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PROFESSOR: OK.

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00:00:22,480 --> 00:00:24,570
Here is the menu for today.

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00:00:24,570 --> 00:00:29,510
We're discussing line
shifts and line broadening.

11
00:00:29,510 --> 00:00:33,700
And I want to finish
up today this chapter

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00:00:33,700 --> 00:00:37,470
by describing collisional
narrowing, also called

13
00:00:37,470 --> 00:00:40,100
Dicke narrowing.

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00:00:40,100 --> 00:00:43,090
Then I want to have
two more shorter topics

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00:00:43,090 --> 00:00:49,310
on two other aspects which
lead to important line shapes

16
00:00:49,310 --> 00:00:50,600
and line broadening.

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00:00:50,600 --> 00:00:53,600
So I want to quickly discuss
the spectrum of emitted light

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00:00:53,600 --> 00:00:55,550
by an atom.

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00:00:55,550 --> 00:00:58,210
And I want to discuss
collisional broadening.

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00:00:58,210 --> 00:01:00,070
None of that will
be done in-depth.

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00:01:00,070 --> 00:01:02,330
The spectrum of emitted
light is really open-ended,

22
00:01:02,330 --> 00:01:05,780
and we will have a more
advanced treatment in 8.422.

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00:01:05,780 --> 00:01:09,650
But I do feel if I show you
all are kinds of line shifts

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00:01:09,650 --> 00:01:11,830
and line broadening,
I should at least

25
00:01:11,830 --> 00:01:14,390
mention here the basic things.

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00:01:14,390 --> 00:01:16,940
And collisional
broadening, I'm not

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00:01:16,940 --> 00:01:18,680
sure how many atomic
physics courses

28
00:01:18,680 --> 00:01:21,910
you will find which teach
about collision broadening,

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00:01:21,910 --> 00:01:26,210
because this is the physics
from gas discharge lamps,

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00:01:26,210 --> 00:01:28,740
the old-fashioned physics.

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00:01:28,740 --> 00:01:31,670
However, I've realized
that a lot of people

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00:01:31,670 --> 00:01:35,150
know now about clock shifts,
and mean field broadening,

33
00:01:35,150 --> 00:01:38,400
and mean field, and
the ultra-cold gases.

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00:01:38,400 --> 00:01:41,310
And they have no idea that
similar physics actually

35
00:01:41,310 --> 00:01:42,950
happens in ordinary gas.

36
00:01:42,950 --> 00:01:46,280
So at least in terms of
broadening your understanding,

37
00:01:46,280 --> 00:01:50,050
I want to talk just 10 minutes
about collisional broadening.

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00:01:50,050 --> 00:01:52,560
I've completely eliminated
from this course

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00:01:52,560 --> 00:01:55,580
the quantitative description
about collisional broadening,

40
00:01:55,580 --> 00:01:57,330
but I want to show
you a few cartoons

41
00:01:57,330 --> 00:01:59,910
and put some pictures
into your mind.

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00:01:59,910 --> 00:02:04,630
And probably, we have time to
start the next chapter, which

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00:02:04,630 --> 00:02:09,490
is actually a pretty short
one, on two-photon transitions.

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00:02:09,490 --> 00:02:11,720
For me, actually,
Dicke narrowing

45
00:02:11,720 --> 00:02:13,620
is really the highlight
of the course,

46
00:02:13,620 --> 00:02:17,900
because it provides
conceptual insight

47
00:02:17,900 --> 00:02:19,840
into what really
line broadening is,

48
00:02:19,840 --> 00:02:22,820
and to realize that collisions
can narrow lines and not

49
00:02:22,820 --> 00:02:25,530
just broaden them.

50
00:02:25,530 --> 00:02:27,910
This is sort of
subtle and insightful.

51
00:02:27,910 --> 00:02:31,470
And similar to photon
transitions, it's short,

52
00:02:31,470 --> 00:02:35,300
but I hope it's also a
highlight, because there's

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00:02:35,300 --> 00:02:39,770
so many people-- you and other
people-- who are sometimes

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00:02:39,770 --> 00:02:43,140
struggling-- a photon is
absorbed and is emitted.

55
00:02:43,140 --> 00:02:44,710
Usually, the photon
is not absorbed,

56
00:02:44,710 --> 00:02:46,090
the photon is scattered.

57
00:02:46,090 --> 00:02:49,280
And whenever you think
about photon in, photon out,

58
00:02:49,280 --> 00:02:51,810
you really should think
about two-photon transitions.

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00:02:51,810 --> 00:02:54,280
So the framework of
two-photon transitions

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00:02:54,280 --> 00:02:59,050
allows me now to give you the
tools how you should really

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00:02:59,050 --> 00:03:02,869
think about whenever you have
atoms interacting with light.

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00:03:02,869 --> 00:03:05,160
The light is not absorbed,
the light is just scattered.

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00:03:05,160 --> 00:03:06,480
And so you need that.

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00:03:06,480 --> 00:03:10,350
On the other hand, based
on all the descriptions

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00:03:10,350 --> 00:03:12,700
I've given you about
light-atom interaction,

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00:03:12,700 --> 00:03:14,120
two-photon
transitions would just

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00:03:14,120 --> 00:03:16,610
mean we need one more order
of perturbation theory,

68
00:03:16,610 --> 00:03:19,060
and then it's the same thing
you have already learned.

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00:03:19,060 --> 00:03:22,850
So it's a highlight, and it's
to some extent also a review.

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00:03:22,850 --> 00:03:25,990
You will actually recognize, in
some situations, the difference

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00:03:25,990 --> 00:03:28,720
between two-photon processes
and one-photon processes.

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00:03:28,720 --> 00:03:31,990
It's not so big, it's just
you have to-- you'll see.

73
00:03:31,990 --> 00:03:34,150
You have to use a
different Rabi frequency

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00:03:34,150 --> 00:03:36,930
and some different concepts.

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00:03:36,930 --> 00:03:37,780
OK.

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00:03:37,780 --> 00:03:41,880
So that's, I think, really
an agenda of highlights.

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00:03:41,880 --> 00:03:48,120
Let's go back to the physical
picture we drew up on Wednesday

78
00:03:48,120 --> 00:03:50,780
about Dicke narrowing.

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00:03:50,780 --> 00:03:53,310
I just explained
to you that when

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00:03:53,310 --> 00:03:57,900
we have an atom which is
trapped and tightly confined,

81
00:03:57,900 --> 00:04:03,470
that the spectrum consists of
a sharp light and sidebands.

82
00:04:03,470 --> 00:04:05,660
And now I was addressing
the situation--

83
00:04:05,660 --> 00:04:07,370
what happens if you
have an atom which

84
00:04:07,370 --> 00:04:09,520
is surrounded by buffer gas?

85
00:04:09,520 --> 00:04:14,270
I would say, well, that's a
cheap trap created by nature,

86
00:04:14,270 --> 00:04:16,350
because when the particle
wants to fly away,

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00:04:16,350 --> 00:04:18,390
it collides with
a buffer gas atom.

88
00:04:18,390 --> 00:04:20,490
So it stays put.

89
00:04:20,490 --> 00:04:23,790
So it's a cheap trap, but
it's also a lousy trap,

90
00:04:23,790 --> 00:04:27,070
because there is some randomness
in the number of collisions.

91
00:04:27,070 --> 00:04:29,910
But I'm waving my
arm, so that maybe you

92
00:04:29,910 --> 00:04:34,050
can go along with the picture
that it is an ensemble of traps

93
00:04:34,050 --> 00:04:36,620
which have very different
trap frequencies.

94
00:04:36,620 --> 00:04:39,280
And then we would expect,
based on our understanding

95
00:04:39,280 --> 00:04:41,860
of the spectrum of
confined particles,

96
00:04:41,860 --> 00:04:44,190
that we have a carrier,
which is always the same,

97
00:04:44,190 --> 00:04:47,600
at the electronic
excitation frequency.

98
00:04:47,600 --> 00:04:51,130
But then we have sidebands
at the trap frequency.

99
00:04:51,130 --> 00:04:53,450
But those sidebands
are now smeared out,

100
00:04:53,450 --> 00:04:56,010
because we do not have a
defined harmonic oscillator

101
00:04:56,010 --> 00:04:57,520
potential here.

102
00:04:57,520 --> 00:05:03,040
So therefore, I sort of
tried to lead you in the way

103
00:05:03,040 --> 00:05:07,340
that that may be one way how you
can think about the situation.

104
00:05:07,340 --> 00:05:10,950
And well, now I want to give you
a different way to look at it.

105
00:05:10,950 --> 00:05:14,110
And let's see how those
things come together.

106
00:05:14,110 --> 00:05:26,260
So if I would ask you, give
me a quantitative estimate

107
00:05:26,260 --> 00:05:29,444
how wide this line
is, do you have

108
00:05:29,444 --> 00:05:30,610
any idea how we can do that?

109
00:05:36,500 --> 00:05:38,284
Or let me even put it this way.

110
00:05:38,284 --> 00:05:39,950
It's again one of
those kind of things--

111
00:05:39,950 --> 00:05:43,080
you have the knowledge, but
to put it together is hard.

112
00:05:43,080 --> 00:05:45,370
But I would guarantee
everybody in this room

113
00:05:45,370 --> 00:05:49,897
has the knowledge to write
down in one line what

114
00:05:49,897 --> 00:05:50,980
is the width of this line.

115
00:06:00,970 --> 00:06:03,630
What is the width of this line?

116
00:06:03,630 --> 00:06:04,980
It's an inverse time.

117
00:06:04,980 --> 00:06:05,690
What time?

118
00:06:09,120 --> 00:06:10,970
Coherence time.

119
00:06:10,970 --> 00:06:11,690
Yeah.

120
00:06:11,690 --> 00:06:16,190
But now, if you have an
atom which is now starting

121
00:06:16,190 --> 00:06:19,010
and it hits buffer
gas, how would you

122
00:06:19,010 --> 00:06:22,970
estimate the coherence time?

123
00:06:22,970 --> 00:06:24,750
AUDIENCE: Mean free pass.

124
00:06:24,750 --> 00:06:29,070
PROFESSOR: Mean free
pass will be important.

125
00:06:29,070 --> 00:06:30,860
But the coherence
time-- how long

126
00:06:30,860 --> 00:06:34,065
will the atom talk to the laser
beam in a phase-coherent way?

127
00:06:36,469 --> 00:06:38,510
So what do we have to
compare to the wavelengths?

128
00:06:38,510 --> 00:06:39,310
AUDIENCE: Mean free pass.

129
00:06:39,310 --> 00:06:40,750
PROFESSOR: The mean free pass.

130
00:06:40,750 --> 00:06:42,422
This will be the
important parameter.

131
00:06:42,422 --> 00:06:44,130
But if the mean free
pass is much shorter

132
00:06:44,130 --> 00:06:47,100
than the wavelengths?

133
00:06:47,100 --> 00:06:51,310
Well, the atom has just
moved a tiny bit, one mean.

134
00:06:51,310 --> 00:06:53,115
Is it still coherent
at this time?

135
00:06:53,115 --> 00:06:54,656
AUDIENCE: It's
shorter than the wave.

136
00:06:54,656 --> 00:06:57,120
PROFESSOR: If the mean free
path is shorter than the wave.

137
00:06:57,120 --> 00:06:59,230
So it will do many collisions.

138
00:06:59,230 --> 00:07:04,790
But since it stays localized, it
will still coherently interact.

139
00:07:04,790 --> 00:07:08,502
When will it stop
coherently interacting?

140
00:07:08,502 --> 00:07:09,870
AUDIENCE: Mean free path.

141
00:07:09,870 --> 00:07:12,036
PROFESSOR: No, the mean
free path is always smaller.

142
00:07:12,036 --> 00:07:14,420
Let's just assume that.

143
00:07:14,420 --> 00:07:15,330
I hear "diffusion."

144
00:07:15,330 --> 00:07:17,610
AUDIENCE: [INAUDIBLE].

145
00:07:17,610 --> 00:07:18,930
PROFESSOR: Yes.

146
00:07:18,930 --> 00:07:21,235
The atom has a short
mean free path,

147
00:07:21,235 --> 00:07:23,500
but it will do a random walk.

148
00:07:23,500 --> 00:07:28,430
And the movement it diffuses
by more than a wavelength,

149
00:07:28,430 --> 00:07:34,630
it has randomly changed its
position by a wavelength.

150
00:07:34,630 --> 00:07:36,670
And that would
mean it experiences

151
00:07:36,670 --> 00:07:39,190
the phase of the drive
field in a random way.

152
00:07:39,190 --> 00:07:40,690
End of coherence.

153
00:07:40,690 --> 00:07:44,801
1 over this time
is its line width.

154
00:07:44,801 --> 00:07:45,300
OK.

155
00:07:45,300 --> 00:07:48,140
So let me write that down.

