:08,199
And the area of the disk
is this cross section.

557
00:34:08,199 --> 00:34:10,469
So in other words, we
want to now understand

558
00:34:10,469 --> 00:34:15,060
how big is the disk of the
atom which will, so to speak,

559
00:34:15,060 --> 00:34:18,719
cast the shadow, which is
synonymous with absorption,

560
00:34:18,719 --> 00:34:21,389
when we illuminate those
atoms with laser light.

561
00:34:21,389 --> 00:34:25,980
So for me, a very
intuitive quantity.

562
00:34:25,980 --> 00:34:32,030
Anyway, so all we do
is we have already

563
00:34:32,030 --> 00:34:34,850
discussed the rate
of excitation,

564
00:34:34,850 --> 00:34:39,110
which is now the
unsaturated rate.

565
00:34:39,110 --> 00:34:42,570
But now we express
the unsaturated rate

566
00:34:42,570 --> 00:34:47,280
by the density of photons
times the cross section.

567
00:34:47,280 --> 00:34:51,860
And the relative velocity
is the speed of light.

568
00:34:51,860 --> 00:34:58,160
And from this equation, we
find-- because everything

569
00:34:58,160 --> 00:35:02,390
is known, we have talked about
that on the last few pages--

570
00:35:02,390 --> 00:35:07,600
we find that the cross section
is-- and this is the result.

571
00:35:07,600 --> 00:35:10,560
6 pi lambda bar square.

572
00:35:10,560 --> 00:35:15,740
Lambda bar is the wavelengths
of light divided by 2 pi.

573
00:35:15,740 --> 00:35:21,150
So we find that for
monochromatic radiation,

574
00:35:21,150 --> 00:35:25,430
the cross sectional
of a two-level system

575
00:35:25,430 --> 00:35:28,130
is independent of the
strength of the transition,

576
00:35:28,130 --> 00:35:30,170
independent of the
matrix element.

577
00:35:30,170 --> 00:35:32,540
It just depends on the
resonant wavelengths.

578
00:35:39,450 --> 00:35:42,610
Now you would say,
well, but what is now

579
00:35:42,610 --> 00:35:47,700
the difference between a
strong and a weak transition?

580
00:35:47,700 --> 00:35:49,750
And this is shown here.

581
00:35:49,750 --> 00:35:54,490
If you take your monochromatic
laser and you scan it,

582
00:35:54,490 --> 00:35:57,020
you scan it through
the cross section.

583
00:35:57,020 --> 00:36:03,980
When you are on resonance, you
have 6 pi lambda bar square.

584
00:36:03,980 --> 00:36:06,960
And the difference between
a narrow transition

585
00:36:06,960 --> 00:36:09,270
with a small Einstein
A coefficient

586
00:36:09,270 --> 00:36:12,400
and a strong transition with
a large Einstein A coefficient

587
00:36:12,400 --> 00:36:15,090
simply means that in
one case, it's narrower.

588
00:36:15,090 --> 00:36:16,350
In the other case, it's wider.

589
00:36:26,700 --> 00:36:34,310
We talked about the
phenomenon of saturation.

590
00:36:34,310 --> 00:36:38,870
6 pi lambda bar squared
is the cross section

591
00:36:38,870 --> 00:36:43,950
in the perturbative limit, or
the unsaturated cross section.

592
00:36:43,950 --> 00:36:46,680
Of course, if you
increase the laser power,

593
00:36:46,680 --> 00:36:49,320
you saturation the transition.

594
00:36:49,320 --> 00:36:53,170
The atom will have a smaller
and smaller cross section.

595
00:36:53,170 --> 00:36:57,680
Actually, that's something
important you should consider.

596
00:36:57,680 --> 00:37:01,080
When you have an atom and
you increase the laser power,

597
00:37:01,080 --> 00:37:02,470
you scatter light.

598
00:37:02,470 --> 00:37:06,770
And the scattered light, or
the absorbed light, saturates.

599
00:37:06,770 --> 00:37:09,700
But with the cross
section, we want

600
00:37:09,700 --> 00:37:13,060
to know what fraction of the
laser light is scattered.

601
00:37:13,060 --> 00:37:15,680
And the fraction of the
laser light scattered

602
00:37:15,680 --> 00:37:18,060
goes to zero, because
you make your laser

603
00:37:18,060 --> 00:37:19,610
light stronger and stronger.

604
00:37:19,610 --> 00:37:22,780
And the total amount of laser
light which is scattered

605
00:37:22,780 --> 00:37:24,340
saturates.

606
00:37:24,340 --> 00:37:26,530
So in other words,
you have a saturation

607
00:37:26,530 --> 00:37:27,600
of the scattered light.

608
00:37:27,600 --> 00:37:31,200
You have a saturation of
the net transfer of atoms

609
00:37:31,200 --> 00:37:35,130
through the excited state in the
limit of infinite laser power.

610
00:37:35,130 --> 00:37:38,500
But since the cross section is
sort of normalized by the laser

611
00:37:38,500 --> 00:37:42,490
power, the cross section has
this dependence, 1 over 1

612
00:37:42,490 --> 00:37:46,440
plus saturation
parameter, and goes to 0.

613
00:37:46,440 --> 00:37:48,490
And that means--
and this is sort

614
00:37:48,490 --> 00:37:52,210
of the language we use-- that
the transition bleaches out.

615
00:37:52,210 --> 00:37:54,300
If you saturation
the transition,

616
00:37:54,300 --> 00:37:58,120
the cross section
becomes smaller.

617
00:37:58,120 --> 00:38:00,980
So when you saturation the
transition in an absorption

618
00:38:00,980 --> 00:38:03,490
imaging experiment,
which many of you do,

619
00:38:03,490 --> 00:38:06,970
the shadow is less
and less black

620
00:38:06,970 --> 00:38:11,940
exactly because the cross
section is bleaching out.

621
00:38:11,940 --> 00:38:13,660
But the amount of
light you would scatter

622
00:38:13,660 --> 00:38:15,450
you would observe
in fluorescence

623
00:38:15,450 --> 00:38:17,510
is not getting
less, it saturates.

624
00:38:17,510 --> 00:38:20,960
This is sort of just the
two flip sides of the coin.

625
00:38:20,960 --> 00:38:23,045
If anybody is confused,
please ask a question.

626
00:38:29,420 --> 00:38:29,920
OK.

627
00:38:33,170 --> 00:38:37,100
So now in this picture,
we can immediately

628
00:38:37,100 --> 00:38:41,170
understand why we
have differences

629
00:38:41,170 --> 00:38:45,190
between monochromatic radiation
and broadband radiation.

630
00:38:45,190 --> 00:38:48,920
If you want to
saturation a transition

631
00:38:48,920 --> 00:38:54,650
with monochromatic radiation,
we have our narrow laser.

632
00:38:54,650 --> 00:38:58,790
We absorb with a cross section
6 pi lambda bar square.

633
00:38:58,790 --> 00:39:02,280
And we have to increase
the intensity of the laser

634
00:39:02,280 --> 00:39:07,534
until the excitation
rate equals gamma over 2.

635
00:39:07,534 --> 00:39:09,075
That's our definition
for saturation.

636
00:39:13,990 --> 00:39:17,730
So therefore, the
laser intensity

637
00:39:17,730 --> 00:39:19,610
is proportional to
gamma, because we

638
00:39:19,610 --> 00:39:22,760
have the cross
section is constant,

639
00:39:22,760 --> 00:39:25,560
but the product of cross
section and laser intensity

640
00:39:25,560 --> 00:39:29,210
has to be equal to gamma over 2.

641
00:39:29,210 --> 00:39:32,445
However, now consider
the case that you

642
00:39:32,445 --> 00:39:34,450
use broadband radiation.

