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PROFESSOR: Let's go
back to our discussion

9
00:00:27,380 --> 00:00:30,430
of spectral broadening.

10
00:00:30,430 --> 00:00:33,780
And what I started
to derive for you

11
00:00:33,780 --> 00:00:39,310
is perturbation theory of
spectral broadening, which

12
00:00:39,310 --> 00:00:42,190
is a very general framework.

13
00:00:42,190 --> 00:00:45,182
I like it, because
it really provides

14
00:00:45,182 --> 00:00:47,530
you insight into
spectral broadening.

15
00:00:47,530 --> 00:00:51,270
But I will also hope it
provides new insight for you

16
00:00:51,270 --> 00:00:54,670
for time-dependent perturbation
theory [INAUDIBLE].

17
00:00:54,670 --> 00:01:00,590
So what we did in
this lecture is

18
00:01:00,590 --> 00:01:05,370
we pretty much did standard
type perturbation theory.

19
00:01:05,370 --> 00:01:07,773
We just did it by
assuming that we

20
00:01:07,773 --> 00:01:10,210
have a general
time-dependent perturbation.

21
00:01:10,210 --> 00:01:12,910
I'm not yet telling you what the
time-dependent perturbation is.

22
00:01:12,910 --> 00:01:14,076
It has actually fluctuating.

23
00:01:14,076 --> 00:01:15,384
It has inhomogeneous.

24
00:01:15,384 --> 00:01:17,760
It has everything in
it which will later

25
00:01:17,760 --> 00:01:21,250
lead to line-broadening
line shifts.

26
00:01:21,250 --> 00:01:28,470
And the important quantity which
is now describing everything

27
00:01:28,470 --> 00:01:31,872
we are interested
is the correlation

28
00:01:31,872 --> 00:01:35,500
function between the
perturbation at time t

29
00:01:35,500 --> 00:01:38,020
pi and t.

30
00:01:38,020 --> 00:01:42,030
And we call this
correlation function G.

31
00:01:42,030 --> 00:01:44,360
And it is either the
correlation function G of t,

32
00:01:44,360 --> 00:01:47,820
or it's Fourier
transform, which tells us

33
00:01:47,820 --> 00:01:49,950
what the excitation
rate of the system is.

34
00:01:59,570 --> 00:02:03,930
I want to give you sort
of two summaries now.

35
00:02:08,154 --> 00:02:10,539
And they're both general
to perturbation theory.

36
00:02:10,539 --> 00:02:14,610
But I have to say, I
myself learned something

37
00:02:14,610 --> 00:02:18,870
about time evolution of quantum
systems from those examples.

38
00:02:18,870 --> 00:02:21,295
So what we get is a
very generic thing,

39
00:02:21,295 --> 00:02:23,753
that when we look
at the probability

40
00:02:23,753 --> 00:02:28,220
to be in the final state,
the amplitude V of t squared,

41
00:02:28,220 --> 00:02:31,226
we start out quadratic, a very
general behavior of any quantum

42
00:02:31,226 --> 00:02:34,990
system, because your linear
equation, which linearly

43
00:02:34,990 --> 00:02:38,430
puts amplitude into the
excited state is quadratic.

44
00:02:38,430 --> 00:02:41,526
And also, this is the beginning
of a Rabi oscillation.

45
00:02:41,526 --> 00:02:45,290
The Rabi oscillation
starts out with 0 slope.

46
00:02:45,290 --> 00:02:49,280
But then you have some
de-coherence type.

47
00:02:49,280 --> 00:02:53,140
The fields are no longer driving
the system in a coherent way.

48
00:02:53,140 --> 00:02:56,710
And then that's where we
enter the regime of Fermi's

49
00:02:56,710 --> 00:02:59,930
golden rule, the probability
to be in the excited state

50
00:02:59,930 --> 00:03:01,770
only goes linear.

51
00:03:01,770 --> 00:03:05,570
And this is when we
have rate equations.

52
00:03:05,570 --> 00:03:10,620
Let me sort of show how it
comes about in equations,

53
00:03:10,620 --> 00:03:12,725
which I think is really nice.

54
00:03:12,725 --> 00:03:21,312
The differential equations
is that in a time delta t,

55
00:03:21,312 --> 00:03:25,150
we add amplitude delta
t to the excited state.

56
00:03:25,150 --> 00:03:28,210
And so the amplitude
we build up is--

57
00:03:28,210 --> 00:03:30,410
I may have called an h bar.

58
00:03:30,410 --> 00:03:34,120
Or maybe I measured
[INAUDIBLE] frequency units.

59
00:03:34,120 --> 00:03:37,600
It's just the matrix
element times delta t.

60
00:03:37,600 --> 00:03:39,790
And we usually call
the matrix element

61
00:03:39,790 --> 00:03:41,350
the Rabi frequency over 2.

62
00:03:41,350 --> 00:03:43,410
So you should think
about it, that we

63
00:03:43,410 --> 00:03:45,800
add sort of this amplitude.

64
00:03:45,800 --> 00:03:47,997
And as long as we
are coherent, I

65
00:03:47,997 --> 00:03:51,744
have to take the amplitude
1/2 the Rabi frequency delta,

66
00:03:51,744 --> 00:03:53,130
then square it.

67
00:03:53,130 --> 00:03:56,650
And here, you get the
quadratic behavior.

68
00:03:56,650 --> 00:04:01,160
This behavior you also have
constructive interference.

69
00:04:01,160 --> 00:04:05,062
All those delta [INAUDIBLE] are
added in a phase-coherent way.

70
00:04:05,062 --> 00:04:09,555
But this behavior ends, of
course, at the coherence time.

71
00:04:19,110 --> 00:04:20,515
Let me just get my notes.

72
00:04:31,030 --> 00:04:36,600
So when the time becomes
comparable to the coherence

73
00:04:36,600 --> 00:04:48,370
time, then we are adding
amplitude not as constructive

74
00:04:48,370 --> 00:04:52,580
interference-- we are
adding B squared [INAUDIBLE]

75
00:04:52,580 --> 00:04:54,420
becomes sort of a
[INAUDIBLE], and we

76
00:04:54,420 --> 00:04:56,560
add things in quadrature.

77
00:04:56,560 --> 00:05:02,925
So what happens is at
the coherence time,

78
00:05:02,925 --> 00:05:08,980
we have created
an amplitude which

79
00:05:08,980 --> 00:05:12,580
is given by this expression.

80
00:05:16,461 --> 00:05:22,760
But now if time goes by,
we add in quadrature t

81
00:05:22,760 --> 00:05:26,350
over tc chunks of that.

82
00:05:26,350 --> 00:05:31,560
So therefore, our B square,
which we build up with time,

83
00:05:31,560 --> 00:05:36,270
is linear in time
and is omega Rabi

84
00:05:36,270 --> 00:05:39,980
squared over 4, the matrix
element times the coherence

85
00:05:39,980 --> 00:05:40,715
time tc.

86
00:05:44,200 --> 00:05:48,720
And this here now is our
rate in the rate equation.

87
00:05:51,440 --> 00:05:56,090
And this is also what
we exactly got out

88
00:05:56,090 --> 00:06:00,680
of the correlation
function formalism.

89
00:06:00,680 --> 00:06:02,490
So this is how you
should think about it.

90
00:06:02,490 --> 00:06:04,200
You go from constructive
interference

91
00:06:04,200 --> 00:06:07,350
in amplitude and adding
things in quadrature.

92
00:06:07,350 --> 00:06:14,560
Let me maybe add one more
discussion to it, which I hope

93
00:06:14,560 --> 00:06:18,500
will help you to
see the big picture.

94
00:06:18,500 --> 00:06:21,900
We have a matrix
element V. And if I just

95
00:06:21,900 --> 00:06:24,720
ask you, think about
Fermi's golden rule,

96
00:06:24,720 --> 00:06:27,890
you would find out
that the rate is

97
00:06:27,890 --> 00:06:34,910
V square times the density
of states at the resonance.

98
00:06:34,910 --> 00:06:38,890
And sometimes instead of
the density of states,

99
00:06:38,890 --> 00:06:43,200
you write a delta function,
which is just a placeholder

100
00:06:43,200 --> 00:06:45,044
that you should do
an energy integral.

101
00:06:45,044 --> 00:06:47,818
And then you get the
density of states.

102
00:06:47,818 --> 00:06:48,730
OK.

103
00:06:48,730 --> 00:06:50,590
So this is sort of
Fermi's golden rule,

104
00:06:50,590 --> 00:06:53,180
and it should be old hat to you.

105
00:06:53,180 --> 00:07:05,940
But what I told you now is
that what is involved here

106
00:07:05,940 --> 00:07:11,340
from the time integration is the
correlation function, V of 0,

107
00:07:11,340 --> 00:07:14,300
V of t as two different types.

108
00:07:14,300 --> 00:07:16,220
But then because of
the type integration

109
00:07:16,220 --> 00:07:21,090
of Schrodinger's equation,
we had a time integral dt.

110
00:07:21,090 --> 00:07:26,590
But V of 0, V of t, remember,
we have a correlation

111
00:07:26,590 --> 00:07:28,150
between the field V of 0.

112
00:07:28,150 --> 00:07:30,470
And then it decays with time.

113
00:07:30,470 --> 00:07:34,700
This here can be written
V of 0 squared times

114
00:07:34,700 --> 00:07:38,190
the correlation time tau c.

115
00:07:38,190 --> 00:07:40,860
So therefore, the
correlation function,

116
00:07:40,860 --> 00:07:42,060
it's a general formalism.

117
00:07:42,060 --> 00:07:44,990
It allows us to deal with
all time dependencies.

118
00:07:44,990 --> 00:07:48,130
But in essence, the
time-integrated correlation

119
00:07:48,130 --> 00:07:51,064
function is nothing
else than your operator,

120
00:07:51,064 --> 00:07:52,480
your perturbation
operator, time t

121
00:07:52,480 --> 00:07:56,240
equals 0 squared times
the correlation time.

122
00:07:56,240 --> 00:07:59,160
And the correlation time, the
inverse of the correlation

123
00:07:59,160 --> 00:08:03,500
time, is the spectral
widths of your drive field.

124
00:08:03,500 --> 00:08:05,390
And this is what
the spectral widths

125
00:08:05,390 --> 00:08:10,310
is in Fermi's golden rule.

126
00:08:10,310 --> 00:08:14,990
On the other hand, if you do
the integration V of t dt,

127
00:08:14,990 --> 00:08:17,610
think about it as
Fourier transform,

128
00:08:17,610 --> 00:08:21,640
it gives us actually
some V of omega.

129
00:08:21,640 --> 00:08:24,560
It gives us the
Fourier transform.

130
00:08:24,560 --> 00:08:26,350
Well, I see it omega.

131
00:08:26,350 --> 00:08:29,440
If you just do V of t
dt, you get the Fourier

132
00:08:29,440 --> 00:08:31,131
transform at 0 frequency.

133
00:08:31,131 --> 00:08:32,714
But if you look
through the derivation

134
00:08:32,714 --> 00:08:35,350
we did that was an
e to the i omega

135
00:08:35,350 --> 00:08:38,559
resonant term, which we
were just spitting out.

136
00:08:38,559 --> 00:08:41,309
So we have sort of shifted
the origin of frequency.

137
00:08:41,309 --> 00:08:43,695
So therefore, the
Fourier transform,

138
00:08:43,695 --> 00:08:46,580
if I correct for this just
offset infrequency which

139
00:08:46,580 --> 00:08:50,240
I introduced for
simplicity, this

140
00:08:50,240 --> 00:08:57,010
is nothing else than asking
whether the drive field has

141
00:08:57,010 --> 00:08:59,280
a Fourier component
which can resonate

142
00:08:59,280 --> 00:09:00,900
the drive to transition.

143
00:09:00,900 --> 00:09:03,850
And we talked last class
about the convolution.

144
00:09:03,850 --> 00:09:05,880
This gives actually
the power spectrum.

145
00:09:05,880 --> 00:09:09,700
It keeps the Fourier
transform of it squared.

146
00:09:09,700 --> 00:09:14,060
But the power spectrum is, of
course-- when you take a field

147
00:09:14,060 --> 00:09:15,560
and fully analyze
it, [INAUDIBLE]

148
00:09:15,560 --> 00:09:20,430
is the power spectrum, the whole
power and the whole intensity

149
00:09:20,430 --> 00:09:22,800
of your laser, of your
field is spread out

150
00:09:22,800 --> 00:09:24,560
over the banquets of the source.

151
00:09:24,560 --> 00:09:28,040
So when you say, the power
spectrum, the power spectrum

152
00:09:28,040 --> 00:09:32,652
is automatically intensity
divided by spectral widths.

153
00:09:32,652 --> 00:09:34,110
And that's where
the delta function

154
00:09:34,110 --> 00:09:37,090
and the spectral widths in the
normal formulation of Fermi's

155
00:09:37,090 --> 00:09:38,840
golden rule come in.

156
00:09:38,840 --> 00:09:41,420
So I'm just emphasizing
I haven't really

157
00:09:41,420 --> 00:09:46,170
done anything new than just
giving you a general notation.

158
00:09:46,170 --> 00:09:48,070
And I would actually
say-- whenever

159
00:09:48,070 --> 00:09:50,640
you were asking yourself
about Fermi's golden rule,

160
00:09:50,640 --> 00:09:52,020
this is the full [INAUDIBLE].

161
00:09:52,020 --> 00:09:54,851
This is about a fully
time-dependent arbitrary time

162
00:09:54,851 --> 00:09:55,350
field.

