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PROFESSOR: OK.

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00:00:21,160 --> 00:00:27,650
So what we are talking about
is actually matrix elements.

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00:00:27,650 --> 00:00:33,060
If you want to do anything
interesting in atomic physics,

11
00:00:33,060 --> 00:00:36,990
you have to copy or induce
transitions from one state

12
00:00:36,990 --> 00:00:37,900
to another state.

13
00:00:43,232 --> 00:00:45,495
Well, maybe that should be Hab.

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00:00:49,320 --> 00:00:51,370
For many phenomenon,
which we will

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00:00:51,370 --> 00:00:55,740
cover throughout the
rest of this semester,

16
00:00:55,740 --> 00:00:58,400
spontaneous
emission, coherences,

17
00:00:58,400 --> 00:01:01,850
and three-level instances, and
super radiance, all we need

18
00:01:01,850 --> 00:01:03,880
is a matrix element.

19
00:01:03,880 --> 00:01:05,690
And this matrix
element will just

20
00:01:05,690 --> 00:01:09,190
run through all the
equations, and be

21
00:01:09,190 --> 00:01:12,230
responsible for a lot of
interesting phenomenon.

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00:01:12,230 --> 00:01:15,890
And for most of the
description of those phenomena,

23
00:01:15,890 --> 00:01:20,280
we don't have to know where
this matrix element comes from.

24
00:01:20,280 --> 00:01:22,140
The only thing we
have to know, there

25
00:01:22,140 --> 00:01:25,320
is a non-seeable matrix element
which drives the process.

26
00:01:25,320 --> 00:01:28,760
And as you know,
the matrix element,

27
00:01:28,760 --> 00:01:31,640
with an external field, is
called the Rabi frequency.

28
00:01:31,640 --> 00:01:34,530
And a lot of physics just
depends on the Rabi frequency.

29
00:01:34,530 --> 00:01:36,320
But what is behind?

30
00:01:36,320 --> 00:01:40,730
The engine behind the Rabi
frequency is a matrix element.

31
00:01:40,730 --> 00:01:47,810
So in the unit I started to
teach the week before spring

32
00:01:47,810 --> 00:01:51,140
break, we talked
about matrix elements.

33
00:01:51,140 --> 00:01:58,080
And for H, for the Hamiltonian,
we used the coupling of an atom

34
00:01:58,080 --> 00:02:00,200
to the electromagnetic field.

35
00:02:00,200 --> 00:02:03,220
And then we calculated,
what is the matrix element

36
00:02:03,220 --> 00:02:05,300
induced by the electric field?

37
00:02:05,300 --> 00:02:07,340
We made the dipole
approximation.

38
00:02:07,340 --> 00:02:10,650
And that's your plain, vanilla,
generic dipole operator

39
00:02:10,650 --> 00:02:12,440
which can connect two states.

40
00:02:12,440 --> 00:02:14,460
But we also consider,
what happens

41
00:02:14,460 --> 00:02:17,460
when we go beyond the
dipole approximation,

42
00:02:17,460 --> 00:02:21,380
and we found extra ways
of copying two states?

43
00:02:21,380 --> 00:02:23,800
For instance, we can
copy two states which

44
00:02:23,800 --> 00:02:27,440
have the same parity with
a quadrupole transition,

45
00:02:27,440 --> 00:02:31,290
or we can couple them with a
magnetic dipole transition.

46
00:02:31,290 --> 00:02:35,090
So these are other ways to
get into matrix elements.

47
00:02:35,090 --> 00:02:37,680
For most of the
course, you don't

48
00:02:37,680 --> 00:02:39,600
have to understand what
is behind the matrix,

49
00:02:39,600 --> 00:02:41,100
and you just know,
there is a number

50
00:02:41,100 --> 00:02:44,430
which drives the process.

51
00:02:44,430 --> 00:02:48,260
So what I want to finish
today is to discuss--

52
00:02:48,260 --> 00:02:53,210
and these are called selection
rules-- which tell us,

53
00:02:53,210 --> 00:02:56,410
when are those numbers,
when is this matrix element

54
00:02:56,410 --> 00:02:59,384
which couples two
states, when is it zero,

55
00:02:59,384 --> 00:03:00,550
or when is it non-vanishing.

56
00:03:08,570 --> 00:03:10,690
And what is helpful
here is, well,

57
00:03:10,690 --> 00:03:14,070
as always in physics,
use symmetry.

58
00:03:14,070 --> 00:03:16,630
And if you have an
operator, let me give you

59
00:03:16,630 --> 00:03:17,880
examples immediately.

60
00:03:17,880 --> 00:03:20,470
But just think for a moment
about the electric dipole.

61
00:03:20,470 --> 00:03:23,350
The electric dipole is
the position operator R.

62
00:03:23,350 --> 00:03:26,970
And you want to know, can the
position operator R induce

63
00:03:26,970 --> 00:03:29,690
a transition between two states.

64
00:03:29,690 --> 00:03:33,810
The way to analyse it is
now in terms of symmetry.

65
00:03:33,810 --> 00:03:36,410
And for symmetry,
which is always

66
00:03:36,410 --> 00:03:40,172
fulfilled for isolated
atoms, is angular momentum.

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00:03:40,172 --> 00:03:42,130
Angular momentum is a
conserved quantum number,

68
00:03:42,130 --> 00:03:44,130
we have rotation symmetry.

69
00:03:44,130 --> 00:03:47,860
So therefore, we want to now
understand matrix elements

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00:03:47,860 --> 00:03:50,680
in the language of
rotation symmetry.

71
00:03:50,680 --> 00:03:54,833
And therefore, we don't want to
use a precision operator x, y,

72
00:03:54,833 --> 00:03:56,200
or z.

73
00:03:56,200 --> 00:03:59,220
x, y, z do not have
the rotation symmetry.

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00:03:59,220 --> 00:04:01,890
We want to use linear
superpositions of x, y,

75
00:04:01,890 --> 00:04:05,950
and z-- I'll give you an example
in a moment-- in such a way

76
00:04:05,950 --> 00:04:10,290
that the operator becomes
an element of aesthetic

77
00:04:10,290 --> 00:04:11,830
of a spherical tensor.

78
00:04:11,830 --> 00:04:14,530
And spherical tensor, I
gave you the definition

79
00:04:14,530 --> 00:04:19,450
in the last lecture, the
element of spherical tensor, Ln,

80
00:04:19,450 --> 00:04:24,520
is defined by-- well, I connect
it with something you know,

81
00:04:24,520 --> 00:04:27,150
that it transforms
on the rotation,

82
00:04:27,150 --> 00:04:29,750
like the spherical
harmonics, ylm.

83
00:04:29,750 --> 00:04:33,880
So it is pretty much for an
operator, what the ylm, what

84
00:04:33,880 --> 00:04:36,870
the spherical harmonics
are for wave functions.

85
00:04:36,870 --> 00:04:38,770
I think I can do it more formal.

86
00:04:38,770 --> 00:04:41,350
And Professor Schwann
knows much more about it.

87
00:04:41,350 --> 00:04:45,690
I think these are elements of
the rotational symmetry group,

88
00:04:45,690 --> 00:04:46,900
But I don't want to go there.

89
00:04:51,230 --> 00:05:06,250
So what I mean by that is the
following, that if you take

90
00:05:06,250 --> 00:05:16,130
the position vector r, you
can expand it into a basis,

91
00:05:16,130 --> 00:05:18,720
which is x and y.

92
00:05:18,720 --> 00:05:32,550
But if you use the spherical
basis, x plus/minus iy.

93
00:05:37,950 --> 00:05:43,525
Then what appears are
the spherical harmonics.

94
00:05:51,410 --> 00:05:54,620
So in that case,
it's rather simple.

95
00:05:54,620 --> 00:05:58,460
The position vector has
actually in this representation

96
00:05:58,460 --> 00:06:01,880
components which
you can even see

97
00:06:01,880 --> 00:06:03,290
are the spherical harmonics.

98
00:06:03,290 --> 00:06:07,510
And therefore, we transform
like the spherical harmonics.

99
00:06:07,510 --> 00:06:14,800
Or just to give you
another example, if you

100
00:06:14,800 --> 00:06:20,110
have the operator, which is
responsible for the quadrupole

101
00:06:20,110 --> 00:06:23,220
transition, well,
you get the gist,

102
00:06:23,220 --> 00:06:26,380
it's a product of
two coordinates.

103
00:06:26,380 --> 00:06:31,640
So therefore, it's a
spherical tensor of rank two.

104
00:06:31,640 --> 00:06:35,950
And it so happens,
but I'm not deriving

105
00:06:35,950 --> 00:06:44,320
that, it is a superposition of
two components with Lm quantum

106
00:06:44,320 --> 00:06:46,970
number, 2 plus 1, 2 minus 1.

107
00:06:49,830 --> 00:06:51,940
So that's how we
should think about it.

108
00:06:51,940 --> 00:06:57,960
So, we want to ask, we want
to extend the operator,

109
00:06:57,960 --> 00:07:02,010
into operators which
have rotational symmetry,

110
00:07:02,010 --> 00:07:05,170
and these are those, or
these are those three.

111
00:07:05,170 --> 00:07:07,860
So instead of using the
vector Cartesian coordinate,

112
00:07:07,860 --> 00:07:10,560
we use its spherical components.

113
00:07:13,410 --> 00:07:25,133
And with that, we can
take this expression

114
00:07:25,133 --> 00:07:45,210
from the last
lecture, and rewrite

115
00:07:45,210 --> 00:07:51,790
it using the Wigner-Eckart
theorem into a way which

116
00:07:51,790 --> 00:07:56,090
allows us to immediately
formulate selection rules.

117
00:07:56,090 --> 00:08:03,600
So [INAUDIBLE] and primer
are the quantum numbers

118
00:08:03,600 --> 00:08:06,330
of the state, except
for angular momentum,

119
00:08:06,330 --> 00:08:09,050
so matricing about principle
quantum number of the hydrogen

120
00:08:09,050 --> 00:08:10,300
atom.

121
00:08:10,300 --> 00:08:14,970
And we want to copy from
a total angular momentum J

122
00:08:14,970 --> 00:08:19,640
prime to total angular
momentum J. Actually,

123
00:08:19,640 --> 00:08:32,210
we want to copy from J prime,
M prime to a state, JM.

124
00:08:32,210 --> 00:08:41,230
And what the Wigner-Eckart
theorem tells us

125
00:08:41,230 --> 00:08:44,440
that we can factor
out the M dependence.

126
00:08:44,440 --> 00:08:48,030
The M dependence just comes
from orientation in space.

127
00:08:48,030 --> 00:08:51,160
So M is just how you orient
wave functions and vectors

128
00:08:51,160 --> 00:08:52,190
and space.

129
00:08:52,190 --> 00:08:58,680
And you can sort of write this
matrix element as a projection.

130
00:08:58,680 --> 00:09:03,356
And this is nothing else than
the familiar Clebsch-Gordan

131
00:09:03,356 --> 00:09:03,855
coefficient.

132
00:09:07,690 --> 00:09:13,930
And the Clebsch-Gordan
coefficient

133
00:09:13,930 --> 00:09:22,240
for coupling the initial
state, JM, or to-- let me

134
00:09:22,240 --> 00:09:24,740
put it this way-- to start with
the initial state J prime, M

135
00:09:24,740 --> 00:09:33,680
prime, we have the L and
the M of our operator.

136
00:09:33,680 --> 00:09:37,280
And that should result
in a total angular

137
00:09:37,280 --> 00:09:39,370
momentum of J and M.

138
00:09:39,370 --> 00:09:42,090
So we retrieve
again the formalism

139
00:09:42,090 --> 00:09:44,935
of the addition of
two angular momenta.

140
00:09:44,935 --> 00:09:46,310
Sometimes, you
have two particles

141
00:09:46,310 --> 00:09:49,554
you couple into angular
momentum and ask,

142
00:09:49,554 --> 00:09:50,970
what is the total
angular momentum

143
00:09:50,970 --> 00:09:52,220
of the composite particles?

144
00:09:52,220 --> 00:09:54,530
But what we do here for
this selection rule,

145
00:09:54,530 --> 00:09:57,230
we have the initial
state, we calculate

146
00:09:57,230 --> 00:09:59,940
with the angular
momentum of the operator.

147
00:09:59,940 --> 00:10:02,030
You can think the
operator is a field which

148
00:10:02,030 --> 00:10:03,720
can transfer angular momentum.

149
00:10:03,720 --> 00:10:06,420
And then, of course,
the final state

150
00:10:06,420 --> 00:10:09,250
has to fulfill angular
momentum conservation.