643
00:39:34,450 --> 00:39:37,200
The spectrum is
completely broad.

644
00:39:37,200 --> 00:39:40,830
Now, if an atom has a
stronger transition,

645
00:39:40,830 --> 00:39:44,880
its cross section is
wider, and the atom

646
00:39:44,880 --> 00:39:49,480
can sort of absorb a wider
part of the incident spectrum.

647
00:39:49,480 --> 00:39:53,530
So therefore, if the atom
has a stronger transition,

648
00:39:53,530 --> 00:39:59,530
it automatically takes, absorbs
more of your spectral profile.

649
00:39:59,530 --> 00:40:05,130
And therefore, the
result for the saturation

650
00:40:05,130 --> 00:40:07,760
and for the spectral
saturation intensity

651
00:40:07,760 --> 00:40:10,230
is independent of
the matrix element

652
00:40:10,230 --> 00:40:11,730
and the strengths
of the transition.

653
00:40:14,560 --> 00:40:18,020
In general, if
you're not in either

654
00:40:18,020 --> 00:40:21,240
of the extreme cases
of monochromatic light

655
00:40:21,240 --> 00:40:23,220
or broadband light,
what you have to do

656
00:40:23,220 --> 00:40:25,750
is you have to use
this cross section

657
00:40:25,750 --> 00:40:28,300
as a function of
frequency, and convolve it

658
00:40:28,300 --> 00:40:31,640
to a convolution with a
spectrum of the incident light.

659
00:40:41,370 --> 00:40:43,200
And this is exactly done here.

660
00:40:43,200 --> 00:40:44,920
You take your
frequency-dependent cross

661
00:40:44,920 --> 00:40:46,890
section.

662
00:40:46,890 --> 00:40:52,110
You do the convolution with the
spectrum of the incident light.

663
00:40:52,110 --> 00:40:58,890
And if you assume the incident
light is spectrally very broad,

664
00:40:58,890 --> 00:41:08,110
you simply integrate over
the Lorentzian line shape

665
00:41:08,110 --> 00:41:09,940
of the cross section.

666
00:41:09,940 --> 00:41:12,750
And then you find
exactly the same result

667
00:41:12,750 --> 00:41:17,230
as we had two slides ago,
that the saturation intensity

668
00:41:17,230 --> 00:41:20,040
is independent of the
strengths of the transition.

669
00:41:25,140 --> 00:41:25,640
OK.

670
00:41:32,510 --> 00:41:36,490
Can you think of a
very intuitive argument

671
00:41:36,490 --> 00:41:45,320
why for spectrally
broad radiation,

672
00:41:45,320 --> 00:41:47,500
all the properties of
the atoms cancel out?

673
00:41:57,620 --> 00:42:02,540
If you think about one physical
example for, let's say,

674
00:42:02,540 --> 00:42:05,910
black body radiation--
this is spectrally broad.

675
00:42:05,910 --> 00:42:08,390
So you have an atom in
a black body cavity.

676
00:42:08,390 --> 00:42:11,535
And the atom experiences
a very broad spectrum.

677
00:42:15,870 --> 00:42:20,130
For what number of photons,
black body photons per mode

678
00:42:20,130 --> 00:42:22,960
would we find saturation?

679
00:42:22,960 --> 00:42:24,190
Think about it.

680
00:42:24,190 --> 00:42:27,370
It's a simple criterion you
can formulate for black body

681
00:42:27,370 --> 00:42:30,860
radiation to saturation
your transition in terms

682
00:42:30,860 --> 00:42:36,832
of the number of
photons per mode.

683
00:43:02,430 --> 00:43:05,000
You crank up the
temperature in your cavity.

684
00:43:05,000 --> 00:43:07,240
How high do you have to
go with the temperature

685
00:43:07,240 --> 00:43:14,140
in order to saturate an atom
which is inside your black body

686
00:43:14,140 --> 00:43:14,640
cavity?

687
00:43:14,640 --> 00:43:15,596
AUDIENCE: One photon.

688
00:43:22,766 --> 00:43:24,010
PROFESSOR: Pretty close.

689
00:43:24,010 --> 00:43:25,626
AUDIENCE: 1 over
[? degenerates. ?]

690
00:43:25,626 --> 00:43:27,170
PROFESSOR:
[? Degenerates. ?] OK.

691
00:43:27,170 --> 00:43:29,430
No [? degenerates. ?] I
hate [? degenerates. ?]

692
00:43:29,430 --> 00:43:32,070
That's your private homework
to put in [? degenerates ?]

693
00:43:32,070 --> 00:43:32,569
afterwards.

694
00:43:35,190 --> 00:43:38,100
The answer I came was
n equals 1/2, I think.

695
00:43:47,390 --> 00:43:49,465
I run the risk that I'm
off by a factor of 2 now.

696
00:43:49,465 --> 00:43:50,590
But the argument was that--

697
00:43:50,590 --> 00:43:55,739
AUDIENCE: The rate equals
[? degeneracy ?] by n by gamma.

698
00:43:55,739 --> 00:43:58,650
So if the rate equals
gamma over 1/2,

699
00:43:58,650 --> 00:44:02,400
that mean that [? degenerates ?]
by n equals 1/2.

700
00:44:05,830 --> 00:44:08,770
And if [? degenerates ?]
equals 1, n equals 1/2.

701
00:44:08,770 --> 00:44:09,445
PROFESSOR: Yes.

702
00:44:09,445 --> 00:44:10,090
OK.

703
00:44:10,090 --> 00:44:14,610
So spontaneous emission, we
know that spontaneous emission--

704
00:44:14,610 --> 00:44:17,510
from our derivation of
spontaneous emission--

705
00:44:17,510 --> 00:44:19,420
corresponds to one
photon per mode.

706
00:44:27,990 --> 00:44:30,910
And our criterion now is that we
want to have an absorption rate

707
00:44:30,910 --> 00:44:34,070
or stimulated rate
which is gamma over 2.

708
00:44:34,070 --> 00:44:37,640
So we get sort of 1/2 the effect
of spontaneous emission when

709
00:44:37,640 --> 00:44:39,010
we have 1/2 a photon per mode.

710
00:44:43,620 --> 00:44:51,200
So therefore, spontaneous
emission absorption

711
00:44:51,200 --> 00:44:53,560
is proportional to n.

712
00:44:53,560 --> 00:44:57,740
And I think if n
equals 1/2, then

713
00:44:57,740 --> 00:45:05,330
we have the unsaturated
rates equal to gamma over 2.

714
00:45:05,330 --> 00:45:07,430
So this is a very
physical argument.

715
00:45:07,430 --> 00:45:11,570
When we put an atom into
a black body cavity,

716
00:45:11,570 --> 00:45:14,870
and we have 1/2 a photon
per mode occupation number,

717
00:45:14,870 --> 00:45:18,440
then we saturate
any atom we put in.

718
00:45:18,440 --> 00:45:23,970
Because using Einstein's
argument, we have now the rate

719
00:45:23,970 --> 00:45:26,730
coefficient for
absorption emission

720
00:45:26,730 --> 00:45:29,660
for stimulated emission
and absorption is just 1/2

721
00:45:29,660 --> 00:45:31,865
of the rate coefficient
for spontaneous emission.

722
00:45:31,865 --> 00:45:34,320
And that explains that
all atomic properties

723
00:45:34,320 --> 00:45:35,150
have to cancel out.

724
00:45:40,010 --> 00:45:44,040
So now question for you.

725
00:45:44,040 --> 00:45:50,170
We talked about the fact that if
you have hyperfine transitions,

726
00:45:50,170 --> 00:45:52,980
that it would take--
what was the value?

727
00:45:52,980 --> 00:45:54,795
1,000 years for
spontaneous emission?