163
00:09:55,350 --> 00:10:00,330
And you realize, what was
re-squaring Fermi's golden rule

164
00:10:00,330 --> 00:10:04,050
is actually just
the simple case,

165
00:10:04,050 --> 00:10:06,360
the more general case--
and the more general case

166
00:10:06,360 --> 00:10:09,950
is it's a correlation
or it's a power spectrum

167
00:10:09,950 --> 00:10:13,820
which creates the rate
in Fermi's golden rule.

168
00:10:16,595 --> 00:10:17,636
Any questions about that?

169
00:10:21,291 --> 00:10:21,790
OK.

170
00:10:21,790 --> 00:10:26,089
So we have this
powerful formalism.

171
00:10:26,089 --> 00:10:28,380
So all we have to do is when
we want to understand what

172
00:10:28,380 --> 00:10:30,720
is the rate and
the spectral widths

173
00:10:30,720 --> 00:10:32,820
of the spectroscopic
features is we

174
00:10:32,820 --> 00:10:38,490
have to understand what is
the field driving the atom.

175
00:10:38,490 --> 00:10:41,500
And then we take the
Fourier transform,

176
00:10:41,500 --> 00:10:43,490
or we figure out what
the correlation is.

177
00:10:43,490 --> 00:10:46,450
We take V of 0 squared
times the correlation time,

178
00:10:46,450 --> 00:10:48,730
and we know what the rate is.

179
00:10:48,730 --> 00:10:51,100
And this will allow
us major insight

180
00:10:51,100 --> 00:10:54,330
into spectral broadening.

181
00:10:54,330 --> 00:10:55,260
OK.

182
00:10:55,260 --> 00:10:59,245
But after introducing
that in general,

183
00:10:59,245 --> 00:11:02,230
I think it's time
for a simple example.

184
00:11:02,230 --> 00:11:04,850
And as a simple
example, I just thought

185
00:11:04,850 --> 00:11:08,590
I'd show you how we get
the natural line widths.

186
00:11:08,590 --> 00:11:11,000
And of course, the
natural line widths--

187
00:11:11,000 --> 00:11:14,000
I've mentioned several times,
we need Optical Bloch equations.

188
00:11:14,000 --> 00:11:15,780
We have to really
kind of capture

189
00:11:15,780 --> 00:11:18,410
the interplay between
the coherent drive

190
00:11:18,410 --> 00:11:20,910
of the atom and the
spontaneous decay.

191
00:11:20,910 --> 00:11:22,900
But just
phenomenologically, if I

192
00:11:22,900 --> 00:11:28,410
say the excited state has a
decay rate of gamma over 2,

193
00:11:28,410 --> 00:11:30,000
I mentioned it
before, we can capture

194
00:11:30,000 --> 00:11:33,740
some aspects of
spontaneous decay.

195
00:11:33,740 --> 00:11:37,990
And that means now,
remember, our spectral widths

196
00:11:37,990 --> 00:11:40,830
comes from the correlation
function of the matrix element.

197
00:11:40,830 --> 00:11:43,160
And now the operator
is constant,

198
00:11:43,160 --> 00:11:44,970
but the state is decaying.

199
00:11:44,970 --> 00:11:49,470
So therefore, the matrix element
has this exponentially decaying

200
00:11:49,470 --> 00:11:50,620
function.

201
00:11:50,620 --> 00:11:53,100
And if you then ask,
what is the correlation

202
00:11:53,100 --> 00:11:56,650
function at time 0 and
time tau, we get this.

203
00:12:02,440 --> 00:12:05,590
Of course, the Fourier transform
of an exponentially decaying

204
00:12:05,590 --> 00:12:07,400
function is a Lorentian.

205
00:12:07,400 --> 00:12:11,630
And I've shown you now with a
different formalism that yes,

206
00:12:11,630 --> 00:12:14,680
if you have dampening in the
excited state, for instance,

207
00:12:14,680 --> 00:12:20,980
you do spontaneous decay,
we get the Lorentian.

208
00:12:20,980 --> 00:12:23,470
Of course, we know there should
be power broadening, too.

209
00:12:23,470 --> 00:12:26,020
But of course, don't
expect power broadening

210
00:12:26,020 --> 00:12:29,180
from a perturbative approach,
because a perturbative approach

211
00:12:29,180 --> 00:12:33,330
is only valid when we
have weak drive fields.

212
00:12:39,586 --> 00:12:42,580
Questions?

213
00:12:42,580 --> 00:12:43,900
OK.

214
00:12:43,900 --> 00:12:48,236
So now I want to-- after
this simple example,

215
00:12:48,236 --> 00:12:50,100
I want to explain to
you Doppler broadening.

216
00:12:57,210 --> 00:12:59,250
I know I will
present it in a way

217
00:12:59,250 --> 00:13:06,143
and draw some conclusions
which are usually not done

218
00:13:06,143 --> 00:13:09,540
in the normal presentation
of Doppler broadening.

219
00:13:09,540 --> 00:13:18,270
So what we have to bring in now
is that we have moving atoms.

220
00:13:18,270 --> 00:13:20,764
That's what Doppler
broadening is about.

221
00:13:24,390 --> 00:13:28,270
We drive the field with
an electromagnetic wave.

222
00:13:28,270 --> 00:13:33,180
And we have usually ignored
the spatial dependence,

223
00:13:33,180 --> 00:13:37,700
assuming the atom was
clamped down in c equals 0.

224
00:13:37,700 --> 00:13:40,960
But now we have to
allow for motion.

225
00:13:40,960 --> 00:13:44,760
So the relevant matrix
element for which

226
00:13:44,760 --> 00:13:49,570
we want to calculate
the correlation function

227
00:13:49,570 --> 00:13:51,065
has now a spatial dependence.

228
00:13:53,740 --> 00:13:58,722
And our correlation
function, G of ba,

229
00:13:58,722 --> 00:14:05,720
involves now-- I call
the Rabi frequency now

230
00:14:05,720 --> 00:14:15,470
x for, well, simplicity
of-- [INAUDIBLE]

231
00:14:15,470 --> 00:14:17,911
took this material,
because they use it.

232
00:14:17,911 --> 00:14:18,410
OK.

233
00:14:18,410 --> 00:14:20,640
So now we have the
correlation function.

234
00:14:20,640 --> 00:14:27,100
We have the temporal
part, t prime minus t.

235
00:14:27,100 --> 00:14:28,880
But then we have a spatial part.

236
00:14:28,880 --> 00:14:31,300
And this is now
the new part which

237
00:14:31,300 --> 00:14:38,586
has to account for that the
atom is moving from one position

238
00:14:38,586 --> 00:14:39,578
to another one.

239
00:14:44,010 --> 00:14:47,390
And when we calculate
the correlation function,

240
00:14:47,390 --> 00:14:50,525
we have to average
now over the velocity

241
00:14:50,525 --> 00:14:51,794
distribution of the atom.

242
00:14:54,450 --> 00:15:00,560
So this new part, which will
account for Doppler broadening,

243
00:15:00,560 --> 00:15:07,370
I called this part I. And
z of t prime minus z of t

244
00:15:07,370 --> 00:15:12,882
is simply the velocity
of the atom times tau.

245
00:15:12,882 --> 00:15:15,250
Tau is t prime minus t.

246
00:15:15,250 --> 00:15:19,530
So atoms move by that.

247
00:15:19,530 --> 00:15:25,940
And v is the z component
of the velocity.

248
00:15:25,940 --> 00:15:28,680
So now we have to
calculate that.

249
00:15:28,680 --> 00:15:31,930
And of course, we
assume that we have

250
00:15:31,930 --> 00:15:33,400
a Maxwell-Boltzmann
distribution.

251
00:15:40,550 --> 00:15:43,560
We assume a Maxwell-Boltzmann
distribution.

252
00:15:46,430 --> 00:15:51,420
And then our
expression I, we have

253
00:15:51,420 --> 00:15:58,900
to convolute-- we have
this term, e to the kv tau.

254
00:15:58,900 --> 00:16:02,380
But now we have to
convolute it with

255
00:16:02,380 --> 00:16:07,110
the one-dimensional
Maxwell-Boltzmann distribution,

256
00:16:07,110 --> 00:16:09,550
where alpha is the
most probable velocity.

257
00:16:12,202 --> 00:16:15,300
And since the velocity
distribution is normalized,

258
00:16:15,300 --> 00:16:24,230
we have this p [? vector. ?]
Alpha is the most probable

259
00:16:24,230 --> 00:16:31,596
speed, and mu is the mass
of the-- sorry, it's not mu,

260
00:16:31,596 --> 00:16:33,193
it's M. And M is the
mass of the atom.

261
00:16:40,600 --> 00:16:44,240
This integral, of course,
can easily be solved.

262
00:16:44,240 --> 00:16:59,165
And we find that provides--
[INAUDIBLE] Gaussian envelope

263
00:16:59,165 --> 00:17:03,530
with time tau, so it's
exponential-decayed with time

264
00:17:03,530 --> 00:17:05,630
tau.

265
00:17:05,630 --> 00:17:16,470
And our rate, which
is the matrix element,

266
00:17:16,470 --> 00:17:24,970
involves now the
temporal integral over--

267
00:17:24,970 --> 00:17:26,060
let me just scroll up.

268
00:17:28,780 --> 00:17:31,680
In our correlation function,
we had a temporal part

269
00:17:31,680 --> 00:17:33,390
and a spatial part.

270
00:17:33,390 --> 00:17:36,318
And the rate-- this was
Fermi's golden rule--

271
00:17:36,318 --> 00:17:38,568
is the time we've taken over
the correlation function.

272
00:17:44,010 --> 00:17:51,120
So therefore, we have the
exponential [? vector ?]

273
00:17:51,120 --> 00:17:52,710
in time.

274
00:17:52,710 --> 00:17:58,970
After convolution with the
Maxwell-Boltzmann distribution,

275
00:17:58,970 --> 00:18:02,870
we have this exponentially
decaying term.

276
00:18:02,870 --> 00:18:06,046
And the result of
that is, well, hooray,

277
00:18:06,046 --> 00:18:10,610
we have re-derived
the Gaussian profile

278
00:18:10,610 --> 00:18:13,112
for a Doppler broad light.

279
00:18:24,230 --> 00:18:25,130
OK.

280
00:18:25,130 --> 00:18:33,080
But yes, we want to look at
this result with some new eyes,

281
00:18:33,080 --> 00:18:35,400
because until now,
you would have said,

282
00:18:35,400 --> 00:18:36,540
OK, that's really trivial.

283
00:18:36,540 --> 00:18:38,150
Each atom has a velocity.

284
00:18:38,150 --> 00:18:41,040
Galilean transformation
into the moving frame

285
00:18:41,040 --> 00:18:43,680
means the frequency has shifted,
and everything falls together.

286
00:18:43,680 --> 00:18:45,356
And yes, that's one
way to look at it.

287
00:18:45,356 --> 00:18:49,400
In inhomogeneous broadening,
each atom has its own velocity.

288
00:18:49,400 --> 00:18:52,220
But now we want to look
at it from the viewpoint

289
00:18:52,220 --> 00:18:56,750
of a correlation function
describing the whole system.

290
00:18:56,750 --> 00:19:01,820
So we had calculated this
correlation function,

291
00:19:01,820 --> 00:19:04,870
which the atomic
ensemble experiences.

292
00:19:04,870 --> 00:19:08,405
And this correlation
function here

293
00:19:08,405 --> 00:19:12,420
decays as a function of time.

294
00:19:12,420 --> 00:19:16,235
And it decays with a
characteristic time.

295
00:19:24,660 --> 00:19:30,510
Tau c, which is 1 over k alpha.

296
00:19:30,510 --> 00:19:35,180
And this is nothing else than
the reduced wavelengths lambda

297
00:19:35,180 --> 00:19:38,210
bar divided by the
most probable velocity.

298
00:19:42,846 --> 00:19:43,600
OK.

299
00:19:43,600 --> 00:19:49,580
So we want to relate
the line widths

300
00:19:49,580 --> 00:19:52,500
to some form of coherence.

301
00:19:52,500 --> 00:19:54,495
And what we realize
is the coherence

302
00:19:54,495 --> 00:19:57,470
time of the correlation
function, which needs Doppler

303
00:19:57,470 --> 00:20:03,740
broadening, is the time it takes
an atom with the most probably

304
00:20:03,740 --> 00:20:07,600
velocity to move one wavelength.

305
00:20:07,600 --> 00:20:10,290
But wait.

306
00:20:10,290 --> 00:20:12,990
What we have is, in the
Maxwell-Boltzmann distribution,

307
00:20:12,990 --> 00:20:15,890
the most probably
velocity is also

308
00:20:15,890 --> 00:20:19,230
the widths of the velocities.

309
00:20:19,230 --> 00:20:23,670
So therefore, what we can say
is, if all the atoms would

310
00:20:23,670 --> 00:20:27,600
start at one position, after
the correlation time tc,

311
00:20:27,600 --> 00:20:33,250
the atoms have spread
out over one wavelength.

312
00:20:33,250 --> 00:20:35,400
So therefore, the
correlation time

313
00:20:35,400 --> 00:20:41,070
tells us how long is the whole
ensemble driven coherently.

314
00:20:41,070 --> 00:20:43,410
But once the atoms,
due to their motion,

315
00:20:43,410 --> 00:20:46,770
spread out compared to
their initial position,

316
00:20:46,770 --> 00:20:49,370
while extra
wavelengths, each atom

317
00:20:49,370 --> 00:20:53,555
experiences now a different
phase of the drive field.

318
00:20:57,130 --> 00:20:59,820
And that sets a limit
to the coherence.