156
00:07:48,140 --> 00:07:52,980
So based on the
concepts we have learned

157
00:07:52,980 --> 00:07:55,450
by looking at Doppler
broadening and all that,

158
00:07:55,450 --> 00:07:57,920
we realize the
important aspect is,

159
00:07:57,920 --> 00:08:03,490
when do atoms in an ensemble
randomly move by a wavelength?

160
00:08:03,490 --> 00:08:10,140
So therefore, our
estimate is now--

161
00:08:10,140 --> 00:08:20,451
estimate the widths
of the sharp peak.

162
00:08:20,451 --> 00:08:22,720
We use a model
which is diffusion.

163
00:08:26,050 --> 00:08:30,110
We know that in diffusion,
the random walk, the RMS

164
00:08:30,110 --> 00:08:33,650
position in atom has moved away.

165
00:08:33,650 --> 00:08:36,370
Ballistic motion
is linear in time.

166
00:08:36,370 --> 00:08:39,330
Diffusive motion is the
square root of time,

167
00:08:39,330 --> 00:08:41,720
or z square is linear in time.

168
00:08:41,720 --> 00:08:43,230
And we have the
diffusion constant.

169
00:08:48,560 --> 00:08:53,030
Diffusion by, well,
lambda or lambda

170
00:08:53,030 --> 00:08:59,055
bar after time, which
is wavelength squared.

171
00:09:06,190 --> 00:09:10,615
So therefore, we would
expect that the full width

172
00:09:10,615 --> 00:09:20,000
at 1/2 maximum of our peak
is k squared-- k, the wave

173
00:09:20,000 --> 00:09:22,905
vector of the light, and D
is the diffusion constant.

174
00:09:37,790 --> 00:09:43,720
Well, since you mentioned
the mean free path, let me,

175
00:09:43,720 --> 00:09:46,930
already at this point-- I
wanted to do it earlier,

176
00:09:46,930 --> 00:09:48,550
but it fits in very well here.

177
00:09:51,660 --> 00:09:56,610
In an ideal gas, the
diffusion constant

178
00:09:56,610 --> 00:10:04,050
is given by some thermal average
speed times the mean free path.

179
00:10:04,050 --> 00:10:06,920
And if you look
up some textbooks,

180
00:10:06,920 --> 00:10:08,160
there's a factor of 3.

181
00:10:13,690 --> 00:10:21,260
So our delta omega,
which is k squared

182
00:10:21,260 --> 00:10:27,800
D. Let me write it
as k times v bar.

183
00:10:27,800 --> 00:10:35,160
And the other k I write
as 1 over lambda bar.

184
00:10:35,160 --> 00:10:38,460
So this is k squared,
and I need l.

185
00:10:38,460 --> 00:10:41,564
So this is nothing
else than k squared--

186
00:10:41,564 --> 00:10:44,540
I take this expression,
k squared l.

187
00:10:44,540 --> 00:10:46,540
This is one k, this
is the other k.

188
00:10:46,540 --> 00:10:51,650
And v bar times l is
the ideal gas expression

189
00:10:51,650 --> 00:10:54,700
for the diffusion constant.

190
00:10:54,700 --> 00:11:00,700
But now you realize that k, dot,
v-- since in a gas, the most

191
00:11:00,700 --> 00:11:03,790
probably velocity is
also the momentum spread,

192
00:11:03,790 --> 00:11:06,980
this is nothing else
than the Doppler width.

193
00:11:10,990 --> 00:11:16,750
So therefore, we find
that the line widths

194
00:11:16,750 --> 00:11:19,840
in Dicke narrowing--
if we have buffer gas

195
00:11:19,840 --> 00:11:28,450
and we have diffusive
motion-- is much smaller--

196
00:11:28,450 --> 00:11:31,370
and this is why it's
called Dicke narrowing--

197
00:11:31,370 --> 00:11:37,690
than the Doppler broadening
if the mean free path is

198
00:11:37,690 --> 00:11:40,970
much smaller than
the wavelength.

199
00:11:40,970 --> 00:11:44,900
So this is where the
mean free path comes in.

200
00:11:44,900 --> 00:11:47,630
What happens if
the mean free path

201
00:11:47,630 --> 00:11:49,703
is much longer than
the wavelength?

202
00:11:54,740 --> 00:11:57,120
What line widths do we then get?

203
00:12:02,676 --> 00:12:04,390
Do we then have a
line width which

204
00:12:04,390 --> 00:12:05,973
is larger than the
Doppler broadening,

205
00:12:05,973 --> 00:12:08,720
or do we get the
Doppler broadening?

206
00:12:08,720 --> 00:12:11,690
Let's have a clicker question.

207
00:12:11,690 --> 00:12:19,070
So if l is much smaller
than lambda bar,

208
00:12:19,070 --> 00:12:28,580
is delta omega-- and this
is your option A-- equal to?

209
00:12:31,410 --> 00:12:34,210
Or option B, larger?

210
00:12:43,294 --> 00:12:43,794
OK.

211
00:12:51,700 --> 00:12:54,570
So what do you think?

212
00:12:54,570 --> 00:12:56,890
It's always a question
when we derive something,

213
00:12:56,890 --> 00:12:59,940
how seriously you should
take what we derived.

214
00:12:59,940 --> 00:13:04,330
So this expression
for delta omega Dicke

215
00:13:04,330 --> 00:13:06,930
shows that the line width
would get larger and larger

216
00:13:06,930 --> 00:13:10,360
the longer the
mean free path is.

217
00:13:10,360 --> 00:13:13,037
And the question is,
is that correct or not?

218
00:13:22,910 --> 00:13:23,410
OK.

219
00:13:23,410 --> 00:13:26,016
Any more takers?

220
00:13:26,016 --> 00:13:27,120
Whoops.

221
00:13:27,120 --> 00:13:27,620
Oops.

222
00:13:27,620 --> 00:13:28,411
Sorry, press again.

223
00:13:28,411 --> 00:13:31,250
I erased it by clicking
the wrong button.

224
00:13:31,250 --> 00:13:33,090
You've already
made up your mind.

225
00:13:33,090 --> 00:13:35,510
You know the answer.

226
00:13:35,510 --> 00:13:39,340
So stop, display.

227
00:13:39,340 --> 00:13:41,950
Yes.

228
00:13:41,950 --> 00:13:43,700
The majority answer
is definitely correct.

229
00:13:43,700 --> 00:13:47,820
What happens until the
first collision happens,

230
00:13:47,820 --> 00:13:49,790
you just have
normal Doppler-- you

231
00:13:49,790 --> 00:13:52,390
don't have diffusive motion,
you have ballistic motion.

232
00:13:52,390 --> 00:13:56,240
And if the line width is already
determined once the atoms have

233
00:13:56,240 --> 00:13:59,150
spread out by a wavelength,
that's it already.

234
00:13:59,150 --> 00:14:02,150
And then if then the particles
collide, it doesn't matter.

235
00:14:02,150 --> 00:14:04,810
So what we have assumed
is we've assumed

236
00:14:04,810 --> 00:14:07,600
that the relevant model for
the spread out to a wavelength

237
00:14:07,600 --> 00:14:08,580
is a diffusive model.

238
00:14:08,580 --> 00:14:11,810
And if we are past that--
because the mean free path is

239
00:14:11,810 --> 00:14:13,310
larger than the
wavelength-- we have

240
00:14:13,310 --> 00:14:15,150
to go back to normal
Doppler broadening.

241
00:14:17,470 --> 00:14:17,970
OK.

242
00:14:20,670 --> 00:14:26,970
But now let me calculate
Dicke narrowing

243
00:14:26,970 --> 00:14:35,500
by using the formalism
we have developed.

244
00:14:35,500 --> 00:14:38,210
So let's use the correlation
function for that.

245
00:14:41,130 --> 00:14:47,850
And we know that the line
width was nothing else

246
00:14:47,850 --> 00:14:51,500
than the Fourier transform
of the correlation function,

247
00:14:51,500 --> 00:14:54,670
how the atoms
experience the drive

248
00:14:54,670 --> 00:14:58,470
field at two different times.

249
00:14:58,470 --> 00:15:00,800
So we had here the
matrix element,

250
00:15:00,800 --> 00:15:03,990
the Rabi frequency squared.

251
00:15:03,990 --> 00:15:10,730
We have between time
t and t plus tau,

252
00:15:10,730 --> 00:15:13,480
the phase of the drive
field accumulates

253
00:15:13,480 --> 00:15:15,850
e to the i omega tau.

254
00:15:15,850 --> 00:15:22,560
But now we have the e
to the minus ikr factor.

255
00:15:22,560 --> 00:15:26,250
And r, or the position--
in diffusion, you

256
00:15:26,250 --> 00:15:30,360
often call it s-- changes now,
because the particle undergoes

257
00:15:30,360 --> 00:15:32,000
a random walk.

258
00:15:32,000 --> 00:15:35,960
So what we have to do is to
describe collisional narrowing,

259
00:15:35,960 --> 00:15:39,520
we have to take this
factor and average it

260
00:15:39,520 --> 00:15:45,080
over our ensemble of
diffusing particles.

261
00:15:45,080 --> 00:15:49,650
And well, diffusion means that
if particles start out at t

262
00:15:49,650 --> 00:15:53,940
equals 0 at the
origin, the probability

263
00:15:53,940 --> 00:15:57,410
that we find the
particle at time t,

264
00:15:57,410 --> 00:16:01,690
a distance s away
from the origin--

265
00:16:01,690 --> 00:16:04,300
so I'm depicting this one here.

266
00:16:04,300 --> 00:16:06,760
The particle does
sort of a random walk.

267
00:16:06,760 --> 00:16:11,840
And after time t, it
is out at a position s.

268
00:16:11,840 --> 00:16:18,400
And the probability for
that is e to the minus

269
00:16:18,400 --> 00:16:22,920
s squared over 4 Dt.

270
00:16:22,920 --> 00:16:26,100
And you see it's s
is quadratic over t.

271
00:16:26,100 --> 00:16:28,840
This is a random walk. s
increases s's square root

272
00:16:28,840 --> 00:16:30,540
of time.

273
00:16:30,540 --> 00:16:35,895
And the probability is
normalized by this expression.

274
00:16:38,460 --> 00:16:49,320
So therefore, this red average
in the correlation function

275
00:16:49,320 --> 00:16:56,720
is calculated by convoluting
with this probability

276
00:16:56,720 --> 00:16:58,710
distribution for
the random walk.

277
00:17:03,150 --> 00:17:10,839
So it is e to the minus
iks, e to the minus

278
00:17:10,839 --> 00:17:16,220
s squared over 4 D. The
time is now called tau.

279
00:17:16,220 --> 00:17:19,420
We used to integrate
over all possibilities.

280
00:17:19,420 --> 00:17:23,115
And e to the i omega
tau is a common factor.

281
00:17:30,140 --> 00:17:33,900
My notes show that the integral
is done from minus infinity

282
00:17:33,900 --> 00:17:34,925
to plus infinity.

283
00:17:38,800 --> 00:17:40,810
Either this is right, or
there's a factor of two

284
00:17:40,810 --> 00:17:45,560
to be accounted, because
well, it depends.

285
00:17:45,560 --> 00:17:47,810
Now, is s a coordinate,
or is s in radians?

286
00:17:47,810 --> 00:17:51,800
In one case, it has a sign,
in the other case, it has not.

287
00:17:51,800 --> 00:17:53,390
I'm not able to
reconstruct it now,

288
00:17:53,390 --> 00:17:55,560
but it's just a
numerical factor,

289
00:17:55,560 --> 00:17:57,820
which would be affected by that.

290
00:17:57,820 --> 00:18:00,360
So the result of
this integral is

291
00:18:00,360 --> 00:18:03,320
that we obtain an
exponential function.

292
00:18:08,840 --> 00:18:15,930
So what we find out
is that there is now

293
00:18:15,930 --> 00:18:19,745
an exponentially
decaying function.

294
00:18:23,100 --> 00:18:25,790
Remember, we had situations
where the correlation function

295
00:18:25,790 --> 00:18:29,140
for how the atom
experiences a coherent field

296
00:18:29,140 --> 00:18:30,600
was decaying
exponentially because

297
00:18:30,600 --> 00:18:32,140
of spontaneous lifetime?

298
00:18:32,140 --> 00:18:33,790
And now we have an
exponential decay,

299
00:18:33,790 --> 00:18:36,060
because the particle
is defusing around.

300
00:18:41,090 --> 00:18:43,850
But we also know, of course,
that exponential decay, when

301
00:18:43,850 --> 00:18:47,630
we Fourier analyze it,
gives us a Lorentzian.

302
00:18:47,630 --> 00:18:52,090
So when we ask what is the
rate, Fermi's golden rule's

303
00:18:52,090 --> 00:18:55,000
rate expression, the
rate for excitation.

304
00:19:00,950 --> 00:19:04,230
We want to do the Fourier
transform of this correlation

305
00:19:04,230 --> 00:19:05,850
function.