728
00:45:58,140 --> 00:46:01,370
So that we can completely
neglect spontaneous emission.

729
00:46:01,370 --> 00:46:05,427
On the other hand, we've just
learned that saturation only

730
00:46:05,427 --> 00:46:06,760
comes from spontaneous emission.

731
00:46:06,760 --> 00:46:09,540
Without spontaneous emission,
we wouldn't have saturation.

732
00:46:09,540 --> 00:46:13,170
But now I'm telling you
that any atom should really

733
00:46:13,170 --> 00:46:18,530
be saturated if we put
it in a black body cavity

734
00:46:18,530 --> 00:46:21,660
where n bar is 1/2.

735
00:46:21,660 --> 00:46:29,030
So what is the story now if we
put an atom into a black body

736
00:46:29,030 --> 00:46:31,890
cavity, and we are asking
about, will we saturate?

737
00:46:35,180 --> 00:46:37,990
The hyperfine transition.

738
00:46:37,990 --> 00:46:40,640
Will we eventually
have-- saturation

739
00:46:40,640 --> 00:46:44,780
means we have [BLOWS AIR], 1/4
of the atoms in the excited

740
00:46:44,780 --> 00:46:47,310
state, 3/4 in the ground state.

741
00:46:47,310 --> 00:46:52,410
So the delta n has been reduced
from 1, which it was initially,

742
00:46:52,410 --> 00:46:54,900
to 3/4 minus 1/4, which is 1/2.

743
00:46:57,580 --> 00:46:58,460
What will happen?

744
00:46:58,460 --> 00:47:01,610
I mean, this was almost like
a thermodynamic argument.

745
00:47:01,610 --> 00:47:06,910
Will we equilibrate and
saturate hyperfine transitions

746
00:47:06,910 --> 00:47:09,535
in a black body cavity
based on this argument

747
00:47:09,535 --> 00:47:13,360
that for n bar equals
1/2, we should really

748
00:47:13,360 --> 00:47:14,225
saturate everything?

749
00:47:20,312 --> 00:47:24,094
AUDIENCE: Yes, but it's
going to take a long time?

750
00:47:24,094 --> 00:47:24,760
PROFESSOR: Yeah.

751
00:47:24,760 --> 00:47:26,960
So for those conditions,
if your black body cavity

752
00:47:26,960 --> 00:47:32,040
was n bar equals 1/2, you should
saturate any two-level system

753
00:47:32,040 --> 00:47:35,010
completely independent
what gamma is.

754
00:47:35,010 --> 00:47:38,490
And if the gamma is 10
nanoseconds or 10,000 years,

755
00:47:38,490 --> 00:47:40,460
you will saturate it.

756
00:47:40,460 --> 00:47:42,560
The value of gamma
has completely

757
00:47:42,560 --> 00:47:44,310
dropped out of the argument.

758
00:47:44,310 --> 00:47:48,320
But of course, if you want to
reach any kind of equilibrium,

759
00:47:48,320 --> 00:47:51,140
it will take a time scale,
which is 1 over gamma.

760
00:47:51,140 --> 00:47:55,686
And then we are
back to 1,000 years.

761
00:47:55,686 --> 00:47:56,185
Questions?

762
00:48:04,550 --> 00:48:05,600
All right.

763
00:48:05,600 --> 00:48:26,110
Then let's conclude this
chapter and start our discussion

764
00:48:26,110 --> 00:48:28,135
about line shifts
and line broadening.

765
00:48:42,780 --> 00:48:45,050
I have a problem with
the tablet computer.

766
00:48:45,050 --> 00:48:48,840
I draw a line, but the computer
draws a line somewhere else.

767
00:48:48,840 --> 00:48:55,540
So maybe I should just go
back to this one and then copy

768
00:48:55,540 --> 00:48:56,070
things over.

769
00:49:00,490 --> 00:49:00,990
OK.

770
00:49:00,990 --> 00:49:03,665
Our next big chapter is
line shifts and broadening.

771
00:49:12,520 --> 00:49:16,780
So the first question
is motivational.

772
00:49:16,780 --> 00:49:21,940
Why should we be interested
in line broadening?

773
00:49:21,940 --> 00:49:25,010
Well, the answer
is almost trivial.

774
00:49:25,010 --> 00:49:28,010
No resonance is
infinitely narrow.

775
00:49:28,010 --> 00:49:31,290
Whenever we want to interpret
any result we obtain

776
00:49:31,290 --> 00:49:34,055
spectroscopically, we are not
observing a delta function,

777
00:49:34,055 --> 00:49:35,450
we are not observing
a resonance,

778
00:49:35,450 --> 00:49:37,420
we are observing a line shape.

779
00:49:37,420 --> 00:49:39,800
And unless we understand
the line shape,

780
00:49:39,800 --> 00:49:43,574
we may not accurately find
the resonance frequency.

781
00:49:43,574 --> 00:49:45,740
You could, of course, assume
that your line shape is

782
00:49:45,740 --> 00:49:50,190
symmetric, which may be the
case but is not always the case.

783
00:49:50,190 --> 00:49:54,300
So without understanding
line broadening,

784
00:49:54,300 --> 00:49:56,500
you cannot interpret
spectroscopic information.

785
00:50:01,420 --> 00:50:06,545
And eventually, as I
mentioned in the first chapter

786
00:50:06,545 --> 00:50:11,460
of this course, the art
of analyzing line shapes

787
00:50:11,460 --> 00:50:16,520
and finding the line center
is very well advanced.

788
00:50:19,910 --> 00:50:29,650
When we have caesium
fountain clocks,

789
00:50:29,650 --> 00:50:35,080
the accuracy how you operate the
clock as a frequency standard

790
00:50:35,080 --> 00:50:39,280
is on the order
of one microhertz.

791
00:50:39,280 --> 00:50:42,500
But those fountain
clocks with you toss up

792
00:50:42,500 --> 00:50:45,460
the atoms for one second
in the atomic fountain,

793
00:50:45,460 --> 00:50:47,720
they fall up and down,
well, like a rock,

794
00:50:47,720 --> 00:50:51,090
which takes about a second for a
rock to go up and down a meter.

795
00:50:51,090 --> 00:50:53,660
So therefore, the line width
is on the order of one Hertz.

796
00:50:56,580 --> 00:51:02,065
So therefore, people have to
understand any single aspect

797
00:51:02,065 --> 00:51:07,570
of the line shape at the
level of 10 to the minus 5,

798
00:51:07,570 --> 00:51:10,005
or 10 to the minus 6 to
have this kind of accuracy.

799
00:51:23,820 --> 00:51:26,550
OK.

800
00:51:26,550 --> 00:51:35,290
So I thought I want to start
this unit by collecting form

801
00:51:35,290 --> 00:51:46,050
you examples for phenomena which
cause broadening and shifting

802
00:51:46,050 --> 00:51:48,780
of lines.

803
00:51:48,780 --> 00:51:52,530
And well, my list
has about 10 of them.

804
00:51:52,530 --> 00:51:54,260
Let's see how many you get.

805
00:51:54,260 --> 00:52:00,310
So what phenomena can lead
to line shifts and line

806
00:52:00,310 --> 00:52:00,810
broadening?

807
00:52:03,750 --> 00:52:05,220
AUDIENCE: Phonons.

808
00:52:05,220 --> 00:52:07,580
PROFESSOR: Phonons?

809
00:52:07,580 --> 00:52:11,590
In terms of-- OK,
AC stark effect.

810
00:52:14,874 --> 00:52:15,374
Pardon?

811
00:52:15,374 --> 00:52:16,793
AUDIENCE: Magnetic field noise.