319
00:20:59,820 --> 00:21:02,750
And this is the point where
when we ask how many atoms

320
00:21:02,750 --> 00:21:07,970
are getting excited, we
can no longer add amplitude

321
00:21:07,970 --> 00:21:09,430
in a constructive, linear way.

322
00:21:09,430 --> 00:21:11,820
We are adding amplitudes
in quadrature.

323
00:21:11,820 --> 00:21:13,639
And this is exactly
what I explained to you

324
00:21:13,639 --> 00:21:15,014
at the beginning
of this lecture.

325
00:21:18,630 --> 00:21:20,310
So let me just summarize it.

326
00:21:20,310 --> 00:21:26,140
Alpha is the thermal
spread of velocities.

327
00:21:37,600 --> 00:21:40,720
So the keyword is here.

328
00:21:40,720 --> 00:21:50,370
Atoms, in a random way,
spread out by lambda

329
00:21:50,370 --> 00:21:53,124
in the coherence time tc.

330
00:22:01,890 --> 00:22:05,310
Well, a question which should
come to your mind is now,

331
00:22:05,310 --> 00:22:09,225
but what happens when the atoms
are in an atom [INAUDIBLE]

332
00:22:09,225 --> 00:22:11,840
and they cannot spread out?

333
00:22:11,840 --> 00:22:14,520
That's what you want to
discuss in a few minutes.

334
00:22:14,520 --> 00:22:17,110
But before I do
that, let me give you

335
00:22:17,110 --> 00:22:28,940
another interpretation,
which is helpful.

336
00:22:28,940 --> 00:22:32,200
You can regard-- if you
have a thermal ensemble,

337
00:22:32,200 --> 00:22:33,640
you can say, OK, I have a box.

338
00:22:33,640 --> 00:22:38,230
Each atom is a plain wave
with a perfect velocity,

339
00:22:38,230 --> 00:22:40,430
and now an ensemble of that.

340
00:22:40,430 --> 00:22:44,420
But very often, especially
if you do localized physics,

341
00:22:44,420 --> 00:22:48,170
you want to regard each
atom as a wave packet.

342
00:22:48,170 --> 00:22:51,210
And if you want to use
a localized description

343
00:22:51,210 --> 00:23:01,710
of your gas, where
atoms are wave packets,

344
00:23:01,710 --> 00:23:07,080
then you assume, in a
consistent description,

345
00:23:07,080 --> 00:23:11,350
that the atoms are spread out
due to the momentum spread

346
00:23:11,350 --> 00:23:13,990
in the Maxwell-Boltzmann
distribution.

347
00:23:13,990 --> 00:23:19,230
This is nothing else
than h bar divided

348
00:23:19,230 --> 00:23:22,786
by the mass and the most
[INAUDIBLE] velocity.

349
00:23:22,786 --> 00:23:25,500
And this is foregoing
[? vectors ?]

350
00:23:25,500 --> 00:23:30,665
on the order of unity, nothing
else than the thermal de

351
00:23:30,665 --> 00:23:33,530
Broglie wavelengths.

352
00:23:33,530 --> 00:23:34,340
OK.

353
00:23:34,340 --> 00:23:37,190
So now we have the
picture that the atom

354
00:23:37,190 --> 00:23:39,840
is a wave packet in
the ground state.

355
00:23:39,840 --> 00:23:42,985
But now we excite
it with a laser.

356
00:23:42,985 --> 00:23:50,450
Well, if we excite it with a
laser, part of the wave packet

357
00:23:50,450 --> 00:23:52,880
goes to the excited state.

358
00:23:52,880 --> 00:23:56,110
But the atoms in
the excited state,

359
00:23:56,110 --> 00:24:00,130
because they have absorbed the
photon recall of the photon,

360
00:24:00,130 --> 00:24:04,320
are now moving away
from the ground state

361
00:24:04,320 --> 00:24:07,625
part of the wave packet,
with the recall velocity,

362
00:24:07,625 --> 00:24:12,100
with this h bar k over M.

363
00:24:12,100 --> 00:24:13,800
So if I regard
the atom as a wave

364
00:24:13,800 --> 00:24:17,450
packet, the natural
question is, when

365
00:24:17,450 --> 00:24:20,770
does the ground state
of the wave packet

366
00:24:20,770 --> 00:24:24,332
lose overlap with the excited
state part of the wave packet?

367
00:24:32,270 --> 00:24:40,716
Lost overlap after time--
well, I derive it for you.

368
00:24:40,716 --> 00:24:43,500
But it can only be
the coherence time.

369
00:24:43,500 --> 00:24:54,550
So the time is the
size of the wave packet

370
00:24:54,550 --> 00:25:01,150
divided by the recall velocity.

371
00:25:01,150 --> 00:25:02,380
The mass cancels out.

372
00:25:02,380 --> 00:25:04,070
H bar cancels out.

373
00:25:04,070 --> 00:25:07,570
And this is 1 over k alpha.

374
00:25:07,570 --> 00:25:11,930
And this was exactly
our coherence time.

375
00:25:11,930 --> 00:25:14,130
So therefore, when
I'm telling you

376
00:25:14,130 --> 00:25:18,250
you should understand
this picture,

377
00:25:18,250 --> 00:25:25,710
Doppler broadening is a loss
of coherence for the ensemble.

378
00:25:25,710 --> 00:25:29,150
You have now two
ways to describe it.

379
00:25:29,150 --> 00:25:34,350
One is you can say, in a
more quantum mechanical way,

380
00:25:34,350 --> 00:25:40,410
after the coherence
time, the recall velocity

381
00:25:40,410 --> 00:25:43,490
has separated the
grounded, excited part

382
00:25:43,490 --> 00:25:44,825
of your wave packet.

383
00:25:49,560 --> 00:25:54,750
Or you can say, when the
atoms in the ensemble

384
00:25:54,750 --> 00:25:57,762
have a velocity
spread of alpha, then

385
00:25:57,762 --> 00:26:02,740
they have spread out by
the optical wavelengths.

386
00:26:02,740 --> 00:26:04,750
So these are two
equivalent picture

387
00:26:04,750 --> 00:26:09,188
to understand why this ensemble
is no longer coherently driven.

388
00:26:12,972 --> 00:26:13,472
Yes?

389
00:26:13,472 --> 00:26:18,169
AUDIENCE: [INAUDIBLE] in both
of the curves, like, why is,

390
00:26:18,169 --> 00:26:20,168
in the first picture, the
optical wavelength not

391
00:26:20,168 --> 00:26:23,500
relevant; in the second one,
[INAUDIBLE] not relevant?

392
00:26:23,500 --> 00:26:26,356
[INAUDIBLE] distance.

393
00:26:26,356 --> 00:26:31,890
PROFESSOR: Well, because these
are two different pictures,

394
00:26:31,890 --> 00:26:33,865
but the results agree.

395
00:26:33,865 --> 00:26:37,314
I mean, the wavelength comes
into the picture with the wave

396
00:26:37,314 --> 00:26:39,510
packets through the
recall velocity,

397
00:26:39,510 --> 00:26:42,666
because the recall
velocity is h bar k over M,

398
00:26:42,666 --> 00:26:44,270
and k is the
inverse wavelengths.

399
00:26:44,270 --> 00:26:45,138
AUDIENCE: Right.

400
00:26:45,138 --> 00:26:48,310
But when we write delta t
there, we write [INAUDIBLE]

401
00:26:48,310 --> 00:26:51,466
the wavelengths off
the packet here,

402
00:26:51,466 --> 00:26:54,418
and we could have also
just thought of it as,

403
00:26:54,418 --> 00:26:58,354
when does the atomic
ensemble become

404
00:26:58,354 --> 00:27:00,322
bigger than the optical rate?

405
00:27:00,322 --> 00:27:02,290
But here, we are [INAUDIBLE].

406
00:27:02,290 --> 00:27:06,245
PROFESSOR: I think
for consistency-- I'll

407
00:27:06,245 --> 00:27:07,250
give you a quick answer.

408
00:27:07,250 --> 00:27:10,260
I should think about it
longer, but what we usually

409
00:27:10,260 --> 00:27:12,450
assume when we describe
atoms by wave packets,

410
00:27:12,450 --> 00:27:15,680
we assume the atoms
are not cooled

411
00:27:15,680 --> 00:27:17,860
below the so-called
recoil limit.

412
00:27:17,860 --> 00:27:20,760
So we assume that the thermal
de Broglie wavelengths

413
00:27:20,760 --> 00:27:23,410
is shorter than the
optical wavelengths.

414
00:27:23,410 --> 00:27:27,232
And so then if you would
say, you would expect also

415
00:27:27,232 --> 00:27:29,440
that there would be something
happening when the wave

416
00:27:29,440 --> 00:27:32,370
packet spreads out by
an optical wavelength,

417
00:27:32,370 --> 00:27:33,670
I want to think about it more.

418
00:27:33,670 --> 00:27:35,800
But the quick answer
is, just assume

419
00:27:35,800 --> 00:27:37,967
what is usually the
semi-classical limit

420
00:27:37,967 --> 00:27:39,745
of these kind of
pictures, where we

421
00:27:39,745 --> 00:27:43,900
assume that we have a hierarchy
that the thermal de Broglie

422
00:27:43,900 --> 00:27:46,242
wavelength is much larger
than the size of the atoms

423
00:27:46,242 --> 00:27:47,950
but smaller than the
optical wavelengths.

424
00:27:47,950 --> 00:27:49,227
AUDIENCE: That makes sense.

425
00:27:49,227 --> 00:27:50,852
PROFESSOR: A few
things happen, really,

426
00:27:50,852 --> 00:27:54,030
in intuitive pictures
when you cool atoms

427
00:27:54,030 --> 00:27:55,440
before the recoil limit.

428
00:27:58,221 --> 00:27:58,720
OK.

429
00:27:58,720 --> 00:28:04,690
There's one reason why I would
like to express it to you.

430
00:28:04,690 --> 00:28:09,270
Armed with that knowledge,
if I would now ask you--

431
00:28:09,270 --> 00:28:14,220
you have a trapped
Bose-Einstein condensate,

432
00:28:14,220 --> 00:28:18,814
and you take the spectrum.

433
00:28:18,814 --> 00:28:20,563
What is the Doppler
widths of the spectrum

434
00:28:20,563 --> 00:28:21,813
of a Bose-Einstein condensate?

435
00:28:25,139 --> 00:28:27,180
[INAUDIBLE] extra
Maxwell-Boltzmann distribution.

436
00:28:27,180 --> 00:28:27,846
No.

437
00:28:27,846 --> 00:28:29,150
The condensate is different.

438
00:28:29,150 --> 00:28:32,430
It's in one quantum state.

439
00:28:32,430 --> 00:28:36,710
But now you can
choose your picture.

440
00:28:36,710 --> 00:28:43,130
One picture you can take is
you can say, the de Broglie

441
00:28:43,130 --> 00:28:48,440
wavelengths here has to be--
you know, the wave packet loses

442
00:28:48,440 --> 00:28:50,590
overlap when the
excited state moves

443
00:28:50,590 --> 00:28:52,250
one de Broglie wavelength.

444
00:28:52,250 --> 00:28:54,360
But the condensate
is fully coherent.

445
00:28:54,360 --> 00:28:56,930
So you would now say,
maybe I should think about

446
00:28:56,930 --> 00:28:59,750
if-- if the
condensate part of it

447
00:28:59,750 --> 00:29:04,850
is coupled to the excited
state, and with a recoil,

448
00:29:04,850 --> 00:29:07,800
the excited state
component has moved

449
00:29:07,800 --> 00:29:10,900
the size of the condensate,
replacing the de Broglie

450
00:29:10,900 --> 00:29:14,380
wavelengths by the
size of the condensate.

451
00:29:14,380 --> 00:29:16,340
And this is a correct answer.

452
00:29:16,340 --> 00:29:20,380
You would then find out
what is, quotation mark,

453
00:29:20,380 --> 00:29:24,206
the Doppler broadening
of a condensate.

454
00:29:24,206 --> 00:29:25,705
Of course, you could
have also said,

455
00:29:25,705 --> 00:29:27,950
the condensate is
a certain size,

456
00:29:27,950 --> 00:29:31,240
h bar divided by the size is
the momentum spread if you

457
00:29:31,240 --> 00:29:33,300
do Heisenberg's
uncertainty relation.

458
00:29:33,300 --> 00:29:35,790
And now I plug in
this momentum spread

459
00:29:35,790 --> 00:29:38,650
into a formula for the
Doppler broadening.

460
00:29:38,650 --> 00:29:40,320
And you would get
the same result.

461
00:29:40,320 --> 00:29:43,000
But especially when you think
in terms of a coherent wave

462
00:29:43,000 --> 00:29:47,400
function, this picture of losing
overlap between the two parts

463
00:29:47,400 --> 00:29:51,190
of the wave packets is very
intuitive, very useful.

464
00:29:51,190 --> 00:29:55,970
And it actually guided
a lot of our intuition

465
00:29:55,970 --> 00:29:59,254
when we looked at the
limitations of super radians

466
00:29:59,254 --> 00:30:02,218
and optical spectroscopy with
Bose-Einstein condensates.

467
00:30:05,680 --> 00:30:06,906
Any questions?

468
00:30:11,590 --> 00:30:12,110
OK.

469
00:30:12,110 --> 00:30:16,315
So now we are ready to
take it to the next level.

470
00:30:16,315 --> 00:30:21,120
When I told you that
the spectral widths

471
00:30:21,120 --> 00:30:23,370
is the inverse of
the coherence time.

472
00:30:23,370 --> 00:30:25,860
And one way to think
about the coherence time

473
00:30:25,860 --> 00:30:30,040
is that the particles spread
out over one wavelength.