151
00:10:09,250 --> 00:10:11,910
But one source of the
momentum is now the operator,

152
00:10:11,910 --> 00:10:14,390
is the external
field, is the photon,

153
00:10:14,390 --> 00:10:18,440
or the microwave drive,
whatever you apply.

154
00:10:18,440 --> 00:10:24,950
And yes, this
Wigner-Eckart theorem

155
00:10:24,950 --> 00:10:36,170
allows us to write the matrix
element as a reduced matrix

156
00:10:36,170 --> 00:10:36,670
element.

157
00:10:40,400 --> 00:10:46,130
Which really decides
whether the transition

158
00:10:46,130 --> 00:10:50,940
is non-vanishing or not,
times a factor which is just

159
00:10:50,940 --> 00:10:53,770
the orientation of the wave
function and of the operator

160
00:10:53,770 --> 00:10:54,280
in space.

161
00:10:57,950 --> 00:11:02,360
So for the Clebsch-Gordan
coefficient,

162
00:11:02,360 --> 00:11:04,750
we have a simple selection rule.

163
00:11:04,750 --> 00:11:10,800
And this is that for
the [INAUDIBLE] number,

164
00:11:10,800 --> 00:11:15,020
the M of the final state has to
be the M of the initial state,

165
00:11:15,020 --> 00:11:18,520
plus a little M of the operator.

166
00:11:18,520 --> 00:11:28,180
And for both the Clebsch-Gordan
and the reduced matrix element,

167
00:11:28,180 --> 00:11:31,160
we have the triangle rule.

168
00:11:31,160 --> 00:11:33,620
Well, if you couple two
angular momentum vectors

169
00:11:33,620 --> 00:11:37,220
to a final angular
momentum, the three vectors

170
00:11:37,220 --> 00:11:39,640
have to form a triangle.

171
00:11:39,640 --> 00:11:48,290
And the triangle rule says that
some-- let me write it down,

172
00:11:48,290 --> 00:12:01,800
and then you recognize it-- that
the angular momentum construct

173
00:12:01,800 --> 00:12:05,750
by the field has to fulfill
the triangle rule that J prime

174
00:12:05,750 --> 00:12:07,920
and J can be connected.

175
00:12:07,920 --> 00:12:08,492
Yes?

176
00:12:08,492 --> 00:12:10,950
AUDIENCE: What is this symbolic
meaning of the double bars?

177
00:12:13,710 --> 00:12:15,790
PROFESSOR: It's just
how, in many textbooks,

178
00:12:15,790 --> 00:12:17,760
the reduced matrix
element is written.

179
00:12:17,760 --> 00:12:21,180
It's nothing else
in a matrix element.

180
00:12:21,180 --> 00:12:29,260
But, you know, plus
the y that I looked

181
00:12:29,260 --> 00:12:30,730
at in the quantum
mechanics book.

182
00:12:30,730 --> 00:12:32,570
But what happens is
these are not states.

183
00:12:32,570 --> 00:12:35,020
J and J prime are not states.

184
00:12:35,020 --> 00:12:36,390
They have an independence.

185
00:12:36,390 --> 00:12:38,390
So we've taken out
the independence.

186
00:12:38,390 --> 00:12:44,340
So this is sort of a matrix
element between a state which

187
00:12:44,340 --> 00:12:46,890
may have been stripped
of its independence.

188
00:12:46,890 --> 00:12:49,830
So maybe, I don't know
if that's 100% correct,

189
00:12:49,830 --> 00:12:52,400
but if you have the
YLM in certain states,

190
00:12:52,400 --> 00:12:56,630
you have an e to
the IM and M part.

191
00:12:56,630 --> 00:12:58,550
And this has probably
been factored out.

192
00:12:58,550 --> 00:13:01,110
So these are not really states,
and the double line just

193
00:13:01,110 --> 00:13:03,850
means it's a reduced matrix
element with the meaning I just

194
00:13:03,850 --> 00:13:04,630
mentioned.

195
00:13:04,630 --> 00:13:07,680
It's a standard way of
factorizing matrix elements.

196
00:13:07,680 --> 00:13:10,470
And yeah, that means
reduced matrix elements.

197
00:13:17,850 --> 00:13:30,970
So in other words, when we
talk about selection rules,

198
00:13:30,970 --> 00:13:33,760
we want to use
the representation

199
00:13:33,760 --> 00:13:37,240
of spherical tensors, because
the spherical tensor, the rank

200
00:13:37,240 --> 00:13:40,100
of the spherical
tensor, just tells us

201
00:13:40,100 --> 00:13:42,974
how much angular momentum
is involved in the photon,

202
00:13:42,974 --> 00:13:44,265
is involved in this transition.

203
00:13:49,430 --> 00:13:53,990
So maybe just to
give you a question,

204
00:13:53,990 --> 00:13:56,390
so if I were to do a
multipole expansion,

205
00:13:56,390 --> 00:13:59,980
and I have an octupole
transition, what is now

206
00:13:59,980 --> 00:14:02,130
the angular momentum
transferred by a photon?

207
00:14:10,832 --> 00:14:12,097
STUDENT: 3?

208
00:14:12,097 --> 00:14:12,763
PROFESSOR: What?

209
00:14:12,763 --> 00:14:13,116
STUDENT: 3?

210
00:14:13,116 --> 00:14:13,616
STUDENT: 3?

211
00:14:13,616 --> 00:14:14,700
PROFESSOR: 3, yeah.

212
00:14:14,700 --> 00:14:16,470
The dipole is L equals 1.

213
00:14:16,470 --> 00:14:18,810
Quadrupole is spherical
tense of rank 2.

214
00:14:18,810 --> 00:14:21,440
L equals 2, so it's L equals 3.

215
00:14:21,440 --> 00:14:28,130
Now, can a photon transfer
three units of angular momentum?

216
00:14:28,130 --> 00:14:32,810
Can an atom get rid of
three units of, let's say,

217
00:14:32,810 --> 00:14:34,830
orbital or spin
angular momentum.

218
00:14:34,830 --> 00:14:38,820
We start in a state which
is J prime equals 3.

219
00:14:38,820 --> 00:14:41,330
You need one photon,
and you go to a state

220
00:14:41,330 --> 00:14:44,190
which is J prime equals 0.

221
00:14:44,190 --> 00:14:45,450
Is that possible or not?

222
00:14:56,249 --> 00:14:57,790
We don't have
[INAUDIBLE], but do you

223
00:14:57,790 --> 00:15:02,920
want to volunteer an answer?

224
00:15:02,920 --> 00:15:05,237
What's the angular
momentum of the photon?

225
00:15:05,237 --> 00:15:07,980
STUDENT: [INAUDIBLE]?

226
00:15:07,980 --> 00:15:11,180
PROFESSOR: Well, be careful.

227
00:15:11,180 --> 00:15:13,650
The photon has an
intrinsic angular momentum,

228
00:15:13,650 --> 00:15:15,380
which is like the
spin of the photon.

229
00:15:15,380 --> 00:15:17,260
That's plus/minus 1.

230
00:15:17,260 --> 00:15:19,500
But just imagine that
you have an atom,

231
00:15:19,500 --> 00:15:22,040
and the photon is not
immediate at the origin.

232
00:15:22,040 --> 00:15:25,480
The photon is emitted a
little bit further out.

233
00:15:25,480 --> 00:15:28,500
Then, with reference
to the origin,

234
00:15:28,500 --> 00:15:31,200
the photon has orbital
angular momentum.

235
00:15:31,200 --> 00:15:32,960
And that's what
we're talking about.

236
00:15:32,960 --> 00:15:37,390
In the multipole expansion,
we fall in powers of x, and z,

237
00:15:37,390 --> 00:15:43,080
and y of the spatial
coordinate of the electron.

238
00:15:43,080 --> 00:15:47,502
And that actually means we're
going away from the origin.

239
00:15:47,502 --> 00:15:50,160
And if you emit something
which is away from the origin,

240
00:15:50,160 --> 00:15:52,590
you have orbital
angular momentum.

241
00:15:52,590 --> 00:15:53,660
So, yes.

242
00:15:53,660 --> 00:15:56,680
An octupole transition
is exactly what I said.

243
00:15:56,680 --> 00:16:01,560
It means a photon is emitted,
and it changes the angular

244
00:16:01,560 --> 00:16:04,982
momentum of the atom left
behind by three units.

245
00:16:04,982 --> 00:16:06,440
That's what we
really mean by that.

246
00:16:06,440 --> 00:16:08,273
And that's what we mean
by those electrodes.

247
00:16:12,660 --> 00:16:14,160
The question that
you should maybe

248
00:16:14,160 --> 00:16:16,690
discuss after class
is, what happens

249
00:16:16,690 --> 00:16:18,720
if you detect this photon?

250
00:16:18,720 --> 00:16:20,650
Is that now a
supercharged photon,

251
00:16:20,650 --> 00:16:22,452
which has three units
of angular momentum?

252
00:16:22,452 --> 00:16:24,410
Is there something strange
in its polarization?

253
00:16:29,380 --> 00:16:31,820
Think about it, and if
you don't find the answer,

254
00:16:31,820 --> 00:16:35,230
we can discuss it
in the next class.

255
00:16:35,230 --> 00:16:35,730
OK.

256
00:16:35,730 --> 00:16:38,120
So this is the classification.

257
00:16:42,510 --> 00:16:46,500
Let's just focus on
the simple examples.

258
00:16:46,500 --> 00:16:50,640
We have discussed electric
dipole and magnetic dipole

259
00:16:50,640 --> 00:16:53,330
radiation.

260
00:16:53,330 --> 00:16:57,660
These are induced by vectors.

261
00:16:57,660 --> 00:17:01,810
Remember, E1 is
the dipole vector.

262
00:17:01,810 --> 00:17:07,425
For M1, the matrix element was
by the angular momentum vector.

263
00:17:09,980 --> 00:17:10,480
So

264
00:17:10,480 --> 00:17:13,220
These are vectors.

265
00:17:13,220 --> 00:17:17,129
And that means
the representation

266
00:17:17,129 --> 00:17:20,130
of the spherical tensors,
or the quantum numbers

267
00:17:20,130 --> 00:17:22,765
of the spherical tensors,
are the same of the Y1n.

268
00:17:28,640 --> 00:17:31,860
And so for dipole
radiation, whether it's

269
00:17:31,860 --> 00:17:35,925
electric or magnetic, we have
now with the dipole selection

270
00:17:35,925 --> 00:17:40,540
rules, which pretty
much save you

271
00:17:40,540 --> 00:17:43,590
at one unit of angular
momentum to state

272
00:17:43,590 --> 00:17:46,130
B. Can you reach
state A with that?

273
00:17:46,130 --> 00:17:49,600
And these selection rules are
that you can change the angular

274
00:17:49,600 --> 00:17:52,900
momentum between initial
state by 0 and 1.

275
00:17:52,900 --> 00:17:55,280
This is the triangle rule.

276
00:17:55,280 --> 00:18:00,480
And delta m can be
0 and plus/minus 1,

277
00:18:00,480 --> 00:18:03,370
depending on polarization, which
we want to discuss in a moment.

278
00:18:06,290 --> 00:18:11,560
So in angular momentum,
electric and magnetic dipoles

279
00:18:11,560 --> 00:18:15,380
have the same
selection rule, where

280
00:18:15,380 --> 00:18:17,880
when it comes to the
question of parity,

281
00:18:17,880 --> 00:18:20,060
we've already discussed that.

282
00:18:20,060 --> 00:18:26,650
That an electric dipole connects
to a state of opposite parity,

283
00:18:26,650 --> 00:18:28,940
whereas, the magnetic
dipole connects

284
00:18:28,940 --> 00:18:32,380
two states of the same parity.

285
00:18:32,380 --> 00:18:36,960
And of course, this comes about
because L is an axial vector,

286
00:18:36,960 --> 00:18:42,355
and R is a polar vector, which
have different symmetry when

287
00:18:42,355 --> 00:18:43,730
you invert the
coordinate system.

288
00:18:48,470 --> 00:18:51,640
The one higher multipole port
transition, which we discussed,

289
00:18:51,640 --> 00:18:53,120
was the electric quadrupole, E2.

290
00:18:55,700 --> 00:18:59,480
And the spherical
tensor operators

291
00:18:59,480 --> 00:19:02,250
for the quadrupole
transition, I gave you

292
00:19:02,250 --> 00:19:07,300
already the example of, let's
say, xz, products of two

293
00:19:07,300 --> 00:19:12,750
coordinates, because we went one
order higher than the dipole.

294
00:19:12,750 --> 00:19:17,700
They transform as Y2m.

