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PROFESSOR: So we want to
discuss line shifts and line

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00:00:24,260 --> 00:00:25,820
broadening.

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00:00:25,820 --> 00:00:31,950
And I assume you'll
remember that last class we

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00:00:31,950 --> 00:00:39,740
did some brainstorming and came
up with quite a number of line

12
00:00:39,740 --> 00:00:44,690
broadening and line shifts
mechanism, temporal lifetime

13
00:00:44,690 --> 00:00:48,040
broadening, motional
broadening, external field

14
00:00:48,040 --> 00:00:50,420
broadening,
collisional broadening.

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00:00:50,420 --> 00:00:58,760
And as I indicated at
the end of last class,

16
00:00:58,760 --> 00:01:00,720
those different
broadening mechanism

17
00:01:00,720 --> 00:01:03,770
have much more in common
than you may think.

18
00:01:03,770 --> 00:01:06,810
And you will see that they
have much more in common

19
00:01:06,810 --> 00:01:09,730
when we look at it from
the fundamental perspective

20
00:01:09,730 --> 00:01:11,020
of coherence.

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00:01:11,020 --> 00:01:13,720
Any line broadening
mechanism comes

22
00:01:13,720 --> 00:01:18,470
because the atom experience, the
environment and the drive field

23
00:01:18,470 --> 00:01:23,000
as a coherent source only
for limited coherence time.

24
00:01:23,000 --> 00:01:25,630
So the concept of
the coherence time

25
00:01:25,630 --> 00:01:29,000
will actually provide
a common denominator

26
00:01:29,000 --> 00:01:32,410
for all those line
broadening mechanisms.

27
00:01:32,410 --> 00:01:36,580
However, before I
present to you this kind

28
00:01:36,580 --> 00:01:40,470
of correlation function approach
to what's line broadening,

29
00:01:40,470 --> 00:01:42,860
I think it's really
important that we

30
00:01:42,860 --> 00:01:48,290
go through some simple cases.

31
00:01:48,290 --> 00:01:52,570
I always want you to learn
phenomena in the simplest

32
00:01:52,570 --> 00:01:55,800
possible manifestation,
in a situation where

33
00:01:55,800 --> 00:01:58,850
without any math you
see what's going on.

34
00:01:58,850 --> 00:02:01,080
And what I particularly
love is if there

35
00:02:01,080 --> 00:02:04,040
is an analytic solution
where everything

36
00:02:04,040 --> 00:02:06,030
is sort of transparent here.

37
00:02:06,030 --> 00:02:10,030
So therefore, before we discuss
in a comprehensive way line

38
00:02:10,030 --> 00:02:11,800
shifts and line
broadening, I want

39
00:02:11,800 --> 00:02:14,550
to go through simple cases.

40
00:02:14,550 --> 00:02:19,350
Now, I decided for this class
that most of today's lecture

41
00:02:19,350 --> 00:02:24,050
uses pre-written slides
because those cases are

42
00:02:24,050 --> 00:02:27,600
so simple that I would be
almost afraid I would bore you

43
00:02:27,600 --> 00:02:29,030
if I would write it out.

44
00:02:29,030 --> 00:02:31,210
Because if you look at
this equation in one second

45
00:02:31,210 --> 00:02:34,580
you get it, and it takes me
10 or 20 seconds to write it.

46
00:02:34,580 --> 00:02:37,050
On the other hand,
I would sort of

47
00:02:37,050 --> 00:02:38,940
give the responsibility to you.

48
00:02:38,940 --> 00:02:41,620
If I'm flooding you with
too much information,

49
00:02:41,620 --> 00:02:43,950
ask me questions,
slow me down, or say,

50
00:02:43,950 --> 00:02:45,570
can you please go over that?

51
00:02:45,570 --> 00:02:47,730
So I don't want
to obscure things

52
00:02:47,730 --> 00:02:50,480
by going through them faster.

53
00:02:50,480 --> 00:02:51,980
I also looked a
little bit forward

54
00:02:51,980 --> 00:02:54,220
to the end of the course.

55
00:02:54,220 --> 00:02:58,820
We are about OK with the
pace of the course, or maybe

56
00:02:58,820 --> 00:03:00,410
one hour behind.

57
00:03:00,410 --> 00:03:05,150
And so in order to make room for
discussions of superradiance,

58
00:03:05,150 --> 00:03:07,740
which I would like to have
with you in a couple of weeks,

59
00:03:07,740 --> 00:03:09,520
I thought I can save
half an hour here

60
00:03:09,520 --> 00:03:12,790
by going over that material
a little bit faster.

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00:03:12,790 --> 00:03:15,620
But these are really
just illustrations

62
00:03:15,620 --> 00:03:16,760
of line broadening.

63
00:03:16,760 --> 00:03:21,120
And you should actually
know most of them already.

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00:03:21,120 --> 00:03:27,870
So the first cases are
actually leading you up

65
00:03:27,870 --> 00:03:30,790
to the Ramsey
resonance, to the method

66
00:03:30,790 --> 00:03:34,350
of separated oscillatory fields,
which is really absolutely

67
00:03:34,350 --> 00:03:36,560
important for high
resolution spectroscopy.

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00:03:36,560 --> 00:03:38,760
But before we can fully
understand the Ramsey

69
00:03:38,760 --> 00:03:42,600
resonance, we have to
understand the Rabi resonance.

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00:03:42,600 --> 00:03:44,500
So what I've written
down here is for you

71
00:03:44,500 --> 00:03:48,600
just the well-known formula
for Rabi oscillations.

72
00:03:48,600 --> 00:03:53,390
And if you now simply assume
we let the atoms interact

73
00:03:53,390 --> 00:03:57,940
with the field by a fixed
amount time tau in such a way

74
00:03:57,940 --> 00:04:01,960
that we have a pi pulse,
then this probability

75
00:04:01,960 --> 00:04:06,960
as a function of detuning
becomes a line shape.

76
00:04:06,960 --> 00:04:10,920
And this line shape
is plotted here.

77
00:04:13,510 --> 00:04:16,329
It has a full width
of half maximum,

78
00:04:16,329 --> 00:04:19,180
which is approximately 1/tau.

79
00:04:19,180 --> 00:04:22,280
And this is-- you can
regard it as the Fourier

80
00:04:22,280 --> 00:04:26,230
limit of finite interaction
time broadening.

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00:04:26,230 --> 00:04:28,590
We are observing the
atoms for time tau

82
00:04:28,590 --> 00:04:31,920
and the line which is
on the order of 1/tau.

83
00:04:31,920 --> 00:04:33,950
But we also know
from the Rabi formula

84
00:04:33,950 --> 00:04:37,940
that we have these side lobes,
so we see some oscillatory line

85
00:04:37,940 --> 00:04:40,302
shape.

86
00:04:40,302 --> 00:04:46,370
OK, now, with the
advent of trapped atoms,

87
00:04:46,370 --> 00:04:48,764
you can often have
the situation that you

88
00:04:48,764 --> 00:04:50,180
have an ensemble
of trapped atoms.

89
00:04:50,180 --> 00:04:55,950
You flash on your drive field
for fixed amount time tau.

90
00:04:55,950 --> 00:05:02,020
At least for most of last
century, this was not possible.

91
00:05:02,020 --> 00:05:04,200
You had atomic beams.

92
00:05:04,200 --> 00:05:07,380
And in an atomic
beam, what is fixed

93
00:05:07,380 --> 00:05:11,730
is not the interaction time tau
but the interaction length l.

94
00:05:11,730 --> 00:05:14,800
And then due to the
velocity distribution,

95
00:05:14,800 --> 00:05:16,730
different velocity
groups of atoms

96
00:05:16,730 --> 00:05:20,140
interact a different
amount of time with that.

97
00:05:20,140 --> 00:05:24,080
So for conceptional reasons but
also for the historic context,

98
00:05:24,080 --> 00:05:28,270
you should sort of have an idea
what the Rabi apparatus is.

99
00:05:31,380 --> 00:05:34,860
I may have mentioned
it, but I regard Rabi

100
00:05:34,860 --> 00:05:38,650
as sort of one of my
ancestors in my family tree

101
00:05:38,650 --> 00:05:41,620
because Rabi was the PhD
advisor of Norman Ramsey, who

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00:05:41,620 --> 00:05:43,540
was the PhD advisor
of Dan Kleppner, who

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00:05:43,540 --> 00:05:45,950
was the PhD advisor
of Dave Pritchard.

104
00:05:45,950 --> 00:05:48,210
And Dave Pritchard was
my postdoctoral mentor.

105
00:05:48,210 --> 00:05:51,690
So I'm really talking here
about my scientific great-great

106
00:05:51,690 --> 00:05:53,770
grandfather.

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00:05:53,770 --> 00:05:57,050
Anyway, so the famous Rabi
apparatus is the following.

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00:05:57,050 --> 00:06:00,400
You use one Stern-Gerlach magnet
to prepare a certain hyperfine

109
00:06:00,400 --> 00:06:02,210
state.

110
00:06:02,210 --> 00:06:04,330
Then you have the
interaction region,

111
00:06:04,330 --> 00:06:06,570
and this is what
we are focusing on.

112
00:06:06,570 --> 00:06:09,590
And then later, if a spin
flip has taken place,

113
00:06:09,590 --> 00:06:11,840
you can figure that
out by running it

114
00:06:11,840 --> 00:06:14,110
through a
Stern-Gerlach analyzer.

115
00:06:14,110 --> 00:06:16,610
So it's this ABC
region, and we are

116
00:06:16,610 --> 00:06:21,500
talking about this middle region
where we have an interaction

117
00:06:21,500 --> 00:06:27,030
time which is now given by the
length divided by the velocity.

118
00:06:27,030 --> 00:06:29,070
So what we have to
do is-- well, that's

119
00:06:29,070 --> 00:06:30,480
why I'm saying
it's simple cases,

120
00:06:30,480 --> 00:06:33,260
and I hope I can
go fast-- you just

121
00:06:33,260 --> 00:06:36,440
take the previous result
with a Rabi probability

122
00:06:36,440 --> 00:06:39,660
and convolute it with a
velocity distribution.

123
00:06:39,660 --> 00:06:46,490
So when we do that,
we find two effects.

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00:06:46,490 --> 00:06:49,600
One is, well, due to the
velocity distribution,

125
00:06:49,600 --> 00:06:52,250
the line width becomes
a factor of 2 broader,

126
00:06:52,250 --> 00:06:54,100
but it's still
proportional to something

127
00:06:54,100 --> 00:06:57,230
on the order of unity divided
by the interaction time.

128
00:06:57,230 --> 00:07:01,770
And because the
different velocity groups

129
00:07:01,770 --> 00:07:04,710
have a different
kind of oscillations

130
00:07:04,710 --> 00:07:08,160
as a function of detuning,
the oscillatory behavior

131
00:07:08,160 --> 00:07:10,462
now disappears because
it becomes averaged out

132
00:07:10,462 --> 00:07:11,670
[? of ?] the velocity groups.

133
00:07:14,240 --> 00:07:18,080
OK so this is the Rabi
method or the Rabi resonance,

134
00:07:18,080 --> 00:07:22,460
where we have one interaction
pulse or one interaction

135
00:07:22,460 --> 00:07:26,190
time for an atom.

136
00:07:26,190 --> 00:07:30,970
The next method is now
what was introduced

137
00:07:30,970 --> 00:07:35,070
by Norman Ramsey as the method
of separate oscillatory fields.

138
00:07:35,070 --> 00:07:38,020
And for that and
other contributions

139
00:07:38,020 --> 00:07:42,010
he was awarded the
Nobel Prize in 1990.

140
00:07:42,010 --> 00:07:44,490
And the difference
is the following.

141
00:07:44,490 --> 00:07:47,410
In the Rabi method, we
let the atoms interact

142
00:07:47,410 --> 00:07:52,010
with the drive field for fixed
lengths l or fixed time tau.

143
00:07:52,010 --> 00:07:55,760
But in the Ramsey
method, we interrogate,

144
00:07:55,760 --> 00:08:02,650
we drive the atoms
with two short pulses,

145
00:08:02,650 --> 00:08:04,880
which are separated.

146
00:08:04,880 --> 00:08:09,060
So in the simplest
case, we do 2 pi

147
00:08:09,060 --> 00:08:14,510
over 2 pulses
separated by a time t.

148
00:08:14,510 --> 00:08:18,470
And now, if you have two
pulses, everything is coherent.

149
00:08:18,470 --> 00:08:22,380
The two pulses can
interfere constructively

150
00:08:22,380 --> 00:08:23,160
or destructively.

151
00:08:31,660 --> 00:08:40,960
So what you would
expect is that-- I

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00:08:40,960 --> 00:08:42,210
want to give you two pictures.

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00:08:42,210 --> 00:08:44,590
One is the Bloch sphere
picture, but let's just sort

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00:08:44,590 --> 00:08:48,270
of play with the
concept of interference.

155
00:08:48,270 --> 00:08:59,860
If this is zero
detuning, the two pulses

156
00:08:59,860 --> 00:09:02,490
are separated by a time t.

157
00:09:02,490 --> 00:09:04,830
And therefore, you
would now observe,

158
00:09:04,830 --> 00:09:09,880
as a function of detuning,
Ramsey fringes and oscillatory

159
00:09:09,880 --> 00:09:16,710
behavior, which
has a spacing which

160
00:09:16,710 --> 00:09:22,631
is 2 pi over the
time between pulses.

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00:09:22,631 --> 00:09:26,060
The envelope of
the whole fringes

162
00:09:26,060 --> 00:09:29,410
is related to the short
time of the duration.

163
00:09:29,410 --> 00:09:30,940
It's like a double
slit experiment.

164
00:09:30,940 --> 00:09:35,160
You know, each slit is
broad, and the width here

165
00:09:35,160 --> 00:09:38,830
is given by 1 over the short
time delta t of the pulse.

166
00:09:38,830 --> 00:09:40,980
But then the two
slits interfere.

167
00:09:40,980 --> 00:09:43,330
And the interference
pattern is the distance

168
00:09:43,330 --> 00:09:46,190
between the slits, which
now in the temporal domain

169
00:09:46,190 --> 00:09:49,110
is the time capital
T between them.

170
00:09:49,110 --> 00:09:51,630
So these are sort of
typical Ramsey fringes.

171
00:09:51,630 --> 00:09:55,380
And now, if you would
average over a broad velocity

172
00:09:55,380 --> 00:10:00,360
distribution, then
you would kind of

173
00:10:00,360 --> 00:10:03,300
average-- you would maybe
see one or two side lobes,

174
00:10:03,300 --> 00:10:05,820
but the other fringes
are averaged out.

175
00:10:13,530 --> 00:10:25,460
So the central peak,
the resolution is again

176
00:10:25,460 --> 00:10:30,640
on the order of 1/T.
It's the total time

177
00:10:30,640 --> 00:10:32,390
of the experiment,
which is setting

178
00:10:32,390 --> 00:10:37,690
the ultimate resolution,
similar to the Rabi resonance.