306
00:19:05,850 --> 00:19:21,540
And what we obtain is, well,
a Lorentzian which looks

307
00:19:21,540 --> 00:19:24,320
like the Lorentzian for
spontaneous emission,

308
00:19:24,320 --> 00:19:30,230
except for that we have
a different width now.

309
00:19:30,230 --> 00:19:48,720
It's a Lorentzian with a
width 2 k squared times D.

310
00:19:48,720 --> 00:19:49,950
So pretty straightforward.

311
00:19:49,950 --> 00:19:52,580
But I hope you've
seen and you've

312
00:19:52,580 --> 00:19:55,200
enjoyed there's a very
intuitive picture.

313
00:19:55,200 --> 00:19:57,250
But the correlation
function-- if you just

314
00:19:57,250 --> 00:19:59,350
look at it from the
perspective of the atoms,

315
00:19:59,350 --> 00:20:01,870
how do I experience
a coherent field?

316
00:20:01,870 --> 00:20:03,970
You put in simply
the diffusive motion.

317
00:20:03,970 --> 00:20:08,620
You can exactly calculate
what the line shape is.

318
00:20:08,620 --> 00:20:11,420
And that's definitely
something you would not

319
00:20:11,420 --> 00:20:14,750
have known how to do it
without this formalism.

320
00:20:14,750 --> 00:20:15,250
OK.

321
00:20:23,520 --> 00:20:25,060
I have a question for you.

322
00:20:25,060 --> 00:20:34,110
And this is, how does the
spectrum for diffusive motion

323
00:20:34,110 --> 00:20:35,860
look like?

324
00:20:35,860 --> 00:20:39,090
What we have just
calculated was that it

325
00:20:39,090 --> 00:20:48,020
is a Lorentzian with the line
widths we've just determined.

326
00:20:48,020 --> 00:20:50,890
But I want to give
you another choice.

327
00:20:50,890 --> 00:20:57,150
And this is what we discussed
at the end of the last class.

328
00:20:57,150 --> 00:21:01,040
This picture where we took
the confined particles

329
00:21:01,040 --> 00:21:04,280
to the ridiculous limit-- where
there was no trap anymore,

330
00:21:04,280 --> 00:21:06,630
just collisions--
suggested actually

331
00:21:06,630 --> 00:21:08,610
that we have some sharp line.

332
00:21:08,610 --> 00:21:12,080
But then this envelope
of those sidebands

333
00:21:12,080 --> 00:21:18,100
give some unresolved pedestal.

334
00:21:18,100 --> 00:21:22,820
So the question is,
what is correct?

335
00:21:22,820 --> 00:21:24,940
We have just done a
quantitative calculation

336
00:21:24,940 --> 00:21:26,650
using the model, a
diffusion propagator.

337
00:21:29,650 --> 00:21:33,030
But we also had this
intuitive picture,

338
00:21:33,030 --> 00:21:37,980
which had more this kind
of bimodal distribution.

339
00:21:37,980 --> 00:21:41,570
A broad pedestal
and a sharp peak.

340
00:21:41,570 --> 00:21:45,510
And I want you to think
about it for a few seconds.

341
00:21:45,510 --> 00:21:47,720
What would you expect to
be the correct answer?

342
00:21:53,720 --> 00:21:55,520
Is the line shape
just a Lorentzian,

343
00:21:55,520 --> 00:21:58,200
or does the line shape have
two different parts to it?

344
00:22:27,850 --> 00:22:28,350
OK.

345
00:22:32,600 --> 00:22:35,930
Does somebody want to speak out
in favor of his or her choice?

346
00:22:43,940 --> 00:22:46,780
Well, one argument is,
when you derive something,

347
00:22:46,780 --> 00:22:50,190
it must be more correct than
when you wave your hands.

348
00:22:50,190 --> 00:22:52,500
Therefore, we
derived a Lorentzian,

349
00:22:52,500 --> 00:22:54,750
and the other picture
was just waving our hands

350
00:22:54,750 --> 00:22:57,350
and using some analogy.

351
00:22:57,350 --> 00:22:59,050
OK.

352
00:22:59,050 --> 00:23:01,180
That would be one
argument to vote

353
00:23:01,180 --> 00:23:07,350
for A. Somebody
wants to defend B?

354
00:23:14,530 --> 00:23:15,030
Pardon?

355
00:23:15,030 --> 00:23:16,200
AUDIENCE: More intuitive.

356
00:23:16,200 --> 00:23:17,920
PROFESSOR: More intuitive.

357
00:23:17,920 --> 00:23:18,750
Yes.

358
00:23:18,750 --> 00:23:21,980
You know, something
must be right about it.

359
00:23:21,980 --> 00:23:25,050
I mean, in some limit,
this must be like a trap,

360
00:23:25,050 --> 00:23:28,600
and there should
be some sidebands.

361
00:23:28,600 --> 00:23:31,280
Yes.

362
00:23:31,280 --> 00:23:31,960
OK.

363
00:23:31,960 --> 00:23:37,970
But now I would ask you-- Nancy,
if you say it's more intuitive,

364
00:23:37,970 --> 00:23:43,570
why don't we get
those broadening?

365
00:23:43,570 --> 00:23:45,620
Why don't we get
this extra pedestal

366
00:23:45,620 --> 00:23:49,088
in our quantitative derivation?

367
00:23:49,088 --> 00:23:50,996
AUDIENCE: [INAUDIBLE].

368
00:23:50,996 --> 00:23:53,430
PROFESSOR: But what?

369
00:23:53,430 --> 00:23:56,260
Didn't we lose the exact
probability distribution

370
00:23:56,260 --> 00:23:57,590
for diffusive process?

371
00:23:57,590 --> 00:24:00,050
AUDIENCE: There's no
harmonic oscillator.

372
00:24:00,050 --> 00:24:03,494
The sidebands [INAUDIBLE]
harmonic oscillator.

373
00:24:07,922 --> 00:24:09,920
PROFESSOR: So that's
an argument also for A.

374
00:24:09,920 --> 00:24:11,590
We don't have a
harmonic oscillator.

375
00:24:11,590 --> 00:24:15,080
And when I said there is sort
of this trap-type feature,

376
00:24:15,080 --> 00:24:17,310
I'm really over-extending
the analogy.

377
00:24:17,310 --> 00:24:19,230
AUDIENCE: The mean
free path is not--

378
00:24:19,230 --> 00:24:20,190
PROFESSOR: Pardon?

379
00:24:20,190 --> 00:24:22,610
AUDIENCE: The mean free path
is not infinitesimally small.

380
00:24:22,610 --> 00:24:25,680
PROFESSOR: The mean free path
is not infinitesimally small.

381
00:24:25,680 --> 00:24:28,980
Now we're getting close.

382
00:24:28,980 --> 00:24:33,720
We assumed diffusive motion.

383
00:24:33,720 --> 00:24:37,930
We put in the exact expression
for diffusive motion.

384
00:24:37,930 --> 00:24:41,840
And of course, you also know,
when you describe a line width,

385
00:24:41,840 --> 00:24:44,310
what happens in the
middle at detuning

386
00:24:44,310 --> 00:24:48,130
0 is more what happens in
the limit of long times.

387
00:24:48,130 --> 00:24:54,050
What happens further and further
out happens at shorter times.

388
00:24:54,050 --> 00:24:56,850
What is the motion of
atoms at short times?

389
00:25:02,370 --> 00:25:05,850
Before the first collision
happens, it moves straight.

390
00:25:05,850 --> 00:25:08,950
So the diffusive
propagator which we put in

391
00:25:08,950 --> 00:25:11,300
is only valid after
the first collision.

392
00:25:11,300 --> 00:25:14,950
Until the first collision
happens, we have free motion.

393
00:25:14,950 --> 00:25:19,670
And free motion should give rise
to simple Doppler broadening.

394
00:25:19,670 --> 00:25:21,380
So until the first
collision happens,

395
00:25:21,380 --> 00:25:22,990
we should get a little
bit of Doppler broadening.

396
00:25:22,990 --> 00:25:24,390
But once the
collisions happened,

397
00:25:24,390 --> 00:25:26,700
we should be very well
within the description

398
00:25:26,700 --> 00:25:28,740
of the diffusion operator.

399
00:25:28,740 --> 00:25:31,525
And to address your
concern, actually,

400
00:25:31,525 --> 00:25:34,340
in the limit of many
sideband, often,

401
00:25:34,340 --> 00:25:36,860
the envelope of
all those sidebands

402
00:25:36,860 --> 00:25:38,960
is actually determined
by the Doppler profile.

403
00:25:38,960 --> 00:25:42,470
There is a limit, if you have
a large modulation index,

404
00:25:42,470 --> 00:25:45,890
that you have many sidebands.

405
00:25:45,890 --> 00:25:47,800
And in some
semi-classical limit,

406
00:25:47,800 --> 00:25:51,310
the sidebands have an envelope
which is the Doppler profile.

407
00:25:51,310 --> 00:25:54,100
So now I think the
two pictures agree.

408
00:25:54,100 --> 00:26:02,560
If we had used a propagator
which would interpolate

409
00:26:02,560 --> 00:26:05,470
between the first moment where
the particle moves straight,

410
00:26:05,470 --> 00:26:09,030
and then diffusion, we would
have gotten little pedestals

411
00:26:09,030 --> 00:26:10,400
here.

412
00:26:10,400 --> 00:26:14,870
And so that sort of tells us
that the smeared outside bands,

413
00:26:14,870 --> 00:26:18,460
yes, strictly speaking, they
are linked to harmonic motion.

414
00:26:18,460 --> 00:26:19,954
But they are sort
of the leftover

415
00:26:19,954 --> 00:26:22,120
of the motional effect,
which is the Doppler effect.

416
00:26:25,410 --> 00:26:25,910
OK.

417
00:26:30,401 --> 00:26:30,900
Good.

418
00:26:30,900 --> 00:26:35,600
So we have the
correct answer here.

419
00:26:35,600 --> 00:26:46,130
And here, it is the model
neglects ballistic motion

420
00:26:46,130 --> 00:26:51,465
until the first collision.

421
00:26:53,950 --> 00:26:54,450
Questions?

422
00:27:03,650 --> 00:27:04,190
OK.

423
00:27:04,190 --> 00:27:12,930
Let's now spend 10
minutes discussing

424
00:27:12,930 --> 00:27:15,500
the fluorescent
spectrum of an atom.

425
00:27:22,980 --> 00:27:27,310
As I pointed out in my overview,
we teach much more about it

426
00:27:27,310 --> 00:27:28,205
in 8.422.

427
00:27:28,205 --> 00:27:30,280
We use a dressed atom picture.

428
00:27:30,280 --> 00:27:34,400
But something would be missing
if I wouldn't do it also

429
00:27:34,400 --> 00:27:38,120
in this course, because
we have discussed

430
00:27:38,120 --> 00:27:41,400
to quite some extent what
happens when we excite atoms

431
00:27:41,400 --> 00:27:43,970
with light.

432
00:27:43,970 --> 00:27:47,380
And what we have
discussed so far

433
00:27:47,380 --> 00:28:00,220
is that we excite atom and scan
the laser, or the excitation

434
00:28:00,220 --> 00:28:01,840
frequency.

435
00:28:01,840 --> 00:28:16,240
And when we are plotting the
intensity of the fluorescence,

436
00:28:16,240 --> 00:28:20,040
we are looking at the
number of scattered photon.

437
00:28:20,040 --> 00:28:24,900
And what we are scanning is
the detuning of the laser.

438
00:28:24,900 --> 00:28:36,310
We would expect, in the case
of a motionless atom, simply

439
00:28:36,310 --> 00:28:41,500
a Lorentzian, or in
the general limit,

440
00:28:41,500 --> 00:28:43,240
a power-broadened Lorentzian.

441
00:28:47,600 --> 00:28:51,090
Just to be clear, what I'm
discussing in those 10 minutes

442
00:28:51,090 --> 00:28:54,950
is I'm completely ignoring
that the atom can move.

443
00:28:54,950 --> 00:28:56,450
So you should think
that it's either

444
00:28:56,450 --> 00:29:00,350
an atom with infinite mass,
or it's an atom tightly

445
00:29:00,350 --> 00:29:02,360
localized in the
Lamb-Dicke limit,

446
00:29:02,360 --> 00:29:07,660
and all we are looking at is the
structure of the central peak.

447
00:29:07,660 --> 00:29:08,820
So no motional effects.

448
00:29:08,820 --> 00:29:12,550
We just look at the pure
kind of intrinsic line

449
00:29:12,550 --> 00:29:16,245
widths of the
electronic transition.

450
00:29:16,245 --> 00:29:16,890
OK.

451
00:29:16,890 --> 00:29:18,500
So we have discussed that.

452
00:29:18,500 --> 00:29:26,940
But now I want to look at
another aspect of spectroscopy.

453
00:29:26,940 --> 00:29:30,290
And this is another
scan we can do.

454
00:29:30,290 --> 00:29:33,160
We want to have
a fixed detuning.