474
00:30:30,040 --> 00:30:34,390
So if you take this
thought seriously and say,

475
00:30:34,390 --> 00:30:38,792
what happens if I confine atoms
in a container or an atom trap

476
00:30:38,792 --> 00:30:41,264
to less than the
optical wavelengths,

477
00:30:41,264 --> 00:30:42,680
then you would
say, they can never

478
00:30:42,680 --> 00:30:45,190
spread out by an
optical wavelength.

479
00:30:45,190 --> 00:30:49,280
Does it mean that the
coherence time is now infinite

480
00:30:49,280 --> 00:30:53,790
and that we can do spectroscopy,
which is no longer affected

481
00:30:53,790 --> 00:30:58,550
in any way by
Doppler broadening.

482
00:30:58,550 --> 00:31:01,240
Well, what I just
motivated in words

483
00:31:01,240 --> 00:31:05,020
is the so-called Lamb-Dicke
limit of tight confinement.

484
00:31:05,020 --> 00:31:08,230
And as I want to show
you now, yes indeed,

485
00:31:08,230 --> 00:31:10,560
you have a very,
very sharp line which

486
00:31:10,560 --> 00:31:14,350
is not broadened by
Doppler broadening, which

487
00:31:14,350 --> 00:31:16,960
is not shifted by
the recoil shift.

488
00:31:16,960 --> 00:31:19,590
It's really the unperturbed
line of the atom which

489
00:31:19,590 --> 00:31:23,030
can be probed by confining
the atoms to less

490
00:31:23,030 --> 00:31:23,890
than a wavelength.

491
00:31:27,730 --> 00:31:32,950
So therefore, let's
now discuss the line

492
00:31:32,950 --> 00:31:37,150
shape of confined particles.

493
00:31:46,645 --> 00:31:48,020
So what I want to
present you now

494
00:31:48,020 --> 00:31:52,070
is we have particles trapped
in a harmonic oscillator.

495
00:31:52,070 --> 00:31:54,130
And in one limit, which
I want to explain you,

496
00:31:54,130 --> 00:32:00,830
we should just find the normal
Gaussian Doppler profile

497
00:32:00,830 --> 00:32:03,910
which we have
obtained for free gas.

498
00:32:03,910 --> 00:32:06,950
This must be the limit when
the [INAUDIBLE] confinement is

499
00:32:06,950 --> 00:32:08,020
very weak.

500
00:32:08,020 --> 00:32:12,490
But for tight confinement,
we should actually find,

501
00:32:12,490 --> 00:32:15,230
unless we assume other
means of line broadening,

502
00:32:15,230 --> 00:32:17,510
a delta function
spectral feature.

503
00:32:17,510 --> 00:32:20,170
And I will explain to
you that this is actually

504
00:32:20,170 --> 00:32:23,360
the same as the
Mossbauer effect.

505
00:32:23,360 --> 00:32:25,660
It's a Mossbauer line
due to the confinement.

506
00:32:30,260 --> 00:32:38,040
So therefore, to have
trapped particles

507
00:32:38,040 --> 00:32:53,854
allows us to go to the
ultimate limit in precision

508
00:32:53,854 --> 00:32:54,395
spectroscopy.

509
00:32:58,080 --> 00:33:02,420
What happens when you
have trapped particles--

510
00:33:02,420 --> 00:33:05,580
the Mossbauer effect,
which I mentioned,

511
00:33:05,580 --> 00:33:09,210
or simply the effect of
confinement-- in other words,

512
00:33:09,210 --> 00:33:13,600
the trapping potential
is completely

513
00:33:13,600 --> 00:33:19,910
eliminating the Doppler effect.

514
00:33:19,910 --> 00:33:21,430
But I want to be specific.

515
00:33:21,430 --> 00:33:23,863
It only eliminates the
first-order Doppler effect.

516
00:33:27,180 --> 00:33:29,940
Everything I just did with
the correlation function

517
00:33:29,940 --> 00:33:32,940
assumed first-order
Doppler effect.

518
00:33:32,940 --> 00:33:49,010
If you want to get rid of the
second-order Doppler effect,

519
00:33:49,010 --> 00:33:51,310
then you need some
form of proving.

520
00:33:51,310 --> 00:33:52,810
But usually, when
you do experiments

521
00:33:52,810 --> 00:33:55,990
with trapped particles, you
do confinement and cooling

522
00:33:55,990 --> 00:33:56,740
at the same time.

523
00:34:05,590 --> 00:34:06,210
OK.

524
00:34:06,210 --> 00:34:09,960
Let me start out with
very basic things.

525
00:34:13,639 --> 00:34:16,635
So let's talk about the spectrum
of an oscillating emitter.

526
00:34:24,630 --> 00:34:29,316
If we have an atom, it undergoes
a transition from excited state

527
00:34:29,316 --> 00:34:33,909
b to excited state a,
this is an internal state.

528
00:34:33,909 --> 00:34:36,440
But now we want
to include motion.

529
00:34:36,440 --> 00:34:39,650
And we have to include the
external degree of freedom.

530
00:34:39,650 --> 00:34:42,930
And for our discussion right
now, the external degree

531
00:34:42,930 --> 00:34:46,090
of freedom is [INAUDIBLE]
trapping potential.

532
00:34:46,090 --> 00:34:50,330
So now we look at the
combined system, combined--

533
00:34:50,330 --> 00:34:52,310
we can say Hubert
space, which combines

534
00:34:52,310 --> 00:34:55,120
external and internal motion.

535
00:34:55,120 --> 00:34:59,090
And of course, the external
motion is now quantized.

536
00:35:03,960 --> 00:35:08,355
I can't assume, but it doesn't
really add anything to it

537
00:35:08,355 --> 00:35:09,980
at this point-- that
the trap frequency

538
00:35:09,980 --> 00:35:13,400
and the ground and excited
state are different.

539
00:35:13,400 --> 00:35:16,755
I simply assume that
the trap frequency omega

540
00:35:16,755 --> 00:35:19,080
trap is the same.

541
00:35:19,080 --> 00:35:23,620
This, of course, is excellently
fulfilled in ion traps.

542
00:35:23,620 --> 00:35:27,920
If you reduce spectroscopy of
neutral atoms in a dipole trap,

543
00:35:27,920 --> 00:35:31,660
of course, the ground
and excited state

544
00:35:31,660 --> 00:35:34,250
may experience a
different AC stark shift.

545
00:35:34,250 --> 00:35:37,220
And then you have to account
for two different frequencies.

546
00:35:37,220 --> 00:35:39,550
But let me just make this
simplifying assumption.

547
00:35:46,010 --> 00:35:53,690
So if you assume that
an atom emits radiation,

548
00:35:53,690 --> 00:35:57,260
it will, for energy
conservation,

549
00:35:57,260 --> 00:36:00,030
emit at the electronic energy.

550
00:36:00,030 --> 00:36:10,530
But then there is an extra
term, which, in general, is

551
00:36:10,530 --> 00:36:12,380
the energy of the
external motion,

552
00:36:12,380 --> 00:36:16,530
or the trapping potential
for the initial state

553
00:36:16,530 --> 00:36:18,500
minus the final state.

554
00:36:18,500 --> 00:36:22,530
And if we now make our
simplifying assumption

555
00:36:22,530 --> 00:36:25,000
that everything is harmonic,
that hyper-potential

556
00:36:25,000 --> 00:36:27,570
is harmonic, and the
trapping frequency

557
00:36:27,570 --> 00:36:31,920
is the same in the excited
state and the ground state,

558
00:36:31,920 --> 00:36:37,852
we simply have the electronic
energy plus n quanta--

559
00:36:37,852 --> 00:36:41,170
n is the change of the number of
quanta of the harmonic motion.

560
00:36:45,470 --> 00:36:48,610
I want to point out,
it looks so trivial.

561
00:36:48,610 --> 00:36:52,280
But you should at least
think for a second

562
00:36:52,280 --> 00:36:58,230
about this statement, that this
formula includes the Doppler

563
00:36:58,230 --> 00:36:59,586
shift and the recoil shift.

564
00:37:14,470 --> 00:37:18,282
And of course, this is trivial,
because we are talking here

565
00:37:18,282 --> 00:37:21,370
about the total energy
of the external state,

566
00:37:21,370 --> 00:37:23,990
the total energy of the final
state after photon emission.

567
00:37:23,990 --> 00:37:27,160
And the energy and
the trapping potential

568
00:37:27,160 --> 00:37:30,790
includes all the kinetic
energy of the particle, which

569
00:37:30,790 --> 00:37:33,270
includes whatever comes
from [INAUDIBLE] velocity

570
00:37:33,270 --> 00:37:35,329
or from the motion of the atom.

571
00:37:39,550 --> 00:37:49,670
So therefore, we obtain what is
called the sideband spectrum.

572
00:37:53,819 --> 00:37:56,910
I'll just show you
a stick diagram.

573
00:37:56,910 --> 00:38:01,220
Here is the
electronic transition.

574
00:38:01,220 --> 00:38:04,550
And then we have sidebands.

575
00:38:08,380 --> 00:38:10,310
And the spacing
of those sidebands

576
00:38:10,310 --> 00:38:13,640
is nothing else than the
harmonic oscillator frequency.

577
00:38:17,600 --> 00:38:18,225
Any questions?

578
00:38:21,800 --> 00:38:23,620
At that level, I want
you to appreciate

579
00:38:23,620 --> 00:38:27,080
that this is radically different
from Doppler broadening.

580
00:38:27,080 --> 00:38:28,580
There is no Doppler broadening.

581
00:38:28,580 --> 00:38:32,365
By fully quantizing the motion
in the harmonic oscillator,

582
00:38:32,365 --> 00:38:36,210
we obtain a discrete spectrum.

583
00:38:36,210 --> 00:38:38,810
And what I want to show
you is when we calculate

584
00:38:38,810 --> 00:38:42,460
the intensity in the peak
for strong confinement,

585
00:38:42,460 --> 00:38:44,770
almost all of the intensity's
in the central peak.

586
00:38:44,770 --> 00:38:48,020
And therefore, there is
no Doppler broadening.

587
00:38:48,020 --> 00:38:50,350
But I want to later
show you how we

588
00:38:50,350 --> 00:38:53,090
can go from the discrete
spectrum back to the Doppler

589
00:38:53,090 --> 00:38:56,235
broadening which we just
described in free space.

590
00:39:07,719 --> 00:39:09,760
You should say, well, but
the motion, the recall,

591
00:39:09,760 --> 00:39:11,440
it must come in.

592
00:39:11,440 --> 00:39:12,570
Where is it?

593
00:39:12,570 --> 00:39:15,270
Well, it's not in
this stick diagram.

594
00:39:15,270 --> 00:39:19,460
But the sticks are
the only possibilities

595
00:39:19,460 --> 00:39:23,459
for the possible photon
frequencies or photon energies.

596
00:39:23,459 --> 00:39:25,250
But the question is,
what is the amplitude?

597
00:39:25,250 --> 00:39:28,554
What is the probability
that this will happen?

598
00:39:28,554 --> 00:39:30,220
And you already see
where I'm aiming to.

599
00:39:30,220 --> 00:39:32,011
In the limit that we
have many, many sticks

600
00:39:32,011 --> 00:39:34,020
and we are not
resolving the sticks,

601
00:39:34,020 --> 00:39:36,650
we will get back to them in the
standard Doppler broadening.

602
00:39:36,650 --> 00:39:39,770
So the big question is, how
many of those sticks do we have?

603
00:39:39,770 --> 00:39:42,570
Are we in the limit where
things are heavily discrete?

604
00:39:42,570 --> 00:39:43,660
This is our new limit?

605
00:39:43,660 --> 00:39:46,380
Or do we have
many, many of them?

606
00:39:46,380 --> 00:39:48,725
So therefore, the recoil
and the velocity, they

607
00:39:48,725 --> 00:39:53,430
really enter when we
calculate the intensity.

608
00:39:53,430 --> 00:39:57,430
And whatever our formulation
is with Fermi's golden rule,

609
00:39:57,430 --> 00:39:59,755
the rate is proportional
to the relevant matrix

610
00:39:59,755 --> 00:40:01,240
element squared.

611
00:40:03,750 --> 00:40:07,218
And now I want to show you
how we calculate those matrix

612
00:40:07,218 --> 00:40:08,194
element.

613
00:40:08,194 --> 00:40:09,170
You had a question?

614
00:40:09,170 --> 00:40:12,336
AUDIENCE: But is it true
that each of these sticks

615
00:40:12,336 --> 00:40:16,816
has the intrinsic line
width of the atom?

616
00:40:16,816 --> 00:40:17,490
PROFESSOR: Yes.

617
00:40:17,490 --> 00:40:19,013
We come to that
in a few minutes.

618
00:40:21,990 --> 00:40:25,890
I ignore here the
spontaneous broadening just

619
00:40:25,890 --> 00:40:27,580
for pedagogical reasons.

620
00:40:27,580 --> 00:40:30,170
But a little bit
later, I will-- I first

621
00:40:30,170 --> 00:40:32,290
want to sort of discuss
the number of sticks.

622
00:40:32,290 --> 00:40:33,790
Do we have a few?

623
00:40:33,790 --> 00:40:36,386
This is sort of new, then
we have only few sidebands,

624
00:40:36,386 --> 00:40:38,300
and we have the
Mossbauer effect.

625
00:40:38,300 --> 00:40:41,831
If we have many, that's sort
of more the continuum, which

626
00:40:41,831 --> 00:40:44,080
we described with the classical
velocity distribution.

627
00:40:44,080 --> 00:40:45,390
That's my message number one.