295
00:19:17,700 --> 00:19:26,680
And therefore, we
have selection rules

296
00:19:26,680 --> 00:19:32,700
for quadrupole transitions,
which tell us now

297
00:19:32,700 --> 00:19:36,120
that we can change the total
angular momentum up to 2.

298
00:19:41,020 --> 00:19:45,310
And also, delta m can
change up to two units.

299
00:19:50,820 --> 00:19:53,170
And again, just to
emphasize, because people get

300
00:19:53,170 --> 00:19:56,080
confused all the time.

301
00:19:56,080 --> 00:19:59,200
When we talk about a
quadrupole transition,

302
00:19:59,200 --> 00:20:03,010
we mean absolutely
positively a transition

303
00:20:03,010 --> 00:20:05,747
where one photon is emitted.

304
00:20:05,747 --> 00:20:07,330
If you fully quantize
the field, there

305
00:20:07,330 --> 00:20:10,880
is one creation
operator of the photon.

306
00:20:10,880 --> 00:20:12,960
It's one photon
which is created,

307
00:20:12,960 --> 00:20:15,555
and this photon carries
away the angular momentum

308
00:20:15,555 --> 00:20:16,430
we've just specified.

309
00:20:23,340 --> 00:20:24,650
Questions about that?

310
00:20:29,440 --> 00:20:36,500
Let me conclude our
discussion of matrix elements

311
00:20:36,500 --> 00:20:40,170
by talking about something
which is experimentally

312
00:20:40,170 --> 00:20:41,720
very relevant.

313
00:20:41,720 --> 00:20:46,070
And this is how
selection rules depend

314
00:20:46,070 --> 00:20:48,190
on the polarization of light.

315
00:20:52,080 --> 00:20:54,705
And I only want to discuss
it for electric dipole

316
00:20:54,705 --> 00:20:55,205
transitions.

317
00:20:57,830 --> 00:21:03,300
So when we wrote down
the coupling of the atom

318
00:21:03,300 --> 00:21:14,700
to electromagnetic radiation,
we had the dipole operator,

319
00:21:14,700 --> 00:21:17,050
but we also had,
of course, the mode

320
00:21:17,050 --> 00:21:19,520
of the electromagnetic
field, which

321
00:21:19,520 --> 00:21:24,560
was characterized by a
polarization epsilon.

322
00:21:24,560 --> 00:21:28,090
So until now, when I talked
about selection rules,

323
00:21:28,090 --> 00:21:30,330
we discussed this part.

324
00:21:30,330 --> 00:21:32,670
But now we want to see how
it effects polarization.

325
00:21:37,146 --> 00:21:44,630
Well, the epsilon, for instance,
for circular polarization--

326
00:21:44,630 --> 00:21:49,930
we'll talk about linear
polarization in a moment--

327
00:21:49,930 --> 00:21:51,170
has this representation.

328
00:21:54,620 --> 00:21:58,810
So this is the unit
vector of the polarization

329
00:21:58,810 --> 00:22:02,950
of the electric field when
it's circularly polarization.

330
00:22:05,690 --> 00:22:20,960
And now remember, we
take this vector r,

331
00:22:20,960 --> 00:22:24,550
and expand it in
the following way.

332
00:22:24,550 --> 00:22:29,620
So if you multiply now the
operator r, or the matrix

333
00:22:29,620 --> 00:22:32,650
elements created by
this vectorate operator,

334
00:22:32,650 --> 00:22:39,150
by the polarization, you see
that one circular polarization

335
00:22:39,150 --> 00:22:40,260
checks out this component.

336
00:22:40,260 --> 00:22:42,559
The other circular
polarization projects this out.

337
00:22:42,559 --> 00:22:45,100
And later, we'll talk about that
linear polarization projects

338
00:22:45,100 --> 00:22:46,550
that out.

339
00:22:46,550 --> 00:22:50,250
So when we said that
we have matrix elements

340
00:22:50,250 --> 00:22:53,050
for dipole transition,
which can change

341
00:22:53,050 --> 00:22:56,680
angular momentum, or the
incremental number, by minus 1,

342
00:22:56,680 --> 00:22:59,610
plus 1, and 0,
this is now related

343
00:22:59,610 --> 00:23:01,620
to the polarization
of the light,

344
00:23:01,620 --> 00:23:03,640
either the photon
which is emitted,

345
00:23:03,640 --> 00:23:06,820
or when we use circularly
polarized light,

346
00:23:06,820 --> 00:23:09,890
we can only drive this
transition, that or that,

347
00:23:09,890 --> 00:23:12,930
because the scale of
product of the polarization

348
00:23:12,930 --> 00:23:21,150
vector and the matrix
element project out

349
00:23:21,150 --> 00:23:23,560
only one component of
the spherical tensor.

350
00:23:30,780 --> 00:23:35,370
So if you look at
the expansion above,

351
00:23:35,370 --> 00:23:42,380
we realize that the left- and
right-handed circular light

352
00:23:42,380 --> 00:23:47,030
projects now out
the spherical tensor

353
00:23:47,030 --> 00:23:51,530
operator, T1 plus minus 1.

354
00:23:51,530 --> 00:23:55,190
And since it's circularly
polarized light,

355
00:23:55,190 --> 00:23:57,670
and therefore, we find
this election rule

356
00:23:57,670 --> 00:24:08,280
that delta m, the Z component,
the Z component of the angular

357
00:24:08,280 --> 00:24:11,550
momentum, changes
by plus/minus 1

358
00:24:11,550 --> 00:24:14,725
when the circular polarized
light is sigma plus or sigma

359
00:24:14,725 --> 00:24:16,183
minus, right-handed
or left-handed.

360
00:24:21,771 --> 00:24:22,270
OK.

361
00:24:22,270 --> 00:24:24,311
So this is responsible
for circular polarization.

362
00:24:24,311 --> 00:24:29,310
These are selection rules for
circularly polarized light.

363
00:24:29,310 --> 00:24:31,952
Let me conclude by
discussing the case

364
00:24:31,952 --> 00:24:32,910
of linear polarization.

365
00:24:36,020 --> 00:24:40,040
Well, when we ask
linear polarization,

366
00:24:40,040 --> 00:24:44,460
if we ask for linear
polarization along x or y,

367
00:24:44,460 --> 00:24:51,220
well, it's linear polarization,
but we should regard it

368
00:24:51,220 --> 00:24:59,570
as the linear superposition
of sigma plus and sigma minus.

369
00:24:59,570 --> 00:25:04,200
So in other words, if you have
the quantization axis along z,

370
00:25:04,200 --> 00:25:08,980
and you use light which
is polarized along x or y,

371
00:25:08,980 --> 00:25:11,600
the way how the light
talks to the atom

372
00:25:11,600 --> 00:25:14,210
with symmetric operators
is that the light

373
00:25:14,210 --> 00:25:16,550
is a superposition of
sigma plus and sigma minus.

374
00:25:21,450 --> 00:25:32,640
So what we have so far is,
so we had here the light key,

375
00:25:32,640 --> 00:25:42,715
the propagation of the
light was along the z-axis.

376
00:25:48,690 --> 00:25:57,040
But now, we want to look at
the other possibility that z,

377
00:25:57,040 --> 00:26:05,300
or the quantization
axis, is parallel

378
00:26:05,300 --> 00:26:10,320
to the polarization
of the electric field,

379
00:26:10,320 --> 00:26:13,910
which would mean that the
quantization axis is usually

380
00:26:13,910 --> 00:26:16,090
defined by an external
magnetic field.

381
00:26:16,090 --> 00:26:19,930
If you're talking about the
situation that the electric

382
00:26:19,930 --> 00:26:23,040
field of the electromagnetic
wave is parallel

383
00:26:23,040 --> 00:26:27,962
to the magnetic field, then
with this polarization,

384
00:26:27,962 --> 00:26:34,150
we peak out this spherical
tensor component, which is z,

385
00:26:34,150 --> 00:26:38,620
which is r times Y1,0.

386
00:26:38,620 --> 00:26:42,940
And that means that this
polarization of the light

387
00:26:42,940 --> 00:26:52,070
induces a transition for
which delta m equals 0.

388
00:26:52,070 --> 00:26:56,790
And this is referred
to as pi light.

389
00:26:56,790 --> 00:27:01,280
So maybe if that got
confusing for you,

390
00:27:01,280 --> 00:27:02,960
let me just help
out with a drawing.

391
00:27:07,210 --> 00:27:14,260
We have our atom here, which is
quantized by a magnetic field

392
00:27:14,260 --> 00:27:32,910
B. And if you shine light on
it, we have the electric field

393
00:27:32,910 --> 00:27:37,120
perpendicular to
the magnetic field.

394
00:27:37,120 --> 00:27:39,290
So this would be x and y.

395
00:27:39,290 --> 00:27:44,790
And the natural way to describe
it is by using x plus/minus IY.

396
00:27:44,790 --> 00:27:51,400
And we have selection rules
where delta m is plus/minus 1.

397
00:27:51,400 --> 00:27:58,670
But alternatively, we
can also shine light

398
00:27:58,670 --> 00:28:01,470
along this direction.

399
00:28:01,470 --> 00:28:05,860
And for the electric field,
which was perpendicular to B,

400
00:28:05,860 --> 00:28:07,400
we retrieve the previous case.

401
00:28:07,400 --> 00:28:10,080
We have superpositions of the
sigma plus and sigma minus.

402
00:28:10,080 --> 00:28:13,790
But the new case now is that
the electric field is parallel

403
00:28:13,790 --> 00:28:18,500
to B. And then, we
drive transitions,

404
00:28:18,500 --> 00:28:21,300
which have delta M equals 0.

405
00:28:23,820 --> 00:28:27,760
So these are sigma plus and
sigma minus transitions.

406
00:28:27,760 --> 00:28:30,360
And this here is what is
called a pi transition.

407
00:28:35,920 --> 00:28:37,510
Anyway, it's a
little bit formal,

408
00:28:37,510 --> 00:28:41,360
but I just wanted to
present it in this context.

409
00:28:46,210 --> 00:28:48,728
Questions?

410
00:28:48,728 --> 00:28:52,102
STUDENT: I have one slightly,
maybe, basic question.

411
00:28:52,102 --> 00:28:55,114
When we talk about polarization
in all these matrix elements--

412
00:28:55,114 --> 00:28:57,404
so for example,
photon [INAUDIBLE],

413
00:28:57,404 --> 00:29:02,390
right-- these are single
photons [INAUDIBLE] elements.

414
00:29:02,390 --> 00:29:05,573
And so when we talk
about shining a laser,

415
00:29:05,573 --> 00:29:06,555
it has a polarization.

416
00:29:06,555 --> 00:29:08,191
But we don't talk
about polarization

417
00:29:08,191 --> 00:29:09,992
for single photons.

418
00:29:09,992 --> 00:29:11,465
Or do we?

419
00:29:14,420 --> 00:29:17,205
PROFESSOR: Actually, we
talk about-- the question

420
00:29:17,205 --> 00:29:18,413
is, what is the polarization?

421
00:29:18,413 --> 00:29:20,420
Do we talk about polarization
of single photons,

422
00:29:20,420 --> 00:29:23,170
or polarization of laser beams?

423
00:29:23,170 --> 00:29:26,020
Well, let me back
up and say, we talk

424
00:29:26,020 --> 00:29:30,160
about polarization of a mode
of the electromagnetic field.

425
00:29:30,160 --> 00:29:33,820
We will always expand
the electromagnetic field

426
00:29:33,820 --> 00:29:34,626
into modes.

427
00:29:34,626 --> 00:29:36,000
And the mode is
the polarization.

428
00:29:36,000 --> 00:29:38,600
It may happen that
at some point,

429
00:29:38,600 --> 00:29:41,050
a photon is emitting a
superposition of modes.

430
00:29:41,050 --> 00:29:45,990
But in the most
straightforward description,

431
00:29:45,990 --> 00:29:48,200
we always do a mode analysis.

432
00:29:48,200 --> 00:29:50,870
And often, we simplify
the case by saying

433
00:29:50,870 --> 00:29:52,650
that the atom
interacts only with one

434
00:29:52,650 --> 00:29:54,860
mode of the
electromagnetic field.

435
00:29:54,860 --> 00:29:56,960
And maybe in the case
of spontaneous emission,

436
00:29:56,960 --> 00:29:59,170
we then sum over all modes.

437
00:29:59,170 --> 00:30:02,490
But for each mode, there's
a specific polarization.

438
00:30:05,220 --> 00:30:08,400
And it doesn't matter if this
mode is filled with one atom,

439
00:30:08,400 --> 00:30:11,760
or with a laser beam, with
a classical electromagnetic

440
00:30:11,760 --> 00:30:15,010
field, which corresponds
to zillions of photons.