455
00:29:37,840 --> 00:29:47,490
And we look at the spectrum
of the emitted light.

456
00:29:52,330 --> 00:29:56,890
So let's assume that
we have a laser which

457
00:29:56,890 --> 00:29:59,590
is at a detuning delta.

458
00:29:59,590 --> 00:30:01,700
The laser is fixed.

459
00:30:01,700 --> 00:30:03,720
The light is emitting.

460
00:30:03,720 --> 00:30:07,070
But we are now dispersing
with a spectrograph.

461
00:30:07,070 --> 00:30:10,600
We are analyzing what is the
frequency of the emitted light.

462
00:30:10,600 --> 00:30:12,090
And we determine the spectrum.

463
00:30:14,800 --> 00:30:19,340
And yes, the question is,
how does it look like?

464
00:30:19,340 --> 00:30:21,680
And I want to give
you four options.

465
00:30:31,010 --> 00:30:37,040
So this is 0 detuning
at resonance.

466
00:30:37,040 --> 00:30:38,010
Oops, I need one more.

467
00:30:43,290 --> 00:30:45,600
Let me relabel it
to make it clear.

468
00:30:54,400 --> 00:30:56,840
So our option one
is our spectrum

469
00:30:56,840 --> 00:31:01,440
is a delta function at
the resonance of the atom.

470
00:31:05,830 --> 00:31:16,010
Option two is it is a Lorentzian
centered around the resonance.

471
00:31:19,630 --> 00:31:31,370
Option three is it is a
delta function at the laser

472
00:31:31,370 --> 00:31:33,110
frequency.

473
00:31:33,110 --> 00:31:44,700
And option four is it is a
Lorentzian with line width

474
00:31:44,700 --> 00:31:46,380
gamma.

475
00:31:46,380 --> 00:31:48,130
So we have two options.

476
00:31:48,130 --> 00:31:51,750
Is it a delta function
at omega 0 or omega l?

477
00:31:51,750 --> 00:31:54,700
Or is it a Lorentzian-broadened
function at either omega 0

478
00:31:54,700 --> 00:31:57,050
or omega l.

479
00:31:57,050 --> 00:32:02,760
And since we want to
keep things simple,

480
00:32:02,760 --> 00:32:08,030
we want to first discuss the
case of the perturbative case

481
00:32:08,030 --> 00:32:12,990
that the laser which excites
the atom has very low power.

482
00:32:12,990 --> 00:32:14,370
So what would you expect?

483
00:32:14,370 --> 00:32:17,040
An atom is excited, it's
a little light bulb.

484
00:32:17,040 --> 00:32:19,890
You analyze the spectrum
of the light bulb.

485
00:32:19,890 --> 00:32:22,720
Which of the four
spectrum will you measure?

486
00:33:04,570 --> 00:33:07,320
OK.

487
00:33:07,320 --> 00:33:08,320
Let's try again.

488
00:33:08,320 --> 00:33:11,470
[LAUGHTER]

489
00:33:11,470 --> 00:33:15,510
And maybe I should, like
our online learning system,

490
00:33:15,510 --> 00:33:17,770
try to give you one hint.

491
00:33:17,770 --> 00:33:22,620
I want you to really think hard
what energy conservation means

492
00:33:22,620 --> 00:33:23,330
in this problem.

493
00:33:46,741 --> 00:33:47,240
OK.

494
00:33:53,530 --> 00:33:55,890
OK, we're getting closer.

495
00:33:55,890 --> 00:33:58,640
Now, the way how I want
you to think about it

496
00:33:58,640 --> 00:34:04,030
is if you take the
limit of low power,

497
00:34:04,030 --> 00:34:06,450
you should really think
about it that there

498
00:34:06,450 --> 00:34:09,210
is only one photon in
your whole laboratory.

499
00:34:09,210 --> 00:34:11,260
This photon is
scattered of an atom,

500
00:34:11,260 --> 00:34:14,380
and then you measure the
frequency of the photon.

501
00:34:14,380 --> 00:34:20,030
And energy conservation clearly
tells us that the frequency

502
00:34:20,030 --> 00:34:23,820
of the scattered photon cannot
be at the resonance frequency.

503
00:34:23,820 --> 00:34:25,880
It has to be at the
laser frequency.

504
00:34:25,880 --> 00:34:29,310
Otherwise, we would
violate energy.

505
00:34:29,310 --> 00:34:31,600
But it seems-- still,
the more subtle thing

506
00:34:31,600 --> 00:34:34,850
is, is it a delta function?

507
00:34:34,850 --> 00:34:39,016
Of course, a delta
function will be broadened.

508
00:34:39,016 --> 00:34:41,320
If you do the experiment
for one second,

509
00:34:41,320 --> 00:34:44,239
you will have a Fourier
line width which is 1 hertz.

510
00:34:44,239 --> 00:34:46,590
But that's, for
practical reasons, almost

511
00:34:46,590 --> 00:34:49,580
like a delta function, because
the spectral line widths,

512
00:34:49,580 --> 00:34:51,884
or the natural line widths
of your favorite atom

513
00:34:51,884 --> 00:34:53,699
is 10 megahertz.

514
00:34:53,699 --> 00:34:55,989
So there will be some
temporal broadening,

515
00:34:55,989 --> 00:34:58,810
which is in case in any
realistic experiment.

516
00:34:58,810 --> 00:35:01,660
But we are talking about, are
you limited by the experiment

517
00:35:01,660 --> 00:35:03,580
and by technical
noise, or are you

518
00:35:03,580 --> 00:35:06,610
limited by the
natural line widths?

519
00:35:06,610 --> 00:35:09,341
Now, I would argue, just
with energy conservation--

520
00:35:09,341 --> 00:35:11,340
but I'll give you another
argument in a second--

521
00:35:11,340 --> 00:35:13,560
that if I have a
monochromatic laser,

522
00:35:13,560 --> 00:35:15,140
the photon which
has to come out has

523
00:35:15,140 --> 00:35:17,370
to be exactly the
same frequency.

524
00:35:17,370 --> 00:35:20,360
Because we talked about
energy conservation.

525
00:35:20,360 --> 00:35:23,500
And if I would start with a
photon at the laser frequency

526
00:35:23,500 --> 00:35:27,630
and I would measure a photon
which has 10 megahertz away,

527
00:35:27,630 --> 00:35:32,310
I would violate energy
conservation by 10 megahertz.

528
00:35:32,310 --> 00:35:34,490
So the argument is
actually correct.

529
00:35:38,010 --> 00:35:41,390
It's a delta function
at the drive frequency.

530
00:35:41,390 --> 00:35:45,730
But I want to give
you another argument.

531
00:35:45,730 --> 00:35:48,600
It's, again, something you
have heard in the course.

532
00:35:48,600 --> 00:35:50,400
But you should really
take it seriously,

533
00:35:50,400 --> 00:35:52,080
and apply to that situation.

534
00:35:52,080 --> 00:35:53,290
And that's the following.

535
00:35:53,290 --> 00:35:56,480
Remember when we talked
about the AC stark effect.

536
00:35:56,480 --> 00:35:58,940
I told you that
you should really

537
00:35:58,940 --> 00:36:04,240
think about the atom as
a harmonic oscillator.

538
00:36:04,240 --> 00:36:07,380
We even introduced for you
the oscillator strengths

539
00:36:07,380 --> 00:36:10,560
that we could even
quantitatively describe

540
00:36:10,560 --> 00:36:16,110
the response of the atom
to the scattering of light

541
00:36:16,110 --> 00:36:18,800
by pretty much a model where we
have a mechanical model where

542
00:36:18,800 --> 00:36:22,680
an electron is attached
with a spring to an origin.

543
00:36:22,680 --> 00:36:24,717
OK.

544
00:36:24,717 --> 00:36:26,175
Let me now paraphrase
the question.

545
00:36:30,550 --> 00:36:32,507
Well, I've given you the answer.

546
00:36:32,507 --> 00:36:34,090
Before giving you
the answer, I should

547
00:36:34,090 --> 00:36:36,165
have asked you the
following-- if you now

548
00:36:36,165 --> 00:36:39,730
make the assumption that you
have a harmonic oscillator,

549
00:36:39,730 --> 00:36:44,590
the harmonic oscillator has a
resonance frequency of omega 0.

550
00:36:44,590 --> 00:36:48,600
And it has a damping
rate of gamma.

551
00:36:48,600 --> 00:36:52,850
But we are driving the harmonic
oscillator not at omega 0;

552
00:36:52,850 --> 00:36:55,780
we were driving it at
a frequency omega l.

553
00:36:59,321 --> 00:37:03,280
At what frequency does the
harmonic oscillator respond?

554
00:37:03,280 --> 00:37:06,502
At the drive frequency
or at its own frequency?

555
00:37:06,502 --> 00:37:07,386
AUDIENCE: Drive.

556
00:37:07,386 --> 00:37:08,290
PROFESSOR: Of course.

557
00:37:08,290 --> 00:37:10,300
It's a drive frequency.

558
00:37:10,300 --> 00:37:13,920
And if you have a CW experiment,
I said let's wait a second,

559
00:37:13,920 --> 00:37:17,600
let's really do a
long experiment,

560
00:37:17,600 --> 00:37:19,440
if you have a driven
harmonic oscillator,

561
00:37:19,440 --> 00:37:21,560
and you drive it
for a long time,

562
00:37:21,560 --> 00:37:26,090
and you analyze the
spectrum of the motion,

563
00:37:26,090 --> 00:37:30,300
is the frequency spectrum of
the driven harmonic oscillator

564
00:37:30,300 --> 00:37:32,920
absolutely sharp at
the drive frequency?

565
00:37:32,920 --> 00:37:35,885
Or is it broadened by
gamma, the damping constant

566
00:37:35,885 --> 00:37:37,010
of the harmonic oscillator?

567
00:37:40,589 --> 00:37:41,563
AUDIENCE: Sharp.

568
00:37:44,485 --> 00:37:46,506
PROFESSOR: Who
thinks it's sharp?

569
00:37:46,506 --> 00:37:49,380
Who thinks it's broad?

570
00:37:49,380 --> 00:37:51,200
Hands up for sharp.

571
00:37:51,200 --> 00:37:53,699
Hands up for broad.

572
00:37:53,699 --> 00:37:54,199
OK.

573
00:37:54,199 --> 00:37:55,185
AUDIENCE: [INAUDIBLE].

574
00:37:55,185 --> 00:37:58,870
PROFESSOR: Look, what you are
overlooking is the following.

575
00:37:58,870 --> 00:38:01,220
There is the difference
between a transient, which

576
00:38:01,220 --> 00:38:03,390
is damped out at a rate gamma.

577
00:38:03,390 --> 00:38:06,000
But then there is
the CW response.

578
00:38:06,000 --> 00:38:10,230
If you have an ultra-weak drive,
you leave it on for an hour,

579
00:38:10,230 --> 00:38:12,730
the harmonic oscillator
will reach steady state.

580
00:38:12,730 --> 00:38:15,910
And it will just oscillate
driven by your drive.

581
00:38:15,910 --> 00:38:18,070
And it's actually monochromatic.

582
00:38:18,070 --> 00:38:20,010
And if you analyze
the motion, it's

583
00:38:20,010 --> 00:38:22,890
a harmonic oscillator in
steady state with a drive,

584
00:38:22,890 --> 00:38:26,480
and it just moves with a fixed
amplitude and fixed frequency.

585
00:38:26,480 --> 00:38:29,580
I mean, the way how
I shake my hands,

586
00:38:29,580 --> 00:38:33,730
this is a delta function
at the drive frequency.

587
00:38:33,730 --> 00:38:36,780
So in this simple
harmonic oscillator model,

588
00:38:36,780 --> 00:38:40,410
gamma and damping
comes in when you

589
00:38:40,410 --> 00:38:47,890
do what I discussed
earlier, when

590
00:38:47,890 --> 00:38:49,890
you change the drive frequency.

591
00:38:49,890 --> 00:38:52,356
But this is a completely
different experiment.

592
00:38:52,356 --> 00:38:53,980
We are not changing
the drive frequency

593
00:38:53,980 --> 00:38:55,680
and looking for the response.

594
00:38:55,680 --> 00:38:58,136
We are a fixed drive
frequency, and we

595
00:38:58,136 --> 00:39:01,750
are analyzing what the
motion is which comes out.

596
00:39:01,750 --> 00:39:04,660
And then, as you know from
differential equation,

597
00:39:04,660 --> 00:39:06,500
you may have a grade transient.

598
00:39:06,500 --> 00:39:08,930
You may have a transient
at the resonance frequency

599
00:39:08,930 --> 00:39:12,750
which dies out
with a rate gamma.

600
00:39:12,750 --> 00:39:14,970
This is when you suddenly
switch on your drive,

601
00:39:14,970 --> 00:39:17,310
and you're not adiabatically
switching on your drive.

602
00:39:17,310 --> 00:39:19,130
But this is just a transient.