628
00:40:45,390 --> 00:40:48,220
But then the next message
is, do we resolve the sticks?

629
00:40:48,220 --> 00:40:50,670
Do we have resolved
sidebands or not?

630
00:40:50,670 --> 00:40:55,410
And for that, the criterion
is, is the natural line widths

631
00:40:55,410 --> 00:40:57,446
large or smaller than
the sideband spacing?

632
00:40:59,730 --> 00:41:01,980
It's not just one parameter;
there are two parameters.

633
00:41:04,982 --> 00:41:06,940
One is which will be the
Lamb-Dicke parameter--

634
00:41:06,940 --> 00:41:08,530
how many sticks do we have?

635
00:41:08,530 --> 00:41:10,780
And the second question is,
do we resolve the sticks?

636
00:41:10,780 --> 00:41:12,821
And you can say there are
four different regimes,

637
00:41:12,821 --> 00:41:16,007
you know-- yes or no for
question one, or yes or no

638
00:41:16,007 --> 00:41:18,810
for questions two.

639
00:41:18,810 --> 00:41:21,350
Other questions?

640
00:41:21,350 --> 00:41:22,850
OK.

641
00:41:22,850 --> 00:41:26,040
So the rate is proportional
to the matrix element squared.

642
00:41:26,040 --> 00:41:30,950
And yes, we have all kind of
the matrix elements involving

643
00:41:30,950 --> 00:41:31,940
the internal degree.

644
00:41:31,940 --> 00:41:35,150
But the new thing is
the matrix element

645
00:41:35,150 --> 00:41:39,710
for the center of mass wave
function of the atom, which

646
00:41:39,710 --> 00:41:42,730
is-- you can see just
the emission where

647
00:41:42,730 --> 00:41:52,040
the polynomials of the
harmonic oscillator.

648
00:41:52,040 --> 00:41:56,890
So we have the eigen functions
for the harmonic oscillator

649
00:41:56,890 --> 00:41:59,540
between initial and final state.

650
00:41:59,540 --> 00:42:04,780
And what acts on the only part
of the electromagnetic field

651
00:42:04,780 --> 00:42:07,450
operator, which acts on
the position of the atom,

652
00:42:07,450 --> 00:42:09,860
is this term, e
to the minus ikr.

653
00:42:13,770 --> 00:42:16,290
OK.

654
00:42:16,290 --> 00:42:21,200
I mentioned already
that the new regime

655
00:42:21,200 --> 00:42:25,040
is that the
confinement is tight.

656
00:42:25,040 --> 00:42:28,200
So let's just look
at this situation.

657
00:42:28,200 --> 00:42:36,080
When kr is much
smaller than 1, then we

658
00:42:36,080 --> 00:42:44,370
can expand this exponential
into 1 minus ikr.

659
00:42:44,370 --> 00:42:47,130
And now I want to remind
you that the position

660
00:42:47,130 --> 00:42:52,170
operator, when we treat
the harmonic oscillator,

661
00:42:52,170 --> 00:42:56,160
is nothing else than
a plus a dagger.

662
00:42:56,160 --> 00:43:01,700
So therefore, if kr is
small and therefore we

663
00:43:01,700 --> 00:43:07,780
can do the first-order expansion
of e to the ikr, our spectrum,

664
00:43:07,780 --> 00:43:13,550
our operator is here-- 1 minus
e [? vector ?] a plus a dagger.

665
00:43:13,550 --> 00:43:20,300
And therefore, the
only possible sidebands

666
00:43:20,300 --> 00:43:23,020
are the one where the
change in harmonic

667
00:43:23,020 --> 00:43:25,630
oscillator quantum number
is plus or minus 1.

668
00:43:28,450 --> 00:43:35,305
So we are already
obtaining the result,

669
00:43:35,305 --> 00:43:39,200
which is called the
Lamb-Dicke limit when

670
00:43:39,200 --> 00:43:44,230
kr is much smaller than 1,
that we have a strong carrier.

671
00:43:44,230 --> 00:43:50,110
We have only sticks-- the
delta n equals plus, minus 1.

672
00:43:50,110 --> 00:43:53,460
And the intensity in
each of those sticks

673
00:43:53,460 --> 00:44:02,290
is actually proportional
to k squared r squared,

674
00:44:02,290 --> 00:44:09,214
which is nothing else than
the square of the extension

675
00:44:09,214 --> 00:44:10,380
of the atomic wave function.

676
00:44:13,740 --> 00:44:16,520
And k is 1 over lambda
bar squared, divided

677
00:44:16,520 --> 00:44:19,880
by the optical
wavelengths we get.

678
00:44:19,880 --> 00:44:24,685
So we see already-- I mean,
without any major formalism

679
00:44:24,685 --> 00:44:27,550
or mathematical
tools-- what happens

680
00:44:27,550 --> 00:44:30,615
in the limit of
tight confinement.

681
00:44:30,615 --> 00:44:33,820
The spectrum of confined
particles is eclectic,

682
00:44:33,820 --> 00:44:36,490
the spontaneous line widths.

683
00:44:36,490 --> 00:44:39,840
A delta function without
any Doppler shift,

684
00:44:39,840 --> 00:44:44,780
without any recoil shift, right
at the resonance frequency,

685
00:44:44,780 --> 00:44:46,220
electronic frequency
of the atom.

686
00:44:48,900 --> 00:44:51,713
The only thing
which is reminiscent

687
00:44:51,713 --> 00:44:53,170
of the motional
degree of freedom

688
00:44:53,170 --> 00:44:55,340
are those small satellites.

689
00:44:55,340 --> 00:44:58,770
But their intensity goes
to 0 with the extension

690
00:44:58,770 --> 00:45:01,570
squared over lambda squared.

691
00:45:01,570 --> 00:45:03,850
So therefore, if you
confine the particle

692
00:45:03,850 --> 00:45:06,790
to less than an
optical wavelength,

693
00:45:06,790 --> 00:45:10,800
the older picture I told
you, it can never spread out

694
00:45:10,800 --> 00:45:12,630
over wavelengths,
can never get out

695
00:45:12,630 --> 00:45:14,740
of coherence with
the drive field.

696
00:45:14,740 --> 00:45:16,840
And here, we have a
quantitative description

697
00:45:16,840 --> 00:45:19,690
that at that moment,
we can obtain

698
00:45:19,690 --> 00:45:23,150
spectroscopic information
about the resonance completely

699
00:45:23,150 --> 00:45:24,550
unperturbed by motional effects.

700
00:45:42,050 --> 00:45:44,711
Questions?

701
00:45:44,711 --> 00:45:45,210
OK.

702
00:45:45,210 --> 00:45:47,350
So that was maybe
the most fun part,

703
00:45:47,350 --> 00:45:50,560
the extreme limit and
you realize what happens.

704
00:45:50,560 --> 00:45:57,980
But now we want to sort
of fill in the gaps.

705
00:45:57,980 --> 00:46:00,637
I first want to sort
of contrast what I just

706
00:46:00,637 --> 00:46:02,595
described to you with a
semi-classical picture.

707
00:46:09,040 --> 00:46:11,770
The semi-classical picture-- if
you have an emitting oscillator

708
00:46:11,770 --> 00:46:17,860
or absorber and we have
electromagnetic plane wave,

709
00:46:17,860 --> 00:46:24,910
we can now ask, what is
the phase of the plane wave

710
00:46:24,910 --> 00:46:27,820
experienced by the atom?

711
00:46:27,820 --> 00:46:31,950
And of course, the
phase is affected

712
00:46:31,950 --> 00:46:35,184
by the motion of the
atom if we assume

713
00:46:35,184 --> 00:46:39,630
the atom moves [INAUDIBLE]
with harmonic frequency omega

714
00:46:39,630 --> 00:46:42,000
t and an amplitude x0.

715
00:46:42,000 --> 00:46:47,475
Then this here is the
phase seen by the atom

716
00:46:47,475 --> 00:46:50,540
in its own reference frame.

717
00:46:50,540 --> 00:46:55,970
And if I then define-- with
another quotation mark,

718
00:46:55,970 --> 00:47:00,900
because it's sort of something
which needs explanation.

719
00:47:00,900 --> 00:47:04,660
If I define an instantaneous
frequency, which

720
00:47:04,660 --> 00:47:07,761
is nothing else than the
derivative of the phase,

721
00:47:07,761 --> 00:47:12,130
then I'll retrieve the
normal Doppler broadening.

722
00:47:12,130 --> 00:47:14,438
So now you see where sort
of normal Doppler broadening

723
00:47:14,438 --> 00:47:16,270
would come in.

724
00:47:16,270 --> 00:47:19,563
But the question is, you
cannot measure an instantaneous

725
00:47:19,563 --> 00:47:20,250
frequency.

726
00:47:20,250 --> 00:47:23,450
It would violate
the Fourier theorem.

727
00:47:23,450 --> 00:47:28,470
But if we can apply-- if the
atom oscillates slowly enough--

728
00:47:28,470 --> 00:47:33,200
the motion is slow enough
that we can apply the concept

729
00:47:33,200 --> 00:47:36,910
that we can look at the
frequency the atom experiences,

730
00:47:36,910 --> 00:47:40,650
you would at least say, before
the atom changes its velocity,

731
00:47:40,650 --> 00:47:42,645
it should see a few cycles.

732
00:47:42,645 --> 00:47:46,960
If it oscillates fairly fast,
this concept, of course,

733
00:47:46,960 --> 00:47:48,070
cannot be applied.

734
00:47:48,070 --> 00:47:52,020
But at least you see where your
normal Doppler shift comes in.

735
00:47:52,020 --> 00:47:56,030
It comes in in the concept of
an instantaneous frequency.

736
00:47:56,030 --> 00:47:59,020
Of course, what we
should do now is

737
00:47:59,020 --> 00:48:02,598
we should not take the concept
of the instantaneous frequency.

738
00:48:02,598 --> 00:48:04,988
We should do rather
do it correctly.

739
00:48:04,988 --> 00:48:08,340
What we have is we see we
have the phase of the atom.

740
00:48:08,340 --> 00:48:13,240
So therefore, we
should-- by using

741
00:48:13,240 --> 00:48:15,736
the motion of the atom
for x, we should have

742
00:48:15,736 --> 00:48:18,470
formed Fourier transform.

743
00:48:18,470 --> 00:48:20,710
The Fourier transform
tells us what

744
00:48:20,710 --> 00:48:23,020
is the tonal spectrum
which the atom experiences.

745
00:48:25,590 --> 00:48:29,702
So therefore, we
take this plane wave,

746
00:48:29,702 --> 00:48:32,770
we put in the oscillatory
behavior of the atom,

747
00:48:32,770 --> 00:48:34,684
and then we take away transform.

748
00:48:41,145 --> 00:48:41,650
OK.

749
00:48:41,650 --> 00:48:46,430
So the electromagnetic
field seen by the atom

750
00:48:46,430 --> 00:48:52,530
is an amplitude which is
the cosine of the phase.

751
00:48:52,530 --> 00:48:56,920
And the phase has a
temporal dependence

752
00:48:56,920 --> 00:49:00,040
that's just the frequency
of the plane wave.

753
00:49:00,040 --> 00:49:03,500
But now the precision
involves the oscillation

754
00:49:03,500 --> 00:49:07,800
of the atom at the
[INAUDIBLE] frequency omega t.

755
00:49:07,800 --> 00:49:13,069
And what I've introduced
here is the data [? vector ?]

756
00:49:13,069 --> 00:49:14,360
is called the modulation index.

757
00:49:14,360 --> 00:49:16,550
This is the relevant quantity.

758
00:49:16,550 --> 00:49:20,480
It is k, which comes from
the plane wave, e to the ikr,

759
00:49:20,480 --> 00:49:22,465
times the amplitude of the atom.

760
00:49:22,465 --> 00:49:26,310
And so the modulation
index is nothing else

761
00:49:26,310 --> 00:49:34,170
than the amplitude of the atomic
motion divided by the reduced

762
00:49:34,170 --> 00:49:35,230
wavelengths of the light.

763
00:49:37,870 --> 00:49:42,400
Just if you remember for a
second, this extreme stick

764
00:49:42,400 --> 00:49:45,720
diagram where we had one big
stick and two smaller sticks,

765
00:49:45,720 --> 00:49:48,690
remember, the intensity
in the smaller sticks

766
00:49:48,690 --> 00:49:51,460
was the extension
of the atom divided

767
00:49:51,460 --> 00:49:53,310
by the wavelength squared.

768
00:49:53,310 --> 00:49:55,230
So we had exactly
the same parameter,

769
00:49:55,230 --> 00:49:59,180
but the previous description
assumed a quantized picture

770
00:49:59,180 --> 00:50:01,010
for the harmonic oscillator.

771
00:50:01,010 --> 00:50:04,120
I really used a and a daggers
in the atomic wave function.

772
00:50:04,120 --> 00:50:06,210
This is a semi-classical
picture where

773
00:50:06,210 --> 00:50:08,410
I treat the
oscillation of the atom

774
00:50:08,410 --> 00:50:10,520
in a classical description.

775
00:50:10,520 --> 00:50:15,410
But in both cases, of course,
the relevant parameter

776
00:50:15,410 --> 00:50:18,722
is the ratio over
which the atom moves--

777
00:50:18,722 --> 00:50:20,450
the amplitude of
the atomic motion

778
00:50:20,450 --> 00:50:23,860
or the size of the atomic
wave function in relation

779
00:50:23,860 --> 00:50:27,651
to the wavelengths of the light.

780
00:50:27,651 --> 00:50:28,150
OK.

781
00:50:28,150 --> 00:50:31,420
So we want to Fourier
transform this function.