812
00:52:16,793 --> 00:52:19,610
PROFESSOR: Magnetic field noise.

813
00:52:19,610 --> 00:52:20,190
OK.

814
00:52:20,190 --> 00:52:26,830
I tried to-- yes, very good.

815
00:52:26,830 --> 00:52:28,564
OK, yes.

816
00:52:28,564 --> 00:52:30,230
Let me just try to
group it a little bit

817
00:52:30,230 --> 00:52:31,940
further, because I
want to discuss it.

818
00:52:31,940 --> 00:52:35,960
So we have external fields.

819
00:52:35,960 --> 00:52:43,730
And external fields can
have AC stark shifts.

820
00:52:47,910 --> 00:52:51,680
If an external field is noisy,
we have noise fluctuations.

821
00:52:54,280 --> 00:52:55,890
All right.

822
00:52:55,890 --> 00:52:57,450
Anything else?

823
00:52:57,450 --> 00:52:58,450
AUDIENCE: Doppler shift.

824
00:52:58,450 --> 00:53:00,920
PROFESSOR: Doppler shift.

825
00:53:00,920 --> 00:53:01,420
Yes.

826
00:53:01,420 --> 00:53:07,580
So we have the
velocity of the atoms.

827
00:53:13,560 --> 00:53:16,515
Doppler shift.

828
00:53:16,515 --> 00:53:17,425
AUDIENCE: Collisions.

829
00:53:17,425 --> 00:53:18,355
PROFESSOR: Collisions.

830
00:53:26,330 --> 00:53:26,830
Very good.

831
00:53:43,160 --> 00:53:44,829
Well, we just talked
about one thing.

832
00:53:44,829 --> 00:53:45,745
AUDIENCE: [INAUDIBLE].

833
00:53:45,745 --> 00:53:46,670
PROFESSOR: Exactly.

834
00:53:46,670 --> 00:53:48,840
When we have external
fields, we can

835
00:53:48,840 --> 00:53:51,070
have external fields
like magnetic fields

836
00:53:51,070 --> 00:53:54,100
or electric fields which
cause shift and broadening.

837
00:53:54,100 --> 00:53:56,160
And if there's noise,
additional shifts.

838
00:53:56,160 --> 00:54:01,400
But when we regard those
fields as drive fields,

839
00:54:01,400 --> 00:54:02,960
they can do power broadening.

840
00:54:11,540 --> 00:54:15,580
Maybe by collisions, I should
add the keyboard "pressure

841
00:54:15,580 --> 00:54:16,600
broadening."

842
00:54:16,600 --> 00:54:19,050
The higher the pressure
in your gas cell is,

843
00:54:19,050 --> 00:54:24,660
the more collisions you have and
the more you have broadening.

844
00:54:24,660 --> 00:54:25,410
Other suggestion?

845
00:54:33,200 --> 00:54:36,255
If you don't have
any of those effects,

846
00:54:36,255 --> 00:54:39,410
do you measure delta function?

847
00:54:39,410 --> 00:54:40,870
What's the line width?

848
00:54:40,870 --> 00:54:41,370
Will?

849
00:54:41,370 --> 00:54:42,380
STUDENT: [INAUDIBLE].

850
00:54:42,380 --> 00:54:43,630
STUDENT: Spontaneous emission.

851
00:54:43,630 --> 00:54:44,788
PROFESSOR: Spontaneous emission.

852
00:54:44,788 --> 00:54:45,288
Yes.

853
00:54:55,540 --> 00:54:59,185
And if you don't have
spontaneous emission,

854
00:54:59,185 --> 00:55:01,040
do we then measure
delta function?

855
00:55:04,442 --> 00:55:05,900
STUDENT: There's
a Fourier limit.

856
00:55:05,900 --> 00:55:07,680
PROFESSOR: The Fourier limit.

857
00:55:07,680 --> 00:55:10,290
You can call it
observation time,

858
00:55:10,290 --> 00:55:12,270
or time of light broadening.

859
00:55:12,270 --> 00:55:15,020
If an atom flies
through your laser beam

860
00:55:15,020 --> 00:55:17,910
and you can interrogate
it only for a finite time,

861
00:55:17,910 --> 00:55:20,990
you have a broadening due
to the Fourier theorem.

862
00:55:20,990 --> 00:55:37,830
And this can be called time of
flight broadening and time of--

863
00:55:37,830 --> 00:55:44,035
or interaction time broadening.

864
00:55:53,254 --> 00:55:54,733
STUDENT: Rotations
and vibrations?

865
00:55:54,733 --> 00:55:58,260
PROFESSOR: Rotations
and vibrations.

866
00:55:58,260 --> 00:55:58,760
Not really.

867
00:55:58,760 --> 00:56:01,650
These are more-- then the
system has more energy levels,

868
00:56:01,650 --> 00:56:03,550
and that's what you
want to find out.

869
00:56:03,550 --> 00:56:06,599
So maybe I'm more asking,
how are those energy levels--

870
00:56:06,599 --> 00:56:08,140
how do they appear
spectroscopically?

871
00:56:13,120 --> 00:56:16,610
Well, I think that's
pretty complete.

872
00:56:16,610 --> 00:56:17,930
Two external fields.

873
00:56:17,930 --> 00:56:20,260
If you want, you
can add gravity.

874
00:56:20,260 --> 00:56:24,750
There is a
gravitational red shift,

875
00:56:24,750 --> 00:56:27,840
which is general relativity.

876
00:56:27,840 --> 00:56:34,810
But anyway, let me look over
that and try to categorize it.

877
00:56:34,810 --> 00:56:41,440
What we had here actually all
comes from a finite observation

878
00:56:41,440 --> 00:56:41,940
time.

879
00:56:46,020 --> 00:56:50,380
Either we do not have the atom
long enough in our laser beam,

880
00:56:50,380 --> 00:56:51,620
and that sets a limit.

881
00:56:51,620 --> 00:56:55,260
Or if you are interested in an
excited state and the excited

882
00:56:55,260 --> 00:56:58,200
state decays, then
the atoms themselves

883
00:56:58,200 --> 00:57:00,500
have terminated our
interrogation time.

884
00:57:03,440 --> 00:57:06,110
The second class
here, velocity, I

885
00:57:06,110 --> 00:57:10,370
would summarize that we
have motion of the atom.

886
00:57:10,370 --> 00:57:16,590
It's a form of
motional broadening.

887
00:57:20,000 --> 00:57:25,810
We will actually discuss,
when we discuss motion, also

888
00:57:25,810 --> 00:57:29,160
the possibility of having
atoms in a harmonic oscillator

889
00:57:29,160 --> 00:57:31,830
potential, ions in an ion trap.

890
00:57:31,830 --> 00:57:35,600
So these are now
trapped particles.

891
00:57:35,600 --> 00:57:41,640
This will actually often give
rise to a splitting of the line

892
00:57:41,640 --> 00:57:44,990
into side bends.

893
00:57:44,990 --> 00:57:46,370
So we want to discuss that.

894
00:57:55,930 --> 00:58:00,975
I've already mentioned external
fields, conditional [INAUDIBLE]

895
00:58:00,975 --> 00:58:03,480
interrogation, power broadening.

896
00:58:03,480 --> 00:58:06,320
Some power broadening
will actually

897
00:58:06,320 --> 00:58:12,700
result into a splitting of
line into [INAUDIBLE] triplet.

898
00:58:12,700 --> 00:58:14,790
So power will not
only broaden the line,

899
00:58:14,790 --> 00:58:16,610
it can also split the line.

900
00:58:16,610 --> 00:58:19,820
And we want to discuss that.

901
00:58:19,820 --> 00:58:32,015
And finally, we have the
effect of atomic interactions.