603
00:39:19,130 --> 00:39:21,910
But what we are asking here
when you do a CW experiment,

604
00:39:21,910 --> 00:39:25,400
you drive it for a long
time, you look what happens.

605
00:39:25,400 --> 00:39:31,970
And the motion of the
driven harmonic oscillator

606
00:39:31,970 --> 00:39:35,580
is a delta function
at the drive.

607
00:39:35,580 --> 00:39:37,970
And what is valid for
a harmonic oscillator

608
00:39:37,970 --> 00:39:39,930
is also valid for
the atom, pretty much

609
00:39:39,930 --> 00:39:40,805
for the same reasons.

610
00:39:44,960 --> 00:39:45,635
Any questions?

611
00:39:49,240 --> 00:39:49,740
Yes.

612
00:39:49,740 --> 00:39:51,781
AUDIENCE: Regarding the
energy conservation in D,

613
00:39:51,781 --> 00:39:56,418
wouldn't the photon processes
explain energy conservation?

614
00:39:56,418 --> 00:39:57,540
PROFESSOR: OK.

615
00:39:57,540 --> 00:39:59,240
You are now really
asking-- Nancy

616
00:39:59,240 --> 00:40:02,060
is asking about
two-photon processes.

617
00:40:02,060 --> 00:40:03,660
Well, we don't
want to stop here.

618
00:40:03,660 --> 00:40:05,689
We want to see something
more interesting.

619
00:40:05,689 --> 00:40:07,730
This is just sort of the
trivial, simple harmonic

620
00:40:07,730 --> 00:40:08,610
oscillator.

621
00:40:08,610 --> 00:40:13,862
The question is now, if you
want to see something richer,

622
00:40:13,862 --> 00:40:15,950
if we want to see a
little bit of broadening,

623
00:40:15,950 --> 00:40:18,650
or we want to see something
which is not as boring as just

624
00:40:18,650 --> 00:40:22,204
a classical harmonic oscillator,
what do we have to do?

625
00:40:22,204 --> 00:40:24,464
AUDIENCE: Increase the strength.

626
00:40:24,464 --> 00:40:27,260
PROFESSOR: Increase the
strengths of the field.

627
00:40:27,260 --> 00:40:30,502
And I heard somebody
saying, two photons.

628
00:40:30,502 --> 00:40:31,210
And that may be--

629
00:40:31,210 --> 00:40:31,780
AUDIENCE: Next-order
perturbation theory.

630
00:40:31,780 --> 00:40:32,720
PROFESSOR: Pardon?

631
00:40:32,720 --> 00:40:33,660
AUDIENCE: It's like next
order perturbation theory.

632
00:40:33,660 --> 00:40:35,660
PROFESSOR: Next-order
perturbation theory.

633
00:40:35,660 --> 00:40:37,770
But just in the sort
of intuitive picture,

634
00:40:37,770 --> 00:40:41,020
if you have two photons
which come quickly enough--

635
00:40:41,020 --> 00:40:44,370
and how quickly the two
photons come will be debated.

636
00:40:44,370 --> 00:40:46,685
Of course, it's parametrized
by the Rabi frequency,

637
00:40:46,685 --> 00:40:47,980
the strengths of the drive.

638
00:40:47,980 --> 00:40:50,859
But if two photons come
quickly, each photon

639
00:40:50,859 --> 00:40:52,900
has to be scattered with
a delta function because

640
00:40:52,900 --> 00:40:54,300
of energy conservation.

641
00:40:54,300 --> 00:40:56,910
But if we have two
photons, suddenly, it's

642
00:40:56,910 --> 00:40:59,920
possible that one is scattered
here and one is scattered here.

643
00:40:59,920 --> 00:41:01,760
And we still conserve energy.

644
00:41:01,760 --> 00:41:04,480
So if you want to see some form
of line broadening, if you want

645
00:41:04,480 --> 00:41:08,870
to see all the things
you have mentioned,

646
00:41:08,870 --> 00:41:11,250
we have to go away from
the low power limit.

647
00:41:15,740 --> 00:41:18,650
And this is what we do next.

648
00:41:22,250 --> 00:41:26,730
So we assume that we
have higher power.

649
00:41:29,316 --> 00:41:30,260
We are on resonance.

650
00:41:35,560 --> 00:41:45,130
And let me assume that the Rabi
frequency is larger than gamma.

651
00:41:49,020 --> 00:41:51,309
So what is the physics now
of the atom if the Rabi

652
00:41:51,309 --> 00:41:52,600
frequency is larger than gamma?

653
00:41:56,530 --> 00:41:59,570
It can do Rabi oscillation
before it's damped.

654
00:41:59,570 --> 00:42:04,420
So now we have a
system which has

655
00:42:04,420 --> 00:42:07,025
an internal dynamics
at the Rabi frequency.

656
00:42:10,940 --> 00:42:12,780
And you know if you
have an emitter,

657
00:42:12,780 --> 00:42:15,150
if you have an antenna
which emits light,

658
00:42:15,150 --> 00:42:18,870
and you move the antenna
around at a frequency--

659
00:42:18,870 --> 00:42:21,680
we had the example of a trapped
particle which has a trapped

660
00:42:21,680 --> 00:42:25,230
frequency of omega
trap-- the spectrum,

661
00:42:25,230 --> 00:42:29,450
if you Fourier transform, it
leads to the basal function,

662
00:42:29,450 --> 00:42:31,520
it leads to the sideband.

663
00:42:31,520 --> 00:42:35,820
So therefore, we know now,
based on all the analogies,

664
00:42:35,820 --> 00:42:38,310
if you have a modulated
emitter, we obtain sidebands.

665
00:42:43,210 --> 00:42:46,310
That's what we would
expect classically.

666
00:42:46,310 --> 00:42:49,490
So let me now ask you the
three following possibilities.

667
00:42:52,270 --> 00:42:55,410
Maybe we still get
a delta function.

668
00:42:55,410 --> 00:43:01,400
Maybe we get three peaks where
the splitting is the Rabi

669
00:43:01,400 --> 00:43:02,950
frequency.

670
00:43:02,950 --> 00:43:12,660
So the two outer
peaks have a-- and the

671
00:43:12,660 --> 00:43:17,810
third option is that yes, we
observe the Rabi frequency,

672
00:43:17,810 --> 00:43:20,740
but the Rabi frequency
is the splitting

673
00:43:20,740 --> 00:43:21,910
between the outer peaks.

674
00:43:26,350 --> 00:43:30,650
So the first and second
answer differ by a factor of 2

675
00:43:30,650 --> 00:43:32,005
in the sideband spacing.

676
00:43:35,730 --> 00:43:37,150
So we have three choices.

677
00:43:37,150 --> 00:43:38,750
Do we expect sidebands?

678
00:43:38,750 --> 00:43:42,890
That's A and B. Or would
you expect that we still

679
00:43:42,890 --> 00:43:45,589
have a delta function,
because, well, maybe energy has

680
00:43:45,589 --> 00:43:47,380
to be conserved at the
single photon level?

681
00:43:51,070 --> 00:43:55,880
And then is the splitting
Rabi frequency on each side,

682
00:43:55,880 --> 00:43:59,020
or Rabi frequency between
the two sidebands?

683
00:44:13,012 --> 00:44:13,512
All right.

684
00:44:18,410 --> 00:44:19,640
Yes.

685
00:44:19,640 --> 00:44:23,030
It's indeed the situation.

686
00:44:23,030 --> 00:44:27,280
For those who picked
A, it's pretty much

687
00:44:27,280 --> 00:44:29,155
the definition of what
the Rabi frequency is.

688
00:44:34,710 --> 00:44:38,300
After one cycle of
the Rabi frequency,

689
00:44:38,300 --> 00:44:41,040
the atom has gone
from the ground state

690
00:44:41,040 --> 00:44:44,390
to the excited state and
back to the ground state.

691
00:44:44,390 --> 00:44:48,520
So the model you should have is
that you have an object which

692
00:44:48,520 --> 00:44:50,050
is emitting light,
but the object

693
00:44:50,050 --> 00:44:53,980
has some internal modulation
at the Rabi frequency.

694
00:44:53,980 --> 00:44:58,260
Whatever factor of 2
you had in the amplitude

695
00:44:58,260 --> 00:45:00,540
versus probability or
something, all this

696
00:45:00,540 --> 00:45:02,620
has been factored into
the definition of the Rabi

697
00:45:02,620 --> 00:45:05,680
frequency in such a way
the atom is really blinking

698
00:45:05,680 --> 00:45:08,230
between ground and excited
state with a frequency

699
00:45:08,230 --> 00:45:09,970
which is the Rabi frequency.

700
00:45:09,970 --> 00:45:14,020
And this frequency
leads-- and the spectrum

701
00:45:14,020 --> 00:45:20,600
is now the sum and difference
of the relevant frequencies.

702
00:45:20,600 --> 00:45:24,850
And the relevant frequency
is the resonance frequency

703
00:45:24,850 --> 00:45:26,780
and the Rabi frequency.

704
00:45:26,780 --> 00:45:28,980
So the answer is B.

705
00:45:28,980 --> 00:45:33,880
But this was sort of more
in form of a stick diagram.

706
00:45:33,880 --> 00:45:38,140
Now I want to bring in
in the next question

707
00:45:38,140 --> 00:45:40,280
the line broadening.

708
00:45:40,280 --> 00:45:47,950
So the question is-- and
there are again four choices.

709
00:45:47,950 --> 00:45:52,905
The question is, we have
now our three sticks.

710
00:45:55,610 --> 00:45:59,660
One is a carrier, and these
are the two sidebands split off

711
00:45:59,660 --> 00:46:02,470
by a Rabi frequency.

712
00:46:02,470 --> 00:46:06,140
Are all three sticks now
sharp delta functions?

713
00:46:10,450 --> 00:46:13,490
Are all three sticks broadened
by the natural line widths?

714
00:46:17,240 --> 00:46:20,470
Is the central part
sharp, and only the

715
00:46:20,470 --> 00:46:22,850
sidebands are broadened?

716
00:46:22,850 --> 00:46:28,102
Or do we have a sharp
stick with a pedestal,

717
00:46:28,102 --> 00:46:29,060
and then two sidebands?

718
00:46:31,810 --> 00:46:34,960
I'm not actually expecting
all of you to know the answer,

719
00:46:34,960 --> 00:46:37,160
because this is
really now getting

720
00:46:37,160 --> 00:46:38,380
into more subtle things.

721
00:46:38,380 --> 00:46:42,680
But just in terms
of show-and-tell,

722
00:46:42,680 --> 00:46:46,040
and attract your curiosity
to the second part

723
00:46:46,040 --> 00:46:49,140
of the course, what
would you expect?

724
00:47:20,045 --> 00:47:20,545
OK.

725
00:47:25,580 --> 00:47:26,130
Yes.

726
00:47:26,130 --> 00:47:26,960
Good.

727
00:47:26,960 --> 00:47:31,150
A is definitely eliminated.

728
00:47:31,150 --> 00:47:34,430
I mean, if we scatter two
photons simultaneously, there

729
00:47:34,430 --> 00:47:38,090
will be sort of
broadening of gamma,

730
00:47:38,090 --> 00:47:39,840
because the atom has
a natural broadening.

731
00:47:39,840 --> 00:47:43,940
So we can't expect that the
two photon processes are sharp.

732
00:47:43,940 --> 00:47:45,330
There's no reason
to expect that.

733
00:47:51,200 --> 00:47:57,120
The answer between B,
C, and D depends now.

734
00:47:57,120 --> 00:48:00,820
If we have very, very little
power, what we observe

735
00:48:00,820 --> 00:48:04,200
is, of course, the
low-power delta function.

736
00:48:04,200 --> 00:48:07,470
And then we observe very,
very small sidebands

737
00:48:07,470 --> 00:48:10,070
which are broadened.

738
00:48:10,070 --> 00:48:12,200
So answer C is correct.

739
00:48:12,200 --> 00:48:14,370
If you just think
about it-- what

740
00:48:14,370 --> 00:48:16,820
is the structure of
infinitesimal peaks?

741
00:48:16,820 --> 00:48:20,290
But when you crank up the
power, and what you have is

742
00:48:20,290 --> 00:48:22,200
you have an elastic
scattering peak,

743
00:48:22,200 --> 00:48:24,530
which has a delta function,
energy conservation

744
00:48:24,530 --> 00:48:26,180
at the single photon level.

745
00:48:26,180 --> 00:48:30,280
And then you have those-- they
are called inelastic peaks.

746
00:48:30,280 --> 00:48:35,030
But when you crank up the
power, then the central feature

747
00:48:35,030 --> 00:48:38,150
has also an inelastic component.

748
00:48:38,150 --> 00:48:41,115
You can sort of argue
that in Rabi oscillation,

749
00:48:41,115 --> 00:48:43,490
you are in the ground state,
excited state, ground state,

750
00:48:43,490 --> 00:48:44,210
excited state.