441
00:30:15,010 --> 00:30:16,511
STUDENT: [INAUDIBLE]
does it always

442
00:30:16,511 --> 00:30:18,010
end up being
electrical polarization

443
00:30:18,010 --> 00:30:20,510
in this case, then?

444
00:30:20,510 --> 00:30:23,010
Like because if
it's many photons,

445
00:30:23,010 --> 00:30:28,010
then there's a lot of
[INAUDIBLE] for each of them,

446
00:30:28,010 --> 00:30:31,107
or each of them
individually-- I don't know.

447
00:30:33,802 --> 00:30:34,885
PROFESSOR: No, it depends.

448
00:30:40,680 --> 00:30:44,650
If you have an atom, and it has
one unit of angular momentum,

449
00:30:44,650 --> 00:30:47,740
and it spontaneously
emits a photon,

450
00:30:47,740 --> 00:30:51,380
if the photon is emitted
along the quantization axis,

451
00:30:51,380 --> 00:30:53,055
it can only be sigma plus.

452
00:30:53,055 --> 00:30:54,680
If it's emitted in
the other direction,

453
00:30:54,680 --> 00:30:56,770
it has to be sigma minus.

454
00:30:56,770 --> 00:31:01,710
Now if you go at strange
angles, then at this angle,

455
00:31:01,710 --> 00:31:04,970
you overlay it with
different modes.

456
00:31:04,970 --> 00:31:09,100
And you may now find
photons in a superposition

457
00:31:09,100 --> 00:31:13,934
of polarizations, because
we have several modes which

458
00:31:13,934 --> 00:31:15,850
are connected with this
direction of emission.

459
00:31:19,180 --> 00:31:21,180
I think if you write it
down, it's pretty clear.

460
00:31:21,180 --> 00:31:23,270
It's just sort of
projection operators.

461
00:31:23,270 --> 00:31:27,300
And for spontaneous emission,
we sum over all modes.

462
00:31:27,300 --> 00:31:38,610
But for me, I always think
about-- we can always

463
00:31:38,610 --> 00:31:43,000
think about what a single
photon does by saying, well,

464
00:31:43,000 --> 00:31:45,060
if I'm getting confused
about a single photon,

465
00:31:45,060 --> 00:31:47,250
let me figure out
what many, many, many

466
00:31:47,250 --> 00:31:48,960
identical photons would be.

467
00:31:48,960 --> 00:31:51,490
And that would mean,
instead of a single photon

468
00:31:51,490 --> 00:31:54,960
in a certain mode, I
release a beam in this mode.

469
00:31:54,960 --> 00:31:57,060
And then, suddenly, I
can think, classically,

470
00:31:57,060 --> 00:31:58,970
I know what the
electric field is such.

471
00:31:58,970 --> 00:32:00,470
And then you go
back to the, what

472
00:32:00,470 --> 00:32:02,178
is the electric field
of a single photon,

473
00:32:02,178 --> 00:32:03,940
and usually make the connection.

474
00:32:03,940 --> 00:32:06,010
So I think at least
for the discussion

475
00:32:06,010 --> 00:32:09,660
of matrix elements,
transitions, angular momentum,

476
00:32:09,660 --> 00:32:12,150
I don't think you ever have
to distinguish between what

477
00:32:12,150 --> 00:32:13,940
single photons do and
what laser beams do.

478
00:32:16,600 --> 00:32:19,910
But there are important
aspects of single photons,

479
00:32:19,910 --> 00:32:25,260
non-classical aspects, which
we'll discuss in a short while.

480
00:32:25,260 --> 00:32:25,995
Other questions?

481
00:32:29,690 --> 00:32:30,250
OK.

482
00:32:30,250 --> 00:32:36,410
That's all I want to say
about selection rules.

483
00:32:36,410 --> 00:32:41,414
So with that now, we can
simply take the matrix element

484
00:32:41,414 --> 00:32:42,080
and run with it.

485
00:32:45,780 --> 00:32:53,820
So in this lecture
and on [INAUDIBLE],

486
00:32:53,820 --> 00:32:56,347
I want to talk
about basic aspects

487
00:32:56,347 --> 00:32:57,430
of atom-light interaction.

488
00:33:02,490 --> 00:33:10,110
And what I want to
talk today about it

489
00:33:10,110 --> 00:33:14,690
is the two important cases
when an atom interacts

490
00:33:14,690 --> 00:33:20,970
with monochromatic
wave, or when it

491
00:33:20,970 --> 00:33:24,370
interacts with a broad spectrum.

492
00:33:24,370 --> 00:33:27,490
In one case, when I
say monochromatic case,

493
00:33:27,490 --> 00:33:30,550
you may just think of the
best laser money can buy.

494
00:33:30,550 --> 00:33:31,520
Very, very sharp.

495
00:33:31,520 --> 00:33:33,130
Very, very monochromatic.

496
00:33:33,130 --> 00:33:35,280
When I talk about
a broad spectrum,

497
00:33:35,280 --> 00:33:37,790
you may just think about
black-body radiation,

498
00:33:37,790 --> 00:33:39,950
which is an ultra
broad spectrum.

499
00:33:39,950 --> 00:33:43,340
And they're two very
different cases.

500
00:33:43,340 --> 00:33:46,320
And some of it is just
related to Nancy's question,

501
00:33:46,320 --> 00:33:47,920
that if you have
a broad spectrum,

502
00:33:47,920 --> 00:33:50,610
we're always talking
about many, many modes,

503
00:33:50,610 --> 00:33:52,600
and they will be
incoherent, and they

504
00:33:52,600 --> 00:33:54,340
will be irreversible physics.

505
00:33:54,340 --> 00:33:57,110
Whereas for monochromatic
light, everything

506
00:33:57,110 --> 00:34:01,810
is a pure, plain wave, and
everything is coherent.

507
00:34:01,810 --> 00:34:05,090
So we want to sort of
talk about that first.

508
00:34:05,090 --> 00:34:08,630
And then later this week,
I think on Wednesday,

509
00:34:08,630 --> 00:34:10,880
we will talk about
spontaneous emission.

510
00:34:10,880 --> 00:34:13,739
But right now, we focus
on the simpler case,

511
00:34:13,739 --> 00:34:18,489
where we drive the system with
electromagnetic radiation,

512
00:34:18,489 --> 00:34:25,290
which is either
narrow-band or broadband.

513
00:34:32,780 --> 00:34:35,580
But let's just start
with a cartoon.

514
00:34:35,580 --> 00:34:37,650
We have an atom.

515
00:34:37,650 --> 00:34:41,230
And for that discussion,
all we need is two levels.

516
00:34:41,230 --> 00:34:42,900
And all we need is
that the two levels

517
00:34:42,900 --> 00:34:44,600
are connected by
some matrix element.

518
00:34:48,780 --> 00:34:52,850
And the basic
phenomenological situation

519
00:34:52,850 --> 00:35:03,950
is that we have one atom,
which sits in a vacuum.

520
00:35:13,070 --> 00:35:16,350
So we have volume, V, of vacuum.

521
00:35:16,350 --> 00:35:22,370
And what is important
now is that the walls

522
00:35:22,370 --> 00:35:27,920
of the imaginary boundary
of what defines our vacuum

523
00:35:27,920 --> 00:35:29,200
is at low temperature.

524
00:35:33,350 --> 00:35:39,630
And low temperature means that
the atom will irreversibly

525
00:35:39,630 --> 00:35:42,070
decay into the ground
state with a lifetime tau.

526
00:36:02,750 --> 00:36:08,200
And that means that
in some picture,

527
00:36:08,200 --> 00:36:16,170
the excited state
is the broadening,

528
00:36:16,170 --> 00:36:20,320
which is broadened by
the natural lifetime.

529
00:36:20,320 --> 00:36:25,815
And in our discussion,
we assume--

530
00:36:25,815 --> 00:36:30,680
and this is what I said with
the cold walls of the vacuum--

531
00:36:30,680 --> 00:36:40,950
that the energy
difference is much,

532
00:36:40,950 --> 00:36:45,070
much larger than the
relevant temperature.

533
00:36:45,070 --> 00:36:53,910
And this is very well fulfilled
for our standard atomic system.

534
00:36:53,910 --> 00:36:56,230
The typical excitation
energy, even

535
00:36:56,230 --> 00:36:59,510
for atoms with loosely bound
electrons, as the alkalis,

536
00:36:59,510 --> 00:37:02,590
is two electron volt,
which corresponds

537
00:37:02,590 --> 00:37:09,840
to a temperature
of 20,000 Kelvin.

538
00:37:09,840 --> 00:37:13,350
And even at the rather hot
temperature, definitely

539
00:37:13,350 --> 00:37:17,340
hot temperatures, in The
Center for Ultracold Atoms,

540
00:37:17,340 --> 00:37:20,970
but the KT at room
temperature corresponds

541
00:37:20,970 --> 00:37:22,220
to 25 milli-electron volt.

542
00:37:31,550 --> 00:37:42,930
So therefore, when we
have an atom in isolation,

543
00:37:42,930 --> 00:37:44,590
this is what we find.

544
00:37:44,590 --> 00:37:47,090
We find an atom which
will irreversibly decay

545
00:37:47,090 --> 00:37:48,960
to the ground state.

546
00:37:48,960 --> 00:37:50,650
And the fact that
it irreversibly

547
00:37:50,650 --> 00:37:52,830
decays to the ground
state is really

548
00:37:52,830 --> 00:37:55,070
an inequality between energies.

549
00:37:55,070 --> 00:37:57,500
If you will talk about
a hyperfine transition

550
00:37:57,500 --> 00:38:01,460
or something, there
may be a possibility

551
00:38:01,460 --> 00:38:05,660
that we have an excited state,
which is thermally excited.

552
00:38:05,660 --> 00:38:09,030
But in the following discussion,
when we drive the atom,

553
00:38:09,030 --> 00:38:11,290
and when we look at
spontaneous decay,

554
00:38:11,290 --> 00:38:17,330
we always assume that the
thermal energies are so small,

555
00:38:17,330 --> 00:38:21,930
that we really assume an atom
sitting in a cold vacuum.

556
00:38:21,930 --> 00:38:24,480
Actually, it's your next
homework assignment,

557
00:38:24,480 --> 00:38:26,350
where you will
consider, what are

558
00:38:26,350 --> 00:38:28,560
the effects of
black-body radiation.

559
00:38:28,560 --> 00:38:32,070
And you will actually
find out in your homework

560
00:38:32,070 --> 00:38:34,440
that they are non-negligible.

561
00:38:34,440 --> 00:38:35,844
So yes, there are corrections.

562
00:38:35,844 --> 00:38:38,260
But you will also find out
that the corrections are rather

563
00:38:38,260 --> 00:38:43,880
small, or it takes a long time
before black-body radiation

564
00:38:43,880 --> 00:38:45,800
induces any
observable transition.

565
00:38:48,440 --> 00:38:48,940
OK.

566
00:38:48,940 --> 00:38:52,750
So I'll just try to be a
little bit formal here.

567
00:38:52,750 --> 00:38:56,420
Give you sort of a sketch
of an atom in a cold vacuum.

568
00:38:56,420 --> 00:38:57,700
Ground state is stable.

569
00:38:57,700 --> 00:39:00,920
Excited state,
irreversibly decays.

570
00:39:00,920 --> 00:39:08,950
And now, we want to bring
life into this situation.

571
00:39:08,950 --> 00:39:10,050
Now we add light.

572
00:39:17,540 --> 00:39:21,300
And the light-- and this
is now our discussion--

573
00:39:21,300 --> 00:39:22,540
has a [INAUDIBLE].

574
00:39:26,870 --> 00:39:31,375
And we want to
distinguish the cases

575
00:39:31,375 --> 00:39:36,580
of narrow-band and
broadband radiation.

576
00:39:36,580 --> 00:39:43,630
So it's clear that
if the bandwidth

577
00:39:43,630 --> 00:39:50,496
of the light, the
only scale-- well,

578
00:39:50,496 --> 00:39:51,620
we have the scale of omega.

579
00:39:51,620 --> 00:39:53,260
But that's a huge scale.

580
00:39:53,260 --> 00:39:56,770
The only smaller scale,
which is given by the atom,

581
00:39:56,770 --> 00:39:59,410
is the natural linewidth.

582
00:39:59,410 --> 00:40:04,900
And depending, in
which case we are,

583
00:40:04,900 --> 00:40:12,510
we talk about
narrow-band excitation

584
00:40:12,510 --> 00:40:15,306
and broadband radiation.

585
00:40:19,040 --> 00:40:25,390
And once the linewidth is
much narrower than gamma,

586
00:40:25,390 --> 00:40:29,270
we don't get any
new physics when

587
00:40:29,270 --> 00:40:31,582
we assume perfectly
monochromatic light.