179
00:10:37,690 --> 00:10:41,170
So this uses the
picture of interference

180
00:10:41,170 --> 00:10:44,040
between two pulses.

181
00:10:44,040 --> 00:10:45,950
But I also want to
sort of give you

182
00:10:45,950 --> 00:10:48,600
the Bloch sphere picture,
because it's beautiful.

183
00:10:48,600 --> 00:10:50,340
For that I need a
little bit of room.

184
00:10:50,340 --> 00:10:53,650
So an atom enters the
first Ramsey region,

185
00:10:53,650 --> 00:10:55,410
and it has to spin down.

186
00:10:55,410 --> 00:10:59,390
You do a pi/2 pulse,
which is at 90 degrees.

187
00:10:59,390 --> 00:11:03,100
And now what happens is
there is a field-free region

188
00:11:03,100 --> 00:11:06,060
between the two pulses,
so the atom is now

189
00:11:06,060 --> 00:11:07,855
precessing at its
resonance frequency.

190
00:11:10,430 --> 00:11:14,670
OK, but the synthesizer which
is attached to your coils

191
00:11:14,670 --> 00:11:17,980
is also sort of precessing
at this frequency.

192
00:11:17,980 --> 00:11:22,290
And when you add zero detuning,
the synthesizer and the atoms

193
00:11:22,290 --> 00:11:23,860
are aligned again.

194
00:11:23,860 --> 00:11:27,880
And the second pi/2 pulse
is now flipping the atoms,

195
00:11:27,880 --> 00:11:29,760
and you've 100% excitation.

196
00:11:29,760 --> 00:11:32,000
And this is what you see here.

197
00:11:32,000 --> 00:11:35,780
But now let's assume you
are slightly detuned.

198
00:11:35,780 --> 00:11:40,280
Then the atom is precessing,
and your synthesizer

199
00:11:40,280 --> 00:11:43,100
is precessing at a slightly
different frequency.

200
00:11:43,100 --> 00:11:46,250
And now in the second
region, the frequency

201
00:11:46,250 --> 00:11:48,960
of the synthesizer
may be different

202
00:11:48,960 --> 00:11:51,700
with-- the phase
of the synthesizer

203
00:11:51,700 --> 00:11:56,340
is 180 degree different
from the phase of the atom.

204
00:11:56,340 --> 00:12:00,560
And then instead of flipping
the atoms up and getting 100%

205
00:12:00,560 --> 00:12:03,570
in the up state, the
atom is now flipped down.

206
00:12:03,570 --> 00:12:06,660
And this explains
the first minimum.

207
00:12:06,660 --> 00:12:10,230
So you should really see
that in this long region,

208
00:12:10,230 --> 00:12:11,430
nothing happens.

209
00:12:11,430 --> 00:12:15,640
But you accrue a relative
phase between the synthesizer

210
00:12:15,640 --> 00:12:19,900
and the atom, which oscillates
at its resonance frequency.

211
00:12:19,900 --> 00:12:22,350
So based on this model,
you could work out

212
00:12:22,350 --> 00:12:25,980
mathematically every aspect of
those fringes I've shown you.

213
00:12:25,980 --> 00:12:30,300
But I decided to take the
equations out of the lecture

214
00:12:30,300 --> 00:12:32,390
and just present you with
this physical picture.

215
00:12:35,250 --> 00:12:36,970
Questions?

216
00:12:36,970 --> 00:12:39,220
AUDIENCE: If you have a
velocity distribution,

217
00:12:39,220 --> 00:12:42,480
would the points of
those minimums change

218
00:12:42,480 --> 00:12:44,879
or is it still this--

219
00:12:44,879 --> 00:12:46,170
PROFESSOR: Well, good question.

220
00:12:46,170 --> 00:12:49,960
The spacing is the time T.
But if you have a velocity

221
00:12:49,960 --> 00:12:53,800
distribution, let's say a
velocity distribution which

222
00:12:53,800 --> 00:12:57,620
has a width delta v over v
of 30%, what we have kept

223
00:12:57,620 --> 00:13:00,480
fixed in the beam
experiment are the lengths,

224
00:13:00,480 --> 00:13:02,210
the interaction lengths.

225
00:13:02,210 --> 00:13:04,700
So time is length over velocity.

226
00:13:04,700 --> 00:13:06,910
So therefore, when
the velocity changes

227
00:13:06,910 --> 00:13:10,660
by plus/minus 30% in
your atomic beam--

228
00:13:10,660 --> 00:13:12,665
unless you take a
supersonic beam with a very

229
00:13:12,665 --> 00:13:14,920
narrow velocity
distribution-- then that

230
00:13:14,920 --> 00:13:19,340
means one velocity class
has this set of fringes.

231
00:13:19,340 --> 00:13:22,890
The other velocity class has
this similar set of fringes,

232
00:13:22,890 --> 00:13:26,130
but like in a
harmonica, everything

233
00:13:26,130 --> 00:13:28,390
is now spread out by 30%.

234
00:13:28,390 --> 00:13:31,440
And if you have now 30%
velocity resolution,

235
00:13:31,440 --> 00:13:33,985
that means you may be able
to see two or three fringes.

236
00:13:36,560 --> 00:13:43,220
But the central fringe is sort
of like a white light fringe.

237
00:13:43,220 --> 00:13:45,180
There is no relative
phase shift.

238
00:13:45,180 --> 00:13:48,240
And therefore all the
different velocity classes

239
00:13:48,240 --> 00:13:50,520
will have a maximum
at zero detuning.

240
00:13:50,520 --> 00:13:53,460
So in the extreme limit
of a very broad velocity

241
00:13:53,460 --> 00:13:56,370
distribution, the only
feature which survives

242
00:13:56,370 --> 00:13:58,230
is the central fringe.

243
00:13:58,230 --> 00:14:01,480
But this is where you obtain
your spectroscopic information

244
00:14:01,480 --> 00:14:02,340
from.

245
00:14:02,340 --> 00:14:03,036
Will?

246
00:14:03,036 --> 00:14:05,952
AUDIENCE: You have explained
this Bloch sphere picture

247
00:14:05,952 --> 00:14:09,354
assuming unbroken coherence
between your synthesizer

248
00:14:09,354 --> 00:14:11,920
and the field-free
evolution of your atom?

249
00:14:11,920 --> 00:14:12,545
PROFESSOR: Yes.

250
00:14:12,545 --> 00:14:15,265
AUDIENCE: Is it equivalent
to if you wrote down the time

251
00:14:15,265 --> 00:14:17,720
evolution operator, you
would say that there's 0 Rabi

252
00:14:17,720 --> 00:14:21,165
frequency, but no
detuning in your field?

253
00:14:21,165 --> 00:14:21,790
PROFESSOR: Yes.

254
00:14:21,790 --> 00:14:25,726
AUDIENCE: So what if
I-- is this necessary,

255
00:14:25,726 --> 00:14:28,702
to have this unbroken coherence
if I unplug my synthesizer

256
00:14:28,702 --> 00:14:30,160
and plug in a new
one [INAUDIBLE]--

257
00:14:30,160 --> 00:14:32,250
PROFESSOR: Yes, you
can do spectroscopy

258
00:14:32,250 --> 00:14:37,230
with a resolution
delta mu, which is 1/T,

259
00:14:37,230 --> 00:14:40,080
only if you have a synthesizer
which has a frequency

260
00:14:40,080 --> 00:14:42,660
stability which is
better than 1/T.

261
00:14:42,660 --> 00:14:44,580
AUDIENCE: OK.

262
00:14:44,580 --> 00:14:47,110
PROFESSOR: Otherwise, we've
done this a little bit

263
00:14:47,110 --> 00:14:50,240
when we talk narrowband
and broadband.

264
00:14:50,240 --> 00:14:53,860
When we talked about
narrowband and broadband cases,

265
00:14:53,860 --> 00:14:56,040
you are always limited
in your resolution

266
00:14:56,040 --> 00:14:58,340
by whatever is broader.

267
00:14:58,340 --> 00:15:01,710
Here I'm discussing about the
width of the atomic system,

268
00:15:01,710 --> 00:15:04,110
assuming a perfect synthesizer.

269
00:15:04,110 --> 00:15:07,750
But then in essence, you
should convolute this result

270
00:15:07,750 --> 00:15:12,320
with the spectral distribution
of your synthesizer.

271
00:15:12,320 --> 00:15:16,390
And then if the synthesizer
has a resolution which

272
00:15:16,390 --> 00:15:20,890
is worse than that, you would
actually blur out the fringes

273
00:15:20,890 --> 00:15:22,710
through the convolution
with the frequency

274
00:15:22,710 --> 00:15:25,480
spectrum of the synthesizer.

275
00:15:25,480 --> 00:15:25,980
[INAUDIBLE]?

276
00:15:25,980 --> 00:15:28,230
AUDIENCE: So that also means
if you add a [INAUDIBLE]

277
00:15:28,230 --> 00:15:36,482
to the second coil, [INAUDIBLE]
just like if you had

278
00:15:36,482 --> 00:15:38,240
[INAUDIBLE] to one of the--

279
00:15:38,240 --> 00:15:43,960
PROFESSOR: Yes, actually,
if you would add a pi phase

280
00:15:43,960 --> 00:15:48,750
shift to the second
coil, then the minimum--

281
00:15:48,750 --> 00:15:51,231
the central feature would
not be maximum or minimal.

282
00:15:51,231 --> 00:15:51,730
Yes.

283
00:15:58,150 --> 00:16:04,690
And that's why when you have
atomic clocks with a beam,

284
00:16:04,690 --> 00:16:06,670
the question of
distributed phase

285
00:16:06,670 --> 00:16:10,840
shifts within the microwave
cavity play a big role.

286
00:16:10,840 --> 00:16:12,820
And this is related
to are the two Ramsey

287
00:16:12,820 --> 00:16:15,870
zones really at the
same phase or not?

288
00:16:15,870 --> 00:16:19,421
So that's an important
issue for ultimately making

289
00:16:19,421 --> 00:16:20,420
resolution spectroscopy.

290
00:16:24,430 --> 00:16:24,950
OK.

291
00:16:24,950 --> 00:16:37,140
Since-- OK.

292
00:16:37,140 --> 00:16:39,360
So we've discussed
two methods now,

293
00:16:39,360 --> 00:16:42,300
the Ramsey method
versus the Rabi method.

294
00:16:42,300 --> 00:16:47,470
And let me discuss advantages
or disadvantages of the method.

295
00:16:47,470 --> 00:16:49,840
So if you're an atomic
physicist and you

296
00:16:49,840 --> 00:16:51,680
have to give advice
to your friends

297
00:16:51,680 --> 00:16:54,740
whether they should use the
Ramsey or the Rabi method,

298
00:16:54,740 --> 00:16:56,950
these are your talking points.

299
00:16:56,950 --> 00:17:02,750
So one point is that I said the
central feature of the Ramsey

300
00:17:02,750 --> 00:17:08,339
fringes is 1/T. The
Rabi feature is 1/T,

301
00:17:08,339 --> 00:17:10,557
so we are both limited
by time resolution,

302
00:17:10,557 --> 00:17:11,890
because this is a Fourier limit.

303
00:17:11,890 --> 00:17:14,050
But if you work out
the details, you

304
00:17:14,050 --> 00:17:28,610
find that the Ramsey fringe
is about 2 times narrower

305
00:17:28,610 --> 00:17:30,530
than the Rabi resonance.

306
00:17:30,530 --> 00:17:32,700
Just how things work
out mathematically--

307
00:17:32,700 --> 00:17:35,340
I can't give you any
real deeper insight why.

308
00:17:35,340 --> 00:17:41,200
It just works out to be a factor
of 2 narrower than the Rabi

309
00:17:41,200 --> 00:17:41,700
resonance.

310
00:17:46,700 --> 00:17:50,810
There's one important aspect,
and this is the following.

311
00:17:50,810 --> 00:17:53,600
When you accrue the
spectroscopic information,

312
00:17:53,600 --> 00:17:57,850
you compare-- I hope you
remember my demonstration-- you

313
00:17:57,850 --> 00:18:01,530
compare the atomic oscillator
to the synthesizer,

314
00:18:01,530 --> 00:18:04,170
but you're not interacting
with each other.

315
00:18:04,170 --> 00:18:06,270
So therefore you don't
have any power broadening.

316
00:18:06,270 --> 00:18:08,950
You're comparing the free
evolution of the atom

317
00:18:08,950 --> 00:18:11,750
with the propagating
phase in your synthesizer.

318
00:18:11,750 --> 00:18:19,370
And therefore, your Rabi signal
has no power broadening at all.

319
00:18:19,370 --> 00:18:25,930
And that means that,
however, the Rabi signal,

320
00:18:25,930 --> 00:18:28,850
at least for small detunings
which are smaller than the Rabi

321
00:18:28,850 --> 00:18:38,790
frequency, will always depend
quadratically on delta.

322
00:18:38,790 --> 00:18:41,050
Remember, generalized
Rabi frequency,

323
00:18:41,050 --> 00:18:44,530
you add or make a Rabi
squared and detuning

324
00:18:44,530 --> 00:18:46,350
squared in quadrature.

325
00:18:46,350 --> 00:18:48,260
So therefore you will
always get effects

326
00:18:48,260 --> 00:18:50,850
for small detunings
which are quadratic,

327
00:18:50,850 --> 00:18:59,980
whereas the way how you set
up the Ramsey experiment, you

328
00:18:59,980 --> 00:19:01,820
can explore a linear dependence.

329
00:19:01,820 --> 00:19:05,450
So you have more
sensitivity here.

330
00:19:05,450 --> 00:19:07,400
Finally-- well, not finally.

331
00:19:07,400 --> 00:19:08,100
There are more.

332
00:19:08,100 --> 00:19:10,330
Next is the Ramsey spectroscopy.

333
00:19:13,220 --> 00:19:25,220
It's done in a
field-free region,

334
00:19:25,220 --> 00:19:27,000
so you're not
driving the system.

335
00:19:27,000 --> 00:19:32,680
You observe the free evolution.

336
00:19:32,680 --> 00:19:38,400
Therefore you have no AC Stark
effects due to the drive field.

337
00:19:49,820 --> 00:19:53,060
Of course, you drive the
system in the two Ramsey zones.

338
00:19:53,060 --> 00:19:58,570
And some form of AC Stark
effect may come in in the way

339
00:19:58,570 --> 00:20:00,810
that you may not
have exactly a pi

340
00:20:00,810 --> 00:20:02,880
pulse due to AC Stark effects.

341
00:20:02,880 --> 00:20:05,710
But this is sort of a
higher order effect.

342
00:20:05,710 --> 00:20:08,640
The basic spectroscopy
is done by comparing

343
00:20:08,640 --> 00:20:11,446
the atomic oscillator with
the frequency synthesizer.

344
00:20:18,510 --> 00:20:23,820
Since it's a field-free
region, this region can now

345
00:20:23,820 --> 00:20:29,150
be used for-- well,
let me say generally

346
00:20:29,150 --> 00:20:30,600
for experimental additions.