751
00:48:44,210 --> 00:48:46,960
But if you do now light
scattering in the excited

752
00:48:46,960 --> 00:48:49,230
state, you
automatically broaden it

753
00:48:49,230 --> 00:48:52,380
by the lifetime of
the excited state.

754
00:48:52,380 --> 00:48:55,870
And if you crank up the
power higher and higher,

755
00:48:55,870 --> 00:48:59,050
the elastic peak will be
more and more suppressed.

756
00:48:59,050 --> 00:49:01,210
And you find actually
a spectrum which

757
00:49:01,210 --> 00:49:03,695
has only three broadened
peaks, and the delta function

758
00:49:03,695 --> 00:49:05,860
has disappeared.

759
00:49:05,860 --> 00:49:10,380
It will actually be something
where we need more knowledge.

760
00:49:10,380 --> 00:49:14,030
The broadening of those
peaks is not gamma anymore.

761
00:49:14,030 --> 00:49:16,410
One of them will be 3/2
gamma, the other one

762
00:49:16,410 --> 00:49:18,720
will be 1/2 gamma.

763
00:49:18,720 --> 00:49:22,820
But this now really requires
a deeper understanding

764
00:49:22,820 --> 00:49:24,860
in terms of the atom picture.

765
00:49:24,860 --> 00:49:27,100
But the scale of
all the broadening

766
00:49:27,100 --> 00:49:31,700
is a factor on the order
of unity times gamma.

767
00:49:31,700 --> 00:49:32,500
Any questions?

768
00:49:39,210 --> 00:49:43,670
So the general answer, sorry,
would be-- the generic picture

769
00:49:43,670 --> 00:49:46,080
which you should keep
in your mind is that.

770
00:49:46,080 --> 00:49:50,010
You have inelastic scattering
on all three peaks.

771
00:49:50,010 --> 00:49:55,060
But the limit of D at
very low power is that.

772
00:49:55,060 --> 00:50:01,655
And at very high power, is that.

773
00:50:01,655 --> 00:50:04,710
The elastic component
of the central peak

774
00:50:04,710 --> 00:50:06,290
is either 100% or 0%.

775
00:50:16,510 --> 00:50:17,010
OK.

776
00:50:33,580 --> 00:50:45,670
So our last topic for line
shapes and line broadening

777
00:50:45,670 --> 00:50:48,270
is pressure broadening.

778
00:50:48,270 --> 00:50:50,890
And as I mentioned
in the introduction,

779
00:50:50,890 --> 00:50:54,600
pressure broadening has
made modern appearances

780
00:50:54,600 --> 00:50:58,290
in the form of clock shift,
and mean field shifts,

781
00:50:58,290 --> 00:51:02,450
and mean field broadenings
in Bose-Einstein condensates.

782
00:51:02,450 --> 00:51:05,360
But let me sort of
tell you how you

783
00:51:05,360 --> 00:51:07,260
should understand
pressure broadening.

784
00:51:10,110 --> 00:51:14,590
I'm using here the
semi-classical picture

785
00:51:14,590 --> 00:51:19,155
that an excited atom
acts as an oscillator.

786
00:51:23,210 --> 00:51:29,660
And what happened is the atom--
you can say the oscillation

787
00:51:29,660 --> 00:51:32,990
is superposition of
ground and excited state.

788
00:51:32,990 --> 00:51:35,150
It oscillates.

789
00:51:35,150 --> 00:51:46,920
But after some time tau, there
may be a quenching collision,

790
00:51:46,920 --> 00:51:48,280
a de-excitation collision.

791
00:51:52,030 --> 00:51:55,950
Then the atom is sort
of in the ground state.

792
00:51:55,950 --> 00:51:59,130
And then it waits until it's
excited again by the laser,

793
00:51:59,130 --> 00:52:03,540
and the atomic dipole
is oscillating again.

794
00:52:03,540 --> 00:52:05,840
In a situation like
this, you would

795
00:52:05,840 --> 00:52:11,730
expect that the total
line widths is actually

796
00:52:11,730 --> 00:52:13,450
the sum of two rates.

797
00:52:13,450 --> 00:52:17,100
One is the spontaneous
emission rate,

798
00:52:17,100 --> 00:52:23,520
and the other one would be the
collision rate, which is this.

799
00:52:28,710 --> 00:52:33,660
And in general, this is sort
of how people looked at it.

800
00:52:33,660 --> 00:52:37,250
You vary the pressure
in your buffer gas cell,

801
00:52:37,250 --> 00:52:40,490
and you find that the line
width has a component which

802
00:52:40,490 --> 00:52:42,520
increases linearly
with pressure.

803
00:52:45,900 --> 00:52:49,390
So this is one model
how you can imagine

804
00:52:49,390 --> 00:52:51,470
what happens in a collision.

805
00:52:51,470 --> 00:52:54,470
These are like
knock-out collisions.

806
00:52:54,470 --> 00:52:57,730
When an atom collides,
the excitation, the energy

807
00:52:57,730 --> 00:52:58,460
disappears.

808
00:52:58,460 --> 00:53:01,940
The atom is quenched
to the ground state.

809
00:53:01,940 --> 00:53:05,770
Well, we can draw
up another model,

810
00:53:05,770 --> 00:53:14,310
where we have an oscillation
of our atomic oscillator.

811
00:53:14,310 --> 00:53:23,240
But then after
the same time tau,

812
00:53:23,240 --> 00:53:26,100
there is now a
hiccup in the phase.

813
00:53:26,100 --> 00:53:28,290
It collides with another atom.

814
00:53:28,290 --> 00:53:30,750
The other atom is
not de-exciting.

815
00:53:30,750 --> 00:53:32,550
It's not removing the energy.

816
00:53:32,550 --> 00:53:34,910
But after the
collision, the atom

817
00:53:34,910 --> 00:53:40,440
continues to oscillate, but with
a very, very different phase.

818
00:53:40,440 --> 00:53:44,780
And sort of what I assumed
here is this time where

819
00:53:44,780 --> 00:53:48,540
the collision happens
is very short.

820
00:53:48,540 --> 00:53:51,480
It's a very short
collision time.

821
00:53:51,480 --> 00:53:54,850
That I can approximate
the collisions

822
00:53:54,850 --> 00:53:59,540
as simply imparting random
phase jumps to the atom.

823
00:54:02,440 --> 00:54:05,240
Well, if you ask,
what is the widths

824
00:54:05,240 --> 00:54:09,830
of the spectroscopic line,
it's exactly the same result.

825
00:54:12,350 --> 00:54:14,710
It just means we have
a different rate.

826
00:54:14,710 --> 00:54:18,420
It's now a de-phasing
rate, which is 1 over tau.

827
00:54:18,420 --> 00:54:20,370
And this is added to the
spectroscopic widths.

828
00:54:26,830 --> 00:54:27,850
OK.

829
00:54:27,850 --> 00:54:33,340
This is sort of just creating
some phenomenological picture.

830
00:54:33,340 --> 00:54:37,116
Let's now ask a little
bit more microscopically.

831
00:54:39,630 --> 00:54:40,683
How can it happen?

832
00:54:47,540 --> 00:54:53,250
How can it happen that there
is a phase change, that there's

833
00:54:53,250 --> 00:54:57,910
a change in the phase of
the atomic oscillator?

834
00:54:57,910 --> 00:55:02,470
And this leads us now to
consider the interaction

835
00:55:02,470 --> 00:55:08,680
potential as a function
of r between two atoms.

836
00:55:08,680 --> 00:55:14,240
We have one atom which is
sort of our active atom, which

837
00:55:14,240 --> 00:55:18,110
has an excited and ground state.

838
00:55:18,110 --> 00:55:21,475
And this atom is now
getting closer to,

839
00:55:21,475 --> 00:55:24,840
let's say, an argon atom,
which acts as buffer gas.

840
00:55:24,840 --> 00:55:27,450
And what I'm plotting
in this graph now

841
00:55:27,450 --> 00:55:32,830
is the interaction potential
between our light-emitting,

842
00:55:32,830 --> 00:55:35,970
or light-absorbing, atoms
and the buffer gas atom.

843
00:55:35,970 --> 00:55:38,090
And let's just
genetically assume there

844
00:55:38,090 --> 00:55:40,140
is sort of something like
a molecular potential.

845
00:55:44,220 --> 00:55:48,840
But in general, the
molecular potentials

846
00:55:48,840 --> 00:55:51,070
will be different in the
ground and excited state.

847
00:55:53,750 --> 00:55:56,850
And of course, if you
do hyperfine transitions

848
00:55:56,850 --> 00:55:58,750
between atoms in a
Bose-Einstein condensate,

849
00:55:58,750 --> 00:56:00,200
you may have a
scattering length,

850
00:56:00,200 --> 00:56:02,170
which is different
in the two states.

851
00:56:02,170 --> 00:56:04,010
And I think you
see the connection.

852
00:56:04,010 --> 00:56:07,364
So in general, if the
interaction environment

853
00:56:07,364 --> 00:56:09,280
is different between
ground and excited state,

854
00:56:09,280 --> 00:56:11,210
we expect that the
interaction potential

855
00:56:11,210 --> 00:56:13,670
causes certain shifts.

856
00:56:13,670 --> 00:56:27,860
So we could use the picture
that we have a phase evolution.

857
00:56:27,860 --> 00:56:30,910
And the phase evolution
is a frequency difference

858
00:56:30,910 --> 00:56:35,580
which is simply given by
the difference of the two

859
00:56:35,580 --> 00:56:38,440
potentials.

860
00:56:38,440 --> 00:56:40,650
So the picture is a little
bit-- which is actually

861
00:56:40,650 --> 00:56:43,530
very valid for cold
collisions, which-- well,

862
00:56:43,530 --> 00:56:45,580
really a new chapter
in atomic physics

863
00:56:45,580 --> 00:56:49,260
opened up by laser cooling
and magneto-optic traps.

864
00:56:49,260 --> 00:56:51,570
An atom, when it emits a
photon, would emit here

865
00:56:51,570 --> 00:56:53,290
at the resonance frequency.

866
00:56:53,290 --> 00:56:56,170
But if the collision
is very, very slow,

867
00:56:56,170 --> 00:56:59,990
it will actually emit,
with a certain probability,

868
00:56:59,990 --> 00:57:04,130
a photon which is shifted
by exactly this expression.

869
00:57:04,130 --> 00:57:07,200
And in that limit, that
could be very interesting,

870
00:57:07,200 --> 00:57:08,760
because by analyzing
the spectrum

871
00:57:08,760 --> 00:57:10,780
of the immediate light,
you learn something

872
00:57:10,780 --> 00:57:12,655
about the interaction
potential of two atoms.

873
00:57:15,530 --> 00:57:24,330
But of course, this argument
has a little bit of a flaw,

874
00:57:24,330 --> 00:57:29,493
because you can observe a
frequency not instantaneously.

875
00:57:37,335 --> 00:57:55,060
You can observe-- if a
collision happens very fast,

876
00:57:55,060 --> 00:57:59,690
you would actually go through
those frequency changes so fast

877
00:57:59,690 --> 00:58:01,120
that you cannot resolve them.

878
00:58:01,120 --> 00:58:04,160
The question is,
which one is larger?

879
00:58:04,160 --> 00:58:10,290
So if the frequency
shift is larger

880
00:58:10,290 --> 00:58:16,250
than the inverse time for the
duration of the collision,

881
00:58:16,250 --> 00:58:17,250
then you can observe it.

882
00:58:17,250 --> 00:58:19,180
Otherwise, you can't.

883
00:58:19,180 --> 00:58:22,420
So the picture we should
draw is now the following.

884
00:58:22,420 --> 00:58:24,230
And this is sort of
a microscopic picture

885
00:58:24,230 --> 00:58:27,070
on those phase jumps,
which I mentioned earlier.

886
00:58:27,070 --> 00:58:29,280
That you have your
atomic oscillator.

887
00:58:29,280 --> 00:58:30,860
The atom is oscillating.

888
00:58:30,860 --> 00:58:33,400
But now it comes
close to another atom.

889
00:58:33,400 --> 00:58:37,260
And let's assume there
is an energy increase

890
00:58:37,260 --> 00:58:39,630
between ground and excited
state due to the presence

891
00:58:39,630 --> 00:58:40,970
of the buffer gas atom.

892
00:58:40,970 --> 00:58:44,180
Then you would say you
get a quick oscillation.

893
00:58:44,180 --> 00:58:46,340
Depending on the
impact parameter,

894
00:58:46,340 --> 00:58:49,980
it can last various
amount of time.

895
00:58:49,980 --> 00:58:52,230
And then later, the
oscillation starts.

896
00:58:52,230 --> 00:58:55,485
But essentially, this
causes a random phase jump.

897
00:59:00,710 --> 00:59:03,380
So this time here is the
collision time tau c.

898
00:59:19,820 --> 00:59:34,030
What we expect now is we
have to now interpolate

899
00:59:34,030 --> 00:59:36,080
between two models.