588
00:40:31,582 --> 00:40:33,290
So once we are much
smaller, we're really

589
00:40:33,290 --> 00:40:35,430
discussing the case
of, well, we can

590
00:40:35,430 --> 00:40:38,790
neglect the spectrum
broadening of the light source.

591
00:40:38,790 --> 00:40:41,270
Or in the other case, when
we have broadband light,

592
00:40:41,270 --> 00:40:43,870
we can pretty much
make the assumption

593
00:40:43,870 --> 00:40:46,020
that the light is
infinitely broad,

594
00:40:46,020 --> 00:40:49,772
and what matters is only the
spectral density of the light.

595
00:40:54,400 --> 00:41:03,430
So in a pictorial
representation,

596
00:41:03,430 --> 00:41:12,590
if this is the
frequency omega, we

597
00:41:12,590 --> 00:41:22,900
have the atom with the
natural linewidth gamma.

598
00:41:26,130 --> 00:41:30,390
Narrow-band means we are
much sharper than that.

599
00:41:37,440 --> 00:41:41,200
And broadband means
really wide distribution.

600
00:41:44,650 --> 00:41:54,620
So if we have
broadband light, it

601
00:41:54,620 --> 00:41:56,910
doesn't really matter
what the total power is.

602
00:41:56,910 --> 00:41:58,365
If the light is
very broad, there

603
00:41:58,365 --> 00:42:00,140
can be infinite
power in the wings,

604
00:42:00,140 --> 00:42:02,000
but the atoms don't care.

605
00:42:02,000 --> 00:42:04,610
What matters when we
have broadband radiation

606
00:42:04,610 --> 00:42:08,410
is the quantity called
the spectral density.

607
00:42:11,190 --> 00:42:14,130
And that's what we
need in the following.

608
00:42:14,130 --> 00:42:18,170
Which is, let me just
give you the units.

609
00:42:18,170 --> 00:42:29,400
Which is energy per volume
and frequency interval.

610
00:42:32,370 --> 00:42:38,030
So we can talk about the
spectral density as of omega.

611
00:42:38,030 --> 00:42:46,055
Or alternatively, when we
have a propagating beam,

612
00:42:46,055 --> 00:42:47,780
we don't want to
talk about energy,

613
00:42:47,780 --> 00:42:52,550
we want to talk about intensity.

614
00:42:52,550 --> 00:43:01,490
So it is intensity per
unit frequency interval.

615
00:43:01,490 --> 00:43:08,200
Which would mean I of omega
is the energy density,

616
00:43:08,200 --> 00:43:09,840
multiplied with
the speed of light.

617
00:43:13,360 --> 00:43:20,900
And that becomes energy
per area and time.

618
00:43:20,900 --> 00:43:24,090
So that's the flow of energy.

619
00:43:24,090 --> 00:43:35,660
But because we are
talking about board light,

620
00:43:35,660 --> 00:43:40,270
it has to be normalized
by the frequency interval.

621
00:43:40,270 --> 00:43:50,200
In contrast,
monochromatic radiation,

622
00:43:50,200 --> 00:43:53,460
it's sort of one
monochromatic electric field.

623
00:43:53,460 --> 00:43:59,210
And we will specify it by
the single frequency, omega,

624
00:43:59,210 --> 00:44:02,410
and the electric
field amplitude.

625
00:44:10,260 --> 00:44:14,810
Which when multiplied
by a matrix element

626
00:44:14,810 --> 00:44:15,980
becomes the Rabi frequency.

627
00:44:20,310 --> 00:44:27,570
Or we can characterize the
light by the intensity I.

628
00:44:27,570 --> 00:44:30,010
But then it's an
intensity which has

629
00:44:30,010 --> 00:44:32,570
the units of energy
per area time.

630
00:44:32,570 --> 00:44:35,050
It's not normalized to
any frequency interval,

631
00:44:35,050 --> 00:44:37,445
because we have assumed that
the frequency interval is 0.

632
00:44:45,900 --> 00:44:54,700
So if you now have a description
how these two forms of light

633
00:44:54,700 --> 00:44:59,020
interact with the
atom, at this point,

634
00:44:59,020 --> 00:45:01,050
and we come to that
later this week,

635
00:45:01,050 --> 00:45:07,330
we have to make an
assumption that we

636
00:45:07,330 --> 00:45:12,670
are looking at
times which are much

637
00:45:12,670 --> 00:45:16,120
smaller than the time
for spontaneous emission.

638
00:45:16,120 --> 00:45:19,440
So if you now, in a
perturbative sense,

639
00:45:19,440 --> 00:45:25,070
expose the atom's monochromatic
or broadband radiation,

640
00:45:25,070 --> 00:45:28,200
unless we have included
in the description

641
00:45:28,200 --> 00:45:30,455
the many, many modes for
spontaneous emission,

642
00:45:30,455 --> 00:45:33,790
we are limiting ourselves
to a very short time.

643
00:45:33,790 --> 00:45:35,850
This is, you would say,
a severe description,

644
00:45:35,850 --> 00:45:38,920
because atoms emit photons
after a short time.

645
00:45:38,920 --> 00:45:42,160
But we already capture,
without considering

646
00:45:42,160 --> 00:45:45,100
spontaneous emission, a
lot of different physics.

647
00:45:45,100 --> 00:45:47,725
And we can nicely
distinguish between features

648
00:45:47,725 --> 00:45:50,825
of monochromatic and features
of broadband excitation.

649
00:46:03,816 --> 00:46:04,316
OK.

650
00:46:07,220 --> 00:46:14,990
So let's start out with the
case of-- give me a second.

651
00:46:22,960 --> 00:46:23,690
OK.

652
00:46:23,690 --> 00:46:28,180
So if you look at the two cases,
in the monochromatic case,

653
00:46:28,180 --> 00:46:31,660
we will discuss the
idealized situation

654
00:46:31,660 --> 00:46:34,870
of an atom interacting
only with a single mode.

655
00:46:34,870 --> 00:46:40,090
And what we will find out is,
we will find out that now,

656
00:46:40,090 --> 00:46:43,370
in the optical domain,
we will find actually

657
00:46:43,370 --> 00:46:45,830
equations for the
two-level system which

658
00:46:45,830 --> 00:46:49,124
are identical to what
we discussed earlier

659
00:46:49,124 --> 00:46:50,540
when we discussed
spin [INAUDIBLE]

660
00:46:50,540 --> 00:46:52,320
in a magnetic field.

661
00:46:52,320 --> 00:46:54,340
So in that sense,
a two level system,

662
00:46:54,340 --> 00:46:57,600
driven by a laser system,
will behave identically

663
00:46:57,600 --> 00:47:00,410
to a spin driven by
a magnetic field.

664
00:47:00,410 --> 00:47:04,180
Shouldn't come as a surprise,
but I will show that to you.

665
00:47:04,180 --> 00:47:07,730
But I can go over
that very quickly.

666
00:47:07,730 --> 00:47:11,630
The board-band
case will actually

667
00:47:11,630 --> 00:47:13,910
follow from the
single mode case,

668
00:47:13,910 --> 00:47:16,150
because what we
assume is broadband

669
00:47:16,150 --> 00:47:18,810
means many, many modes.

670
00:47:18,810 --> 00:47:22,370
And then we do an averaging
over many single modes

671
00:47:22,370 --> 00:47:25,440
by assuming random freeze.

672
00:47:25,440 --> 00:47:27,400
But I also want
to show it to you

673
00:47:27,400 --> 00:47:29,550
because I picked
my verbs carefully.

674
00:47:29,550 --> 00:47:32,610
You have many, many
more things, but we

675
00:47:32,610 --> 00:47:34,930
assume that there
is a random phase.

676
00:47:34,930 --> 00:47:37,940
When we talked about one
photon emitted into a angle--

677
00:47:37,940 --> 00:47:43,170
it maybe responds to
a question earlier--

678
00:47:43,170 --> 00:47:45,430
this photon may be in a
coherent superposition.

679
00:47:45,430 --> 00:47:48,260
This is not many modes
in a broadband wave.

680
00:47:48,260 --> 00:47:50,750
Many modes in
broadband wave means

681
00:47:50,750 --> 00:47:54,060
that there is no correlation
whatsoever between the modes,

682
00:47:54,060 --> 00:47:56,220
and all we will be
able to talk about is

683
00:47:56,220 --> 00:47:59,360
an IMS value of
an electric field.

684
00:47:59,360 --> 00:48:01,830
But anyway, the result
is sort of predictable,

685
00:48:01,830 --> 00:48:04,760
and I wanted to tell
you what I'm aiming for.

686
00:48:04,760 --> 00:48:09,340
But it's now really worthwhile
to go through those exercises

687
00:48:09,340 --> 00:48:13,140
and look at what
happens in perturbation

688
00:48:13,140 --> 00:48:23,749
theory for short times when we
have monochromatic radiation,

689
00:48:23,749 --> 00:48:25,290
and when we have
broadband radiation.

690
00:48:33,820 --> 00:48:41,780
So the first discussion
will show Rabi flopping.

691
00:48:41,780 --> 00:48:43,430
I don't know how
many times we have

692
00:48:43,430 --> 00:48:45,080
looked at Rabi oscillation.

693
00:48:45,080 --> 00:48:46,810
But these are now
Rabi oscillations

694
00:48:46,810 --> 00:48:51,130
between two electronic states
covered by a laser beam.

695
00:48:51,130 --> 00:48:53,980
And I want to show you
how this comes about.

696
00:48:53,980 --> 00:48:56,295
And when I said strong
driving, well, we

697
00:48:56,295 --> 00:48:59,840
have only a limited time window
before spontaneous emission

698
00:48:59,840 --> 00:49:00,520
happens.

699
00:49:00,520 --> 00:49:03,100
We have to discuss
the physics we

700
00:49:03,100 --> 00:49:05,350
want to discuss in
this shot time window.

701
00:49:05,350 --> 00:49:06,860
And if you want
to excite an atom,

702
00:49:06,860 --> 00:49:09,390
and see Rabi oscillation
in a short time,

703
00:49:09,390 --> 00:49:11,120
you better have a
strong laser beam.

704
00:49:11,120 --> 00:49:16,680
So this is why the monochromatic
excitation that we discussed

705
00:49:16,680 --> 00:49:19,860
will pretty much automatically
be in this strong coupling

706
00:49:19,860 --> 00:49:21,620
limit.

707
00:49:21,620 --> 00:49:22,200
OK.

708
00:49:22,200 --> 00:49:24,320
So what do we have?

709
00:49:24,320 --> 00:49:28,220
We have a ground,and we
have an excited state.

710
00:49:28,220 --> 00:49:29,380
We have a matrix element.

711
00:49:29,380 --> 00:49:31,870
We know now where it comes from.

712
00:49:31,870 --> 00:49:33,875
And we have a monochromatic
time dependence.

713
00:49:44,290 --> 00:49:47,490
In perturbation
theory, we build up

714
00:49:47,490 --> 00:49:51,480
time-dependent [INAUDIBLE]
amplitude in the excited

715
00:49:51,480 --> 00:49:55,470
state, because we
couple the ground

716
00:49:55,470 --> 00:49:59,000
state with the
off-diagonal matrix

717
00:49:59,000 --> 00:50:01,910
element to the excited state.

718
00:50:01,910 --> 00:50:05,490
And we have to integrate
from the initial time

719
00:50:05,490 --> 00:50:06,405
to the final time.

720
00:50:11,270 --> 00:50:18,190
We have the time dependence
of the electromagnetic field,

721
00:50:18,190 --> 00:50:20,100
and we also need
the time dependence

722
00:50:20,100 --> 00:50:21,220
of the excited state.

723
00:50:25,280 --> 00:50:28,780
So when I integrate
now over t prime,

724
00:50:28,780 --> 00:50:31,410
I take out the ground
state amplitude,

725
00:50:31,410 --> 00:50:33,250
because we're doing
perturbation theory,

726
00:50:33,250 --> 00:50:36,730
and we assume that for
short times, needing order,

727
00:50:36,730 --> 00:50:39,610
the ground state amplitude is
one, as prepared initially.

728
00:50:43,390 --> 00:50:49,430
So this in integral can
be solved analytically.

729
00:51:06,170 --> 00:51:11,790
Some of you may remember
that the minus 1

730
00:51:11,790 --> 00:51:15,270
has something to do with the
lower bound of the integral.

731
00:51:15,270 --> 00:51:20,070
And when we discuss the easy
polarizability, we said,

732
00:51:20,070 --> 00:51:23,760
this is a transient, and we
neglected it for good reasons.