347
00:20:35,357 --> 00:20:37,690
And of course, what should
immediately come to your mind

348
00:20:37,690 --> 00:20:39,725
is the Nobel-Prize-winning
experiments

349
00:20:39,725 --> 00:20:42,830
of Serge Haroche where
he had two Ramsey

350
00:20:42,830 --> 00:20:44,790
regions with microwaves.

351
00:20:44,790 --> 00:20:47,450
And in between in the
field-free region,

352
00:20:47,450 --> 00:20:50,990
the atom passed
through another cavity.

353
00:20:50,990 --> 00:20:57,280
And in between the
Ramsey zones, the atom

354
00:20:57,280 --> 00:21:00,590
experienced a phase
shift due to the presence

355
00:21:00,590 --> 00:21:02,990
of a single photon.

356
00:21:02,990 --> 00:21:04,950
So the field-free
region could now

357
00:21:04,950 --> 00:21:09,250
be used to put in a cavity
which was filled with one or two

358
00:21:09,250 --> 00:21:10,030
photons.

359
00:21:10,030 --> 00:21:14,470
And the atom was in a
non-destructive way reading out

360
00:21:14,470 --> 00:21:16,277
how many photons were there.

361
00:21:16,277 --> 00:21:18,610
So this is another advantage
of the Ramsey spectroscopy,

362
00:21:18,610 --> 00:21:20,626
that you can now use
the field-free region

363
00:21:20,626 --> 00:21:21,500
to measure something.

364
00:21:21,500 --> 00:21:24,100
You can introduce a phase
shift, which can then

365
00:21:24,100 --> 00:21:27,210
be read out through the
Ramsey interference.

366
00:21:30,220 --> 00:21:37,010
Similarly, if you just think of
how the resonance comes along,

367
00:21:37,010 --> 00:21:43,850
if you had a slightly
fluctuating magnetic field

368
00:21:43,850 --> 00:21:46,280
between the two
Ramsey regions, this

369
00:21:46,280 --> 00:21:50,090
would not necessarily
broaden your signal.

370
00:21:50,090 --> 00:21:53,170
Because what you measure is
the integrated phase evolution

371
00:21:53,170 --> 00:21:55,720
of the atomic oscillator.

372
00:21:55,720 --> 00:21:59,310
So in other words, what you
get is shift of the resonance,

373
00:21:59,310 --> 00:22:03,600
which is the average over the
inhomogeneous magnetic field.

374
00:22:03,600 --> 00:22:08,430
Whereas when you do
a Rabi resonance,

375
00:22:08,430 --> 00:22:10,670
whenever you have a
field inhomogeneity,

376
00:22:10,670 --> 00:22:13,115
you broaden and shift the
resonance to this field value.

377
00:22:15,770 --> 00:22:24,390
So in other words,
the Ramsey resonance

378
00:22:24,390 --> 00:22:34,500
depends only on the average
along the mean energy

379
00:22:34,500 --> 00:22:42,040
separation between
the two levels.

380
00:22:47,580 --> 00:22:51,950
And therefore
inhomogeneous fluctuations,

381
00:22:51,950 --> 00:22:57,880
spatial fluctuations, can
completely average out.

382
00:22:57,880 --> 00:23:01,780
Whereas in the Rabi method,
any kind of fluctuations

383
00:23:01,780 --> 00:23:03,100
leads to line broadening.

384
00:23:07,240 --> 00:23:18,630
And finally, I will explain
that in more detail later on.

385
00:23:18,630 --> 00:23:27,770
But what happens in
the Ramsey method

386
00:23:27,770 --> 00:23:34,650
if the separation
between the two regions

387
00:23:34,650 --> 00:23:37,527
is much longer than the
spontaneous lifetime of one

388
00:23:37,527 --> 00:23:38,110
of the levels?

389
00:23:41,250 --> 00:23:43,170
Do you now get a
resolution which

390
00:23:43,170 --> 00:23:50,680
is 1/T, the temporal separation
between the two interrogations?

391
00:23:50,680 --> 00:23:56,210
Or do you get a
resolution which is 1/tau,

392
00:23:56,210 --> 00:23:58,506
the lifetime of
the excited level?

393
00:23:58,506 --> 00:23:59,690
What do you think?

394
00:23:59,690 --> 00:24:02,670
AUDIENCE: 1/tau.

395
00:24:02,670 --> 00:24:04,860
PROFESSOR: It would be
a good clicker question.

396
00:24:04,860 --> 00:24:06,350
So who thinks it's 1/tau?

397
00:24:06,350 --> 00:24:07,240
Who thinks it's 1/T?

398
00:24:10,140 --> 00:24:13,260
So who thinks it's
limited by lifetime?

399
00:24:13,260 --> 00:24:17,190
Who thinks it's limited
by the interrogation time?

400
00:24:17,190 --> 00:24:18,290
OK.

401
00:24:18,290 --> 00:24:21,230
So that's a minority
for the Ramsey method.

402
00:24:21,230 --> 00:24:22,900
The minority is correct.

403
00:24:22,900 --> 00:24:24,960
And the picture
you should have is

404
00:24:24,960 --> 00:24:28,010
that you remember the sort of
picture of those oscillators.

405
00:24:28,010 --> 00:24:32,310
But if some atoms decay away,
it diminishes your signal.

406
00:24:32,310 --> 00:24:35,620
But the interference comes
only from the survivors.

407
00:24:35,620 --> 00:24:40,580
And the survivors have survived
in the exponential tail

408
00:24:40,580 --> 00:24:43,640
of the natural decay, but
they are longer lived.

409
00:24:43,640 --> 00:24:46,850
And therefore, you can
actually do spectroscopy

410
00:24:46,850 --> 00:24:51,560
which is narrower than
the natural line width,

411
00:24:51,560 --> 00:24:53,930
using Ramsey spectroscopy.

412
00:24:53,930 --> 00:24:56,180
But if you do Rabi
spectroscopy, you're

413
00:24:56,180 --> 00:24:58,860
limited by the
spontaneous lifetime.

414
00:24:58,860 --> 00:25:01,530
And this is probably what was
in mind of the other people who

415
00:25:01,530 --> 00:25:05,740
raised their hand
for option A. But why

416
00:25:05,740 --> 00:25:07,730
there is a difference
between the Rabi

417
00:25:07,730 --> 00:25:10,670
and the Ramsey method,
that's something

418
00:25:10,670 --> 00:25:11,670
I want to discuss later.

419
00:25:14,320 --> 00:25:19,390
OK, so Ramsey has
the possibility,

420
00:25:19,390 --> 00:25:28,340
sub-natural line
width is possible

421
00:25:28,340 --> 00:25:30,650
when the interaction
time is larger

422
00:25:30,650 --> 00:25:33,590
than the inverse
natural line width.

423
00:25:37,476 --> 00:25:37,975
OK.

424
00:25:42,880 --> 00:25:44,980
Questions about Ramsey method?

425
00:25:49,656 --> 00:25:50,155
Good.

426
00:25:53,890 --> 00:25:57,780
Physics Today has a wonderful
article written by Ramsey

427
00:25:57,780 --> 00:26:01,150
which was reprinted
recently, I think

428
00:26:01,150 --> 00:26:04,370
on the occasion of his death.

429
00:26:04,370 --> 00:26:06,320
I will post this
article to our website.

430
00:26:06,320 --> 00:26:09,080
Then you can really
read about it

431
00:26:09,080 --> 00:26:11,800
in the language
of Norman Ramsey.

432
00:26:15,350 --> 00:26:16,200
OK, let's move on.

433
00:26:16,200 --> 00:26:19,160
I said we are discussing
simple examples.

434
00:26:19,160 --> 00:26:22,480
So we have discussed the example
of Rabi resonance and Ramsey

435
00:26:22,480 --> 00:26:24,750
resonance.

436
00:26:24,750 --> 00:26:29,250
Now I want to talk about line
shape with exponential decay.

437
00:26:32,562 --> 00:26:34,020
One reason why I
wanted to give you

438
00:26:34,020 --> 00:26:35,740
a simple model for
exponential decay

439
00:26:35,740 --> 00:26:38,930
because in the end,
everything is exponentially

440
00:26:38,930 --> 00:26:41,750
decaying because of the
finite lifetime of levels.

441
00:26:41,750 --> 00:26:45,380
And with this very,
very simple model,

442
00:26:45,380 --> 00:26:50,600
I want to convey to you that
not all exponential decays are

443
00:26:50,600 --> 00:26:51,320
equal.

444
00:26:51,320 --> 00:26:53,120
You have to be a
little bit careful.

445
00:26:53,120 --> 00:26:55,430
And this is just sort
of the simplest example.

446
00:26:55,430 --> 00:26:58,190
And you learn something
by figuring out

447
00:26:58,190 --> 00:27:00,680
what is different from
spontaneous decay here

448
00:27:00,680 --> 00:27:03,399
and what are the
consequences of that.

449
00:27:03,399 --> 00:27:04,440
So remember where we are.

450
00:27:04,440 --> 00:27:06,070
We have the Rabi resonance.

451
00:27:06,070 --> 00:27:08,940
I gave you the simple example
that the Rabi resonance

452
00:27:08,940 --> 00:27:11,820
is applied for fixed time tau.

453
00:27:11,820 --> 00:27:14,930
And then we did one extension
we averaged over the velocity

454
00:27:14,930 --> 00:27:16,200
distribution.

455
00:27:16,200 --> 00:27:22,680
But now we can just say, OK, we
have our Rabi resonance here.

456
00:27:22,680 --> 00:27:25,910
But we assume that
while we drive the atom,

457
00:27:25,910 --> 00:27:27,995
they decay away.

458
00:27:27,995 --> 00:27:30,700
If you want, you can think
these are radioactive atoms

459
00:27:30,700 --> 00:27:33,050
and they decay
[? radioactively. ?]

460
00:27:33,050 --> 00:27:35,580
For that situation,
this model is exact.

461
00:27:35,580 --> 00:27:41,780
So therefore, instead of having
a fixed interaction time tau,

462
00:27:41,780 --> 00:27:46,530
you have a distribution, which
is an exponential distribution.

463
00:27:46,530 --> 00:27:49,500
So all we have to do is we
have to take our result, which

464
00:27:49,500 --> 00:27:53,270
I discussed 10 or 20 minutes
ago, with the fixed interaction

465
00:27:53,270 --> 00:27:57,860
time and convolute it with
the distribution of times

466
00:27:57,860 --> 00:28:01,000
the atom experiences
the drive field.

467
00:28:01,000 --> 00:28:05,740
And what I've introduced here
is the exponential is gamma.

468
00:28:05,740 --> 00:28:09,740
And the mean interaction time
over which an atom interacts

469
00:28:09,740 --> 00:28:12,240
with the drive field
is just 1 over gamma,

470
00:28:12,240 --> 00:28:15,100
and I called that tau.

471
00:28:15,100 --> 00:28:18,590
So now we had the
Rabi probability here,

472
00:28:18,590 --> 00:28:20,720
but now we convolute it
with the distribution

473
00:28:20,720 --> 00:28:22,020
of interaction times.

474
00:28:22,020 --> 00:28:26,240
And this is now the probability
that after the Rabi pulse

475
00:28:26,240 --> 00:28:29,630
the spin was moved from
spin down to spin up.

476
00:28:29,630 --> 00:28:31,660
Or if you have an
electronic transition,

477
00:28:31,660 --> 00:28:35,090
from ground to excited state.

478
00:28:35,090 --> 00:28:42,240
So this integral can
be analytically solved.

479
00:28:42,240 --> 00:28:43,990
That's why it's
worth presenting.

480
00:28:43,990 --> 00:28:47,330
And what you get is a
Lorentzian line shape.

481
00:28:47,330 --> 00:28:49,620
And this Lorentzian
line shape shows

482
00:28:49,620 --> 00:28:53,190
power broadening, which
actually you should find nice,

483
00:28:53,190 --> 00:28:55,020
because we will
sometimes [INAUDIBLE]

484
00:28:55,020 --> 00:28:57,820
in power broadening you can't
get out of perturbation theory.

485
00:28:57,820 --> 00:29:00,290
And a lot what we have
done and what I actually

486
00:29:00,290 --> 00:29:03,450
want to do for the remainder of
this chapter on line broadening

487
00:29:03,450 --> 00:29:05,040
is a perturbative approach.

488
00:29:05,040 --> 00:29:07,820
So that's another reason I
want to present it to you here.

489
00:29:07,820 --> 00:29:09,750
These are some
non-perturbative results,

490
00:29:09,750 --> 00:29:12,390
and they show the physics of
power broadening, saturation

491
00:29:12,390 --> 00:29:14,530
broadening.

492
00:29:14,530 --> 00:29:17,010
But there are two things
which are noteworthy.

493
00:29:17,010 --> 00:29:23,140
One is the full
width at half maximum

494
00:29:23,140 --> 00:29:25,190
is not gamma but 2 gamma.

495
00:29:27,710 --> 00:29:33,620
So if we had natural
decay at a rate gamma,

496
00:29:33,620 --> 00:29:38,200
the Lorentzian which we
get is only half as wide.

497
00:29:38,200 --> 00:29:39,650
But you can
immediately say, well,

498
00:29:39,650 --> 00:29:43,190
that can be understood because
here I assumed the atoms just

499
00:29:43,190 --> 00:29:45,370
decay away no matter whether
they are in the ground

500
00:29:45,370 --> 00:29:46,860
or in the excited state.

501
00:29:46,860 --> 00:29:49,450
And I gave you a model that
you assume there is maybe

502
00:29:49,450 --> 00:29:52,810
radioactive decay independent
of the internal state.

503
00:29:52,810 --> 00:29:58,590
And now you can wave
your hands and say, OK,

504
00:29:58,590 --> 00:30:00,900
if you have only decay
in the excited state

505
00:30:00,900 --> 00:30:02,390
and not in the
ground state, this

506
00:30:02,390 --> 00:30:04,470
should give you a factor
of 2 in the width.

507
00:30:04,470 --> 00:30:09,440
And this may explain why we have
a full width of half maximum

508
00:30:09,440 --> 00:30:13,220
here of 2 gamma and in
spontaneous decay it is gamma.

509
00:30:13,220 --> 00:30:18,040
But there's another thing
which is interesting, maybe

510
00:30:18,040 --> 00:30:19,090
more interesting.

511
00:30:19,090 --> 00:30:23,700
And this is the
power broadening.

512
00:30:23,700 --> 00:30:26,500
If you take the power-broadened
Lorentzian line width

513
00:30:26,500 --> 00:30:28,830
and we look at it in
the limit of high power,

514
00:30:28,830 --> 00:30:31,840
it is 2 times the
Rabi frequency.

515
00:30:31,840 --> 00:30:37,710
Well, if we have a system
which has spontaneous decay

516
00:30:37,710 --> 00:30:40,520
and we would go to
the high power limit--

517
00:30:40,520 --> 00:30:42,430
we've discussed it before.