900
00:59:36,080 --> 00:59:39,360
I'm not taking it any
further than that.

901
00:59:39,360 --> 00:59:44,130
We have an interesting
line shape.

902
00:59:44,130 --> 00:59:48,880
There may be a central part
which is simply collisionally

903
00:59:48,880 --> 00:59:54,780
broadened by the
interruptions of the phase,

904
00:59:54,780 --> 00:59:57,980
by the number of collisions
with the collision time tau.

905
01:00:03,310 --> 01:00:12,040
But if when the atoms collide,
there is a huge frequency shift

906
01:00:12,040 --> 01:00:19,230
momentarily, that this will
actually affect the wings.

907
01:00:19,230 --> 01:00:23,540
So for frequency shifts which
come from the interaction

908
01:00:23,540 --> 01:00:29,840
potential which are much
larger than the collision time,

909
01:00:29,840 --> 01:00:34,560
than the time
between collisions,

910
01:00:34,560 --> 01:00:38,260
we will observe
something which goes

911
01:00:38,260 --> 01:00:42,580
by the name "far
wing broadening."

912
01:00:42,580 --> 01:00:46,580
And also, the central part
of the potential is simply,

913
01:00:46,580 --> 01:00:50,647
you can say, it's a
Lorentzian which is simply

914
01:00:50,647 --> 01:00:51,980
broadened by the collision rate.

915
01:00:51,980 --> 01:00:54,160
There is no microscopic
information--

916
01:00:54,160 --> 01:00:56,070
what is the nature,
what is going on

917
01:00:56,070 --> 01:00:57,660
during the phase jumps.

918
01:00:57,660 --> 01:01:00,634
The wings will have
actually information

919
01:01:00,634 --> 01:01:01,883
about the molecular potential.

920
01:01:15,710 --> 01:01:18,680
And I just wanted to present
it to you in this way

921
01:01:18,680 --> 01:01:21,480
to sort of show how you
actually have, in a line shape,

922
01:01:21,480 --> 01:01:22,980
often two effects.

923
01:01:22,980 --> 01:01:25,850
One is pretty much
just the interruption

924
01:01:25,850 --> 01:01:28,990
of the coherence which
gives the central line.

925
01:01:28,990 --> 01:01:32,070
But you still have, at
least in some limit,

926
01:01:32,070 --> 01:01:36,100
information about what
causes those phase jumps.

927
01:01:36,100 --> 01:01:39,770
And this appears in the wings.

928
01:01:39,770 --> 01:01:42,440
20 years ago in the
atomic physics course,

929
01:01:42,440 --> 01:01:45,550
we taught you a theory
how to describe that.

930
01:01:45,550 --> 01:01:47,990
It could be probably
taught in one hour.

931
01:01:47,990 --> 01:01:49,750
There are some
links on the Wiki,

932
01:01:49,750 --> 01:01:52,780
but I don't want to
carry that further.

933
01:01:52,780 --> 01:01:55,114
Any questions about that?

934
01:01:55,114 --> 01:01:57,030
AUDIENCE: So why has the
importance decreased?

935
01:02:05,191 --> 01:02:07,020
PROFESSOR: The
importance has decreased

936
01:02:07,020 --> 01:02:08,965
of that, because
I mean, who of you

937
01:02:08,965 --> 01:02:12,280
is studying atoms and gas cells?

938
01:02:12,280 --> 01:02:13,380
Nobody.

939
01:02:13,380 --> 01:02:15,030
The frontier has moved on.

940
01:02:15,030 --> 01:02:17,300
We are much more in a
regime where we are not

941
01:02:17,300 --> 01:02:19,940
observing atoms in
certain environment.

942
01:02:19,940 --> 01:02:23,510
We are creating an interesting
system out of atoms

943
01:02:23,510 --> 01:02:26,445
by putting the atoms in a
well-defined environment where

944
01:02:26,445 --> 01:02:28,940
those things are absent.

945
01:02:28,940 --> 01:02:31,820
Or in the ultra-cold
domain, and de Broglie

946
01:02:31,820 --> 01:02:36,270
wavelengths is so long that this
kind of model for collisions

947
01:02:36,270 --> 01:02:38,000
is no longer applicable.

948
01:02:38,000 --> 01:02:41,250
We are in the extreme case of
a single partial wave, where

949
01:02:41,250 --> 01:02:43,320
a single parameter,
the scattering lengths,

950
01:02:43,320 --> 01:02:44,840
describes all of it.

951
01:02:49,111 --> 01:02:49,610
OK.

952
01:02:59,355 --> 01:02:59,855
Cory?

953
01:02:59,855 --> 01:03:00,648
AUDIENCE: Yeah.

954
01:03:00,648 --> 01:03:03,038
Could you talk for a
little bit about how

955
01:03:03,038 --> 01:03:05,428
this model of collisions
breaks down and allows

956
01:03:05,428 --> 01:03:08,236
for Dicke narrowing?

957
01:03:08,236 --> 01:03:09,610
PROFESSOR: Yes.

958
01:03:09,610 --> 01:03:10,310
Thank you.

959
01:03:10,310 --> 01:03:12,660
That was one comment I forgot.

960
01:03:12,660 --> 01:03:17,360
What happens is you have Dicke--
thanks for this question.

961
01:03:17,360 --> 01:03:21,370
You have Dicke-- I drew up
the two potential curves.

962
01:03:21,370 --> 01:03:25,310
If the two potential curves
are absolutely identical,

963
01:03:25,310 --> 01:03:28,812
the atoms can approach
each other and can collide.

964
01:03:28,812 --> 01:03:30,270
But there is never
any perturbation

965
01:03:30,270 --> 01:03:33,270
to the atomic oscillator,
because the frequency

966
01:03:33,270 --> 01:03:36,480
between ground and
excited state is omega 0,

967
01:03:36,480 --> 01:03:39,682
no matter whether the atom
undergoes a collision or not.

968
01:03:39,682 --> 01:03:46,140
In this limit, which is
often realized in collisions

969
01:03:46,140 --> 01:03:48,550
with rare gases,
in this limit, you

970
01:03:48,550 --> 01:03:52,140
do not have any phase
interruption by the collision.

971
01:03:52,140 --> 01:03:54,490
And this is the prerequisite
for Dicke narrowing.

972
01:03:59,170 --> 01:03:59,670
OK.

973
01:03:59,670 --> 01:04:02,070
I think we have the
full picture now.

974
01:04:02,070 --> 01:04:05,915
And that means we can move
on to two-photon excitations.

975
01:04:13,080 --> 01:04:15,600
So whenever you
start a new chapter,

976
01:04:15,600 --> 01:04:17,050
you should motivate it.

977
01:04:17,050 --> 01:04:21,270
And the question is, why
should you be interested?

978
01:04:21,270 --> 01:04:24,270
Well, I can also turn it
around and said, why not?

979
01:04:24,270 --> 01:04:26,250
Because those things happen.

980
01:04:26,250 --> 01:04:29,430
And if those things happen,
we want to learn about them.

981
01:04:29,430 --> 01:04:32,060
And they actually
happen very naturally.

982
01:04:32,060 --> 01:04:34,960
The moment you go to
higher laser power

983
01:04:34,960 --> 01:04:37,820
and you go beyond the lowest
order of perturbation theory,

984
01:04:37,820 --> 01:04:41,730
you may actually excite two
photons at the same time.

985
01:04:41,730 --> 01:04:44,060
And actually, what we just
discussed about the emission

986
01:04:44,060 --> 01:04:48,150
spectrum of an atom, with the
sidebands, with the broadening,

987
01:04:48,150 --> 01:04:51,080
these are actually
examples where we really

988
01:04:51,080 --> 01:04:54,000
have to think about it in
a two-photon picture, not

989
01:04:54,000 --> 01:04:55,050
a single-photon picture.

990
01:05:00,886 --> 01:05:05,240
The second motivation
is very practical.

991
01:05:05,240 --> 01:05:09,770
We may want to excite an
atom from a low-lying level

992
01:05:09,770 --> 01:05:11,180
to a high-lying level.

993
01:05:11,180 --> 01:05:14,700
But the only laser we can
buy, or the only laser we have

994
01:05:14,700 --> 01:05:17,600
in the laboratory
has lower frequency.

995
01:05:17,600 --> 01:05:19,190
That doesn't mean
that the case is

996
01:05:19,190 --> 01:05:22,890
lost by just stacking two
photons on top of each other.

997
01:05:22,890 --> 01:05:25,340
We can bridge the
gap and still excite

998
01:05:25,340 --> 01:05:28,580
the atom, which has
transitions only far in the UV.

999
01:05:28,580 --> 01:05:30,390
And all we have is
these visible lasers.

1000
01:05:32,970 --> 01:05:35,680
There's another
thing which changes.

1001
01:05:35,680 --> 01:05:40,850
Maybe we want to excite hydrogen
from the 1s to the 2s level.

1002
01:05:40,850 --> 01:05:43,230
1s and 2s have the same parity.

1003
01:05:43,230 --> 01:05:46,260
And if we try to excite
it with one photon,

1004
01:05:46,260 --> 01:05:50,040
because of the dipole
operator which has odd parity,

1005
01:05:50,040 --> 01:05:51,960
this is not allowed.

1006
01:05:51,960 --> 01:05:54,040
So there are maybe
selection rules

1007
01:05:54,040 --> 01:05:57,250
where we can go to
desirable final state

1008
01:05:57,250 --> 01:06:00,790
only with two photons
and not with one photon.

1009
01:06:12,730 --> 01:06:21,990
Finally, I will show you
on Wednesday-- remember,

1010
01:06:21,990 --> 01:06:25,630
there is no class on Monday,
because the whole CUA

1011
01:06:25,630 --> 01:06:29,610
will go crazy with the NSF
site visit on Monday, Tuesday.

1012
01:06:29,610 --> 01:06:31,216
So the next class on Wednesday.

1013
01:06:31,216 --> 01:06:32,590
And on Wednesday,
I will actually

1014
01:06:32,590 --> 01:06:38,030
show you if the two photons come
from counter-propagating beams,

1015
01:06:38,030 --> 01:06:41,020
the net momentum transfer
to the atom is 0.

1016
01:06:41,020 --> 01:06:45,050
And net momentum transfer of
0 means zero Doppler shift.

1017
01:06:45,050 --> 01:06:48,470
So two photons give
us an opportunity,

1018
01:06:48,470 --> 01:06:50,860
which doesn't exist
with the single photon.

1019
01:06:50,860 --> 01:06:53,445
We can excite the atom
without transferring momentum.

1020
01:06:56,220 --> 01:07:00,190
And this is the basis of a
Doppler free spectroscopy

1021
01:07:00,190 --> 01:07:02,240
technique.

1022
01:07:02,240 --> 01:07:05,350
And finally, for purely
conceptional reasons,

1023
01:07:05,350 --> 01:07:08,140
we've talked so much about
excitation of an atom,

1024
01:07:08,140 --> 01:07:11,410
and then ask, what is the
rate of light scattering?

1025
01:07:11,410 --> 01:07:15,630
So we have actually used
two-photon processes

1026
01:07:15,630 --> 01:07:18,606
in the course many, many
times without actually

1027
01:07:18,606 --> 01:07:19,730
adequately addressing them.

1028
01:07:23,865 --> 01:07:25,240
I didn't tell you
anything wrong.

1029
01:07:25,240 --> 01:07:27,050
I was sort of choosing
my words carefully,

1030
01:07:27,050 --> 01:07:29,840
that everything I told you about
light emission and absorption

1031
01:07:29,840 --> 01:07:32,520
is comparable with
the correct picture.

1032
01:07:32,520 --> 01:07:35,140
And the correct picture
is if a photon goes in

1033
01:07:35,140 --> 01:07:37,650
and a photon goes out,
you have to describe it

1034
01:07:37,650 --> 01:07:41,050
as one process
involving two photons.

1035
01:07:41,050 --> 01:07:43,840
So I think we have
enough reasons why

1036
01:07:43,840 --> 01:07:47,790
we should be interested.

1037
01:07:47,790 --> 01:07:54,200
And if I take this picture,
that I start in one state

1038
01:07:54,200 --> 01:08:06,720
and have a two-photon
absorption process,

1039
01:08:06,720 --> 01:08:13,110
I actually want to
redraw it, because

1040
01:08:13,110 --> 01:08:17,560
in the dipole approximation,
which I want to use here,

1041
01:08:17,560 --> 01:08:23,979
we have the dipole operator,
and second quantization

1042
01:08:23,979 --> 01:08:25,490
is a plus a dagger.

1043
01:08:25,490 --> 01:08:28,090
The electric field
is a plus a dagger.

1044
01:08:28,090 --> 01:08:31,090
So each application
of the operator

1045
01:08:31,090 --> 01:08:35,560
takes us from one state to
the next with a single photon.