902
00:58:38,130 --> 00:58:43,720
So for interactions, I think
we should add something

903
00:58:43,720 --> 00:58:49,960
like mean field shifts,
which also goes sometimes

904
00:58:49,960 --> 00:58:51,370
by the name of clock shift.

905
00:58:51,370 --> 00:58:54,820
If you're not at zero
density, your transition

906
00:58:54,820 --> 00:58:59,320
can be shifted by the
presence of other atoms.

907
00:58:59,320 --> 00:58:59,820
Will?

908
00:58:59,820 --> 00:59:02,280
STUDENT: Isn't collisional
broadening or pressure

909
00:59:02,280 --> 00:59:04,248
broadening sort of
just an ensemble

910
00:59:04,248 --> 00:59:05,724
average of a stark effect?

911
00:59:05,724 --> 00:59:07,692
So that's sort of
an external field?

912
00:59:12,612 --> 00:59:15,040
PROFESSOR: That depends now.

913
00:59:15,040 --> 00:59:20,170
Collisions is one of the richest
phenomena on the list here.

914
00:59:23,559 --> 00:59:24,350
You're ahead of me.

915
00:59:24,350 --> 00:59:26,610
But in the next few minutes,
I wanted to actually

916
00:59:26,610 --> 00:59:32,320
see, well, maybe we should--
those categories are not

917
00:59:32,320 --> 00:59:36,100
mutually exclusive, because
one part of collisions is.

918
00:59:36,100 --> 00:59:38,830
An atom is in the excited
state, it collides,

919
00:59:38,830 --> 00:59:40,370
it gets de-excited.

920
00:59:40,370 --> 00:59:43,850
So then collisions
have no other effect

921
00:59:43,850 --> 00:59:47,140
than sort of give us a
finite observation time,

922
00:59:47,140 --> 00:59:51,027
where there is an effective
lifetime, which is just

923
00:59:51,027 --> 00:59:52,360
the time between two collisions.

924
00:59:52,360 --> 00:59:54,370
So it can be this.

925
00:59:54,370 --> 00:59:56,210
There's another
aspect of collisions,

926
00:59:56,210 --> 00:59:58,280
that every time
there is a collision,

927
00:59:58,280 --> 01:00:01,800
an atom feels the electric
field of another atom.

928
01:00:01,800 --> 01:00:07,480
And then we have some form
of collisional broadening,

929
01:00:07,480 --> 01:00:13,090
because we do some statistical
averaging over stark effects,

930
01:00:13,090 --> 01:00:16,280
over level shifts.

931
01:00:16,280 --> 01:00:20,020
Now, there is a third
aspect of collisions,

932
01:00:20,020 --> 01:00:23,120
which is maybe surprising
to many of you.

933
01:00:23,120 --> 01:00:28,900
And this is actually-- I
put it here under motion.

934
01:00:28,900 --> 01:00:33,118
It is collisional narrowing,
or [? diche ?] narrowing.

935
01:00:40,420 --> 01:00:45,110
There's one limiting case
when you have collisions,

936
01:00:45,110 --> 01:00:48,605
that collisions lead to a
narrower line and not only

937
01:00:48,605 --> 01:00:50,334
to a broader line.

938
01:00:53,180 --> 01:00:56,910
the reason is a little bit-- if
you put an atom in a buffer gas

939
01:00:56,910 --> 01:00:59,250
and it collides with
all the buffer gas atom,

940
01:00:59,250 --> 01:01:00,450
it cannot fly away.

941
01:01:03,080 --> 01:01:06,090
So buffer gas and
collisions can sort of

942
01:01:06,090 --> 01:01:10,280
help to increase your
observation time.

943
01:01:10,280 --> 01:01:14,370
But only if the other effects
of collisions are absent.

944
01:01:14,370 --> 01:01:16,600
So anyway, I thought
this is a number

945
01:01:16,600 --> 01:01:18,200
of really interesting effects.

946
01:01:18,200 --> 01:01:21,530
And you already see from my
presentation and discussion

947
01:01:21,530 --> 01:01:26,050
that it makes perfect sense to
discuss them not one by one,

948
01:01:26,050 --> 01:01:28,050
as they appear in
other chapters,

949
01:01:28,050 --> 01:01:30,880
but try to have comprehensive
discussions of those.

950
01:01:34,710 --> 01:01:44,630
Let me talk about one
classification of those shifts

951
01:01:44,630 --> 01:01:46,630
and broadening.

952
01:01:46,630 --> 01:02:01,580
And one is the distinction
between homogeneous

953
01:02:01,580 --> 01:02:10,580
and inhomogeneous broadening.

954
01:02:10,580 --> 01:02:15,390
So the picture
here is that if you

955
01:02:15,390 --> 01:02:19,210
have-- let me just give
you the cartoon picture.

956
01:02:19,210 --> 01:02:27,710
If you have different atoms,
atom 1, atom 2, atom 3.

957
01:02:27,710 --> 01:02:29,650
A homogeneous
broadening situation

958
01:02:29,650 --> 01:02:34,020
is if the line
has been broadened

959
01:02:34,020 --> 01:02:36,710
for each atom in the same way.

960
01:02:36,710 --> 01:02:39,350
An inhomogeneous
broadening situation

961
01:02:39,350 --> 01:02:45,600
is that atom 1 has a line
here, atom 2 has a line here,

962
01:02:45,600 --> 01:02:47,710
atom 3 has a line there.

963
01:02:47,710 --> 01:02:50,940
And if you look at the
statistical ensemble,

964
01:02:50,940 --> 01:02:53,940
you may find the
same line widths

965
01:02:53,940 --> 01:02:59,530
as on the left-hand side, but
the situation and the mechanism

966
01:02:59,530 --> 01:03:01,058
is a very different one.

967
01:03:05,180 --> 01:03:11,690
So the different characteristics
are that here, we

968
01:03:11,690 --> 01:03:15,290
have a mechanism which
broadens or widens

969
01:03:15,290 --> 01:03:22,200
the line for each atom.

970
01:03:22,200 --> 01:03:25,890
Whereas here, there is
maybe not even any line

971
01:03:25,890 --> 01:03:27,450
broadening for the atom.

972
01:03:27,450 --> 01:03:38,280
It's more a random shift
to individual atoms.

973
01:03:38,280 --> 01:03:46,030
And the widening happens
for the ensemble.

974
01:03:54,240 --> 01:03:56,340
Another very
important distinction

975
01:03:56,340 --> 01:04:01,420
is in the left case, if you
have one powerful laser,

976
01:04:01,420 --> 01:04:03,800
it can talk to all the atoms.

977
01:04:03,800 --> 01:04:05,610
Whereas in the
right-hand side, you

978
01:04:05,610 --> 01:04:07,940
may have a laser with
a certain frequency,

979
01:04:07,940 --> 01:04:12,370
and it may only excite one
group of atoms in your ensemble.

980
01:04:30,010 --> 01:04:31,790
So this is the opposite here.

981
01:04:35,450 --> 01:04:40,505
In many situations do we have
a physical picture where,

982
01:04:40,505 --> 01:04:43,870
in homogeneous broadening,
we can understand it

983
01:04:43,870 --> 01:04:49,390
as random interruptions of the
phase's evolution of the atom.

984
01:04:49,390 --> 01:04:52,620
For instance, through
spontaneous emission,

985
01:04:52,620 --> 01:04:56,300
or you can see certain
collisions-- just mean

986
01:04:56,300 --> 01:04:59,730
the phase of the excited
state is suddenly perturbed.

987
01:04:59,730 --> 01:05:03,060
And therefore, the
phase is randomized.