518
00:30:42,430 --> 00:30:48,960
What you get is square root
2 of the Rabi frequency.

519
00:30:48,960 --> 00:30:51,330
So the message I
can give you here

520
00:30:51,330 --> 00:30:56,570
is that saturation
broadening, power broadening

521
00:30:56,570 --> 00:31:03,120
depends sensitively on the
exact nature of the decay

522
00:31:03,120 --> 00:31:05,790
and of the lifetime
broadening involved.

523
00:31:05,790 --> 00:31:08,050
And if you really
want to do it right,

524
00:31:08,050 --> 00:31:10,411
you have to use the
optical Bloch equations.

525
00:31:15,710 --> 00:31:17,480
So let me just write that down.

526
00:31:17,480 --> 00:31:20,610
So what we learned from that is,
yes, we get power broadening.

527
00:31:20,610 --> 00:31:23,240
We have a simple model
for power broadening here.

528
00:31:23,240 --> 00:31:36,320
But power broadening
depends sensitively

529
00:31:36,320 --> 00:31:42,890
on the nature of
the decay process.

530
00:31:42,890 --> 00:31:52,120
And so if you want to get this
result without any assumptions

531
00:31:52,120 --> 00:31:55,040
or approximations, you
should use the optical Bloch

532
00:31:55,040 --> 00:31:55,540
equations.

533
00:32:02,410 --> 00:32:04,750
And yes, your
homework assignment

534
00:32:04,750 --> 00:32:06,960
looked at the optical
Bloch equations.

535
00:32:06,960 --> 00:32:13,310
And I think you also found
out that some results

536
00:32:13,310 --> 00:32:15,400
for the optical Bloch
equations really

537
00:32:15,400 --> 00:32:20,310
depend on the ratio of gamma
1 gamma 2 or T1 and T2.

538
00:32:20,310 --> 00:32:23,580
So that is with a more
mathematical formalism

539
00:32:23,580 --> 00:32:27,450
shows you that the way
how you introduce decay

540
00:32:27,450 --> 00:32:31,300
into the atomic system, it's not
just there is one time constant

541
00:32:31,300 --> 00:32:34,042
and the result just depends
on the time constant.

542
00:32:34,042 --> 00:32:35,125
There are some subtleties.

543
00:32:42,515 --> 00:32:43,390
Questions about that?

544
00:32:48,620 --> 00:32:49,210
OK.

545
00:32:49,210 --> 00:32:53,220
Now-- and many people
have asked me about it--

546
00:32:53,220 --> 00:32:55,670
I think for the first
time in this course

547
00:32:55,670 --> 00:32:59,420
we bring in motion of the atoms.

548
00:32:59,420 --> 00:33:04,740
So the atoms are now not
pinned down at the origin.

549
00:33:04,740 --> 00:33:06,980
Maybe you can imagine
you have an atom which

550
00:33:06,980 --> 00:33:11,790
is held in a solid state
lattice with a nanometer,

551
00:33:11,790 --> 00:33:13,970
and we just look at
the internal structure.

552
00:33:13,970 --> 00:33:16,980
We have maybe some ions
implanted into a material.

553
00:33:16,980 --> 00:33:19,970
And these ions are fluorescent,
and we are probing them.

554
00:33:19,970 --> 00:33:23,640
Or we're doing spin
flip on nuclei,

555
00:33:23,640 --> 00:33:28,512
which are nuclei of atoms which
are part of a condensed metal

556
00:33:28,512 --> 00:33:29,520
lattice.

557
00:33:29,520 --> 00:33:33,570
Or if you want to think
more [? with ?] the methods

558
00:33:33,570 --> 00:33:37,140
of atomic physics, you have the
most tightly confining ion trap

559
00:33:37,140 --> 00:33:37,850
in the world.

560
00:33:37,850 --> 00:33:40,520
You are deeply in
the Lamb-Dicke limit,

561
00:33:40,520 --> 00:33:42,400
and your ion just cannot move.

562
00:33:42,400 --> 00:33:45,620
It's always in the ground
state of the ion trap,

563
00:33:45,620 --> 00:33:50,317
and all you are dealing with is
the internal degree of freedom.

564
00:33:50,317 --> 00:33:51,650
Actually, let me make a comment.

565
00:33:51,650 --> 00:33:54,070
I often see when people
approach me and ask

566
00:33:54,070 --> 00:33:56,240
me question that they
are not necessarily

567
00:33:56,240 --> 00:33:58,570
making the separation.

568
00:33:58,570 --> 00:34:00,440
When they think
about what happens

569
00:34:00,440 --> 00:34:02,790
to the internal
structure of atoms, what

570
00:34:02,790 --> 00:34:05,940
is in their head is, but there
is motion, there is recoil.

571
00:34:05,940 --> 00:34:09,790
You can, in my experience,
always separate the two.

572
00:34:09,790 --> 00:34:12,440
You can create a
situation where you only

573
00:34:12,440 --> 00:34:14,520
probe the internal
degree of freedom

574
00:34:14,520 --> 00:34:17,699
by tightly confining
the atom, and then you

575
00:34:17,699 --> 00:34:21,480
can relax the condition that
the atoms is tightly confined.

576
00:34:21,480 --> 00:34:25,094
Now the atoms can
move, and then all

577
00:34:25,094 --> 00:34:27,260
of the things we want to
discuss now come into play.

578
00:34:31,330 --> 00:34:34,610
Sometimes people assume, yes,
but if you confine an atom,

579
00:34:34,610 --> 00:34:38,190
doesn't the atom always have
a de Broglie wavelengths?

580
00:34:38,190 --> 00:34:40,270
And isn't that
another length scale?

581
00:34:40,270 --> 00:34:42,739
The answer is no,
because you need

582
00:34:42,739 --> 00:34:46,000
a coupling between the internal
degree and the external degree,

583
00:34:46,000 --> 00:34:51,340
or you need some way of exciting
the external degree of motion.

584
00:34:51,340 --> 00:34:54,469
And if you have a tightly
confining ion trap,

585
00:34:54,469 --> 00:34:58,110
h bar omega, the next vibration
or level in the ion trap

586
00:34:58,110 --> 00:35:02,300
is so high, you may not excite
it with the recoil of a photon.

587
00:35:02,300 --> 00:35:05,930
But there is another limit
which I often find very useful,

588
00:35:05,930 --> 00:35:07,590
and this is the following.

589
00:35:07,590 --> 00:35:12,880
When we talk about
spectroscopy, spin flips,

590
00:35:12,880 --> 00:35:16,210
electronic transitions,
we have not really

591
00:35:16,210 --> 00:35:18,550
talked about the
mass of the atom.

592
00:35:18,550 --> 00:35:22,800
The mass of the nucleus only
appeared in the reduced mass.

593
00:35:22,800 --> 00:35:24,970
Remember when we did last
class the hydrogen atom?

594
00:35:24,970 --> 00:35:27,260
We had the reduced mass,
which was slightly different

595
00:35:27,260 --> 00:35:29,030
from the electron mass.

596
00:35:29,030 --> 00:35:33,400
So if you want to completely
exclude the motion of the atom,

597
00:35:33,400 --> 00:35:37,380
just work in the limit that
the nucleus has infinite mass.

598
00:35:37,380 --> 00:35:39,330
If the nucleus
has infinite mass,

599
00:35:39,330 --> 00:35:41,860
its de Broglie wavelength is 0.

600
00:35:41,860 --> 00:35:44,830
It's confined in a
harmonic oscillator

601
00:35:44,830 --> 00:35:47,590
to sort of a delta function.

602
00:35:47,590 --> 00:35:51,330
So by just assuming that
you deal with infinite mass,

603
00:35:51,330 --> 00:35:55,870
you automatically neglect
all possible motions.

604
00:35:55,870 --> 00:35:58,710
And as you will actually
see from the next formula,

605
00:35:58,710 --> 00:36:01,900
working in the infinite mass
limit means that your Doppler

606
00:36:01,900 --> 00:36:06,650
shift is 0, your recoil
shift is 0, everything is 0.

607
00:36:06,650 --> 00:36:08,940
So either way,
I've given you now

608
00:36:08,940 --> 00:36:12,600
two ways how I recommend
that you think about all

609
00:36:12,600 --> 00:36:14,330
the physics we have
discussed which

610
00:36:14,330 --> 00:36:16,510
deals with the internal
degree of freedom

611
00:36:16,510 --> 00:36:20,620
by either saying the atom
is tightly organized.

612
00:36:20,620 --> 00:36:22,260
But then some people
say, ah, but then

613
00:36:22,260 --> 00:36:23,737
it has Heisenberg Uncertainty.

614
00:36:23,737 --> 00:36:24,820
There's a lot of momentum.

615
00:36:24,820 --> 00:36:25,569
It doesn't matter.

616
00:36:25,569 --> 00:36:27,370
If it's localized,
it's localized.

617
00:36:27,370 --> 00:36:30,120
But you can also assume just
the infinite mass limit.

618
00:36:30,120 --> 00:36:35,250
And in both cases, the
result is you can completely

619
00:36:35,250 --> 00:36:38,410
talk about so many aspects
of internal excitations

620
00:36:38,410 --> 00:36:43,419
without even considering
what happens externally.

621
00:36:43,419 --> 00:36:44,460
Any questions about that?

622
00:36:46,970 --> 00:36:51,560
So now we take the mass from
infinity to finite value,

623
00:36:51,560 --> 00:36:53,930
and now we want
to allow the atom

624
00:36:53,930 --> 00:36:56,270
to move and have kinetic energy.

625
00:36:56,270 --> 00:36:58,170
So let's start out very simple.

626
00:36:58,170 --> 00:37:02,220
We have an atom which is
addressed in the excited state,

627
00:37:02,220 --> 00:37:04,900
and it emits a photon.

628
00:37:04,900 --> 00:37:09,940
Before the emission
of the photon,

629
00:37:09,940 --> 00:37:12,560
the total energy is
the excitation energy.

630
00:37:12,560 --> 00:37:14,990
But after the emission
of the photon,

631
00:37:14,990 --> 00:37:16,900
the atom is in the ground state.

632
00:37:16,900 --> 00:37:19,700
The photon has been emitted.

633
00:37:19,700 --> 00:37:23,490
But now the atom, due to
the photon recoil h bar k,

634
00:37:23,490 --> 00:37:25,850
has kinetic energy.

635
00:37:25,850 --> 00:37:33,340
So therefore the frequency
of emission of the photon

636
00:37:33,340 --> 00:37:37,110
does not happen at the
resonance frequency of the atom

637
00:37:37,110 --> 00:37:40,160
because some part of
the electronic energy

638
00:37:40,160 --> 00:37:42,490
goes into the kinetic energy
of the recoiling atom.

639
00:37:46,330 --> 00:37:48,470
And this is called
the recoil shift.

640
00:37:48,470 --> 00:37:50,080
We start at 0 velocity.

641
00:37:50,080 --> 00:37:52,650
At 0 velocity, you don't
have any Doppler shift.

642
00:37:52,650 --> 00:37:54,085
But you do have recoil shift.

643
00:37:57,270 --> 00:37:59,660
Well, we can play the same game.

644
00:37:59,660 --> 00:38:00,990
We have an atom addressed.

645
00:38:00,990 --> 00:38:02,390
It absorbs the photon.

646
00:38:02,390 --> 00:38:04,450
And after the atom has
absorbed the photon,

647
00:38:04,450 --> 00:38:06,610
it's in the excited state.

648
00:38:06,610 --> 00:38:09,330
But now you have
to excite the atom

649
00:38:09,330 --> 00:38:11,500
if you want to transfer
the atom to the excited

650
00:38:11,500 --> 00:38:14,780
state with a frequency
which is slightly

651
00:38:14,780 --> 00:38:16,780
higher than the
resonance frequency.

652
00:38:16,780 --> 00:38:19,860
Because the laser has
to provide not only

653
00:38:19,860 --> 00:38:22,110
the energy for the
electronic excitation

654
00:38:22,110 --> 00:38:25,360
but also the energy for
the kinetic energy at which

655
00:38:25,360 --> 00:38:26,400
the atom recoils.

656
00:38:29,050 --> 00:38:36,340
So therefore we find that, due
to the recoil of the photon,

657
00:38:36,340 --> 00:38:39,220
the absorption line
and the emission

658
00:38:39,220 --> 00:38:43,040
line for an atom
addressed are shifted.

659
00:38:43,040 --> 00:38:53,600
The shift is the
recoil energy, h bar--

660
00:38:53,600 --> 00:38:56,880
the momentum of the photon
squared, divided by 2 times

661
00:38:56,880 --> 00:38:58,360
the mass of the atom.

662
00:38:58,360 --> 00:39:01,390
And the shifts are opposite
for absorption and emission.

663
00:39:01,390 --> 00:39:07,180
So therefore, if you look at
the two processes for absorption

664
00:39:07,180 --> 00:39:13,990
and emission, there is a recoil
splitting between the two.

665
00:39:13,990 --> 00:39:16,660
This recoil splitting between
emission and absorption

666
00:39:16,660 --> 00:39:18,720
is just a few kilohertz.

667
00:39:18,720 --> 00:39:21,300
And it was really one of the
wonderful accomplishments

668
00:39:21,300 --> 00:39:24,040
when high resolution
spectroscopy came along

669
00:39:24,040 --> 00:39:27,860
and John Hall at
Boulder managed to have

670
00:39:27,860 --> 00:39:31,230
lasers stabilized
to sub kilohertz.

671
00:39:31,230 --> 00:39:36,550
For the first time, this recoil
splitting could be resolved.

672
00:39:36,550 --> 00:39:39,420
So he had set up some
intracavity absorption,

673
00:39:39,420 --> 00:39:44,190
and he saw sort of two peaks
in some kind of spectrum.

674
00:39:44,190 --> 00:39:46,870
I don't remember the
details, but two peaks

675
00:39:46,870 --> 00:39:49,320
split by a few
kilohertz were really

676
00:39:49,320 --> 00:39:56,110
the hallmark of the photon
recoil shifting the lines away

677
00:39:56,110 --> 00:39:59,070
from resonance.

678
00:39:59,070 --> 00:40:03,600
OK, so now we know what the
kinetic energy of the atom

679
00:40:03,600 --> 00:40:04,110
does.

680
00:40:04,110 --> 00:40:07,090
If it emits a photon,
there is recoil.

681
00:40:07,090 --> 00:40:10,110
But now, in addition, we
can drop the assumption

682
00:40:10,110 --> 00:40:13,550
that the atom is initially at
rest when it absorbs or emits.

683
00:40:13,550 --> 00:40:15,230
Now the atom is moving.

684
00:40:15,230 --> 00:40:17,840
But for that, we don't
need any new concept.

685
00:40:17,840 --> 00:40:21,190
Because the moving atom-- we
can just do a transformation

686
00:40:21,190 --> 00:40:23,990
into the frame of the atom
where the atom is at rest.