733
00:51:23,760 --> 00:51:26,010
But now, we're really
interested in the time evolution

734
00:51:26,010 --> 00:51:27,975
of the system, so now
we have to keep it.

735
00:51:33,460 --> 00:51:33,960
OK.

736
00:51:33,960 --> 00:51:36,750
We are interested in the
probability in the excited

737
00:51:36,750 --> 00:51:37,520
state.

738
00:51:37,520 --> 00:51:42,880
So we take the above
expression and square it.

739
00:51:42,880 --> 00:51:59,080
And we find the well-known
result, with sine squared,

740
00:51:59,080 --> 00:52:02,580
divided by omega minus omega eg.

741
00:52:08,890 --> 00:52:09,390
OK.

742
00:52:14,620 --> 00:52:16,540
So this is pretty much
just straightforward,

743
00:52:16,540 --> 00:52:19,280
writing down an
analytic expression.

744
00:52:19,280 --> 00:52:23,690
But now, let's discuss it.

745
00:52:23,690 --> 00:52:31,310
For very short times, and this
is an important limiting case,

746
00:52:31,310 --> 00:52:34,880
the probability in
the excited state

747
00:52:34,880 --> 00:52:38,000
is proportionate
to times squared.

748
00:52:49,654 --> 00:52:50,570
And this is important.

749
00:52:50,570 --> 00:52:54,650
We're not getting a rate
which is proportional to time.

750
00:52:54,650 --> 00:52:57,450
We're obtaining something
which is time square.

751
00:52:57,450 --> 00:53:00,850
And the proportionality
to t square

752
00:53:00,850 --> 00:53:03,070
means it's a fully
coherent process.

753
00:53:07,980 --> 00:53:10,132
So whenever somebody
asks you, you

754
00:53:10,132 --> 00:53:12,730
switch on a strong
coupling from a ground

755
00:53:12,730 --> 00:53:15,830
to the excited state, what is
the probability in the excited

756
00:53:15,830 --> 00:53:16,510
state?

757
00:53:16,510 --> 00:53:19,210
It starts out quadratically.

758
00:53:19,210 --> 00:53:22,790
The linear dependence-- famous
golden rule, [INAUDIBLE]

759
00:53:22,790 --> 00:53:24,460
or such-- only come later.

760
00:53:24,460 --> 00:53:27,090
This is a very
universal feature.

761
00:53:27,090 --> 00:53:33,440
And even if you use
broadened light,

762
00:53:33,440 --> 00:53:37,870
for a time window,
delta T, which

763
00:53:37,870 --> 00:53:43,720
is shorter than the inverse
bandwidth of the light,

764
00:53:43,720 --> 00:53:45,620
talking about
Fourier's theory, you

765
00:53:45,620 --> 00:53:47,480
don't have time
to even figure out

766
00:53:47,480 --> 00:53:49,850
that your light is broad
and not monochromatic.

767
00:53:49,850 --> 00:53:52,590
For very short times,
the Fourier limit

768
00:53:52,590 --> 00:53:54,010
does not allow
you to distinguish

769
00:53:54,010 --> 00:53:55,950
whether the light is
broad or monochromatic.

770
00:53:55,950 --> 00:53:58,640
So what I just derived for
you, an initial quadratic

771
00:53:58,640 --> 00:54:01,580
dependence, is the
universal behavior

772
00:54:01,580 --> 00:54:06,090
of a quantum system
at very short times.

773
00:54:06,090 --> 00:54:08,770
Because it simply says the
amplitude in the excited state

774
00:54:08,770 --> 00:54:11,020
goes linearly in time, and
the probability, quadratic.

775
00:54:16,000 --> 00:54:17,540
OK.

776
00:54:17,540 --> 00:54:20,650
So this is for a
very short times.

777
00:54:20,650 --> 00:54:34,670
But if you look at it
now for longer times,

778
00:54:34,670 --> 00:54:38,960
we have actually-- we'll
see the atomic behavior,

779
00:54:38,960 --> 00:54:42,610
and these are Rabi oscillations.

780
00:54:42,610 --> 00:54:43,680
But there is one caveat.

781
00:54:46,999 --> 00:54:47,790
So we have derived.

782
00:54:54,370 --> 00:55:01,930
However, we have derived them
only perturbatively by assuming

783
00:55:01,930 --> 00:55:04,390
that the ground state
has always a population

784
00:55:04,390 --> 00:55:20,835
close to 100%, which
means we have assumed

785
00:55:20,835 --> 00:55:26,370
that the probability
in the excited state

786
00:55:26,370 --> 00:55:27,470
is much smaller than 1.

787
00:55:27,470 --> 00:55:30,120
Otherwise, we wouldn't
keep the ground state.

788
00:55:30,120 --> 00:55:34,430
And this is only fulfilled
if you inspect the solution.

789
00:55:34,430 --> 00:55:36,810
The solution is
only self-consistent

790
00:55:36,810 --> 00:55:39,590
if you have an off-resonant
case, where the Rabi

791
00:55:39,590 --> 00:55:42,992
oscillation only comes from a
small fraction of the ground

792
00:55:42,992 --> 00:55:45,000
state population of
the excited state.

793
00:55:45,000 --> 00:55:50,270
Of course, you all know
that Rabi oscillations,

794
00:55:50,270 --> 00:55:52,870
this formula, is also
varied on-resonance.

795
00:55:52,870 --> 00:55:55,240
And you can have
full Rabi flopping.

796
00:55:55,240 --> 00:55:59,090
But I want to make a case
here, distinguish carefully

797
00:55:59,090 --> 00:56:03,020
between monochromatic radiation
and broadband radiation.

798
00:56:03,020 --> 00:56:05,380
For that, I need for
perturbation theory.

799
00:56:05,380 --> 00:56:08,490
And therefore, I'm telling you
what perturbation theory gives

800
00:56:08,490 --> 00:56:11,880
us at short times, and in
terms of Rabi oscillations.

801
00:56:21,120 --> 00:56:23,910
STUDENT: So you're
saying we assume

802
00:56:23,910 --> 00:56:27,940
strong coupling with respect
to the atomic linewidth,

803
00:56:27,940 --> 00:56:35,368
but weak coupling with respect
to the resonance, for instance,

804
00:56:35,368 --> 00:56:36,350
in [INAUDIBLE]?

805
00:56:39,822 --> 00:56:41,280
PROFESSOR: It's
simple, but subtle.

806
00:56:41,280 --> 00:56:41,780
Yes.

807
00:56:41,780 --> 00:56:43,980
So what we have is,
we assume we switch

808
00:56:43,980 --> 00:56:45,760
on a monochromatic laser.

809
00:56:45,760 --> 00:56:48,900
Since we do not include
spontaneous emission, which

810
00:56:48,900 --> 00:56:52,200
will actually damp out Rabi
oscillation-- we'll talk about

811
00:56:52,200 --> 00:56:55,790
that later-- we
are only limited,

812
00:56:55,790 --> 00:57:00,020
we are limited here
to short times, which

813
00:57:00,020 --> 00:57:04,160
are shorter than the
spontaneous decay.

814
00:57:04,160 --> 00:57:07,070
And now, I gave you
one universal thing.

815
00:57:07,070 --> 00:57:10,080
At very, very short times,
it's always quadratic.

816
00:57:10,080 --> 00:57:11,800
It's a coherent process.

817
00:57:11,800 --> 00:57:14,790
So that's one simple,
limiting, exact case

818
00:57:14,790 --> 00:57:16,560
you should keep in your mind.

819
00:57:16,560 --> 00:57:20,680
But now the question is, if
you let the time go longer,

820
00:57:20,680 --> 00:57:22,770
something will happen.

821
00:57:22,770 --> 00:57:25,290
And there are several options.

822
00:57:25,290 --> 00:57:28,610
One is, if times go longer,
spontaneous emission happens.

823
00:57:28,610 --> 00:57:29,110
OK.

824
00:57:29,110 --> 00:57:30,230
We are invalid.

825
00:57:30,230 --> 00:57:35,700
The other possibility is,
when time gets longer,

826
00:57:35,700 --> 00:57:40,220
and we are on-resonance, we
deplete the ground state,

827
00:57:40,220 --> 00:57:43,260
or [INAUDIBLE] perturbation
theory doesn't deal with that.

828
00:57:43,260 --> 00:57:46,940
But if we are off-resonance,
we can allow time

829
00:57:46,940 --> 00:57:51,240
to go over many Rabi periods
and observe perturbative Rabi

830
00:57:51,240 --> 00:57:52,210
oscillations.

831
00:57:52,210 --> 00:57:54,540
So this is how we
have formulated it.

832
00:57:54,540 --> 00:57:58,520
We do perturbation
theory of the system

833
00:57:58,520 --> 00:58:00,440
without spontaneous emission.

834
00:58:00,440 --> 00:58:02,960
And eventually, we
violate our assumptions,

835
00:58:02,960 --> 00:58:05,970
either because spontaneous
emission kicks in,

836
00:58:05,970 --> 00:58:08,740
or because we deplete
the ground state

837
00:58:08,740 --> 00:58:13,650
when we drive it too hard, or
if we go too close to resonance.

838
00:58:13,650 --> 00:58:15,620
But the later
assumption, of course,

839
00:58:15,620 --> 00:58:19,200
that we can't drive it hard,
as you know, is artificial.

840
00:58:19,200 --> 00:58:23,540
We can actually discuss
the monochromatic case.

841
00:58:23,540 --> 00:58:26,620
Not just in perturbation theory,
but we can do it exactly.

842
00:58:30,050 --> 00:58:32,020
STUDENT: I want to
go back again to--

843
00:58:32,020 --> 00:58:33,895
PROFESSOR: And this is
what I want to do now.

844
00:58:33,895 --> 00:58:35,532
But first, we can go back.

845
00:58:38,508 --> 00:58:42,630
STUDENT: So when we are
talking about non-B resonance

846
00:58:42,630 --> 00:58:47,320
and B-resonance, so if
we decrease the detuning,

847
00:58:47,320 --> 00:58:49,700
then we are getting
close to resonance.

848
00:58:49,700 --> 00:58:52,680
So again, this gets invalid.

849
00:58:52,680 --> 00:58:54,970
But if we increase
the detuning, we

850
00:58:54,970 --> 00:58:57,230
could exceed the
spontaneous emission rate.

851
00:58:57,230 --> 00:59:01,010
So then, we won't see any
Rabi oscillations again,

852
00:59:01,010 --> 00:59:04,200
because, at those time
periods, this oscillation

853
00:59:04,200 --> 00:59:06,650
would [INAUDIBLE] detuning.

854
00:59:06,650 --> 00:59:09,623
So to observe Rabi
oscillations, we

855
00:59:09,623 --> 00:59:11,414
have to be at times
more than the detuning,

856
00:59:11,414 --> 00:59:14,517
or more than
[INAUDIBLE] detuning.

857
00:59:14,517 --> 00:59:15,350
PROFESSOR: Oh, yeah.

858
00:59:15,350 --> 00:59:16,340
Of course.

859
00:59:16,340 --> 00:59:20,450
STUDENT: So the detuning has
to be less than [INAUDIBLE],

860
00:59:20,450 --> 00:59:24,957
but more [INAUDIBLE] that we
are still [INAUDIBLE] resonant.

861
00:59:24,957 --> 00:59:25,540
PROFESSOR: No.

862
00:59:25,540 --> 00:59:33,630
The detuning has to be larger
than the natural linewidth,

863
00:59:33,630 --> 00:59:36,410
because then the Rabi
oscillations are fast,

864
00:59:36,410 --> 00:59:38,160
and we have Rabi
oscillations which

865
00:59:38,160 --> 00:59:42,160
are faster than any damping
due to spontaneous decay.

866
00:59:42,160 --> 00:59:43,950
That's an image we
are talking about.

867
00:59:43,950 --> 00:59:47,280
So in the limit of our detuning,
you can detune very, very far,

868
00:59:47,280 --> 00:59:51,084
and you never reach the limit
of our perturbative abode.

869
00:59:51,084 --> 00:59:52,080
STUDENT: Yes.

870
00:59:52,080 --> 00:59:53,842
OK.

871
00:59:53,842 --> 00:59:55,675
PROFESSOR: Anyway, I
want to do perturbation

872
00:59:55,675 --> 00:59:57,960
theory of the broadband case.

873
00:59:57,960 --> 01:00:00,650
And the broadband case
will be an incoherent sum

874
01:00:00,650 --> 01:00:02,200
over the single mode case.

875
01:00:02,200 --> 01:00:04,882
So this is why I had
to bore you with,

876
01:00:04,882 --> 01:00:06,590
what do we get out of
perturbation theory

877
01:00:06,590 --> 01:00:08,900
for the monochromatic case?