988
01:05:03,060 --> 01:05:12,250
So if the physical picture
is a random interruption

989
01:05:12,250 --> 01:05:20,490
of the phase evolution,
well, a random interruption

990
01:05:20,490 --> 01:05:23,300
of a phase evolution
means that there

991
01:05:23,300 --> 01:05:25,500
is an exponential
decay of coherence.

992
01:05:32,010 --> 01:05:34,510
And the line shape,
the Fourier transform

993
01:05:34,510 --> 01:05:36,940
of an exponential
decay is a Lorentzian.

994
01:05:39,790 --> 01:05:43,890
Whereas the physical picture
behind inhomogeneous broadening

995
01:05:43,890 --> 01:05:46,390
is that you have
random perturbations.

996
01:05:49,490 --> 01:05:52,630
And if you have many random
or small perturbations,

997
01:05:52,630 --> 01:05:54,870
they often follow a
normal distribution,

998
01:05:54,870 --> 01:05:56,092
which is a Gaussian.

999
01:06:09,710 --> 01:06:13,290
There's one other aspect of
an inhomogeneous broadening.

1000
01:06:13,290 --> 01:06:15,020
If it's an inhomogeneous
broadening,

1001
01:06:15,020 --> 01:06:18,390
it is as if the individual
atom is not broadened,

1002
01:06:18,390 --> 01:06:20,520
the individual atom
is actually sharp,

1003
01:06:20,520 --> 01:06:22,290
it has a longer coherence.

1004
01:06:22,290 --> 01:06:25,170
And you can-- there
are techniques to make

1005
01:06:25,170 --> 01:06:26,400
that visible.

1006
01:06:26,400 --> 01:06:28,920
And one famous technique,
for those of you

1007
01:06:28,920 --> 01:06:31,570
who have heard about it,
are an echo technique.

1008
01:06:37,240 --> 01:06:46,660
So having explained to
you in a general way

1009
01:06:46,660 --> 01:06:49,270
the difference between
inhomogeneous and homogeneous

1010
01:06:49,270 --> 01:06:55,570
broadening, how would you
classify the line broadening

1011
01:06:55,570 --> 01:06:57,745
mechanisms we have
collected before?

1012
01:07:02,185 --> 01:07:03,810
Which one are
inhomogeneous broadening?

1013
01:07:08,714 --> 01:07:09,880
STUDENT: Doppler broadening.

1014
01:07:09,880 --> 01:07:11,450
PROFESSOR: Doppler broadening.

1015
01:07:11,450 --> 01:07:14,450
We exploit that when we
do saturation spectroscopy

1016
01:07:14,450 --> 01:07:17,900
in the lab, when we just talk
to one component of the velocity

1017
01:07:17,900 --> 01:07:20,430
distribution.

1018
01:07:20,430 --> 01:07:21,850
What else?

1019
01:07:21,850 --> 01:07:22,830
STUDENT: [INAUDIBLE].

1020
01:07:29,200 --> 01:07:30,117
STUDENT: Collisions.

1021
01:07:30,117 --> 01:07:31,033
PROFESSOR: Collisions.

1022
01:07:34,110 --> 01:07:36,720
That's actually a good one.

1023
01:07:36,720 --> 01:07:38,840
Usually, collisions
are classified

1024
01:07:38,840 --> 01:07:41,340
as homogeneous broadening,
because the simplest

1025
01:07:41,340 --> 01:07:43,500
model for collisions
is collisions

1026
01:07:43,500 --> 01:07:46,950
are sort of just hard-core
collisions which just de-excite

1027
01:07:46,950 --> 01:07:50,400
the atom, completely change
the coherent phase evolution.

1028
01:07:50,400 --> 01:07:54,970
And therefore, collisions
would broaden the transition

1029
01:07:54,970 --> 01:08:00,000
for all atoms to a
line widths which

1030
01:08:00,000 --> 01:08:02,080
is 1 over the collision rates.

1031
01:08:02,080 --> 01:08:05,610
However-- and this shows that
the distinction cannot always

1032
01:08:05,610 --> 01:08:09,630
be made-- you can actually have
collision rate which depends

1033
01:08:09,630 --> 01:08:10,700
on the velocity.

1034
01:08:10,700 --> 01:08:13,400
The faster atoms may have
a smaller collision cross

1035
01:08:13,400 --> 01:08:15,810
section than the slower atoms.

1036
01:08:15,810 --> 01:08:19,350
And now you have an
inhomogeneous aspect

1037
01:08:19,350 --> 01:08:20,350
of the collision rate.

1038
01:08:20,350 --> 01:08:22,516
And therefore, collision
rate becomes inhomogeneous.

1039
01:08:25,960 --> 01:08:28,649
I mean, the standard example
for inhomogeneous fields

1040
01:08:28,649 --> 01:08:33,270
would-- if you have an
inhomogeneous magnetic field,

1041
01:08:33,270 --> 01:08:37,170
you have stationary atoms--
well, not in an atomic gas,

1042
01:08:37,170 --> 01:08:39,680
but maybe in [INAUDIBLE]
or in a solid.

1043
01:08:39,680 --> 01:08:43,034
And you have an
inhomogeneous magnetic field.

1044
01:08:43,034 --> 01:08:44,450
This is actually
the standard case

1045
01:08:44,450 --> 01:08:47,140
of nuclear magnetic
resonance, that each atom

1046
01:08:47,140 --> 01:08:49,630
possesses at its
local magnetic field.

1047
01:08:49,630 --> 01:08:53,130
And the line shape is
inhomogeneously broadened.

1048
01:08:53,130 --> 01:08:53,629
Colin?

1049
01:08:53,629 --> 01:08:56,885
STUDENT: [INAUDIBLE]
clock shift sometimes,

1050
01:08:56,885 --> 01:08:59,265
in some circumstances.

1051
01:08:59,265 --> 01:09:01,370
PROFESSOR: If the
density is constant,

1052
01:09:01,370 --> 01:09:03,130
you would actually
say the mean field is

1053
01:09:03,130 --> 01:09:05,680
the same for all
atoms in the ensemble.

1054
01:09:05,680 --> 01:09:07,620
But if you have a
trapped atom sample

1055
01:09:07,620 --> 01:09:10,430
where the density
drops at the edge,

1056
01:09:10,430 --> 01:09:12,670
you may actually
have a sharper line,

1057
01:09:12,670 --> 01:09:17,529
and less broadening, or less
shift at the edge of the cloud.

1058
01:09:17,529 --> 01:09:20,760
Anyway, so I think you have
all the tools to classify it.

1059
01:09:20,760 --> 01:09:23,510
And you see from the
discussion that sometimes it's

1060
01:09:23,510 --> 01:09:25,859
not so obvious.

1061
01:09:25,859 --> 01:09:27,560
Or you may have
a mechanism which

1062
01:09:27,560 --> 01:09:30,649
has [? both ?] inhomogeneous
that it does something

1063
01:09:30,649 --> 01:09:31,979
to all atoms.

1064
01:09:31,979 --> 01:09:34,890
So for instance, collisions
broaden all the atoms,

1065
01:09:34,890 --> 01:09:37,540
but then different atoms are
more broadened than others.

1066
01:09:37,540 --> 01:09:39,570
So there may be also an
inhomogeneous aspect.

1067
01:09:44,350 --> 01:09:50,990
But finally, let me
ask you the following.

1068
01:09:50,990 --> 01:09:56,890
It seems the first
items on our list

1069
01:09:56,890 --> 01:09:59,040
had sort of a very
natural explanation

1070
01:09:59,040 --> 01:10:03,660
in terms of the Fourier theorem,
that, well, we only talk

1071
01:10:03,660 --> 01:10:06,400
to the atoms for a finite time.

1072
01:10:06,400 --> 01:10:11,100
Or the atoms decide not
to talk to us for longer,

1073
01:10:11,100 --> 01:10:14,000
because they
spontaneously decay.