687
00:40:23,990 --> 00:40:28,600
And then just using the
relativistic transformation,

688
00:40:28,600 --> 00:40:35,500
we are now transferring
the laser frequency

689
00:40:35,500 --> 00:40:39,720
from the atomic frame
into the lab frame.

690
00:40:39,720 --> 00:40:42,410
So what I've
written down here is

691
00:40:42,410 --> 00:40:45,990
the general special
relativity formula

692
00:40:45,990 --> 00:40:48,320
for the frequency shift.

693
00:40:48,320 --> 00:40:51,600
And I've assumed that
the photon is emitted

694
00:40:51,600 --> 00:40:55,129
at an angle phi with respect
to the motion of the atom.

695
00:41:07,850 --> 00:41:13,870
So now we obtain-- OK, let me
do a second-order expansion.

696
00:41:13,870 --> 00:41:16,290
Usually, our atoms
are non-relativistic,

697
00:41:16,290 --> 00:41:18,700
so it's the first- and
second-order term which

698
00:41:18,700 --> 00:41:20,130
are most important.

699
00:41:20,130 --> 00:41:22,910
And if we are now
looking in the lab frame,

700
00:41:22,910 --> 00:41:27,680
what is the frequency where
we emit and absorb photons.

701
00:41:27,680 --> 00:41:34,510
It's a resonance frequency
minus/plus the recoil shift,

702
00:41:34,510 --> 00:41:36,900
which we have already
discussed in isolation

703
00:41:36,900 --> 00:41:40,380
by assuming we have
atoms at 0 velocity.

704
00:41:40,380 --> 00:41:43,290
But now the velocity
of the atom leads

705
00:41:43,290 --> 00:41:46,420
to a first-order and
second-order Doppler shift.

706
00:41:50,110 --> 00:41:53,880
If v/c is small, what
is most important

707
00:41:53,880 --> 00:41:56,000
is the first-order
Doppler shift.

708
00:41:56,000 --> 00:42:01,370
And in almost all cases
[? where we ?] do spectroscopy,

709
00:42:01,370 --> 00:42:03,770
dominant line
broadening effect comes

710
00:42:03,770 --> 00:42:05,050
from the first-order shift.

711
00:42:07,650 --> 00:42:16,280
However, let me point out
that the first-order shift can

712
00:42:16,280 --> 00:42:16,865
be suppressed.

713
00:42:21,410 --> 00:42:24,800
One simple way to suppress
it-- interrogate the atoms

714
00:42:24,800 --> 00:42:27,270
at an angle of 90 degrees.

715
00:42:27,270 --> 00:42:28,570
Have an atomic beam.

716
00:42:28,570 --> 00:42:33,510
And if you interrogate them
with a laser beam at 90 degrees,

717
00:42:33,510 --> 00:42:35,100
the cosine phi is 0.

718
00:42:35,100 --> 00:42:36,240
k dot v is 0.

719
00:42:40,140 --> 00:42:43,900
And this is the oldest method
to do Doppler-free spectroscopy.

720
00:42:46,650 --> 00:42:48,620
In your new homework
assignment, you

721
00:42:48,620 --> 00:42:53,930
will discuss saturation
spectroscopy.

722
00:42:53,930 --> 00:42:56,440
If you have a broad
velocity distribution,

723
00:42:56,440 --> 00:43:00,470
but you find a way of labeling
atoms with a certain velocity

724
00:43:00,470 --> 00:43:04,640
class, then you have created
your own narrow velocity class

725
00:43:04,640 --> 00:43:06,650
where the Doppler
broadening is absent

726
00:43:06,650 --> 00:43:09,660
because you've only
one velocity class.

727
00:43:09,660 --> 00:43:13,570
And these methods of nonlinear
spectroscopy, the concept

728
00:43:13,570 --> 00:43:17,290
will be developed in
the homework assignment.

729
00:43:17,290 --> 00:43:20,220
Finally-- and this will
be the next chapter

730
00:43:20,220 --> 00:43:23,890
which we talk in class here
in about two weeks-- this

731
00:43:23,890 --> 00:43:25,780
is by having two-photon
spectroscopy.

732
00:43:28,510 --> 00:43:30,280
To give you the
appetizer, if you

733
00:43:30,280 --> 00:43:33,560
have two photons from opposite
direction, the Doppler shifts.

734
00:43:33,560 --> 00:43:36,640
One has a positive, one has
a negative Doppler shift.

735
00:43:36,640 --> 00:43:38,720
And if you stack
up the two photons,

736
00:43:38,720 --> 00:43:41,130
the sum of the two
Doppler shifts is 0.

737
00:43:41,130 --> 00:43:46,370
So two-photon spectroscopy
provides you an opportunity

738
00:43:46,370 --> 00:43:48,255
to completely eliminate
the Doppler shift.

739
00:43:52,300 --> 00:43:59,530
However, no matter what you
pick here for the angle,

740
00:43:59,530 --> 00:44:02,870
there is a part of the
second-order Doppler

741
00:44:02,870 --> 00:44:05,680
shift you can never get rid of.

742
00:44:05,680 --> 00:44:08,400
And this is something
important to keep in mind.

743
00:44:08,400 --> 00:44:12,050
The second-order Doppler
shift, at least one part of it,

744
00:44:12,050 --> 00:44:16,570
comes from the relativistic
transformation of time.

745
00:44:16,570 --> 00:44:19,900
So if you have atoms moving
at different velocities,

746
00:44:19,900 --> 00:44:25,210
time in the frame of the atom
ticks slightly differently

747
00:44:25,210 --> 00:44:27,700
depending what the velocity is.

748
00:44:27,700 --> 00:44:30,900
And therefore, if
you do spectroscopy,

749
00:44:30,900 --> 00:44:32,530
you measure time
in the lab frame,

750
00:44:32,530 --> 00:44:35,320
but the atoms measure time
in the resonance frequency

751
00:44:35,320 --> 00:44:36,860
in their own frame.

752
00:44:36,860 --> 00:44:39,690
Then there is
inevitably broadening.

753
00:44:39,690 --> 00:44:43,470
So for instance, when people
did Ted Hansch's experiment,

754
00:44:43,470 --> 00:44:45,430
the famous two-photon
spectroscopy

755
00:44:45,430 --> 00:44:48,580
on hydrogen, high resolution,
determination of the Lamb

756
00:44:48,580 --> 00:44:52,260
shift, the Rydberg constant--
one of the flagship experiments

757
00:44:52,260 --> 00:44:55,410
of high resolution
laser spectroscopy.

758
00:44:55,410 --> 00:44:59,660
A limit is the
second-order Doppler effect

759
00:44:59,660 --> 00:45:01,660
because of its
relativistic nature.

760
00:45:01,660 --> 00:45:05,090
And the only way to suppress
the relativistic Doppler effect

761
00:45:05,090 --> 00:45:08,390
is by cooling the atoms,
reducing their velocity.

762
00:45:16,840 --> 00:45:18,930
So let me just write that down.

763
00:45:18,930 --> 00:45:25,070
So when we suppress
the first-order effect,

764
00:45:25,070 --> 00:45:30,290
then the limit is given
by the second-order term.

765
00:45:33,910 --> 00:45:38,140
And just repeat, the
second-order term cannot be

766
00:45:38,140 --> 00:45:40,460
eliminated by playing
geometric tricks,

767
00:45:40,460 --> 00:45:44,190
90-degree angles and such,
because it's fundamentally

768
00:45:44,190 --> 00:45:46,900
rooted in the relativistic
definition of time.

769
00:45:55,500 --> 00:45:56,000
OK.

770
00:45:59,860 --> 00:46:06,325
Any questions about recoil
shifts, Doppler shifts?

771
00:46:06,325 --> 00:46:06,825
Yes.

772
00:46:06,825 --> 00:46:08,571
AUDIENCE: What are
the correct way

773
00:46:08,571 --> 00:46:11,910
to determine whether
we're in tightly confined

774
00:46:11,910 --> 00:46:12,864
[? regime ?] or not?

775
00:46:16,220 --> 00:46:19,330
PROFESSOR: OK, the
question is what

776
00:46:19,330 --> 00:46:22,640
is the criteria on whether we
are tightly confined or not.

777
00:46:22,640 --> 00:46:25,690
I will give a full answer
to that in about a week when

778
00:46:25,690 --> 00:46:29,040
I discuss with you
in great detail

779
00:46:29,040 --> 00:46:31,760
the spectrum of a
confined particle.

780
00:46:31,760 --> 00:46:34,690
And what we will
introduce is it's

781
00:46:34,690 --> 00:46:37,590
the frequency of
harmonic confinement.

782
00:46:37,590 --> 00:46:39,490
And we have to
compare the frequency

783
00:46:39,490 --> 00:46:43,270
of harmonic confinement
to two other relative,

784
00:46:43,270 --> 00:46:45,890
to two other
important frequencies.

785
00:46:45,890 --> 00:47:00,970
One frequency is the
recoil frequency.

786
00:47:00,970 --> 00:47:05,280
And another one may be
the natural line width.

787
00:47:05,280 --> 00:47:08,360
So in other words, when
we discuss the spectrum

788
00:47:08,360 --> 00:47:11,190
of confined particles,
we can discuss it

789
00:47:11,190 --> 00:47:13,710
as a function of three
parameters-- confinement

790
00:47:13,710 --> 00:47:17,210
frequency, recoil frequency,
and natural line widths.

791
00:47:17,210 --> 00:47:22,020
And based on the hierarchy
of those three frequencies,

792
00:47:22,020 --> 00:47:24,140
we find limiting cases.

793
00:47:24,140 --> 00:47:26,840
And we will find, then--
and this is probably

794
00:47:26,840 --> 00:47:30,070
what you are aiming for-- at
some limit when you reduce

795
00:47:30,070 --> 00:47:33,020
the confinement in your
harmonic oscillator,

796
00:47:33,020 --> 00:47:35,880
you will actually retrieve
the free gas limit.

797
00:47:39,020 --> 00:47:42,350
Or to be very brief,
confinement, you

798
00:47:42,350 --> 00:47:45,020
have the benefit of confinement.

799
00:47:45,020 --> 00:47:47,600
Confinement means the
motion of the atom

800
00:47:47,600 --> 00:47:49,967
is quantized in
units of h bar omega.

801
00:47:49,967 --> 00:47:51,550
So you shouldn't
think about velocity.

802
00:47:51,550 --> 00:47:53,900
You should think
about discrete levels.

803
00:47:53,900 --> 00:47:57,060
And I will show you that the
spectrum which is broadened

804
00:47:57,060 --> 00:48:01,100
becomes a spectrum with
discrete levels and side bands.

805
00:48:01,100 --> 00:48:04,620
As long as you can resolve the
side bands, you can see them,

806
00:48:04,620 --> 00:48:06,980
you can actually
address the line

807
00:48:06,980 --> 00:48:10,440
in the middle which has
no motion blurring at all.

808
00:48:10,440 --> 00:48:11,990
But you have to resolve it.

809
00:48:11,990 --> 00:48:15,200
So one condition here now is
that the harmonic oscillator

810
00:48:15,200 --> 00:48:18,390
frequency is larger than
the natural broadening

811
00:48:18,390 --> 00:48:19,870
of each line.

812
00:48:19,870 --> 00:48:23,146
If the lines blur, you're
pretty much back to free space

813
00:48:23,146 --> 00:48:25,020
and you've lost the
advantage of confinement.

814
00:48:28,600 --> 00:48:32,160
But maybe we can discuss
some of those aspects

815
00:48:32,160 --> 00:48:34,660
after I've introduced the line
shape of confined particles.

816
00:48:40,200 --> 00:48:40,700
OK.

817
00:48:46,380 --> 00:48:46,880
OK.

818
00:48:56,340 --> 00:48:58,980
Let me now discuss
briefly, or use

819
00:48:58,980 --> 00:49:01,190
what we have just discussed
about Doppler shift

820
00:49:01,190 --> 00:49:04,570
to discuss the line
shape in a gas.

821
00:49:04,570 --> 00:49:07,140
Well, of course, line
shapes in a gas--

822
00:49:07,140 --> 00:49:10,150
that's what all people observed
when they did spectroscopy

823
00:49:10,150 --> 00:49:12,910
before the advent of
trapping and cooling.

824
00:49:12,910 --> 00:49:15,940
But even now, we
often have a situation

825
00:49:15,940 --> 00:49:18,780
that we have a thermal clouded
microkelvin temperature.

826
00:49:18,780 --> 00:49:20,940
And what we see is
still broadening

827
00:49:20,940 --> 00:49:23,120
due to the thermal motion.

828
00:49:23,120 --> 00:49:27,420
So therefore, let
me just tell you

829
00:49:27,420 --> 00:49:30,990
a few aspects of that which
you might find interesting.

830
00:49:30,990 --> 00:49:38,120
So one is when we have
an non-degenerate gas,

831
00:49:38,120 --> 00:49:41,575
this is described by a
Boltzmann distribution, sort

832
00:49:41,575 --> 00:49:44,340
of a Gaussian
distribution of velocity.

833
00:49:44,340 --> 00:49:49,850
And therefore, if the
first-order Doppler shift

834
00:49:49,850 --> 00:49:53,310
is relevant, the
first-order Doppler shift

835
00:49:53,310 --> 00:49:59,380
is proportionate to v,
so therefore the spectrum

836
00:49:59,380 --> 00:50:05,420
we observe is nothing else
than the spectrum in velocity

837
00:50:05,420 --> 00:50:07,410
multiplied with a k vector.

838
00:50:07,410 --> 00:50:10,270
The Doppler shift is k dot
v, and therefore the velocity

839
00:50:10,270 --> 00:50:12,560
distribution by
multiplying it with k

840
00:50:12,560 --> 00:50:14,790
is turned into a
frequency distribution.

841
00:50:14,790 --> 00:50:17,420
And so the classic
frequency distribution

842
00:50:17,420 --> 00:50:19,770
you would expect
in a gas is simply

843
00:50:19,770 --> 00:50:21,890
the Gaussian distribution.

844
00:50:21,890 --> 00:50:25,210
And the Doppler width,
the spectroscopic width,

845
00:50:25,210 --> 00:50:29,640
is nothing else than whatever
the characteristic speed

846
00:50:29,640 --> 00:50:32,690
in your Boltzmann
distribution is, typically

847
00:50:32,690 --> 00:50:36,480
the most probable speed, 2 times
the temperature over the mass

848
00:50:36,480 --> 00:50:40,130
multiplied with the k vector.

849
00:50:40,130 --> 00:50:48,880
And in many cases, it is this
Doppler width which dominates.

850
00:50:48,880 --> 00:50:52,950
I've just given you
typical examples here.

851
00:50:52,950 --> 00:50:55,640
If you've stabilized your laser
to a room temperature vapor

852
00:50:55,640 --> 00:50:58,700
cell, you will encounter
typical Doppler widths

853
00:50:58,700 --> 00:51:00,970
on the order of 1 gigahertz.

854
00:51:00,970 --> 00:51:05,000
This is 100 times larger than
the natural line width, which

855
00:51:05,000 --> 00:51:07,970
is 100 times larger
than the recoil shift.