1046
01:08:35,560 --> 01:08:40,430
So therefore, if you
talk about two photons,

1047
01:08:40,430 --> 01:08:48,870
we can only absorb
them if we have

1048
01:08:48,870 --> 01:08:54,050
an intermediate state,
which I will call f.

1049
01:08:54,050 --> 01:08:58,640
So we should really think
about two-photon absorption

1050
01:08:58,640 --> 01:09:07,261
as a two-step process which
involves an intermediate state.

1051
01:09:11,500 --> 01:09:13,510
And it has to be
like that if we want

1052
01:09:13,510 --> 01:09:16,100
to use, as the
operator for atom light

1053
01:09:16,100 --> 01:09:18,470
interaction, the
dipole operator,

1054
01:09:18,470 --> 01:09:21,000
because a dipole
operator is creating

1055
01:09:21,000 --> 01:09:23,620
and-- the dipole operator
involves electric field.

1056
01:09:23,620 --> 01:09:27,850
And that creates or annihilates
one photon at a time.

1057
01:09:27,850 --> 01:09:29,819
Let me just make a
side remark, but this

1058
01:09:29,819 --> 01:09:32,229
will be addressed in 8.422.

1059
01:09:32,229 --> 01:09:35,330
If you use the description
of the interaction

1060
01:09:35,330 --> 01:09:37,189
with the electromagnetic
field, which

1061
01:09:37,189 --> 01:09:44,260
is the p minus e Hamiltonian,
and you square it,

1062
01:09:44,260 --> 01:09:48,240
if you square that, you
get an A squared term.

1063
01:09:48,240 --> 01:09:51,550
The A squared term is
actually the product

1064
01:09:51,550 --> 01:09:54,580
of a plus a dagger squared.

1065
01:09:54,580 --> 01:09:56,980
So you can actually,
with the A squared term,

1066
01:09:56,980 --> 01:10:02,740
scatter two photons by going
from one state to another one.

1067
01:10:02,740 --> 01:10:06,060
But I'm not discussing it here.

1068
01:10:06,060 --> 01:10:10,690
And I leave a detailed
comparison and discussion

1069
01:10:10,690 --> 01:10:13,120
of that to 8.422.

1070
01:10:13,120 --> 01:10:16,030
Now we are strictly adhering
to the dipole approximation.

1071
01:10:16,030 --> 01:10:17,940
Oh, actually, let me
make a comment about it,

1072
01:10:17,940 --> 01:10:22,220
because I just realized if I
say something and don't say it

1073
01:10:22,220 --> 01:10:24,080
fully, I may confuse people.

1074
01:10:24,080 --> 01:10:27,730
So what I want to say is,
we will show, in 8.422

1075
01:10:27,730 --> 01:10:31,350
that the two pictures, the
pap minus a Hamiltonian

1076
01:10:31,350 --> 01:10:35,780
and the dipole Hamiltonian
are equivalent.

1077
01:10:35,780 --> 01:10:39,670
They're connected by
canonical transformation.

1078
01:10:39,670 --> 01:10:43,720
So therefore, it is not a
fundamental aspect of nature,

1079
01:10:43,720 --> 01:10:45,920
whether you can scatter
only one photon when

1080
01:10:45,920 --> 01:10:48,140
you go from one
state to the other.

1081
01:10:48,140 --> 01:10:50,200
There are two
equivalent descriptions.

1082
01:10:50,200 --> 01:10:52,790
In one description, you
sometimes scatter two photons.

1083
01:10:52,790 --> 01:10:55,330
In the other description,
you scatter only one photon.

1084
01:10:55,330 --> 01:10:57,830
And you will all
get a PSET in 8.422

1085
01:10:57,830 --> 01:11:00,860
where you show for one
example that the results are

1086
01:11:00,860 --> 01:11:04,500
the same when you sum up
over all possibilities.

1087
01:11:04,500 --> 01:11:06,790
So therefore, I can
maybe here should rather

1088
01:11:06,790 --> 01:11:11,400
take the position, the generic
description of light scattering

1089
01:11:11,400 --> 01:11:13,440
uses the dipole approximation.

1090
01:11:13,440 --> 01:11:16,170
And in the dipole
approximation, we

1091
01:11:16,170 --> 01:11:18,860
are describing the
atoms, that they only

1092
01:11:18,860 --> 01:11:22,510
exchange one photon when they
go from one state to the next.

1093
01:11:22,510 --> 01:11:26,190
And this is a full description
but not the only description.

1094
01:11:28,900 --> 01:11:33,020
So therefore, we need
an intermediate state.

1095
01:11:33,020 --> 01:11:36,170
But assuming that the
first laser is not

1096
01:11:36,170 --> 01:11:39,420
resonant with the transition
to the intermediate state,

1097
01:11:39,420 --> 01:11:42,000
we need this dashed line.

1098
01:11:42,000 --> 01:11:46,820
This dashed line is sometimes
called a virtual state.

1099
01:11:46,820 --> 01:11:49,485
And in the following
discussion, we

1100
01:11:49,485 --> 01:11:52,570
will really learn what is the
nature of the virtual state

1101
01:11:52,570 --> 01:11:54,750
and what does it mean.

1102
01:11:54,750 --> 01:12:00,680
But we need a stepping stone
for a two-photon process

1103
01:12:00,680 --> 01:12:02,584
in the form of
intermediate states.

1104
01:12:08,430 --> 01:12:12,750
So maybe I should
just use three minutes

1105
01:12:12,750 --> 01:12:15,650
and show you how you
would calculate it.

1106
01:12:15,650 --> 01:12:18,117
I use pre-written
slides here, because I'm

1107
01:12:18,117 --> 01:12:20,075
getting a little bit
bored of just writing down

1108
01:12:20,075 --> 01:12:23,230
the same or similar
perturbative expressions.

1109
01:12:23,230 --> 01:12:25,600
And I just step you through.

1110
01:12:25,600 --> 01:12:31,260
So I said we use
the dipole operator.

1111
01:12:31,260 --> 01:12:33,390
But now-- and this
is a new thing--

1112
01:12:33,390 --> 01:12:35,650
the electric field
has not only one

1113
01:12:35,650 --> 01:12:37,470
component and one frequency.

1114
01:12:37,470 --> 01:12:40,270
We want to look at two
photons, so therefore, it

1115
01:12:40,270 --> 01:12:42,520
has two frequencies.

1116
01:12:42,520 --> 01:12:44,940
So therefore, our
perturbation Hamiltonian

1117
01:12:44,940 --> 01:12:48,100
is what we had so far
for monochromatic field.

1118
01:12:48,100 --> 01:12:50,450
But then we have
an additional term

1119
01:12:50,450 --> 01:12:53,630
where we just change the
index from one to two

1120
01:12:53,630 --> 01:12:57,710
to have the second
laser field described.

1121
01:12:57,710 --> 01:13:01,910
So let me now introduce
matrix elements.

1122
01:13:01,910 --> 01:13:12,170
We have to go from state a
to some intermediate state.

1123
01:13:12,170 --> 01:13:15,080
And then we go to state b.

1124
01:13:15,080 --> 01:13:16,980
So therefore, we
need matrix elements

1125
01:13:16,980 --> 01:13:20,870
which take us from
state a to state f.

1126
01:13:20,870 --> 01:13:25,720
And again, we have two
such possibilities--

1127
01:13:25,720 --> 01:13:28,410
one at omega 1,
one at omega 2, one

1128
01:13:28,410 --> 01:13:32,175
driven by field 1, the
other one driven by field 2.

1129
01:13:36,140 --> 01:13:41,530
And yeah, we all have
the counter-rotating and

1130
01:13:41,530 --> 01:13:42,280
co-rotating terms.

1131
01:13:46,010 --> 01:13:49,860
What happens is we have, of
course, a complex conjugate.

1132
01:13:49,860 --> 01:13:53,290
But I told you
already several times

1133
01:13:53,290 --> 01:13:56,320
that there is an e
to the minus i omega

1134
01:13:56,320 --> 01:13:58,530
1, which is responsible
for absorption.

1135
01:13:58,530 --> 01:14:00,830
The plus i omega
1 does emission.

1136
01:14:00,830 --> 01:14:04,090
And let me just-- I want
to do the rotating wave

1137
01:14:04,090 --> 01:14:06,500
approximation, only
keep the relevant term.

1138
01:14:06,500 --> 01:14:08,900
So I look at
two-photon absorption.

1139
01:14:08,900 --> 01:14:12,290
And that means I only keep
forward the minus terms.

1140
01:14:12,290 --> 01:14:15,690
You could, if you
want, duplicate

1141
01:14:15,690 --> 01:14:17,750
the lengths of each
formula and carry forward

1142
01:14:17,750 --> 01:14:18,940
the counter-rotating term.

1143
01:14:18,940 --> 01:14:20,450
You're not learning
anything new.

1144
01:14:20,450 --> 01:14:21,825
You get additional
Bloch-Siegerts

1145
01:14:21,825 --> 01:14:23,100
and AC stark shifts.

1146
01:14:23,100 --> 01:14:28,180
This is not the new feature
I want to implement.

1147
01:14:28,180 --> 01:14:30,480
So let's focus on
the new aspect.

1148
01:14:30,480 --> 01:14:31,930
And this is the following.

1149
01:14:31,930 --> 01:14:34,880
We do first-order
perturbation theory,

1150
01:14:34,880 --> 01:14:38,210
which takes us in the first
step to the intermediate state.

1151
01:14:38,210 --> 01:14:42,290
And you have seen this
expression many, many times.

1152
01:14:42,290 --> 01:14:46,860
The only thing is in addition to
what we had for one laser beam,

1153
01:14:46,860 --> 01:14:50,158
we have to add a second
possibility, which

1154
01:14:50,158 --> 01:14:51,324
comes from the second field.

1155
01:14:55,040 --> 01:14:57,630
And now we are at the
intermediate step.

1156
01:14:57,630 --> 01:15:00,900
And we want to take the second
step to the final state.

1157
01:15:00,900 --> 01:15:03,940
So what we are now doing is we
are writing down Schrodinger's

1158
01:15:03,940 --> 01:15:06,410
equation, derivative
of the wave function

1159
01:15:06,410 --> 01:15:09,410
is Hamiltonian times
the wave function.

1160
01:15:09,410 --> 01:15:11,450
And we are especially
interested in how

1161
01:15:11,450 --> 01:15:16,930
do we accumulate probability or
amplitude in the final state.

1162
01:15:16,930 --> 01:15:21,200
But what we are doing is
on the right-hand side,

1163
01:15:21,200 --> 01:15:23,090
because we can't start
in the ground state,

1164
01:15:23,090 --> 01:15:24,870
we've gone to the
intermediate state,

1165
01:15:24,870 --> 01:15:28,040
we are now plugging in
the previous result,

1166
01:15:28,040 --> 01:15:31,530
the first-order result for
the intermediate state.

1167
01:15:31,530 --> 01:15:34,212
So therefore, by using
second-order perturbation

1168
01:15:34,212 --> 01:15:36,420
theory, we want the second-order
perturbation theory,

1169
01:15:36,420 --> 01:15:39,660
so we integrate this equation
with respect to time.

1170
01:15:39,660 --> 01:15:43,450
But on the right side, we
use our first-order result,

1171
01:15:43,450 --> 01:15:46,900
which we had derived earlier.

1172
01:15:46,900 --> 01:15:49,180
And then we just write
down the integral.

1173
01:15:49,180 --> 01:15:51,780
Everything is just
exponential function.

1174
01:15:51,780 --> 01:15:53,910
And we get the result here.

1175
01:15:53,910 --> 01:16:04,770
So now with two steps, we
have obtained an expression

1176
01:16:04,770 --> 01:16:08,650
for-- I know time
is over, but let

1177
01:16:08,650 --> 01:16:10,790
me just finish the argument.

1178
01:16:10,790 --> 01:16:13,280
So we have now an expression
in second-order perturbation

1179
01:16:13,280 --> 01:16:14,260
theory.

1180
01:16:14,260 --> 01:16:17,150
What is the amplitude
in the final state?

1181
01:16:17,150 --> 01:16:19,150
And things look a
little bit messy,

1182
01:16:19,150 --> 01:16:21,230
because we have four terms.

1183
01:16:21,230 --> 01:16:23,430
And we should get
four terms, because we

1184
01:16:23,430 --> 01:16:24,930
have two interactions.

1185
01:16:24,930 --> 01:16:27,480
We can take one
photon of one laser

1186
01:16:27,480 --> 01:16:30,870
beam-- first step, omega
1; second step, omega two.

1187
01:16:30,870 --> 01:16:32,960
We can switch the
order of photons.

1188
01:16:32,960 --> 01:16:35,560
And of course, if you write
down everything correctly,

1189
01:16:35,560 --> 01:16:38,600
nothing is forbidding the atom
of taking both photons out

1190
01:16:38,600 --> 01:16:40,912
of the same laser beam.

1191
01:16:48,200 --> 01:16:50,350
Yeah, let me stop here.