1074
01:10:14,000 --> 01:10:18,500
Now, maybe you want to
give me some arguments why

1075
01:10:18,500 --> 01:10:22,160
some of the other
mechanisms are actually

1076
01:10:22,160 --> 01:10:31,904
also due to some form of
finite time of interrogation.

1077
01:10:46,530 --> 01:10:51,410
Well, if I would say,
can we regard collisions

1078
01:10:51,410 --> 01:10:55,820
as an effect of finite
observation time?

1079
01:10:55,820 --> 01:11:00,370
Well, if I rephrase "observation
time" to "finite coherence

1080
01:11:00,370 --> 01:11:03,950
time," that something interrupts
the coherent evolution

1081
01:11:03,950 --> 01:11:05,830
of the wave function,
I think we would

1082
01:11:05,830 --> 01:11:09,690
say the collision time sets
a time limit to the coherence

1083
01:11:09,690 --> 01:11:12,750
time and therefore,
should also be regarded

1084
01:11:12,750 --> 01:11:15,679
as due to the
finite time, we can

1085
01:11:15,679 --> 01:11:17,220
drive the atom in
a coherent fashion.

1086
01:11:23,390 --> 01:11:29,340
If I take power
broadening, we just

1087
01:11:29,340 --> 01:11:30,726
discussed power broadening.

1088
01:11:33,340 --> 01:11:41,090
Well, what is the rate--
or 1 over the rate

1089
01:11:41,090 --> 01:11:42,016
of power broadening?

1090
01:11:47,850 --> 01:11:51,870
We just discussed
that that's maybe

1091
01:11:51,870 --> 01:11:53,620
nice to take it
out of the context.

1092
01:11:53,620 --> 01:11:56,530
We discussed before that
power-broadened line widths

1093
01:11:56,530 --> 01:12:00,610
is gamma over 2 times S
plus 1 square root of it--

1094
01:12:00,610 --> 01:12:02,270
the saturation parameter.

1095
01:12:02,270 --> 01:12:04,990
But when does power
broadening happening?

1096
01:12:04,990 --> 01:12:10,761
And what is the real time scale
for what is the physical--

1097
01:12:10,761 --> 01:12:12,132
STUDENT: Spontaneous emission.

1098
01:12:12,132 --> 01:12:15,650
PROFESSOR: Spontaneous--
so we had a criterion

1099
01:12:15,650 --> 01:12:21,100
that the unsaturated rate has
to be comparable to gamma.

1100
01:12:21,100 --> 01:12:22,940
Let's forget about
factors of 2 now.

1101
01:12:22,940 --> 01:12:26,160
But that means that
the Rabi frequency

1102
01:12:26,160 --> 01:12:28,610
has to be comparable to gamma.

1103
01:12:28,610 --> 01:12:32,840
The Rabi frequency tells
us a time of Rabi flopping.

1104
01:12:32,840 --> 01:12:35,440
So actually, power
broadening can

1105
01:12:35,440 --> 01:12:40,090
be understood as a finite
observation time broadening,

1106
01:12:40,090 --> 01:12:45,520
but the atom is leaving
the excited state

1107
01:12:45,520 --> 01:12:49,130
not by spontaneous emission,
but by stimulated emission.

1108
01:12:49,130 --> 01:12:51,950
So in other words,
stimulated emission

1109
01:12:51,950 --> 01:12:55,220
interrupts our ability
to observe the atoms

1110
01:12:55,220 --> 01:12:56,760
in the excited state.

1111
01:12:56,760 --> 01:13:00,730
And so again, we see that
there is a process coming

1112
01:13:00,730 --> 01:13:04,180
in which interrupts
our observation

1113
01:13:04,180 --> 01:13:06,603
time of the unperturbed
atomic levels.

1114
01:13:11,540 --> 01:13:14,510
Well, let me go
one step further.

1115
01:13:14,510 --> 01:13:18,390
Let me ask you, do
you have any idea

1116
01:13:18,390 --> 01:13:23,890
how we could discuss
the Doppler shift as

1117
01:13:23,890 --> 01:13:26,518
due to some finite time scale?

1118
01:13:30,190 --> 01:13:32,780
You would say, well, yeah,
that's dimensional analysis.

1119
01:13:32,780 --> 01:13:35,670
If you have a broadening, a
broadening is a frequency,

1120
01:13:35,670 --> 01:13:37,940
1 over the frequency
is the time,

1121
01:13:37,940 --> 01:13:40,020
and there is a time
scale associated

1122
01:13:40,020 --> 01:13:42,050
with Doppler broadening.

1123
01:13:42,050 --> 01:13:42,560
Sure.

1124
01:13:42,560 --> 01:13:44,690
But now my question is,
what is the physical time

1125
01:13:44,690 --> 01:13:46,074
scale with Doppler broadening?

1126
01:13:54,690 --> 01:13:55,190
Yes?

1127
01:13:55,190 --> 01:13:56,838
STUDENT: [? Collisions. ?]

1128
01:13:56,838 --> 01:13:59,030
PROFESSOR: No.

1129
01:13:59,030 --> 01:14:01,230
We have an ideal gas without
any collisions-- just

1130
01:14:01,230 --> 01:14:02,659
a [INAUDIBLE] distribution.

1131
01:14:02,659 --> 01:14:03,200
You're right.

1132
01:14:03,200 --> 01:14:04,640
In practice, yes.

1133
01:14:04,640 --> 01:14:07,470
But I try to create an
idealized situation.

1134
01:14:07,470 --> 01:14:12,667
So what is the time scale
of Doppler broadening?

1135
01:14:12,667 --> 01:14:14,250
You may have never
heard the question.

1136
01:14:14,250 --> 01:14:16,960
But this is for me, what
I want to really teach you

1137
01:14:16,960 --> 01:14:20,377
when I teach all these
different line shifts

1138
01:14:20,377 --> 01:14:21,210
and line broadening.

1139
01:14:21,210 --> 01:14:22,460
There is a common denominator.

1140
01:14:22,460 --> 01:14:25,930
STUDENT: You could think of
the atom [INAUDIBLE] emission.

1141
01:14:25,930 --> 01:14:29,360
And then you would have
velocity [INAUDIBLE] emission.

1142
01:14:41,120 --> 01:14:43,360
PROFESSOR: You're talking
about recoil shifts,

1143
01:14:43,360 --> 01:14:46,765
and the atom is changing
its velocity due to recoil.

1144
01:14:51,020 --> 01:14:52,790
This would something
in addition,

1145
01:14:52,790 --> 01:14:55,880
but it's not necessarily
the case here.

1146
01:14:59,310 --> 01:15:00,760
I give you a physical argument.

1147
01:15:00,760 --> 01:15:03,720
If I make the atom heavier
and heavier and heavier,

1148
01:15:03,720 --> 01:15:05,980
the effect of the recoil vanish.

1149
01:15:05,980 --> 01:15:08,110
But then I can heat
up the heavier atom,

1150
01:15:08,110 --> 01:15:10,870
that it moves with the same
velocity as the slow atom.

1151
01:15:10,870 --> 01:15:13,454
So there is an effect which
you can associate just

1152
01:15:13,454 --> 01:15:15,495
to the velocity and to
the velocity distribution.

1153
01:15:15,495 --> 01:15:17,710
And that's what I
want to discuss now.

1154
01:15:17,710 --> 01:15:20,050
But there is another
effect with the recoil.

1155
01:15:20,050 --> 01:15:23,000
But I can say the
recoil is a finite mass

1156
01:15:23,000 --> 01:15:24,540
effect, for that purpose.

1157
01:15:24,540 --> 01:15:27,460
The mass is sort of my
handle, whether the recoil

1158
01:15:27,460 --> 01:15:29,628
of a singular photon
is important or not.