856
00:51:07,970 --> 00:51:13,087
So that is the usual
hierarchy of shifts

857
00:51:13,087 --> 00:51:14,170
and broadening mechanisms.

858
00:51:17,400 --> 00:51:28,740
So therefore, you would
think that if the Gaussian

859
00:51:28,740 --> 00:51:31,920
line widths due to the
velocity distribution is

860
00:51:31,920 --> 00:51:36,080
100 times larger than the
natural line widths which

861
00:51:36,080 --> 00:51:39,010
is described by Lorentzian,
you can completely

862
00:51:39,010 --> 00:51:41,970
neglect the Lorentzian.

863
00:51:41,970 --> 00:51:46,140
But that's not the
case, and that's

864
00:51:46,140 --> 00:51:48,000
what I want to discuss now.

865
00:51:48,000 --> 00:51:50,540
What happens is you
have a Gaussian which

866
00:51:50,540 --> 00:51:51,850
is much, much broader.

867
00:51:51,850 --> 00:51:55,470
But the Gaussian
decays exponentially,

868
00:51:55,470 --> 00:52:02,090
whereas your narrow Lorentzian
decays with a power law.

869
00:52:02,090 --> 00:52:04,220
So just to give you
the example, if you

870
00:52:04,220 --> 00:52:08,100
go two full
half-line widths away

871
00:52:08,100 --> 00:52:10,400
from the center of a
Gaussian, the Gaussian

872
00:52:10,400 --> 00:52:12,540
has dropped to 0.2%.

873
00:52:12,540 --> 00:52:16,110
The Lorentzian has still 6%.

874
00:52:16,110 --> 00:52:18,390
So therefore, what
happens is if you

875
00:52:18,390 --> 00:52:25,180
have your gigahertz
broadened line in a gas,

876
00:52:25,180 --> 00:52:28,370
but you go further
and further away,

877
00:52:28,370 --> 00:52:31,740
at some point what
you encounter are not

878
00:52:31,740 --> 00:52:36,560
the Gaussian wings but
the Lorentzian wings.

879
00:52:36,560 --> 00:52:39,980
And that's maybe also sort of
intellectually interesting.

880
00:52:39,980 --> 00:52:42,740
I've talked to you about
homogeneous, inhomogeneous line

881
00:52:42,740 --> 00:52:43,440
widths.

882
00:52:43,440 --> 00:52:46,390
The bulk part of the
Gaussian is inhomogeneous

883
00:52:46,390 --> 00:52:48,490
because you can talk
to different atoms

884
00:52:48,490 --> 00:52:49,790
at different velocities.

885
00:52:49,790 --> 00:52:52,940
Each atom resonates
with its own Lorentzian,

886
00:52:52,940 --> 00:52:56,070
and it is inhomogeneously
broadened.

887
00:52:56,070 --> 00:53:01,010
But in the line widths which
is due to the Lorentzian,

888
00:53:01,010 --> 00:53:04,980
the homogeneous
broadening dominates.

889
00:53:04,980 --> 00:53:08,460
And since the Lorentzian
has information

890
00:53:08,460 --> 00:53:10,100
about either the
natural line widths

891
00:53:10,100 --> 00:53:12,900
or in a gas about
collisional broadening,

892
00:53:12,900 --> 00:53:19,040
you can actually, far away in
the wings of your line shape,

893
00:53:19,040 --> 00:53:22,530
retrieve information about
collisional physics, which,

894
00:53:22,530 --> 00:53:24,736
in the center of the
line, is completely

895
00:53:24,736 --> 00:53:26,402
masked by the first-order
Doppler shift.

896
00:53:36,734 --> 00:53:37,240
OK.

897
00:53:37,240 --> 00:53:44,060
So I've already written it down.

898
00:53:44,060 --> 00:53:47,230
So far wing spectroscopy
in the gas phase

899
00:53:47,230 --> 00:53:49,320
can give you
interesting information

900
00:53:49,320 --> 00:53:52,920
about atomic collisions
and atomic interactions.

901
00:53:52,920 --> 00:53:56,390
So having started
out by telling you

902
00:53:56,390 --> 00:54:01,180
that it's a first-order Doppler
shift which usually dominates,

903
00:54:01,180 --> 00:54:04,150
but then telling you that if
you go far away from the line

904
00:54:04,150 --> 00:54:10,780
center, the Lorentzian actually
dominates, that means now there

905
00:54:10,780 --> 00:54:14,980
are situations where
we want actually both.

906
00:54:14,980 --> 00:54:19,015
And the general
solution, of course,

907
00:54:19,015 --> 00:54:22,150
is you do a convolution.

908
00:54:22,150 --> 00:54:25,980
Each atom with a given
velocity has a Lorentzian.

909
00:54:25,980 --> 00:54:28,370
And then you have to do the
convolution with the velocity

910
00:54:28,370 --> 00:54:29,960
distribution.

911
00:54:29,960 --> 00:54:35,550
So therefore, the general
situation for gas phase

912
00:54:35,550 --> 00:54:43,420
spectroscopy is the
convolution of the Lorentzian

913
00:54:43,420 --> 00:54:50,650
for each atom and the Gaussian
velocity distribution.

914
00:54:50,650 --> 00:54:55,010
And since this was the standard
case which people encountered

915
00:54:55,010 --> 00:54:58,170
when they did spectroscopy
in the gas phase,

916
00:54:58,170 --> 00:55:02,520
this convoluted line
shape has its own name.

917
00:55:02,520 --> 00:55:03,860
It's called the Voigt profile.

918
00:55:14,210 --> 00:55:16,247
Colin.

919
00:55:16,247 --> 00:55:18,742
AUDIENCE: It's not
obvious to me what exactly

920
00:55:18,742 --> 00:55:21,736
about the atomic
collisions and interaction,

921
00:55:21,736 --> 00:55:23,732
the wings of the
Lorentzian [INAUDIBLE].

922
00:55:23,732 --> 00:55:28,223
Don't you just learn
about the bare line?

923
00:55:28,223 --> 00:55:30,219
Like the un-Doppler
shifted line itself?

924
00:55:33,001 --> 00:55:35,500
PROFESSOR: OK, that's a good
question, as it is not obvious.

925
00:55:35,500 --> 00:55:38,670
And yes, the literature
is full of it.

926
00:55:38,670 --> 00:55:42,970
Because if you don't have
Doppler-free spectroscopy,

927
00:55:42,970 --> 00:55:46,530
if you are always limited
by the Doppler broadening,

928
00:55:46,530 --> 00:55:48,660
put yourself back
into the last century.

929
00:55:48,660 --> 00:55:50,510
But you really want
to learn something

930
00:55:50,510 --> 00:55:52,890
about how atoms
interact and collide,

931
00:55:52,890 --> 00:55:55,470
this was the way to do it.

932
00:55:55,470 --> 00:55:56,970
I don't want to go
into many details

933
00:55:56,970 --> 00:55:58,870
because it's a little
bit old fashioned.

934
00:55:58,870 --> 00:56:00,690
We have other ways to do it now.

935
00:56:00,690 --> 00:56:03,060
But I find it
intellectually interesting

936
00:56:03,060 --> 00:56:05,910
when we talk about line shapes
to realize there are maybe

937
00:56:05,910 --> 00:56:09,020
some subtleties we wouldn't have
thought about it by ourselves.

938
00:56:09,020 --> 00:56:12,000
But just to give you one
example, for whatever reason,

939
00:56:12,000 --> 00:56:13,760
you're very, very interested.

940
00:56:13,760 --> 00:56:17,790
What is the rate of
collision of atoms?

941
00:56:17,790 --> 00:56:21,950
Let's just assume the
simplest model for collisions

942
00:56:21,950 --> 00:56:25,980
that when two atoms collide,
the excited state is de-excited.

943
00:56:25,980 --> 00:56:27,850
Then the lifetime
of the excited state

944
00:56:27,850 --> 00:56:31,040
is no longer 1 over
gamma, the lifetime

945
00:56:31,040 --> 00:56:35,830
is the collision time defined
as a de-excitation time.

946
00:56:35,830 --> 00:56:38,050
And you're really interested
for whatever reason,

947
00:56:38,050 --> 00:56:41,250
because you have the world's
best theory on this object,

948
00:56:41,250 --> 00:56:46,980
that you have a theory what is
the collision time for excited

949
00:56:46,980 --> 00:56:50,930
sodium with argon,
with helium, kind

950
00:56:50,930 --> 00:56:53,264
of with all different elements.

951
00:56:53,264 --> 00:56:54,680
And you have an
interesting theory

952
00:56:54,680 --> 00:56:59,170
which actually reflects how
sodium in the excited state

953
00:56:59,170 --> 00:57:00,830
would interact with noble gases.

954
00:57:00,830 --> 00:57:03,610
And you really want
to test your theory.

955
00:57:03,610 --> 00:57:07,070
Well, now what happens is
the situation is simple.

956
00:57:07,070 --> 00:57:11,300
You have a Lorentzian,
but the Lorentzian

957
00:57:11,300 --> 00:57:14,290
has a broadening which
is the collision rate.

958
00:57:14,290 --> 00:57:19,770
And by carefully analyzing the
wings of your Doppler profile,

959
00:57:19,770 --> 00:57:25,700
you find the collision rate--
as a functional of buffer

960
00:57:25,700 --> 00:57:30,230
gas density, as a function of
noble gas, whatever you pick.

961
00:57:30,230 --> 00:57:31,930
That's one example.

962
00:57:31,930 --> 00:57:34,330
Another example would
be, there may be also

963
00:57:34,330 --> 00:57:37,610
somewhere nontrivial shifts
when the atoms collide.

964
00:57:37,610 --> 00:57:39,760
We'll talk about it
also a little bit later.

965
00:57:39,760 --> 00:57:43,880
The experience-- de-excitation
is one possibility.

966
00:57:43,880 --> 00:57:45,460
But more subtle
things can happen.

967
00:57:45,460 --> 00:57:48,200
For instance, just a
phase perturbation,

968
00:57:48,200 --> 00:57:50,510
or when atoms come
close to each other,

969
00:57:50,510 --> 00:57:52,720
they feel the electric field.

970
00:57:52,720 --> 00:57:56,030
And the electric field
causes an AC Stark shift.

971
00:57:56,030 --> 00:57:59,410
But by understanding the
AC Stark shift or AC Stark

972
00:57:59,410 --> 00:58:01,480
broadening which
comes along with that,

973
00:58:01,480 --> 00:58:03,990
you can maybe map out
the interaction potential

974
00:58:03,990 --> 00:58:05,560
between two atoms.

975
00:58:05,560 --> 00:58:08,570
So people were really
ingenious in trying

976
00:58:08,570 --> 00:58:11,220
to learn something about
atomic interactions

977
00:58:11,220 --> 00:58:13,870
from spectroscopic
information, and that

978
00:58:13,870 --> 00:58:15,310
was one of the few
tools they had.

979
00:58:18,749 --> 00:58:19,415
Other questions?

980
00:58:23,650 --> 00:58:24,460
OK.

981
00:58:24,460 --> 00:58:29,010
We have covered the
simple examples.

982
00:58:29,010 --> 00:58:38,240
And now I want to give you a
more comprehensive framework

983
00:58:38,240 --> 00:58:42,070
called perturbation theory
of spectral broadening.

984
00:58:42,070 --> 00:58:47,950
And in the last class, I
mentioned to you already

985
00:58:47,950 --> 00:58:50,410
that, by using
this framework, we

986
00:58:50,410 --> 00:58:56,950
can more deeply understand
line broadening mechanisms.

987
00:58:56,950 --> 00:59:02,130
And one highlight actually
will be that next week,

988
00:59:02,130 --> 00:59:06,370
using these concepts, we
will actually understand that

989
00:59:06,370 --> 00:59:10,550
collisions cannot only cause
collisional broadening,

990
00:59:10,550 --> 00:59:13,940
they can also cause
collisional narrowing.

991
00:59:13,940 --> 00:59:16,920
So some things which
are counterintuitive

992
00:59:16,920 --> 00:59:21,690
have a very clear description
using this method.

993
00:59:21,690 --> 00:59:25,310
So I think what I'm able to do
in the remaining 20 minutes is

994
00:59:25,310 --> 00:59:28,910
to step you through
the derivation.

995
00:59:28,910 --> 00:59:32,530
And then we apply it to a
number of interesting physical

996
00:59:32,530 --> 00:59:35,750
situations next week.

997
00:59:35,750 --> 00:59:39,220
Now, in a way, I
have to apologize

998
00:59:39,220 --> 00:59:42,880
that what I present to you is
time dependent perturbation

999
00:59:42,880 --> 00:59:45,410
theory again.

1000
00:59:45,410 --> 00:59:50,350
And again with a slightly
different notation, so

1001
00:59:50,350 --> 00:59:55,280
I think I go rather quickly for
the part which is just review.

1002
00:59:55,280 --> 00:59:57,980
But repeating something
is also a good thing.

1003
00:59:57,980 --> 01:00:02,530
But then I will
tell you when we go

1004
01:00:02,530 --> 01:00:05,635
beyond what we have
discussed and beyond what you

1005
01:00:05,635 --> 01:00:06,760
may have seen in textbooks.

1006
01:00:10,880 --> 01:00:18,790
So in other words, we do time
dependent perturbation theory.

1007
01:00:18,790 --> 01:00:22,860
We have a wave function which
is in two states, A and B.

1008
01:00:22,860 --> 01:00:27,810
And there's a time dependent
perturbation v. Schrodinger's

1009
01:00:27,810 --> 01:00:32,500
equation tells us that it's
probably the interaction

1010
01:00:32,500 --> 01:00:36,140
picture, that the rate of
change of the amplitude B

1011
01:00:36,140 --> 01:00:38,880
comes because we
start in the state A

1012
01:00:38,880 --> 01:00:42,430
and the perturbation
couples from state A to B.

1013
01:00:42,430 --> 01:00:44,186
And so we can solve it.

1014
01:00:44,186 --> 01:00:47,470
At this point, it's not even
perturbative, it's general.

1015
01:00:47,470 --> 01:00:50,270
And we are interested
in spectroscopy

1016
01:00:50,270 --> 01:00:53,930
in the rate of a transition,
because we do spectroscopy

1017
01:00:53,930 --> 01:00:56,930
and we measure what is the
population in the excited

1018
01:00:56,930 --> 01:00:57,590
state.

1019
01:00:57,590 --> 01:00:59,780
Because there was
a rate at which

1020
01:00:59,780 --> 01:01:02,860
atoms were transferred from
the calm to the excited state.

1021
01:01:02,860 --> 01:01:05,370
So the rate is the
probability to be

1022
01:01:05,370 --> 01:01:08,770
in the excited
state per unit time.

1023
01:01:08,770 --> 01:01:12,679
So what we are interested now
is, what is the amplitude B?

1024
01:01:12,679 --> 01:01:14,220
And what is the
probability B squared

1025
01:01:14,220 --> 01:01:17,440
to be in the excited state?