878
01:00:08,900 --> 01:00:12,320
Of course, you know already
that in a two-level system,

879
01:00:12,320 --> 01:00:14,100
we can do it exactly.

880
01:00:14,100 --> 01:00:16,235
And I just want to
outline it, mainly

881
01:00:16,235 --> 01:00:17,440
to introduce some notation.

882
01:00:20,330 --> 01:00:27,460
So our Hamiltonian
here, which couples

883
01:00:27,460 --> 01:00:39,270
the ground in the excited state
is given by the dipole matrix

884
01:00:39,270 --> 01:00:45,220
element, the electric
field vector,

885
01:00:45,220 --> 01:00:48,310
and we call this
the Rabi frequency.

886
01:00:48,310 --> 01:00:51,850
And then we have a sinusoidal
or co-sinusoidal frequency

887
01:00:51,850 --> 01:00:53,990
dependence.

888
01:00:53,990 --> 01:00:56,690
And all I want to
do is to show you

889
01:00:56,690 --> 01:01:00,500
that a two-level system driven
by an electromagnetic field

890
01:01:00,500 --> 01:01:04,090
is identical to spin 1/2,
which we discussed earlier,

891
01:01:04,090 --> 01:01:04,970
and then we are done.

892
01:01:07,620 --> 01:01:11,300
There is one technical or
little trick we have to do,

893
01:01:11,300 --> 01:01:14,320
which is trivial, but
I want to mention it.

894
01:01:14,320 --> 01:01:23,730
So if you want to compare
directly with spin 1/2,

895
01:01:23,730 --> 01:01:32,690
we are now shifting
the ground state

896
01:01:32,690 --> 01:01:36,180
to half the
excitation frequency.

897
01:01:36,180 --> 01:01:39,690
In other words, just to make
the key analogy with the spin,

898
01:01:39,690 --> 01:01:42,710
usually we say for an electronic
transition, we start at 0,

899
01:01:42,710 --> 01:01:43,940
and we go up.

900
01:01:43,940 --> 01:01:47,550
But now we shift things
that the zero of energy

901
01:01:47,550 --> 01:01:50,610
is in the middle between the
ground and the excited state.

902
01:01:50,610 --> 01:01:53,740
And then, it looks like the
excited state, we spin up,

903
01:01:53,740 --> 01:01:56,430
the ground state, we spin down.

904
01:01:56,430 --> 01:02:10,000
So with that, our Hamiltonian
is now excited, excited,

905
01:02:10,000 --> 01:02:15,045
minus-- so all I've done
is I've shifted the origin.

906
01:02:17,980 --> 01:02:23,060
And the coupling, using
our definition of the Rabi

907
01:02:23,060 --> 01:02:36,700
frequency is couples
ground and excited state.

908
01:02:41,480 --> 01:02:42,530
And excited ground state.

909
01:02:42,530 --> 01:02:45,050
These are the two
off-diagonal matrix element.

910
01:02:45,050 --> 01:02:50,450
And the time dependence
is cosine omega t.

911
01:02:50,450 --> 01:02:55,280
So we are now very close to
exploit the correspondence

912
01:02:55,280 --> 01:02:57,790
with spin 1/2.

913
01:02:57,790 --> 01:03:01,360
Because after shifting
the ground state energy,

914
01:03:01,360 --> 01:03:05,460
this is the z component
of the spin operator,

915
01:03:05,460 --> 01:03:07,720
the [INAUDIBLE] matrix.

916
01:03:07,720 --> 01:03:12,230
And this here is
the x component.

917
01:03:16,090 --> 01:03:22,600
So therefore, for driving
an electronic transition

918
01:03:22,600 --> 01:03:30,660
with a laser beam,
we have actually

919
01:03:30,660 --> 01:03:42,691
spin Hamiltonian, which
has the standard form.

920
01:03:42,691 --> 01:03:44,440
So let me just write
it down, because it's

921
01:03:44,440 --> 01:03:46,620
an important result.

922
01:03:46,620 --> 01:03:51,220
The Hamiltonian for driving
and dipole transition

923
01:03:51,220 --> 01:03:58,260
with a linearly
polarized laser beam

924
01:03:58,260 --> 01:04:06,260
corresponds, or is identical,
to the Hamiltonian for spin 1/2,

925
01:04:06,260 --> 01:04:13,000
in a static magnetic field
along the Z direction,

926
01:04:13,000 --> 01:04:16,370
which causes a splitting
between spin up and spin down.

927
01:04:16,370 --> 01:04:24,030
And the splitting
is now omega eg

928
01:04:24,030 --> 01:04:28,550
plus a linearly
polarized oscillating

929
01:04:28,550 --> 01:04:45,449
field along the x direction.

930
01:04:52,790 --> 01:04:59,470
And you probably remember that
when we discussed the spin

931
01:04:59,470 --> 01:05:01,940
problem, what we
liked actually most

932
01:05:01,940 --> 01:05:06,500
was that we had a rotating
magnetic field, because it

933
01:05:06,500 --> 01:05:08,590
made everything simpler.

934
01:05:08,590 --> 01:05:16,200
And we are doing that
now by formally writing

935
01:05:16,200 --> 01:05:18,880
the [INAUDIBLE] polarized
field as a superposition

936
01:05:18,880 --> 01:05:22,660
of left-handed and right-handed,
or counter-rotating

937
01:05:22,660 --> 01:05:25,370
and co-rotating magnetic field.

938
01:05:25,370 --> 01:05:26,550
So let me just do that.

939
01:05:32,920 --> 01:05:34,690
So we have the Z part.

940
01:05:39,980 --> 01:05:47,160
And now, instead of having
just sigma x, cosine omega t,

941
01:05:47,160 --> 01:05:52,450
I add sigma y,
sine omega t, and I

942
01:05:52,450 --> 01:05:56,400
subtract sigma y, sine omega t.

943
01:06:16,000 --> 01:06:20,560
So now we have shown
that there is something

944
01:06:20,560 --> 01:06:22,800
in addition to the spin problem.

945
01:06:22,800 --> 01:06:26,780
We discover when we had a
rotating magnetic field,

946
01:06:26,780 --> 01:06:34,110
that we have two components
here which rotate.

947
01:06:34,110 --> 01:06:51,017
And these are the co and
counter rotating magnetic fields

948
01:06:51,017 --> 01:06:51,850
in the spin problem.

949
01:06:56,940 --> 01:07:12,040
And the counter rotating, you
remember in the spin problem,

950
01:07:12,040 --> 01:07:16,540
we solved the problem
exactly by going

951
01:07:16,540 --> 01:07:21,150
into a frame which rotated at
the Larmor frequency, which

952
01:07:21,150 --> 01:07:24,960
becomes now omega eg.

953
01:07:24,960 --> 01:07:30,970
And the co-rotating term
became stationary on resonance

954
01:07:30,970 --> 01:07:36,490
in this rotating frame, whereas
the counter-rotating term

955
01:07:36,490 --> 01:07:46,830
rotates at a very high frequency
in this frame at the Larmor

956
01:07:46,830 --> 01:07:47,330
frequency.

957
01:07:54,870 --> 01:08:05,140
So if this frequency, if
you fulfill the inequalities

958
01:08:05,140 --> 01:08:14,200
that the co-rotating term
is close to resonance,

959
01:08:14,200 --> 01:08:18,560
or in other words, we
are close to resonance,

960
01:08:18,560 --> 01:08:23,450
and we are not using an infinite
intensity of the laser beam,

961
01:08:23,450 --> 01:08:25,630
that we broaden everything
in co and counter

962
01:08:25,630 --> 01:08:28,310
rotating terms of
boson resonance.

963
01:08:28,310 --> 01:08:34,979
So if you fulfill
those two conditions,

964
01:08:34,979 --> 01:08:42,189
then we can neglect
the last term.

965
01:08:42,189 --> 01:08:46,671
And this is the rotating
wave approximation.

966
01:08:55,029 --> 01:08:57,359
So in other words,
in the spin problem,

967
01:08:57,359 --> 01:09:00,120
we can always assume
we haven't circularly

968
01:09:00,120 --> 01:09:02,590
polarized the rotating
magnetic field,

969
01:09:02,590 --> 01:09:06,200
and we have an exact solution.

970
01:09:06,200 --> 01:09:08,899
I say a little bit
more about it later.

971
01:09:08,899 --> 01:09:13,140
But in many situations, when
you excite an atom with a laser

972
01:09:13,140 --> 01:09:15,029
beam, you get both terms.

973
01:09:15,029 --> 01:09:19,294
And usually, you proceed
by neglecting one term,

974
01:09:19,294 --> 01:09:23,359
and by making the rotating
wave approximation.

975
01:09:23,359 --> 01:09:28,350
will, in one or two
lectures, discuss

976
01:09:28,350 --> 01:09:30,920
whether there are situations
where the counter-rotating term

977
01:09:30,920 --> 01:09:33,720
is exactly 0 due to angular
momentum selection rules,

978
01:09:33,720 --> 01:09:35,170
but that's a
separate discussion.

979
01:09:35,170 --> 01:09:38,840
In many situations, it cannot be
avoided, and it's always there.

980
01:09:38,840 --> 01:09:40,700
It's actually always
there to the point

981
01:09:40,700 --> 01:09:43,399
that when I talk to
some colleagues and say,

982
01:09:43,399 --> 01:09:47,000
I can create a
situation, an atom, where

983
01:09:47,000 --> 01:09:50,620
the counter-rotating
term is exactly 0,

984
01:09:50,620 --> 01:09:53,040
some colleagues
reacted with disbelief,

985
01:09:53,040 --> 01:09:55,630
and then eventually
felt that the situation

986
01:09:55,630 --> 01:09:57,830
I created for angular
momentum conservation

987
01:09:57,830 --> 01:09:59,090
was somewhat artificial.

988
01:09:59,090 --> 01:10:00,050
But we'll get there.

989
01:10:00,050 --> 01:10:01,840
It's an interesting discussion.

990
01:10:01,840 --> 01:10:07,020
But anyway, just remember that
for magnetic drive, if you

991
01:10:07,020 --> 01:10:08,850
use a rotating
magnetic field, you

992
01:10:08,850 --> 01:10:10,760
don't need a rotating
wave approximation.

993
01:10:10,760 --> 01:10:12,650
Everything rotates
at one frequency.

994
01:10:12,650 --> 01:10:16,970
But usually, when you drive a
two-level system with lasers,

995
01:10:16,970 --> 01:10:22,150
we usually have an extra term
which needs to be neglected.

996
01:10:22,150 --> 01:10:22,820
OK.

997
01:10:22,820 --> 01:10:26,170
But if you do the rotating
wave vapor approximation,

998
01:10:26,170 --> 01:10:29,160
we have now exactly
the situation

999
01:10:29,160 --> 01:10:35,360
we discussed for spin 1/2 in
a rotating magnetic field.

1000
01:10:35,360 --> 01:10:38,430
And then, the same equation
has the same results.

1001
01:10:42,130 --> 01:10:47,500
And then, our results for
spin 1/2 are now as expected.

1002
01:10:51,230 --> 01:10:56,550
Rabi oscillations without
making any assumptions

1003
01:10:56,550 --> 01:10:58,640
about perturbation theory.

1004
01:10:58,640 --> 01:11:04,720
So this is an exact result
for the initial conditions

1005
01:11:04,720 --> 01:11:07,610
that we start in
the ground state,

1006
01:11:07,610 --> 01:11:12,450
and the initial population
of the excited state is 0.

1007
01:11:12,450 --> 01:11:20,650
And as usual, I have used here
the generalized Rabi frequency,

1008
01:11:20,650 --> 01:11:23,770
which is the quadrature sum
of these matrix elements

1009
01:11:23,770 --> 01:11:27,360
squared and the detuning.

1010
01:11:46,730 --> 01:11:47,230
OK.

1011
01:11:47,230 --> 01:11:51,985
A lot of it was to get ready
for the broadband case.

1012
01:11:54,880 --> 01:11:58,440
So that's-- yes, we have a
little bit more than five

1013
01:11:58,440 --> 01:11:58,940
minutes.

1014
01:12:14,390 --> 01:12:18,090
So, so far, we have discussed
the monochromatic case.

1015
01:12:18,090 --> 01:12:22,130
What I really needed
as a new result,

1016
01:12:22,130 --> 01:12:24,220
because I carried over
for the board-band case,

1017
01:12:24,220 --> 01:12:26,040
was a perturbative result.

1018
01:12:26,040 --> 01:12:29,340
But I also wanted to show you
that the perturbative result is

1019
01:12:29,340 --> 01:12:33,020
one limiting case of the
exact solution, which I just

1020
01:12:33,020 --> 01:12:36,520
derived by analogy to spin 1/2.