1159
01:15:29,628 --> 01:15:30,128
Yes?

1160
01:15:30,128 --> 01:15:31,076
STUDENT: [INAUDIBLE]?

1161
01:15:35,816 --> 01:15:37,780
PROFESSOR: Yes,
but this is really

1162
01:15:37,780 --> 01:15:39,687
a more trivial finite
observation time.

1163
01:15:39,687 --> 01:15:41,270
When you heat the
wall of the chamber,

1164
01:15:41,270 --> 01:15:43,100
it's a collision
with the chamber.

1165
01:15:43,100 --> 01:15:47,290
It means we have only a
finite interaction size.

1166
01:15:47,290 --> 01:15:49,170
Now, let me sort of
guide you to that.

1167
01:15:49,170 --> 01:15:51,850
The secret here is
when we say, you

1168
01:15:51,850 --> 01:15:54,820
have a finite lifetime, a
finite observation time,

1169
01:15:54,820 --> 01:15:57,070
what matters when
we do spectroscopy

1170
01:15:57,070 --> 01:16:01,690
is the time we can observe
the atoms coherently.

1171
01:16:01,690 --> 01:16:05,420
If the atoms de-phase,
if the atoms get out

1172
01:16:05,420 --> 01:16:08,480
of coherence-- for instance,
if you have collisions--

1173
01:16:08,480 --> 01:16:10,850
if collision de-excite
the atom-- we'll

1174
01:16:10,850 --> 01:16:13,520
talk about it later-- it's
like spontaneous emission.

1175
01:16:13,520 --> 01:16:15,300
But then there are
collisions which

1176
01:16:15,300 --> 01:16:18,040
just create a phase
hiccup, that the excited

1177
01:16:18,040 --> 01:16:20,030
state gets a random phase.

1178
01:16:20,030 --> 01:16:23,230
So an interruption of the
phase, an interruption

1179
01:16:23,230 --> 01:16:26,790
of the coherent
evolution is, in effect,

1180
01:16:26,790 --> 01:16:32,070
an interruption of us probing
the atoms in a coherent way.

1181
01:16:32,070 --> 01:16:35,420
And then the Fourier transform
just tells us, this time, or 1

1182
01:16:35,420 --> 01:16:38,680
over this time, is the
line which we observe.

1183
01:16:38,680 --> 01:16:41,110
And you would say,
but how does it

1184
01:16:41,110 --> 01:16:44,000
come into play with atoms
with a velocity distribution?

1185
01:16:44,000 --> 01:16:45,570
In the following way.

1186
01:16:45,570 --> 01:16:48,980
If you line up several atoms
and they interact with a laser

1187
01:16:48,980 --> 01:16:53,616
beam, some atoms are faster,
some atoms are slower.

1188
01:16:53,616 --> 01:16:57,600
If some of the atoms have
moved compared to the slower

1189
01:16:57,600 --> 01:17:00,910
atoms, one additional
wavelength,

1190
01:17:00,910 --> 01:17:04,110
then your ensemble
of atoms is no longer

1191
01:17:04,110 --> 01:17:07,190
interacting with the laser
beam in a phase-coherent way.

1192
01:17:07,190 --> 01:17:10,950
Because of the
different velocities,

1193
01:17:10,950 --> 01:17:13,386
they are now talking to
random phases of the laser.

1194
01:17:17,580 --> 01:17:19,820
So therefore,
Doppler broadening is

1195
01:17:19,820 --> 01:17:23,200
nothing else as a
loss of the atoms

1196
01:17:23,200 --> 01:17:27,070
to coherently interact with
a laser, because some of them

1197
01:17:27,070 --> 01:17:31,235
have moved an additional
wavelength in the laser beam.

1198
01:17:34,880 --> 01:17:38,920
Well, if that is true-- but what
happens if the laser beam is

1199
01:17:38,920 --> 01:17:43,510
like this, with the wavelengths,
and the atoms go perpendicular?

1200
01:17:43,510 --> 01:17:44,320
What happens then?

1201
01:17:47,044 --> 01:17:48,210
STUDENT: There's no Doppler.

1202
01:17:48,210 --> 01:17:50,780
PROFESSOR: Then there
is no Doppler effect.

1203
01:17:50,780 --> 01:17:52,710
So what I'm saying
is fully consistent

1204
01:17:52,710 --> 01:17:55,550
with every single thing you
know about the Doppler effect.

1205
01:18:04,240 --> 01:18:04,740
OK.

1206
01:18:09,270 --> 01:18:14,630
So I think there's not
much more we can do today.

1207
01:18:14,630 --> 01:18:18,070
But let me give you the
summary of this discussion.

1208
01:18:18,070 --> 01:18:31,130
To the best of my knowledge,
all line broadening mechanisms

1209
01:18:31,130 --> 01:18:43,270
can be described by using the
concept of coherence time.

1210
01:18:43,270 --> 01:18:46,645
And it's a coherence time
of a correlation function.

1211
01:18:49,777 --> 01:18:51,485
It's pretty much the
correlation function

1212
01:18:51,485 --> 01:18:54,640
of the phase which
the atom experiences.

1213
01:18:54,640 --> 01:18:58,590
At t equals 0, it experiences
one phase of your drive field.

1214
01:18:58,590 --> 01:19:02,581
And a later time, how
long does it stay coherent

1215
01:19:02,581 --> 01:19:04,830
with the coherent evolution
of the phase of your drive

1216
01:19:04,830 --> 01:19:06,505
field of a correlation function?

1217
01:19:09,710 --> 01:19:14,510
However, in the case
of inhomogeneous

1218
01:19:14,510 --> 01:19:19,410
broadening-- and
this is what I just

1219
01:19:19,410 --> 01:19:21,140
discussed with the
different atoms

1220
01:19:21,140 --> 01:19:24,240
starting together and
having different velocities.

1221
01:19:24,240 --> 01:19:32,090
In the case of
inhomogeneous broadening,

1222
01:19:32,090 --> 01:19:35,260
I have to include in the
description of the correlation

1223
01:19:35,260 --> 01:19:36,819
function ensemble averaging.

1224
01:19:47,300 --> 01:19:49,070
So this is our agenda.

1225
01:19:49,070 --> 01:19:55,070
On Wednesday, I will
start to discuss with you

1226
01:19:55,070 --> 01:19:57,010
very simple cases.

1227
01:19:57,010 --> 01:19:59,900
I sort of like, before I
introduce correlation function,

1228
01:19:59,900 --> 01:20:01,590
we have the
generalized discussion

1229
01:20:01,590 --> 01:20:05,950
to summarize for you the
phenomenological description

1230
01:20:05,950 --> 01:20:11,810
of just Rabi resonance, Ramsey
resonance, exponential decay,

1231
01:20:11,810 --> 01:20:14,630
simple Doppler broadening,
the recoil effect,

1232
01:20:14,630 --> 01:20:16,570
that you have a clear
physical picture of what

1233
01:20:16,570 --> 01:20:18,300
the different phenomena are.

1234
01:20:18,300 --> 01:20:22,278
And then we describe them
with a common language,

1235
01:20:22,278 --> 01:20:23,652
with a common
formalism, which is

1236
01:20:23,652 --> 01:20:26,147
a formalism of
correlation functions.

1237
01:20:26,147 --> 01:20:26,730
Any questions?

1238
01:20:32,950 --> 01:20:34,790
One obvious
question-- the chapter

1239
01:20:34,790 --> 01:20:39,390
on line shifts and broadening
will not be on the mid-term.

1240
01:20:39,390 --> 01:20:39,890
OK.

1241
01:20:39,890 --> 01:20:41,740
See you on Wednesday.