1026
01:01:17,440 --> 01:01:19,560
And now comes
perturbation theory.

1027
01:01:19,560 --> 01:01:21,900
If we take
Schrodinger's equation

1028
01:01:21,900 --> 01:01:26,670
and we integrate it with respect
to time-- so we go from B dot

1029
01:01:26,670 --> 01:01:30,030
to B-- we are integrating
here with respect to time.

1030
01:01:30,030 --> 01:01:32,450
And in first-order
perturbation theory,

1031
01:01:32,450 --> 01:01:36,690
we assume the initial
state is undepleted.

1032
01:01:36,690 --> 01:01:41,110
And we replace the amplitude
A of t by its value at time t

1033
01:01:41,110 --> 01:01:42,995
equals 0, which is
assumed to be unity.

1034
01:01:45,920 --> 01:01:46,510
OK.

1035
01:01:46,510 --> 01:02:02,592
So with that-- oh yeah,
and this may be something.

1036
01:02:02,592 --> 01:02:04,800
I'm not doing anything which
goes beyond perturbation

1037
01:02:04,800 --> 01:02:07,430
theory, but I'm using a
slightly different formulation.

1038
01:02:07,430 --> 01:02:10,890
Because you will see that
I need it in a moment.

1039
01:02:10,890 --> 01:02:14,630
The rate is dB squared dt.

1040
01:02:14,630 --> 01:02:17,250
And if I take the
derivative of B squared,

1041
01:02:17,250 --> 01:02:21,630
I get B or B star times B dot.

1042
01:02:21,630 --> 01:02:24,010
So now I'm using
the perturbative--

1043
01:02:24,010 --> 01:02:28,990
I'm inserting this function
B into this expression

1044
01:02:28,990 --> 01:02:31,090
for the rate.

1045
01:02:31,090 --> 01:02:32,250
And this is what I obtain.

1046
01:02:36,540 --> 01:02:39,060
So I can take the
time derivative of it,

1047
01:02:39,060 --> 01:02:44,550
but the time derivative is only
affecting the upper integral.

1048
01:02:44,550 --> 01:02:48,920
So therefore what I get
is B is the integral.

1049
01:02:48,920 --> 01:02:51,570
B dot is the integrand.

1050
01:02:51,570 --> 01:02:53,740
And now I get this
expression, which

1051
01:02:53,740 --> 01:02:55,915
has the product of the
two matrix elements.

1052
01:02:59,070 --> 01:03:01,210
Simple mathematics,
plug and play.

1053
01:03:01,210 --> 01:03:03,320
No new concept.

1054
01:03:03,320 --> 01:03:07,180
But what it leads us now is,
and this is what is usually

1055
01:03:07,180 --> 01:03:09,090
not so much emphasized
in perturbation

1056
01:03:09,090 --> 01:03:11,910
theory, that everything
which happens to the atom,

1057
01:03:11,910 --> 01:03:17,190
and this is the rate at
which we excite the atom,

1058
01:03:17,190 --> 01:03:20,920
is now involving a
correlation function.

1059
01:03:20,920 --> 01:03:28,680
It involves sort of an
integral over the drive field v

1060
01:03:28,680 --> 01:03:32,240
at time t and time t prime.

1061
01:03:32,240 --> 01:03:33,690
And this is the
important concept

1062
01:03:33,690 --> 01:03:35,810
when you want to
explain and understand

1063
01:03:35,810 --> 01:03:37,510
line broadening and such.

1064
01:03:37,510 --> 01:03:42,440
You are driving the system
with an external field.

1065
01:03:42,440 --> 01:03:43,960
And often in
perturbation theory,

1066
01:03:43,960 --> 01:03:47,870
you assume the external field
is just e to the i omega t,

1067
01:03:47,870 --> 01:03:51,720
and this correlation function
is just e to the i omega t.

1068
01:03:51,720 --> 01:03:54,870
And it's so trivial that you
don't even recognize that e

1069
01:03:54,870 --> 01:03:59,060
to the i omega t is not the
time dependence of your field.

1070
01:03:59,060 --> 01:04:04,170
e to the i omega t is sort of
the product of the field at t

1071
01:04:04,170 --> 01:04:10,240
equals 0 and the field at time
t and the field at time t prime.

1072
01:04:10,240 --> 01:04:12,430
But if you have a more
general field with lots

1073
01:04:12,430 --> 01:04:16,120
of Fourier components, the
difference between whether it's

1074
01:04:16,120 --> 01:04:20,342
a correlation function or the
field itself becomes important.

1075
01:04:20,342 --> 01:04:21,800
In other words,
I'm now telling you

1076
01:04:21,800 --> 01:04:24,700
whenever you did perturbation
theory, this is what you did.

1077
01:04:24,700 --> 01:04:28,550
Maybe you didn't notice it,
but what you had was actually

1078
01:04:28,550 --> 01:04:32,040
the correlation function
between the drive field

1079
01:04:32,040 --> 01:04:34,224
at two different
times, t and t prime.

1080
01:04:38,180 --> 01:04:51,300
OK, so our rate is now given
by the correlation function

1081
01:04:51,300 --> 01:04:52,300
of the field.

1082
01:04:52,300 --> 01:04:55,440
And then we so-to-speak
Fourier transform it

1083
01:04:55,440 --> 01:04:56,960
with e to the i omega 0.

1084
01:05:11,230 --> 01:05:12,980
OK.

1085
01:05:12,980 --> 01:05:16,670
Let's just streamline
the expressions,

1086
01:05:16,670 --> 01:05:18,950
make them look nicer.

1087
01:05:18,950 --> 01:05:21,930
We integrate between
time 0 and t.

1088
01:05:25,590 --> 01:05:30,490
But let's now assume--
which is actually

1089
01:05:30,490 --> 01:05:33,250
the situation for many
fields of interest--

1090
01:05:33,250 --> 01:05:37,790
that the field is invariant
against translation in time t.

1091
01:05:37,790 --> 01:05:40,030
So therefore this
correlation function

1092
01:05:40,030 --> 01:05:41,970
does not depend on two times.

1093
01:05:41,970 --> 01:05:46,370
It only depends on the
time difference tau.

1094
01:05:46,370 --> 01:05:52,850
Finally, because of the complex
character of the Schrodinger

1095
01:05:52,850 --> 01:05:55,730
equation, I had an
expression but I

1096
01:05:55,730 --> 01:05:59,590
had to add the
complex conjugate.

1097
01:05:59,590 --> 01:06:03,290
Remember, the rate was the
derivative of B square.

1098
01:06:03,290 --> 01:06:07,660
And the derivative of B
square is B star B dot

1099
01:06:07,660 --> 01:06:09,330
times B star dot times B.

1100
01:06:09,330 --> 01:06:10,720
You get two terms.

1101
01:06:10,720 --> 01:06:13,260
And this is carried forward
with the complex conjugate.

1102
01:06:13,260 --> 01:06:18,050
But if the correlation
function has the proper t,

1103
01:06:18,050 --> 01:06:23,000
that complex conjugation means
you can go to negative time.

1104
01:06:23,000 --> 01:06:25,430
e to the i omega t
complex conjugate

1105
01:06:25,430 --> 01:06:27,720
is e to the minus i omega t.

1106
01:06:27,720 --> 01:06:31,660
That means now that we can
absorb the complex conjugate

1107
01:06:31,660 --> 01:06:35,430
by integrating not
from 0 to t, but having

1108
01:06:35,430 --> 01:06:37,960
the integral from
minus t to plus t.

1109
01:06:55,140 --> 01:06:58,700
And this will be the next
step for most situations

1110
01:06:58,700 --> 01:07:00,570
of interest.

1111
01:07:00,570 --> 01:07:03,300
This correlation, if you
drive it with the field,

1112
01:07:03,300 --> 01:07:05,850
the field has a
finite coherence time.

1113
01:07:05,850 --> 01:07:08,880
So therefore this
integral will not

1114
01:07:08,880 --> 01:07:11,170
have any contribution
when the times are

1115
01:07:11,170 --> 01:07:13,270
longer than the coherence time.

1116
01:07:13,270 --> 01:07:17,150
And then we can set minus
t and plus t to infinity.

1117
01:07:17,150 --> 01:07:20,450
So that will be our
final expression

1118
01:07:20,450 --> 01:07:23,310
which we will use to discuss
line broadening and line

1119
01:07:23,310 --> 01:07:25,760
shifts.

1120
01:07:25,760 --> 01:07:28,320
But we are not yet there.

1121
01:07:28,320 --> 01:07:36,700
We need-- so far, I've just done
ordinary perturbation theory.

1122
01:07:36,700 --> 01:07:39,360
The one extra thing
is I'm stressing

1123
01:07:39,360 --> 01:07:42,070
that when we have
a product of-- when

1124
01:07:42,070 --> 01:07:44,740
we had a matrix element
squared in perturbation theory,

1125
01:07:44,740 --> 01:07:48,130
this is really a correlation
function between the

1126
01:07:48,130 --> 01:07:51,690
[? external, ?] the drive
field at two different times.

1127
01:07:51,690 --> 01:07:55,000
We come back to that when
I discuss the result.

1128
01:07:55,000 --> 01:07:58,030
But the second thing I
want to introduce now

1129
01:07:58,030 --> 01:08:02,270
is that this framework,
which I have formulated,

1130
01:08:02,270 --> 01:08:06,930
allows me now to
include the fact

1131
01:08:06,930 --> 01:08:09,610
that different
atoms in my ensemble

1132
01:08:09,610 --> 01:08:14,790
may experience a
different drive field.

1133
01:08:14,790 --> 01:08:18,691
For instance, I gave you
the example in last class,

1134
01:08:18,691 --> 01:08:20,149
if you have Doppler
broadening, you

1135
01:08:20,149 --> 01:08:22,550
have atoms which start
out at the same point.

1136
01:08:22,550 --> 01:08:27,140
But the faster ones move faster
and experience the laser field

1137
01:08:27,140 --> 01:08:28,810
now with a different phase.

1138
01:08:28,810 --> 01:08:30,634
So different atoms
now experience

1139
01:08:30,634 --> 01:08:34,500
the perturbation v
in a different way.

1140
01:08:34,500 --> 01:08:37,939
So what I've done here so far
is I've pretty much written down

1141
01:08:37,939 --> 01:08:40,330
Schrodinger's equation
for a single particle.

1142
01:08:40,330 --> 01:08:42,339
But now we have to do
an ensemble average.

1143
01:08:46,149 --> 01:09:02,880
So therefore I introduce
now an ensemble average

1144
01:09:02,880 --> 01:09:06,100
by just taking that
expression and averaging

1145
01:09:06,100 --> 01:09:09,000
over all atoms in the ensemble.

1146
01:09:09,000 --> 01:09:12,640
So then I get the
ensemble averaged rate.

1147
01:09:12,640 --> 01:09:15,970
All the correlation
functions we discussed,

1148
01:09:15,970 --> 01:09:19,859
our ensemble averaged
correlation functions

1149
01:09:19,859 --> 01:09:23,564
and our final result will
also have an ensemble average.

1150
01:09:32,950 --> 01:09:34,410
OK.

1151
01:09:34,410 --> 01:09:47,090
So this correlation function
between v of 0 and v of t

1152
01:09:47,090 --> 01:09:49,939
will go to 0 for
very long times.

1153
01:09:49,939 --> 01:09:54,600
Even the most expensive laser
in the world, the electric field

1154
01:09:54,600 --> 01:09:57,380
which is emitted
now is not related

1155
01:09:57,380 --> 01:10:00,660
to the electric field which
is emitted in an hour.

1156
01:10:00,660 --> 01:10:03,610
Because the phase
relationship has been lost,

1157
01:10:03,610 --> 01:10:06,590
and therefore the correlation
function has decayed to 0.

1158
01:10:12,400 --> 01:10:12,900
OK.

1159
01:10:12,900 --> 01:10:17,270
So this is the ensemble average.

1160
01:10:21,320 --> 01:10:24,200
So therefore, what I'm
naturally drawn to now

1161
01:10:24,200 --> 01:10:27,670
is that if I take this
correlation function,

1162
01:10:27,670 --> 01:10:31,040
and I know any
correlation function has

1163
01:10:31,040 --> 01:10:36,100
a characteristic time called the
coherence time where it decays.

1164
01:10:36,100 --> 01:10:39,930
And therefore I can now
discuss two limiting cases.

1165
01:10:39,930 --> 01:10:43,660
One is where the time
evolution of the system

1166
01:10:43,660 --> 01:10:46,915
is started for times much
shorter than the coherence time

1167
01:10:46,915 --> 01:10:50,650
or much longer than
the coherence time.

1168
01:10:50,650 --> 01:10:53,270
And if what I'm
telling you right now

1169
01:10:53,270 --> 01:10:57,180
reminds you of my discussion of
Rabi oscillation versus Fermi's

1170
01:10:57,180 --> 01:11:00,800
golden rule-- yes,
this is actually

1171
01:11:00,800 --> 01:11:02,030
a very analogous discussion.

1172
01:11:04,950 --> 01:11:06,950
OK, so there are the
two limiting cases.

1173
01:11:11,350 --> 01:11:17,400
If the time is much shorter
than the coherence time-- let

1174
01:11:17,400 --> 01:11:23,170
me give you the example of an
oscillating single mode field.

1175
01:11:23,170 --> 01:11:27,370
The perturbation v of
t is just oscillating

1176
01:11:27,370 --> 01:11:30,540
with one frequency, omega.

1177
01:11:30,540 --> 01:11:34,390
And that means if I look
at the correlation function

1178
01:11:34,390 --> 01:11:38,970
at time t and time
t plus tau, it

1179
01:11:38,970 --> 01:11:41,090
is simply the
amplitude of the field

1180
01:11:41,090 --> 01:11:45,260
squared times e
to the i omega t.

1181
01:11:45,260 --> 01:11:50,260
And now I can take this
correlation function;

1182
01:11:50,260 --> 01:11:52,380
put it into my
integral, which has just

1183
01:11:52,380 --> 01:11:55,800
disappeared from the screen;
do the integration with e

1184
01:11:55,800 --> 01:11:57,540
to the i omega 0 t.

1185
01:11:57,540 --> 01:11:59,188
And this is the result I obtain.

1186
01:12:05,740 --> 01:12:08,430
And of course, this
is nothing else

1187
01:12:08,430 --> 01:12:09,950
than what you have
always obtained

1188
01:12:09,950 --> 01:12:13,820
in time dependent perturbation
theory with a sinusoidal field.

1189
01:12:13,820 --> 01:12:17,860
It is this characteristic
sine detuning

1190
01:12:17,860 --> 01:12:20,720
t over detuning,
which in the limit

1191
01:12:20,720 --> 01:12:23,430
and in the limit of when
you square it and go

1192
01:12:23,430 --> 01:12:26,600
to the limit of long times, it
turns into a delta function.

1193
01:12:26,600 --> 01:12:29,080
This gives us
Fermi's golden rule.