1021
01:12:36,520 --> 01:12:37,180
OK.

1022
01:12:37,180 --> 01:12:52,440
So we just had the result
that in perturbation theory,

1023
01:12:52,440 --> 01:12:55,870
for sufficiently short
times, we discussed all that,

1024
01:12:55,870 --> 01:13:13,582
that the excited state amplitude
has the following dependence.

1025
01:13:21,760 --> 01:13:25,510
So this is nothing
else than-- I want

1026
01:13:25,510 --> 01:13:27,450
to make sure you
recognize it-- Rabi

1027
01:13:27,450 --> 01:13:31,630
oscillations at the
generalized Rabi frequency.

1028
01:13:31,630 --> 01:13:35,380
The generalized Rabi frequency
is simply the detuning,

1029
01:13:35,380 --> 01:13:37,505
because it's a
perturbative result.

1030
01:13:37,505 --> 01:13:38,880
In perturbation
theory, you don't

1031
01:13:38,880 --> 01:13:41,630
get power broadening, because
you assume that your drive

1032
01:13:41,630 --> 01:13:43,620
field is perturbatively weak.

1033
01:13:43,620 --> 01:13:45,870
So therefore, the
Rabi oscillation,

1034
01:13:45,870 --> 01:13:49,340
our now Rabi oscillation
where the Rabi frequency,

1035
01:13:49,340 --> 01:13:53,940
the generalized Rabi frequency,
is delta the detuning.

1036
01:13:53,940 --> 01:13:55,962
And this is just rewriting.

1037
01:13:55,962 --> 01:13:56,878
Let me just scroll up.

1038
01:14:04,200 --> 01:14:05,870
This is this result here.

1039
01:14:05,870 --> 01:14:07,270
I wasn't commenting on it.

1040
01:14:07,270 --> 01:14:11,450
But this is nothing
else than the detuning.

1041
01:14:11,450 --> 01:14:12,504
Look.

1042
01:14:12,504 --> 01:14:15,045
I'm just reminding you what you
get from perturbation theory.

1043
01:14:15,045 --> 01:14:19,550
Power broadening is not
part of perturbation theory.

1044
01:14:19,550 --> 01:14:20,100
OK.

1045
01:14:20,100 --> 01:14:24,280
So this is our
perturbative result.

1046
01:14:24,280 --> 01:14:27,020
And now, we want
to integrate over

1047
01:14:27,020 --> 01:14:33,864
that because we have a broadband
distribution of the light.

1048
01:14:38,040 --> 01:14:45,010
So what we have to use now is
the energy density, W of omega.

1049
01:14:56,190 --> 01:15:04,670
The electric field is
related, the energy

1050
01:15:04,670 --> 01:15:08,340
of the electromagnetic field, is
1/2 epsilon the energy density

1051
01:15:08,340 --> 01:15:11,380
of the electromagnetic field,
is 1/2 epsilon naught times

1052
01:15:11,380 --> 01:15:14,660
the electric field squared.

1053
01:15:14,660 --> 01:15:19,500
Well, if you have many modes,
we add the different modes

1054
01:15:19,500 --> 01:15:20,500
in quadrature.

1055
01:15:20,500 --> 01:15:22,430
And we still have
the same reaction

1056
01:15:22,430 --> 01:15:26,990
between the electric field
squared and the total energy.

1057
01:15:26,990 --> 01:15:31,950
But the total energy is now
an integral over d omega.

1058
01:15:31,950 --> 01:15:34,979
We integrate over frequency
over the spectral distribution

1059
01:15:34,979 --> 01:15:35,520
of the light.

1060
01:15:39,660 --> 01:15:44,800
So this is how we go from energy
density to electric feels.

1061
01:15:44,800 --> 01:15:48,370
But now, we want to
evaluate this expression.

1062
01:15:48,370 --> 01:15:50,700
And what appears
in this expression

1063
01:15:50,700 --> 01:15:53,920
is the Rabi frequency.

1064
01:15:53,920 --> 01:15:57,270
Well, what we have
to do now is we

1065
01:15:57,270 --> 01:16:01,280
have to go back from
the Rabi frequency.

1066
01:16:01,280 --> 01:16:05,700
We assume linearly polarized
light in the x direction

1067
01:16:05,700 --> 01:16:06,950
to the electric field.

1068
01:16:13,670 --> 01:16:37,890
And that means, now,
that when we-- OK.

1069
01:16:37,890 --> 01:16:39,960
We want to now take
this expression,

1070
01:16:39,960 --> 01:16:45,860
and sum it up over all modes,
which means we integrate over,

1071
01:16:45,860 --> 01:16:49,280
we write the Rabi
frequency squared

1072
01:16:49,280 --> 01:16:51,250
as an electric field squared.

1073
01:16:51,250 --> 01:16:53,040
And the electric
field squared is

1074
01:16:53,040 --> 01:16:55,609
obtained as an integral over
the spectral distribution

1075
01:16:55,609 --> 01:16:56,150
of the light.

1076
01:16:58,870 --> 01:17:06,410
So this means we will
replace the Rabi frequency

1077
01:17:06,410 --> 01:17:21,880
in this formula by an
integral over the energy

1078
01:17:21,880 --> 01:17:23,175
density of the radiation.

1079
01:17:26,200 --> 01:17:30,996
We have the matrix element
squared as a prefactor.

1080
01:17:30,996 --> 01:17:33,070
I just try to
re-derive it, but I

1081
01:17:33,070 --> 01:17:37,280
think the prefactor is
2 over epsilon naught.

1082
01:17:37,280 --> 01:17:39,930
So, yes.

1083
01:17:39,930 --> 01:17:46,070
With that, in
perturbation theory,

1084
01:17:46,070 --> 01:17:48,670
the probability to be
in the excited state

1085
01:17:48,670 --> 01:17:52,770
is-- let's just take
all of the prefactors.

1086
01:18:02,640 --> 01:18:05,950
Now, I change the integration
variable from omega

1087
01:18:05,950 --> 01:18:10,320
to detuning, we just
go from resonance--

1088
01:18:10,320 --> 01:18:14,680
we integrate relative
to the resonance.

1089
01:18:14,680 --> 01:18:18,360
So our energy density is now
at the resonance, omega naught

1090
01:18:18,360 --> 01:18:20,710
plus the detuning.

1091
01:18:20,710 --> 01:18:23,380
And we have this Rabi
oscillation term.

1092
01:18:34,470 --> 01:18:35,510
OK.

1093
01:18:35,510 --> 01:18:41,020
So this is nothing else than
taking our perturbative Rabi

1094
01:18:41,020 --> 01:18:45,960
oscillation formula,
which is coherent physics,

1095
01:18:45,960 --> 01:18:47,750
and indicate over many moles. .

1096
01:18:50,730 --> 01:18:52,660
I'm one step away
from the final result.

1097
01:18:58,120 --> 01:19:06,720
If the energy density is
flat, is broadband-- so

1098
01:19:06,720 --> 01:19:09,510
for the extreme
broadband case, we

1099
01:19:09,510 --> 01:19:11,508
can pull that out
of the integral.

1100
01:19:15,870 --> 01:19:19,370
And then, we are left only
with this function, F of t.

1101
01:19:37,950 --> 01:19:46,460
And you can discuss
this function, F of t,

1102
01:19:46,460 --> 01:19:48,190
is a standard result.

1103
01:19:48,190 --> 01:19:50,670
And we have seen
many discussions

1104
01:19:50,670 --> 01:19:51,890
in perturbation theory.

1105
01:19:56,990 --> 01:20:11,590
If I plot this
function, versus delta,

1106
01:20:11,590 --> 01:20:14,035
we have something
which has wiggles.

1107
01:20:14,035 --> 01:20:16,920
Then, there is a maximum,
and it has wiggles.

1108
01:20:20,030 --> 01:20:23,515
The width here is
t to the minus 1.

1109
01:20:28,962 --> 01:20:31,300
And the amplitude is t squared.

1110
01:20:33,900 --> 01:20:37,965
And this is the excited
state amplitude squared.

1111
01:20:51,260 --> 01:20:59,980
So if we integrate
that over delta,

1112
01:20:59,980 --> 01:21:01,970
we get something
which is linear in t.

1113
01:21:01,970 --> 01:21:05,020
Something which goes as
t-square, and has a width 1/t.

1114
01:21:15,650 --> 01:21:16,870
Yes, time is over.

1115
01:21:20,140 --> 01:21:28,460
So the function F of t,
which is under the integrand,

1116
01:21:28,460 --> 01:21:35,660
starts out at short
times, proportion

1117
01:21:35,660 --> 01:21:38,560
to t squared, as we discussed.

1118
01:21:38,560 --> 01:21:41,056
Maybe my drawing
should reflect that.

1119
01:21:45,540 --> 01:21:48,230
But then it becomes linear.

1120
01:21:48,230 --> 01:21:57,920
So for long times, the function
F of t becomes linear in t

1121
01:21:57,920 --> 01:22:00,670
and the delta function
in the detuning delta.

1122
01:22:03,210 --> 01:22:04,870
This is what you
have seen many times

1123
01:22:04,870 --> 01:22:07,390
in the derivation of
Fermi's golden rule.

1124
01:22:07,390 --> 01:22:08,940
I'm running out of time now.

1125
01:22:08,940 --> 01:22:10,880
I'll pick up the
ball on Wednesday,

1126
01:22:10,880 --> 01:22:14,230
and we'll discuss that result
and put it into context.

1127
01:22:14,230 --> 01:22:16,300
But the take-home
message-- and what I

1128
01:22:16,300 --> 01:22:20,470
really wanted to show you is
that we do have coherent Rabi

1129
01:22:20,470 --> 01:22:22,170
oscillations.

1130
01:22:22,170 --> 01:22:24,950
And by just performing
the integral

1131
01:22:24,950 --> 01:22:29,190
over this broad
spectrum of the light,

1132
01:22:29,190 --> 01:22:31,870
we lose the Rabi
oscillations, and we

1133
01:22:31,870 --> 01:22:36,220
find rate equations,
Fermi's golden rule,

1134
01:22:36,220 --> 01:22:40,620
and excitation probability
proportional to t.

1135
01:22:40,620 --> 01:22:45,880
And we have done the transition
from coherent physics

1136
01:22:45,880 --> 01:22:47,870
to irreversible physics.

1137
01:22:47,870 --> 01:22:50,720
This is all hidden
in this one formula,

1138
01:22:50,720 --> 01:22:54,230
but I want to fully explain
it when we start on Wednesday.

1139
01:22:54,230 --> 01:22:56,830
Any last second
question about that?

1140
01:22:56,830 --> 01:22:57,560
Cody?

1141
01:22:57,560 --> 01:22:59,360
STUDENT: It looks
like we're integrating

1142
01:22:59,360 --> 01:23:02,060
right over the point to where
perturbation theory becomes

1143
01:23:02,060 --> 01:23:04,220
an exact, because we're
integrating over delta

1144
01:23:04,220 --> 01:23:05,032
equals 0.

1145
01:23:05,032 --> 01:23:06,490
And that's the most
important part.

1146
01:23:13,220 --> 01:23:14,940
PROFESSOR: We are
integrating over it,

1147
01:23:14,940 --> 01:23:17,620
but we are integrating over
it with the [INAUDIBLE].

1148
01:23:17,620 --> 01:23:22,130
So therefore, since
we have-- perturbation

1149
01:23:22,130 --> 01:23:24,040
theory remains valid, actually.

1150
01:23:24,040 --> 01:23:25,550
Perturbation theory
remains valid,

1151
01:23:25,550 --> 01:23:28,760
as long as the excitation
probability is less than one.

1152
01:23:28,760 --> 01:23:31,660
So I have not put a
scale on it, but we

1153
01:23:31,660 --> 01:23:33,330
can go from a
quadratic dependence

1154
01:23:33,330 --> 01:23:34,230
to linear dependence.

1155
01:23:37,150 --> 01:23:41,610
As long as the probability
of being in the excited state

1156
01:23:41,610 --> 01:23:45,960
is smaller than 1, perturbation
theory is exactly valid.

1157
01:23:45,960 --> 01:23:47,780
So I think what
confuses you here

1158
01:23:47,780 --> 01:23:50,430
is, we can do
resonant excitation.

1159
01:23:50,430 --> 01:23:54,080
The broadband includes
resonant excitation.

1160
01:23:54,080 --> 01:23:56,570
But for sufficiently
short times,

1161
01:23:56,570 --> 01:23:58,870
we reach the rate
equation before we run out

1162
01:23:58,870 --> 01:24:01,020
of [INAUDIBLE]
perturbation theory.