782
00:50:31,420 --> 00:50:32,950
And this gives us the spectrum.

783
00:50:38,385 --> 00:50:40,005
I can show you
that the result is

784
00:50:40,005 --> 00:50:52,730
that we can have a Fourier
expansion of this amplitude,

785
00:50:52,730 --> 00:50:54,240
and it involves basal functions.

786
00:50:54,240 --> 00:50:59,160
So it is the basal
function Jm which tells us

787
00:50:59,160 --> 00:51:03,610
what the intensity is
in the n's sideband.

788
00:51:03,610 --> 00:51:06,420
And the argument of
the basal function,

789
00:51:06,420 --> 00:51:08,550
whether we take the basal
function at the origin

790
00:51:08,550 --> 00:51:16,024
or at a finite argument, is
given by beta modulation index.

791
00:51:16,024 --> 00:51:18,545
You can't immediately
[INAUDIBLE] in just two lines.

792
00:51:18,545 --> 00:51:20,960
We write the Fourier transform.

793
00:51:20,960 --> 00:51:24,755
All you have to use
is those identities

794
00:51:24,755 --> 00:51:28,671
which involve the
basal function.

795
00:51:28,671 --> 00:51:31,170
Whenever you have the cosine
of a sine or the sine of a sine

796
00:51:31,170 --> 00:51:34,960
and you Fourier
transform, you get,

797
00:51:34,960 --> 00:51:37,312
naturally, basal functions.

798
00:51:41,431 --> 00:51:41,930
OK.

799
00:51:41,930 --> 00:51:45,320
So let's just look
at the result.

800
00:51:45,320 --> 00:51:48,590
We talked about
the stick diagram.

801
00:51:48,590 --> 00:51:52,060
The stick diagram was actually
motivated quantum mechanically.

802
00:51:52,060 --> 00:51:56,585
But now we also find a stick
diagram purely classically

803
00:51:56,585 --> 00:52:00,630
because of the periodic
motion in the harmonic trap.

804
00:52:00,630 --> 00:52:04,010
But now what we obtain in
the semi-classical limit,

805
00:52:04,010 --> 00:52:07,850
we obtain what is the
height of each stick.

806
00:52:07,850 --> 00:52:11,960
Well, the height of the stick,
which is n sidebands away,

807
00:52:11,960 --> 00:52:15,590
is given by the square of
the n's basal function.

808
00:52:15,590 --> 00:52:18,690
And beta is the
modulation index.

809
00:52:18,690 --> 00:52:22,570
And if beta goes
to 0, that means

810
00:52:22,570 --> 00:52:24,330
the atom is not moving at all.

811
00:52:24,330 --> 00:52:26,130
The amplitude x0 is 0.

812
00:52:26,130 --> 00:52:28,770
This is the limit of
tight confinement.

813
00:52:28,770 --> 00:52:33,730
Then all basal functions
are 0 except the 0's order.

814
00:52:33,730 --> 00:52:36,870
And that means we are back
to a single delta function,

815
00:52:36,870 --> 00:52:39,023
a single stick in
our stick diagram.

816
00:52:55,320 --> 00:53:00,420
So let's now take the
result and discuss it.

817
00:53:05,890 --> 00:53:09,280
So how does the
spectrum look like?

818
00:53:09,280 --> 00:53:15,300
Well, I want to give you
two limits where the atom is

819
00:53:15,300 --> 00:53:17,050
extended much larger
than the wavelengths.

820
00:53:17,050 --> 00:53:21,940
This means large modulation
index beta and small modulation

821
00:53:21,940 --> 00:53:25,510
index beta.

822
00:53:25,510 --> 00:53:52,750
But first, in the limit where
beta is large-- for large beta,

823
00:53:52,750 --> 00:53:59,150
the argument of
the basal function

824
00:53:59,150 --> 00:54:11,235
becomes just cosine
beta minus phase.

825
00:54:15,490 --> 00:54:36,870
So if we use that-- if we assume
a thermal distribution where

826
00:54:36,870 --> 00:54:43,380
beta over kt d beta squared.

827
00:54:43,380 --> 00:54:46,240
So we take our spectrum,
and we just convolute it

828
00:54:46,240 --> 00:54:51,650
with a thermal distribution
of amplitudes x0, which

829
00:54:51,650 --> 00:54:54,850
is, of course, just
the Boltzmann factor.

830
00:54:54,850 --> 00:55:04,660
And in that limit, we
actually obtain a spectrum

831
00:55:04,660 --> 00:55:11,180
where the envelope of the sticks
looks like the Doppler width.

832
00:55:11,180 --> 00:55:16,330
So if those two
conditions are fulfilled,

833
00:55:16,330 --> 00:55:21,002
then we obtain the
normal Doppler widths.

834
00:55:21,002 --> 00:55:26,420
If, in addition, we assume
that the sidebands are not

835
00:55:26,420 --> 00:55:28,700
resolved, because the
natural line width

836
00:55:28,700 --> 00:55:31,090
is larger than the spacing.

837
00:55:31,090 --> 00:55:33,390
So if we assume
that in addition,

838
00:55:33,390 --> 00:55:35,060
then we just find
the normal Doppler

839
00:55:35,060 --> 00:55:45,107
broadening how we have
derived it as in free space.

840
00:55:48,000 --> 00:55:50,780
So normal Doppler
broadening is the limit

841
00:55:50,780 --> 00:55:54,404
of the large modulation
index, a thermal distribution

842
00:55:54,404 --> 00:55:59,240
of modulation index in the
case of not resolved sidebands.

843
00:56:05,110 --> 00:56:08,460
The opposite limit,
of course, is

844
00:56:08,460 --> 00:56:15,350
when beta is much
smaller than 1,

845
00:56:15,350 --> 00:56:19,150
it means the amplitude of
the oscillating particle

846
00:56:19,150 --> 00:56:20,960
is smaller than the wavelengths.

847
00:56:23,530 --> 00:56:27,050
I mentioned it a few times,
so I should write it down.

848
00:56:27,050 --> 00:56:30,320
This is the Lamb-Dicke regime.

849
00:56:30,320 --> 00:56:37,980
And then the relevant
approximation

850
00:56:37,980 --> 00:56:43,896
for the basal function
is in the limit.

851
00:56:43,896 --> 00:56:52,450
And this limit is 1 over n
factorial times beta over 2 n.

852
00:56:52,450 --> 00:56:58,620
And for the case that n
is 1, this is just beta.

853
00:56:58,620 --> 00:57:02,050
And the amplitude
squared is beta squared.

854
00:57:02,050 --> 00:57:04,650
It's x0 over lambda
squared, as we

855
00:57:04,650 --> 00:57:08,480
had discussed before in
the quantum mechanic unit.

856
00:57:08,480 --> 00:57:10,940
So you'll see that
to use matrix element

857
00:57:10,940 --> 00:57:15,430
for the harmonic oscillator or
use a semi-classical Fourier

858
00:57:15,430 --> 00:57:18,170
transformed with a basal
function, both lead

859
00:57:18,170 --> 00:57:21,080
to the same result, that
deep in the Lamb-Dicke limit,

860
00:57:21,080 --> 00:57:25,250
we have essentially three peaks.

861
00:57:30,380 --> 00:57:34,280
And the satellites
are quadratically

862
00:57:34,280 --> 00:57:38,062
small in the modulation
index squared.

863
00:57:41,930 --> 00:57:52,390
So this regime is particularly
interesting for atomic locks

864
00:57:52,390 --> 00:57:55,010
and for meteorological
applications

865
00:57:55,010 --> 00:57:56,720
if the sidebands are resolved.

866
00:57:59,950 --> 00:58:01,540
If the sidebands
are not resolved,

867
00:58:01,540 --> 00:58:04,880
you have sort of a line
shape which depends,

868
00:58:04,880 --> 00:58:06,885
let's say, on temperature,
because x0 squared,

869
00:58:06,885 --> 00:58:11,970
if it's the thermal excitation,
is proportional to temperature.

870
00:58:11,970 --> 00:58:14,690
So as you cool it
down, you will actually

871
00:58:14,690 --> 00:58:16,940
see that your line
shape changes.

872
00:58:16,940 --> 00:58:23,609
But the good thing is that
once you resolve the sidebands,

873
00:58:23,609 --> 00:58:25,150
the line shape of
the central carrier

874
00:58:25,150 --> 00:58:27,090
does not change anymore.

875
00:58:27,090 --> 00:58:29,390
And once you can
resolve the sidebands,

876
00:58:29,390 --> 00:58:31,640
you can just observe
the central carrier,

877
00:58:31,640 --> 00:58:34,430
and you obtain
spectroscopic information

878
00:58:34,430 --> 00:58:37,673
which is no longer blurred,
which is no longer affected

879
00:58:37,673 --> 00:58:39,770
by motion or by temperature.

880
00:58:39,770 --> 00:58:43,970
And that's, of course, a regime
where the atomic frequency

881
00:58:43,970 --> 00:58:45,844
standards want to operate.

882
00:58:51,740 --> 00:58:52,240
Questions?

883
00:59:07,550 --> 00:59:08,050
OK.

884
00:59:10,942 --> 00:59:12,388
Yes.

885
00:59:12,388 --> 00:59:15,930
Let me just write that down,
because this is important.

886
00:59:15,930 --> 00:59:34,420
For resolved sidebands,
you have sharp lines,

887
00:59:34,420 --> 00:59:35,720
no motional broadening.

888
00:59:45,460 --> 00:59:47,580
And the physics I
described to you

889
00:59:47,580 --> 00:59:53,610
is actually analogous
to the Mossbauer effect.

890
01:00:00,894 --> 01:00:05,580
In the Mossbauer effect, the
intensity of the recoil-less

891
01:00:05,580 --> 01:00:09,580
emission of this recoil-less
line is described

892
01:00:09,580 --> 01:00:16,060
by the [INAUDIBLE]
[? vector. ?] So in this case,

893
01:00:16,060 --> 01:00:21,076
the [INAUDIBLE] [? vector ?]
is 1 minus the probability

894
01:00:21,076 --> 01:00:22,840
for the two sidebands.

895
01:00:22,840 --> 01:00:26,170
So the same concept of
Mossbauer line and [INAUDIBLE]

896
01:00:26,170 --> 01:00:31,950
[? vector ?]
describes the physics

897
01:00:31,950 --> 01:00:33,294
for tightly confined particles.

898
01:00:43,020 --> 01:00:48,250
So if I use the analogy
to the Mossbauer effect,

899
01:00:48,250 --> 01:00:52,010
the Mossbauer effect is called
the recoil-less emission

900
01:00:52,010 --> 01:00:54,190
and absorption of x-rays.

901
01:00:54,190 --> 01:01:04,310
So what we have here is we
have a recoil-less absorption

902
01:01:04,310 --> 01:01:06,284
and emission of photons.

903
01:01:09,630 --> 01:01:11,850
Of course, the photon which
is emitted and absorbed

904
01:01:11,850 --> 01:01:13,160
has momentum.

905
01:01:13,160 --> 01:01:15,040
There should be momentum recoil.

906
01:01:15,040 --> 01:01:18,140
So the question is, where
does the momentum recoil

907
01:01:18,140 --> 01:01:23,600
go when the confined
particle emits or absorbs

908
01:01:23,600 --> 01:01:26,310
on the carrier of
the central line.

909
01:01:26,310 --> 01:01:31,620
How do we reconcile
the result I derived

910
01:01:31,620 --> 01:01:33,928
for you with momentum
conservation?

911
01:01:42,680 --> 01:01:45,000
A trapped particle,
tightly confined.

912
01:01:45,000 --> 01:01:48,270
And photon comes, has
momentum, the atom absorbs it.

913
01:01:52,380 --> 01:01:54,200
But the spectrum does
not show any evidence

914
01:01:54,200 --> 01:01:55,712
for recoil shifts and such.

915
01:02:01,465 --> 01:02:01,965
Colin.

916
01:02:01,965 --> 01:02:03,923
AUDIENCE: Must be absorbed
by the trap somehow.

917
01:02:03,923 --> 01:02:06,930
PROFESSOR: Must be
absorbed by the trap, yes.

918
01:02:06,930 --> 01:02:09,380
Your trap is anchored
in the laboratory.

919
01:02:09,380 --> 01:02:11,200
And you transfer the momentum.

920
01:02:11,200 --> 01:02:12,990
The atom is attached
to the trap.

921
01:02:12,990 --> 01:02:14,830
The trap is attached
to the laboratory.

922
01:02:14,830 --> 01:02:19,020
So the object which
takes over the momentum

923
01:02:19,020 --> 01:02:21,290
is your whole apparatus
or, in the extreme case,

924
01:02:21,290 --> 01:02:23,110
a whole building.

925
01:02:23,110 --> 01:02:26,140
And there is, of
course, a kinetic energy

926
01:02:26,140 --> 01:02:29,820
associated-- momentum
squared over 2 m.

927
01:02:29,820 --> 01:02:32,300
But the mass is now the
mass of the building.

928
01:02:32,300 --> 01:02:35,340
So therefore, there is
no energy associated

929
01:02:35,340 --> 01:02:38,180
with absorbing the recoil.

930
01:02:38,180 --> 01:02:42,260
So it is as if-- and this comes
back to some earlier remarks

931
01:02:42,260 --> 01:02:45,280
I've made-- is
that when you have

932
01:02:45,280 --> 01:02:48,850
this absorption in
the Lamb-Dicke regime

933
01:02:48,850 --> 01:02:51,170
is it is as if your
two-level system has

934
01:02:51,170 --> 01:02:53,850
an infinite mass behind it.