1194
01:12:29,080 --> 01:12:32,850
And of course, it has the
same behavior at short times.

1195
01:12:32,850 --> 01:12:47,910
At short times, the
probability for the atom

1196
01:12:47,910 --> 01:12:50,670
to be in the excited
state is quadratic.

1197
01:12:50,670 --> 01:12:54,230
Quadratic is like an
incipient Rabi oscillation.

1198
01:12:54,230 --> 01:12:56,900
And in perturbation theory,
we never get higher up.

1199
01:12:56,900 --> 01:12:59,800
We just look at the
beginning Rabi oscillation.

1200
01:12:59,800 --> 01:13:03,010
So therefore, the
probability is quadratic.

1201
01:13:03,010 --> 01:13:05,760
But I'm talking about
the rate, and the rate

1202
01:13:05,760 --> 01:13:08,080
is probability divided by time.

1203
01:13:08,080 --> 01:13:11,540
So that means the rate is
linearly increasing in time.

1204
01:13:11,540 --> 01:13:14,580
So I'm just saying this is
nothing else than rewriting

1205
01:13:14,580 --> 01:13:18,560
the physics of
Rabi oscillations.

1206
01:13:18,560 --> 01:13:19,490
OK.

1207
01:13:19,490 --> 01:13:24,070
If the time is longer
than the coherence time,

1208
01:13:24,070 --> 01:13:30,970
then we integrate the integral,
not from minus t to plus t.

1209
01:13:30,970 --> 01:13:33,690
We can take the
limits to infinity.

1210
01:13:33,690 --> 01:13:36,290
And that means we
obtain a result

1211
01:13:36,290 --> 01:13:39,160
which is now
independent of time.

1212
01:13:39,160 --> 01:13:42,960
And that means since
Wba is the rate,

1213
01:13:42,960 --> 01:13:45,920
we retrieve a constant
rate, and this

1214
01:13:45,920 --> 01:13:48,148
is what we have done
in Fermi's golden rule.

1215
01:13:52,830 --> 01:13:57,260
So therefore, when we
look at the time evolution

1216
01:13:57,260 --> 01:14:03,410
of a system driven by an
external field, the moment

1217
01:14:03,410 --> 01:14:06,671
we look for the time evolution
longer than the coherence

1218
01:14:06,671 --> 01:14:08,420
time-- and this is
where the main interest

1219
01:14:08,420 --> 01:14:12,570
is in spectroscopy-- we
have a Fermi's golden rule

1220
01:14:12,570 --> 01:14:18,690
result that the system is
excited at a constant rate.

1221
01:14:18,690 --> 01:14:25,020
And I want to now
reinterpret this rate.

1222
01:14:25,020 --> 01:14:29,400
This rate is nothing else
than the Fourier transform

1223
01:14:29,400 --> 01:14:32,660
of a correlation function.

1224
01:14:32,660 --> 01:14:33,810
It makes a lot of sense.

1225
01:14:36,990 --> 01:14:40,020
You apply time-dependent
magnetic fields, perturbations,

1226
01:14:40,020 --> 01:14:43,660
fluctuating magnetic fields,
whatever, vibration and noise

1227
01:14:43,660 --> 01:14:44,410
in your lab.

1228
01:14:44,410 --> 01:14:47,620
You just apply that to
the atom, and the atom

1229
01:14:47,620 --> 01:14:50,580
is nothing else than a
little Fourier analyzer.

1230
01:14:50,580 --> 01:14:53,590
It says, my resonance
frequency is omega 0.

1231
01:14:53,590 --> 01:14:56,540
And all of what matters for
me to make a real transition

1232
01:14:56,540 --> 01:14:59,020
is what you offer me at omega 0.

1233
01:14:59,020 --> 01:15:01,380
And I now Fourier
analyze whatever

1234
01:15:01,380 --> 01:15:04,570
acts on me, the correlation
function of the perturbation

1235
01:15:04,570 --> 01:15:07,160
which acts on me, and
I fully analyze it.

1236
01:15:07,160 --> 01:15:10,220
And what matters for my rate
to go to the excited state

1237
01:15:10,220 --> 01:15:13,670
is the Fourier component
at the resonance frequency.

1238
01:15:13,670 --> 01:15:15,252
It's just a
generalization of what

1239
01:15:15,252 --> 01:15:16,710
we have done in
perturbation theory

1240
01:15:16,710 --> 01:15:20,227
when we assumed that we
have a drive field only

1241
01:15:20,227 --> 01:15:20,935
at one frequency.

1242
01:15:27,930 --> 01:15:29,860
So I've written it
down here for you.

1243
01:15:29,860 --> 01:15:32,190
The rate of excitation
is nothing else

1244
01:15:32,190 --> 01:15:37,760
than the Fourier transform
of this correlation function.

1245
01:15:37,760 --> 01:15:42,880
But let me now take
it one step further,

1246
01:15:42,880 --> 01:15:45,270
which also makes a lot of sense.

1247
01:15:45,270 --> 01:15:47,400
The Fourier transform
of the correlation

1248
01:15:47,400 --> 01:15:50,970
function-- the
correlation function

1249
01:15:50,970 --> 01:15:54,170
is the convolution of
the time-dependent fields

1250
01:15:54,170 --> 01:15:59,720
with themselves, v of
t with v of t plus tau.

1251
01:15:59,720 --> 01:16:03,160
The Fourier transform
of the convolution

1252
01:16:03,160 --> 01:16:07,460
is the product of the Fourier
transform of the field itself.

1253
01:16:07,460 --> 01:16:13,510
So therefore, I
can take whatever

1254
01:16:13,510 --> 01:16:18,580
perturbation the atom
experiences in its own frame--

1255
01:16:18,580 --> 01:16:21,240
external fields, moving around.

1256
01:16:21,240 --> 01:16:24,860
Whatever the atom
is exposed to, I

1257
01:16:24,860 --> 01:16:27,090
have to calculate
the power spectrum

1258
01:16:27,090 --> 01:16:30,000
of what the atom feels.

1259
01:16:30,000 --> 01:16:35,210
And this power spectrum
provides me the excitation rate.

1260
01:16:35,210 --> 01:16:37,340
It's nothing else than
Fermi's golden rule

1261
01:16:37,340 --> 01:16:40,800
but generalized to the
concept of an arbitrary

1262
01:16:40,800 --> 01:16:43,569
spectrum of the driving field.

1263
01:16:56,900 --> 01:16:57,830
Questions so far?

1264
01:17:09,540 --> 01:17:12,510
You can also say
that's a wonderful way

1265
01:17:12,510 --> 01:17:16,152
to look at energy conservation.

1266
01:17:16,152 --> 01:17:21,140
If an atom is exposed to
any kind of environment,

1267
01:17:21,140 --> 01:17:24,550
it goes from the ground
to the excited state only

1268
01:17:24,550 --> 01:17:32,330
to the extent that whatever
acts on the atom has a Fourier

1269
01:17:32,330 --> 01:17:35,070
component at the
resonance frequency.

1270
01:17:35,070 --> 01:17:38,820
And it is only the power
of the fluctuating drive

1271
01:17:38,820 --> 01:17:42,950
field at the resonance
frequency which

1272
01:17:42,950 --> 01:17:46,370
is responsible for
driving the atom.

1273
01:17:46,370 --> 01:17:49,570
And this is energy conservation.

1274
01:17:49,570 --> 01:17:52,870
The frequency component
has to be omega 0

1275
01:17:52,870 --> 01:17:55,360
to take the atom from the
calm to the excited state.

1276
01:17:55,360 --> 01:17:58,010
All the other
frequency components

1277
01:17:58,010 --> 01:18:01,490
take the atom to the virtual
state and take it down again.

1278
01:18:01,490 --> 01:18:04,150
They create maybe line shifts
or something like this.

1279
01:18:04,150 --> 01:18:05,760
But a real transition,
a transition

1280
01:18:05,760 --> 01:18:08,610
where the atom stays
in the excited state,

1281
01:18:08,610 --> 01:18:13,290
requires photons at the
resonance frequency.

1282
01:18:13,290 --> 01:18:15,960
And so to speak, this
measures how many

1283
01:18:15,960 --> 01:18:20,320
photons are acting on the atoms.

1284
01:18:20,320 --> 01:18:26,460
Let me now give you one
or two general features

1285
01:18:26,460 --> 01:18:31,380
of such correlation
functions which I just

1286
01:18:31,380 --> 01:18:35,080
find very, very useful.

1287
01:18:35,080 --> 01:18:37,650
And then I think
your time is over.

1288
01:18:45,102 --> 01:18:49,760
If we have G of w
is now the spectrum

1289
01:18:49,760 --> 01:18:51,460
of the fluctuating fields.

1290
01:18:51,460 --> 01:18:53,780
And let's assume,
yes, eventually we

1291
01:18:53,780 --> 01:18:55,920
have a fluctuating
field which is somewhere

1292
01:18:55,920 --> 01:18:57,940
centered at the
resonance frequency.

1293
01:18:57,940 --> 01:19:00,390
After all, we use a
laser, but the atom

1294
01:19:00,390 --> 01:19:02,290
may now move around
in the laser beam.

1295
01:19:02,290 --> 01:19:03,920
The mirrors may be vibrating.

1296
01:19:03,920 --> 01:19:06,760
So the spectrum seen
by the atom is sort of

1297
01:19:06,760 --> 01:19:09,730
broadened around the
resonance frequency.

1298
01:19:09,730 --> 01:19:13,530
And the broadening is
nothing else than 1

1299
01:19:13,530 --> 01:19:18,570
over the coherence time
of the environment.

1300
01:19:18,570 --> 01:19:22,100
So let me just normalize
the correlation function

1301
01:19:22,100 --> 01:19:24,520
that the integral is unity.

1302
01:19:24,520 --> 01:19:29,620
And then this is
trivial but important.

1303
01:19:29,620 --> 01:19:32,280
The value at the
resonance frequency

1304
01:19:32,280 --> 01:19:37,650
is 1 over the broadening and is
therefore the coherence time.

1305
01:19:37,650 --> 01:19:40,220
It's sort of subtle
but important.

1306
01:19:40,220 --> 01:19:42,140
If you have a
normalized spectrum,

1307
01:19:42,140 --> 01:19:44,160
the more coherent
your source is,

1308
01:19:44,160 --> 01:19:47,060
the larger is the value
of the correlation

1309
01:19:47,060 --> 01:19:48,770
function in the center.

1310
01:19:52,850 --> 01:20:00,330
Let me do the Fourier transform.

1311
01:20:00,330 --> 01:20:02,300
If you Fourier transform
something like this,

1312
01:20:02,300 --> 01:20:04,950
it gives an oscillating function
at the resonance frequency.

1313
01:20:04,950 --> 01:20:08,350
But let me just multiply by
e to the i omega t and sort

1314
01:20:08,350 --> 01:20:10,980
of shift everything
to 0 frequency,

1315
01:20:10,980 --> 01:20:14,390
then we would find that
the temporal correlation

1316
01:20:14,390 --> 01:20:15,830
function decays.

1317
01:20:15,830 --> 01:20:19,930
And it decays over
characteristic time tau

1318
01:20:19,930 --> 01:20:23,310
coherence, which is nothing
else than the inverse line

1319
01:20:23,310 --> 01:20:25,173
bits of the Fourier transform.

1320
01:20:28,549 --> 01:20:30,340
This has nothing to do
with atomic physics.

1321
01:20:30,340 --> 01:20:36,340
It's just properties of the
function is Fourier transform.

1322
01:20:36,340 --> 01:20:39,980
But now we have the situation.

1323
01:20:39,980 --> 01:20:46,210
Our rate was the temporal
correlation function times e

1324
01:20:46,210 --> 01:20:47,835
to the i omega 0 t.

1325
01:20:53,710 --> 01:20:56,960
The integrand here is exactly
what I'm plotting here.

1326
01:20:56,960 --> 01:21:00,400
And so if I perform
the integration,

1327
01:21:00,400 --> 01:21:03,690
at least without getting
the last numerical factor,

1328
01:21:03,690 --> 01:21:08,640
I can approach the result by the
correlation function at time t

1329
01:21:08,640 --> 01:21:12,650
equals 0 times-- if I
do the integration--

1330
01:21:12,650 --> 01:21:15,420
times the temporal
rates of this curve,

1331
01:21:15,420 --> 01:21:19,190
so this is the
correlation time tau c.

1332
01:21:19,190 --> 01:21:23,090
So therefore, if I have,
for instance, my operator

1333
01:21:23,090 --> 01:21:24,930
is the electric
field and I drive

1334
01:21:24,930 --> 01:21:28,020
the atom with a
dipole operator, what

1335
01:21:28,020 --> 01:21:30,880
I find is the
correlation function at t

1336
01:21:30,880 --> 01:21:34,490
equals 0 is nothing else than
the electric field squared,

1337
01:21:34,490 --> 01:21:38,400
which is what we have called the
Rabi frequency squared so far.

1338
01:21:38,400 --> 01:21:41,714
But now I multiply it
with a coherence time.

1339
01:21:48,690 --> 01:21:51,500
So this result should
come very naturally to you

1340
01:21:51,500 --> 01:21:53,370
because when we have
Fermi's golden rule,

1341
01:21:53,370 --> 01:21:57,200
we have a matrix element squared
times the delta function.

1342
01:21:57,200 --> 01:22:00,900
But I've emphasized it again
and again-- the delta function

1343
01:22:00,900 --> 01:22:05,700
is representative for spectral
widths for density of states.

1344
01:22:05,700 --> 01:22:08,870
And if we have an
environment which

1345
01:22:08,870 --> 01:22:13,860
causes spectral broadening,
1 over the coherence time

1346
01:22:13,860 --> 01:22:21,970
is nothing else than the
spectral widths here.

1347
01:22:21,970 --> 01:22:24,820
And so I've done here exactly
what the delta function

1348
01:22:24,820 --> 01:22:26,680
Fermi's golden rule
asked me to do.

1349
01:22:26,680 --> 01:22:28,630
So if I had wanted, I
could have just said,

1350
01:22:28,630 --> 01:22:30,330
look, here is
Fermi's golden rule.

1351
01:22:30,330 --> 01:22:33,550
And by interpreting Fermi's
golden rule the way I just did,

1352
01:22:33,550 --> 01:22:37,500
I could have written down this
result for you right away.

1353
01:22:37,500 --> 01:22:41,660
OK, I think time is over.

1354
01:22:41,660 --> 01:22:45,670
And maybe the
summary which I could

1355
01:22:45,670 --> 01:22:47,450
give you now is a
good starting point

1356
01:22:47,450 --> 01:22:51,010
for our lecture on Monday.

1357
01:22:51,010 --> 01:22:55,120
Reminder, Friday we have the
midterm in this other lecture

1358
01:22:55,120 --> 01:22:57,120
hall in the other building.

1359
01:22:57,120 --> 01:22:59,140
We start at the
normal class time.

1360
01:22:59,140 --> 01:23:01,590
Please be there on time.