935
01:02:53,850 --> 01:02:56,530
And that's sort of the situation
how I told you you should often

936
01:02:56,530 --> 01:02:58,680
think about it,
you should separate

937
01:02:58,680 --> 01:03:00,890
effects of the internal
degree of freedom

938
01:03:00,890 --> 01:03:02,690
and the external
degree of freedom

939
01:03:02,690 --> 01:03:05,020
by just assuming
I can assume first

940
01:03:05,020 --> 01:03:06,980
that the atom has infinite mass.

941
01:03:06,980 --> 01:03:08,950
And once the atom
has infinite mass,

942
01:03:08,950 --> 01:03:11,650
the motion degree of
freedom doesn't matter.

943
01:03:11,650 --> 01:03:14,090
What I just explained to
you is a practical way

944
01:03:14,090 --> 01:03:17,570
to endow your particle
with infinite mass.

945
01:03:17,570 --> 01:03:21,690
Just connect it with type
confinement to your apparatus.

946
01:03:21,690 --> 01:03:24,160
And then for the
momentum exchange,

947
01:03:24,160 --> 01:03:26,860
there is a photon
field, it is actually

948
01:03:26,860 --> 01:03:30,550
the mass of the whole
apparatus [INAUDIBLE].

949
01:03:30,550 --> 01:03:31,240
Questions?

950
01:03:31,240 --> 01:03:33,039
AUDIENCE: Is there
a more direct way

951
01:03:33,039 --> 01:03:36,641
to think about this for maybe
the example of a magnetic trap,

952
01:03:36,641 --> 01:03:43,024
the mechanism by which this
momentum is transferred?

953
01:03:43,024 --> 01:03:45,479
Or absorbed.

954
01:03:45,479 --> 01:03:46,590
PROFESSOR: OK.

955
01:03:46,590 --> 01:03:48,530
Magnetic traps for
neutral particles

956
01:03:48,530 --> 01:03:49,660
are usually not in
the Lamb-Dicke regime,

957
01:03:49,660 --> 01:03:51,285
so you have to be a
little bit careful.

958
01:03:58,880 --> 01:04:00,790
But the explanation
would be the same.

959
01:04:00,790 --> 01:04:05,840
The magnetic fields
are like tight springs

960
01:04:05,840 --> 01:04:10,050
which connect the neutral
atom to your coins.

961
01:04:10,050 --> 01:04:12,810
And therefore, you
should just think

962
01:04:12,810 --> 01:04:16,430
about the magnetic trap
in a mechanical model.

963
01:04:16,430 --> 01:04:19,430
Your neutral atom is connected
to your coils with strings.

964
01:04:19,430 --> 01:04:24,670
And if you now hit the
atom with a photon,

965
01:04:24,670 --> 01:04:28,200
because of the quantization
of the discreteness

966
01:04:28,200 --> 01:04:31,540
of this specturm, the
photon, in most cases,

967
01:04:31,540 --> 01:04:34,240
does not have enough
recoil to create

968
01:04:34,240 --> 01:04:38,445
a mechanical oscillation
of your particle.

969
01:04:38,445 --> 01:04:41,010
The momentum goes
through the sphinx

970
01:04:41,010 --> 01:04:42,660
to the support structure.

971
01:04:42,660 --> 01:04:45,870
But occasionally,
with a probability

972
01:04:45,870 --> 01:04:48,410
which is a modulation
index squared,

973
01:04:48,410 --> 01:04:50,910
you will actually
promote the particle

974
01:04:50,910 --> 01:04:53,410
to the first state
of harmonic motion.

975
01:04:53,410 --> 01:04:55,680
And then the atom has
acquired some kinetic energy.

976
01:04:55,680 --> 01:04:57,830
But this probability
can be made as small

977
01:04:57,830 --> 01:05:01,266
as you want by going to a
smaller and smaller modulation

978
01:05:01,266 --> 01:05:03,790
index.

979
01:05:03,790 --> 01:05:05,574
That's the way how I
would think about it.

980
01:05:08,510 --> 01:05:09,215
Nancy?

981
01:05:09,215 --> 01:05:11,690
AUDIENCE: About
the multiple lines

982
01:05:11,690 --> 01:05:17,135
that we get, I was wondering
if the levels of the harmonic

983
01:05:17,135 --> 01:05:21,590
traps itself are blurred,
which could happen in the lab

984
01:05:21,590 --> 01:05:26,045
if the trap depth is
moving, for example--

985
01:05:26,045 --> 01:05:28,025
the levels of the trap
would get blurred.

986
01:05:28,025 --> 01:05:30,500
Would that result in
additional broadening of this?

987
01:05:30,500 --> 01:05:32,480
Or how would that affect?

988
01:05:38,915 --> 01:05:40,290
PROFESSOR: It depends.

989
01:05:40,290 --> 01:05:44,420
I think if you have some
temporal broadening, you know,

990
01:05:44,420 --> 01:05:47,200
you just plug it into
your correlation function,

991
01:05:47,200 --> 01:05:49,690
whatever shakes your system.

992
01:05:49,690 --> 01:05:52,235
If that means the
atom sees some shaking

993
01:05:52,235 --> 01:05:54,812
in the phase of the
electromagnetic field,

994
01:05:54,812 --> 01:05:58,040
it affects it.

995
01:05:58,040 --> 01:06:00,980
If you have an ensemble of atoms
or if you do the experiment

996
01:06:00,980 --> 01:06:03,500
repeatedly, and your
measurements are the ensemble,

997
01:06:03,500 --> 01:06:05,940
and every time you
do the measurement,

998
01:06:05,940 --> 01:06:09,426
your magnetic trap has a
slightly different [INAUDIBLE],

999
01:06:09,426 --> 01:06:12,540
and therefore, a slightly
different confinement, well,

1000
01:06:12,540 --> 01:06:15,686
what you would see is that
those sidebands fluctuate,

1001
01:06:15,686 --> 01:06:19,462
that the carrier is independent
of the trap frequency.

1002
01:06:19,462 --> 01:06:22,430
So therefore, the carrier is
actually the central peak--

1003
01:06:22,430 --> 01:06:25,150
would not be broadened
by fluctuations

1004
01:06:25,150 --> 01:06:26,430
in the harmonic confinement.

1005
01:06:33,440 --> 01:06:36,240
Let me just make one comment.

1006
01:06:36,240 --> 01:06:39,330
I derived the result for
you-- at least that one--

1007
01:06:39,330 --> 01:06:44,080
by taking the amplitude of the
atomic motion, the amplitude

1008
01:06:44,080 --> 01:06:46,940
of the phase, and doing
the Fourier analysis.

1009
01:06:46,940 --> 01:06:48,990
I mean, this is
exactly what we learned

1010
01:06:48,990 --> 01:06:53,260
from the formalism of
correlation function.

1011
01:06:53,260 --> 01:06:55,620
You should take the amplitude
of the perturbing field

1012
01:06:55,620 --> 01:06:56,940
and fully analyze it.

1013
01:06:56,940 --> 01:06:59,150
I didn't phrase it
here in the language

1014
01:06:59,150 --> 01:07:00,220
of correlation functions.

1015
01:07:00,220 --> 01:07:04,248
But what I did was exactly what
we learned from the correlation

1016
01:07:04,248 --> 01:07:05,246
function formalism.

1017
01:07:14,727 --> 01:07:15,230
OK.

1018
01:07:15,230 --> 01:07:19,110
Let me maybe summarize
in words what we learned.

1019
01:07:19,110 --> 01:07:21,500
So what we learned
from this discussion

1020
01:07:21,500 --> 01:07:25,980
is that what matters for
line broadening and obtaining

1021
01:07:25,980 --> 01:07:29,130
spectroscopic information
is the accumulated phase,

1022
01:07:29,130 --> 01:07:32,715
the phase which the
atom accumulates.

1023
01:07:32,715 --> 01:07:35,250
And if different
atoms in the ensemble

1024
01:07:35,250 --> 01:07:40,330
accumulate a phase which
is different by 2 pi,

1025
01:07:40,330 --> 01:07:43,060
at that moment, we
have reached what

1026
01:07:43,060 --> 01:07:47,450
we call the coherence time
of the correlation function.

1027
01:07:47,450 --> 01:07:51,270
And the inverse of this
time is a line broadening.

1028
01:07:51,270 --> 01:07:54,850
But now we also discussed the
case of tight confinement.

1029
01:07:54,850 --> 01:07:58,320
The atom can be in
very, very rapid motion.

1030
01:07:58,320 --> 01:08:01,760
And the phi dot,
the change of phase

1031
01:08:01,760 --> 01:08:04,810
can be very rapid due to
the instantaneous velocity.

1032
01:08:04,810 --> 01:08:07,520
But if the atom turns
around because it's

1033
01:08:07,520 --> 01:08:11,060
in a singulatory motion,
positive and negative Doppler

1034
01:08:11,060 --> 01:08:15,105
shifts completely
cancel, because you never

1035
01:08:15,105 --> 01:08:20,420
allow the atom, in this periodic
motion, to acquire a net phase.

1036
01:08:20,420 --> 01:08:26,510
And therefore, the motional
broadening is absent.

1037
01:08:26,510 --> 01:08:28,950
So one way to think
about this carrier

1038
01:08:28,950 --> 01:08:32,765
is that the atom rapidly
oscillates through plus kv

1039
01:08:32,765 --> 01:08:35,240
and minus kv Doppler shifts.

1040
01:08:35,240 --> 01:08:38,140
And the two cancel.

1041
01:08:38,140 --> 01:08:40,248
So this is the reason for that.

1042
01:08:50,130 --> 01:08:50,630
OK.

1043
01:08:50,630 --> 01:08:53,898
I think we are now very well
prepared for Dicke narrowing.

1044
01:09:09,029 --> 01:09:12,710
Actually, I have to say, it's
the first class this semester

1045
01:09:12,710 --> 01:09:16,620
that I was teaching a little
bit faster than I assumed.

1046
01:09:16,620 --> 01:09:19,750
So I'm now right at the end
what I prepared for today.

1047
01:09:19,750 --> 01:09:21,770
But I know my notes
sufficiently well

1048
01:09:21,770 --> 01:09:24,380
that I can go on for 10 minutes.

1049
01:09:24,380 --> 01:09:26,640
So the Dicke narrowing--
the last time I looked at it

1050
01:09:26,640 --> 01:09:27,859
was two years ago.

1051
01:09:30,550 --> 01:09:32,750
But let's get the
physical picture.

1052
01:09:32,750 --> 01:09:38,500
So I want to now apply
what we have learned

1053
01:09:38,500 --> 01:09:43,189
not to a trapped atom,
but to an atom which

1054
01:09:43,189 --> 01:09:45,240
is embedded in buffer gas.

1055
01:09:45,240 --> 01:09:48,790
So just think one
rubidium, or one sodium,

1056
01:09:48,790 --> 01:09:50,600
or one lithium atom.

1057
01:09:50,600 --> 01:09:55,710
And it is surrounded by a
buffer gas of argon or neon.

1058
01:09:55,710 --> 01:10:00,000
And I know sometimes when we
do saturation spectroscopy

1059
01:10:00,000 --> 01:10:03,340
to stabilize our lasers, we
have a little glass cell, which

1060
01:10:03,340 --> 01:10:05,450
has sodium or rubidium in it.

1061
01:10:05,450 --> 01:10:08,336
But we also put an argon
buffer gas into it.

1062
01:10:08,336 --> 01:10:10,550
So that's the situation
I want to describe now.

1063
01:10:10,550 --> 01:10:12,520
But Colin, you had a question.

1064
01:10:12,520 --> 01:10:13,520
AUDIENCE: Yeah.

1065
01:10:13,520 --> 01:10:16,246
You wrote down the condition
for resolving the sidebands

1066
01:10:16,246 --> 01:10:23,394
as your trapped frequency being
larger than your natural line

1067
01:10:23,394 --> 01:10:24,640
width.

1068
01:10:24,640 --> 01:10:27,890
An alkalizer-- if you had,
say it were 10 megahertz line

1069
01:10:27,890 --> 01:10:28,390
width.

1070
01:10:28,390 --> 01:10:32,756
How do you actually resolve
this without a ridiculously--

1071
01:10:32,756 --> 01:10:35,732
because people do
sideband [INAUDIBLE],

1072
01:10:35,732 --> 01:10:39,204
and they don't have 10 megahertz
trap frequencies, do they?

1073
01:10:39,204 --> 01:10:40,320
PROFESSOR: OK.

1074
01:10:40,320 --> 01:10:41,460
So good question.

1075
01:10:41,460 --> 01:10:43,640
The question is, the
resolved sideband

1076
01:10:43,640 --> 01:10:47,924
limit, how can we reach it?

1077
01:10:47,924 --> 01:10:49,610
Well, it can be
reached in ion traps.

1078
01:10:49,610 --> 01:10:53,910
In ion traps, because you can
put kilovolts on electrodes,

1079
01:10:53,910 --> 01:10:56,165
you can really create harmonic
oscillator frequencies

1080
01:10:56,165 --> 01:10:58,190
which are many, many megahertz.

1081
01:10:58,190 --> 01:11:00,350
And then you are at the
resolved sideband limit,

1082
01:11:00,350 --> 01:11:03,164
assuming that the natural
lifetime, if that's the case,

1083
01:11:03,164 --> 01:11:05,912
of many ions is in the
megahertz [INAUDIBLE].

1084
01:11:05,912 --> 01:11:08,036
For neutral atoms,
it looks like, you

1085
01:11:08,036 --> 01:11:11,240
know, mission impossible.

1086
01:11:11,240 --> 01:11:14,990
However, there's a way out,
and this is the following.

1087
01:11:14,990 --> 01:11:18,330
One is you can maybe [INAUDIBLE]
one-dimensional optical latice.

1088
01:11:18,330 --> 01:11:20,700
So in the latices, you at
least have tight confinement

1089
01:11:20,700 --> 01:11:23,390
of many, many kilohertz.

1090
01:11:23,390 --> 01:11:28,260
But now you want to use
a very narrow transition.

1091
01:11:28,260 --> 01:11:31,070
If you use a very,
very narrow transition,

1092
01:11:31,070 --> 01:11:35,910
then even for 10 kilohertz
external harmonic oscillator

1093
01:11:35,910 --> 01:11:40,240
potential, [INAUDIBLE]
resolved sideband [INAUDIBLE].

1094
01:11:40,240 --> 01:11:44,310
Now, for alkalis, you
won't find an excited state

1095
01:11:44,310 --> 01:11:47,035
which has a natural line
width of 10 kilohertz.

1096
01:11:49,630 --> 01:11:52,020
And this will be our
discussion on Wednesday.

1097
01:11:54,600 --> 01:11:56,757
If you use a Raman
transition, we

1098
01:11:56,757 --> 01:11:59,980
go from one count state
with an off-resonant laser

1099
01:11:59,980 --> 01:12:01,640
to another ground
state, [INAUDIBLE]

1100
01:12:01,640 --> 01:12:03,740
will be two photon transitions.

1101
01:12:03,740 --> 01:12:07,980
But those two photons, since
there's no intermediate state,

1102
01:12:07,980 --> 01:12:11,287
can be regarded as, click-click,
you absorb two photons.

1103
01:12:11,287 --> 01:12:12,870
And in some picture--
that's a message

1104
01:12:12,870 --> 01:12:16,810
I will give you next week-- the
equivalent to a single photon.

1105
01:12:16,810 --> 01:12:20,190
So now you have a
two-photon transition,

1106
01:12:20,190 --> 01:12:23,230
which transfers
recall to the atom.

1107
01:12:23,230 --> 01:12:25,792
So the effective wavelengths
of the two-photon transition

1108
01:12:25,792 --> 01:12:29,175
is because you have
twice the photon energy.

1109
01:12:29,175 --> 01:12:31,680
You have two photons involved
which both have a recoil.

1110
01:12:31,680 --> 01:12:36,440
So the effective k vector is two
times the k vector of an atom.

1111
01:12:36,440 --> 01:12:40,070
But the spontaneous line
width is close to 0,

1112
01:12:40,070 --> 01:12:42,420
literally 0, because you
have a Raman transition

1113
01:12:42,420 --> 01:12:46,260
between two common states.

1114
01:12:46,260 --> 01:12:50,130
If you do Raman sideband
cooling of neutral atoms,

1115
01:12:50,130 --> 01:12:51,790
then you reach the
Lamb-Dicke limit,

1116
01:12:51,790 --> 01:12:55,600
you reach the limit
of strong confinement.

1117
01:12:55,600 --> 01:12:57,535
But you need a better
[INAUDIBLE] transition.

1118
01:12:57,535 --> 01:12:59,562
And there are, of
course, some atoms

1119
01:12:59,562 --> 01:13:01,540
which have a very
narrow clock transition.

1120
01:13:01,540 --> 01:13:04,660
But for many, many atoms which
have hyperfine structure,

1121
01:13:04,660 --> 01:13:06,749
you can resolve to
Raman transition.

1122
01:13:10,751 --> 01:13:11,250
OK.

1123
01:13:15,460 --> 01:13:20,104
Let's now talk about
Dicke narrowing.

1124
01:13:23,820 --> 01:13:28,640
So we have a situation that
we have an atom in buffer gas.

1125
01:13:28,640 --> 01:13:33,579
And in most situations, when
you put an atom into buffer gas,

1126
01:13:33,579 --> 01:13:35,495
you get what is called
collisional broadening.

1127
01:13:38,170 --> 01:13:41,500
I will talk about collisional
broadening on Friday.

1128
01:13:41,500 --> 01:13:43,780
Just a reminder, we have
class on Friday but not

1129
01:13:43,780 --> 01:13:44,831
on the following Monday.

1130
01:13:44,831 --> 01:13:47,080
So on Friday, we'll talk
about collisional broadening.

1131
01:13:47,080 --> 01:13:49,406
And I will discuss, for
instance, the model-- an atom

1132
01:13:49,406 --> 01:13:53,624
in the excited state, when it
collides, it gets de-excited.

1133
01:13:53,624 --> 01:13:55,332
And then you have
pretty much a situation

1134
01:13:55,332 --> 01:14:00,150
where you have, in effect,
a shortened lifetime

1135
01:14:00,150 --> 01:14:01,659
of the excited state.

1136
01:14:01,659 --> 01:14:03,200
And what you get is
a Lorentian which

1137
01:14:03,200 --> 01:14:07,766
is broader, which has a width
not of the natural line width,

1138
01:14:07,766 --> 01:14:10,310
but a width which
is [INAUDIBLE].

1139
01:14:10,310 --> 01:14:12,510
But there are
situations-- and that's

1140
01:14:12,510 --> 01:14:15,060
what I want to discuss
here-- that we have

1141
01:14:15,060 --> 01:14:18,630
atoms in a more
benign buffer gas.

1142
01:14:27,110 --> 01:14:30,010
Where we can assume that
this is actually fulfilled,

1143
01:14:30,010 --> 01:14:33,635
that collisions do not change.

1144
01:14:37,310 --> 01:14:40,390
Well, they're not de-exciting
the excited state.

1145
01:14:40,390 --> 01:14:42,905
But they're not even
changing the coherence

1146
01:14:42,905 --> 01:14:45,280
between counter-excited state.

1147
01:14:45,280 --> 01:14:47,720
So the phase evolution,
the internal state,

1148
01:14:47,720 --> 01:14:51,174
grounded excited
state just-- if you

1149
01:14:51,174 --> 01:14:53,340
can assume you have a Bloch
vector which oscillates,

1150
01:14:53,340 --> 01:14:55,940
and the Bloch vector
superposition [INAUDIBLE]

1151
01:14:55,940 --> 01:14:58,350
excited state oscillates
at the natural frequency,

1152
01:14:58,350 --> 01:15:00,600
and this Bloch
vector just rotates,

1153
01:15:00,600 --> 01:15:02,855
it doesn't have a hiccup,
it doesn't change its phase

1154
01:15:02,855 --> 01:15:05,860
when the atom collides
with the buffer gas atom.

1155
01:15:05,860 --> 01:15:09,020
So we assume that we have such
a buffer gas where collisions

1156
01:15:09,020 --> 01:15:11,600
don't change the
internal coherence.

1157
01:15:17,480 --> 01:15:23,420
And by internal coherence,
I mean the phase

1158
01:15:23,420 --> 01:15:31,370
between the ground state
and the excited state.

1159
01:15:31,370 --> 01:15:33,700
So in this situation--
but it's actually

1160
01:15:33,700 --> 01:15:36,320
a very important situation
which has been reached,

1161
01:15:36,320 --> 01:15:37,390
in many cases.

1162
01:15:37,390 --> 01:15:41,060
In this situation,
we have-- thus,

1163
01:15:41,060 --> 01:15:48,100
the buffer gas acts only on the
external motion of the atom.

1164
01:15:48,100 --> 01:15:51,950
And now you can
say, in some way,

1165
01:15:51,950 --> 01:15:54,360
the buffer gas acts like a trap.

1166
01:15:54,360 --> 01:15:56,277
The particle wants
to fly away, but it

1167
01:15:56,277 --> 01:15:58,160
collides with the
buffer gas atom.

1168
01:15:58,160 --> 01:16:01,840
And with a certain probability
or after a few collisions,

1169
01:16:01,840 --> 01:16:03,768
it returns back to the origin.

1170
01:16:12,450 --> 01:16:15,475
However, it's a lousy
trap, because there

1171
01:16:15,475 --> 01:16:17,920
is some randomness
and effusive motion.

1172
01:16:17,920 --> 01:16:21,850
So if you want to
describe it as a trap,

1173
01:16:21,850 --> 01:16:25,906
it would be a trap with a wide
spread of trap frequencies.

1174
01:16:41,201 --> 01:16:41,700
OK.

1175
01:16:45,760 --> 01:16:49,710
So if we use this
picture now, what

1176
01:16:49,710 --> 01:16:52,355
we have learned from
ion traps-- remember,

1177
01:16:52,355 --> 01:16:55,220
we had an ion trap
with a sharp carrier.

1178
01:16:55,220 --> 01:17:00,150
And then we had sidebands
at the trap frequency.

1179
01:17:00,150 --> 01:17:04,350
But if we have sort of
now a lousy trap which

1180
01:17:04,350 --> 01:17:06,745
each realization, each
moment has a different trap

1181
01:17:06,745 --> 01:17:08,760
frequency when
waving all my arms,

1182
01:17:08,760 --> 01:17:11,740
I would say for the other
part of the ensemble,

1183
01:17:11,740 --> 01:17:14,620
we get a carrier, and
we have something else.

1184
01:17:14,620 --> 01:17:19,160
And for another realization,
we have another trap frequency.

1185
01:17:19,160 --> 01:17:23,000
So if I use a little bit
of artistic intuition here,

1186
01:17:23,000 --> 01:17:26,310
I would expect, based
on what we learned

1187
01:17:26,310 --> 01:17:32,896
from the previous discussion,
that in such a buffer gas,

1188
01:17:32,896 --> 01:17:37,490
I would have a sharp
carrier, and then I

1189
01:17:37,490 --> 01:17:40,170
would have sort of
a pedestal, which

1190
01:17:40,170 --> 01:17:43,740
is the envelope of
many trap frequencies.

1191
01:17:43,740 --> 01:17:48,440
And we know sort of that the
envelope of all our sticks--

1192
01:17:48,440 --> 01:17:53,150
this was actually given
by the Doppler effect.

1193
01:17:53,150 --> 01:18:00,865
So what we may expect now
is that in this situation

1194
01:18:00,865 --> 01:18:04,330
with buffer gas, we
get a sharp line.

1195
01:18:04,330 --> 01:18:07,260
And then we have
this broad pedestal,

1196
01:18:07,260 --> 01:18:09,860
which you can think
[INAUDIBLE] intuitive picture

1197
01:18:09,860 --> 01:18:12,720
as smeared outside bands.

1198
01:18:12,720 --> 01:18:16,890
And I may call that
the Doppler pedestal.

1199
01:18:16,890 --> 01:18:22,385
And we would expect--
and I discussed that

1200
01:18:22,385 --> 01:18:23,935
with the basal
function, that there

1201
01:18:23,935 --> 01:18:27,330
is one limit where the
envelope of all those sticks

1202
01:18:27,330 --> 01:18:30,855
eventually looks like a
Doppler-broadened line.

1203
01:18:38,780 --> 01:18:40,200
Anyway, time is over.

1204
01:18:40,200 --> 01:18:44,290
But let me just give an outlook.

1205
01:18:44,290 --> 01:18:46,206
On Friday, what
I want to do is I

1206
01:18:46,206 --> 01:18:48,820
want to calculate the width
of this line with you.

1207
01:18:48,820 --> 01:18:52,510
And remember, all
we have to do is

1208
01:18:52,510 --> 01:18:55,450
we have to calculate the
correlation function.

1209
01:18:55,450 --> 01:18:59,510
Previously, when I derived
for you Doppler broadening,

1210
01:18:59,510 --> 01:19:05,620
the correlation function
was the kx becoming kvt.

1211
01:19:05,620 --> 01:19:09,710
kvt, where x became vt
how much the atom moves

1212
01:19:09,710 --> 01:19:11,500
with the velocity v.

1213
01:19:11,500 --> 01:19:15,960
By simply replacing the
linear motion, V times t,

1214
01:19:15,960 --> 01:19:21,150
with a diffusive model, we
can calculate the line shape.

1215
01:19:21,150 --> 01:19:25,670
And we will actually find
that the central line is not

1216
01:19:25,670 --> 01:19:28,550
infinitely sharp,
but it has a width

1217
01:19:28,550 --> 01:19:30,760
which is given by the
diffusion constant.

1218
01:19:30,760 --> 01:19:32,950
And if the diffusion
constant is very, very small,

1219
01:19:32,950 --> 01:19:34,838
we approach a very sharp line.

1220
01:19:37,780 --> 01:19:42,550
And the final comment is, and
this is called Dicke narrowing.

1221
01:19:42,550 --> 01:19:45,120
It is this
counterintuitive result

1222
01:19:45,120 --> 01:19:49,190
that collisions, if they
have those properties,

1223
01:19:49,190 --> 01:19:50,890
are not broadening the line.

1224
01:19:50,890 --> 01:19:54,650
They actually narrow the line
from the [INAUDIBLE] Doppler

1225
01:19:54,650 --> 01:19:56,790
width to something
which is much sharper.

1226
01:19:56,790 --> 01:20:01,590
And this has been useful for
high precision spectroscopy.

1227
01:20:01,590 --> 01:20:04,065
But I think with the concept
which we discussed today

1228
01:20:04,065 --> 01:20:09,910
of confinement, you realize
why collisions can actually

1229
01:20:09,910 --> 01:20:12,940
reduce the line widths,
namely by preventing

1230
01:20:12,940 --> 01:20:17,924
the atoms from acquiring
random phases with respect

1231
01:20:17,924 --> 01:20:20,620
to the drive field [INAUDIBLE].

1232
01:20:20,620 --> 01:20:23,260
Any questions?

1233
01:20:23,260 --> 01:20:25,960
So see you on Friday
in this other building,

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this other lecture
hall that's just

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been announced on the website.