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00:00:20,690 --> 00:00:24,590
PROFESSOR: So, good afternoon.

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00:00:24,590 --> 00:00:26,500
Today we want to wrap
up our discussion

10
00:00:26,500 --> 00:00:29,400
on two-photon processes.

11
00:00:29,400 --> 00:00:35,400
And just to repeat my motivation
that in almost all cases

12
00:00:35,400 --> 00:00:39,390
when you address atoms,
you do two photon courses

13
00:00:39,390 --> 00:00:42,650
because a photon is scattered.

14
00:00:42,650 --> 00:00:44,920
You may think it's
absorbed and emitted,

15
00:00:44,920 --> 00:00:49,240
but in reality, it is a
two photon process and not

16
00:00:49,240 --> 00:00:51,840
two single photon processes.

17
00:00:51,840 --> 00:00:54,620
So therefore, you should
really pay attention.

18
00:00:54,620 --> 00:00:57,860
If you have any doubts about
some subtleties about how

19
00:00:57,860 --> 00:01:00,700
is light absorbed and
emitted, the correct answer

20
00:01:00,700 --> 00:01:03,640
is always obtained from
the two-photon picture.

21
00:01:03,640 --> 00:01:05,780
Now, I'm using
pre-written slides

22
00:01:05,780 --> 00:01:09,560
because we treat two-photon
absorption in perturbation

23
00:01:09,560 --> 00:01:10,450
theory.

24
00:01:10,450 --> 00:01:14,490
And it is exactly the
same perturbation theory

25
00:01:14,490 --> 00:01:16,330
we've used before.

26
00:01:16,330 --> 00:01:18,730
It's just-- there's
one difference.

27
00:01:18,730 --> 00:01:23,560
Namely, we have two optical
fields at frequency omega 1

28
00:01:23,560 --> 00:01:24,380
and omega 2.

29
00:01:27,160 --> 00:01:32,810
So for the case of two-photon
absorption-- that means

30
00:01:32,810 --> 00:01:40,200
both photons are used or
stacked up to go up in energy.

31
00:01:40,200 --> 00:01:45,730
We derived this result, and
this was the end of lecture

32
00:01:45,730 --> 00:01:47,720
last week.

33
00:01:47,720 --> 00:01:52,380
And what we obtained
in perturbation theory

34
00:01:52,380 --> 00:01:56,070
for the excited
state, it's exactly

35
00:01:56,070 --> 00:01:59,400
the same structure
you have seen before.

36
00:01:59,400 --> 00:02:04,210
But the only difference
is we have now four terms

37
00:02:04,210 --> 00:02:07,540
because we have combinations
of omega 1 and omega 2,

38
00:02:07,540 --> 00:02:11,580
or we can take two photons
out of the same laser beam.

39
00:02:11,580 --> 00:02:13,340
Just sort of a question,
just to sort of

40
00:02:13,340 --> 00:02:17,250
indicate to you
how many terms you

41
00:02:17,250 --> 00:02:20,870
would expect when you do
the most basic light atom

42
00:02:20,870 --> 00:02:22,310
interaction.

43
00:02:22,310 --> 00:02:25,860
Just one photon, a plus a dega.

44
00:02:25,860 --> 00:02:28,660
How many terms do you get if
you write down the Hamiltonian?

45
00:02:32,518 --> 00:02:34,190
No approximation without.

46
00:02:34,190 --> 00:02:35,230
How many do you get?

47
00:02:42,905 --> 00:02:43,530
AUDIENCE: Four.

48
00:02:43,530 --> 00:02:45,310
PROFESSOR: Four.

49
00:02:45,310 --> 00:02:49,720
You have a plus dega for the
electric field, sigma plus plus

50
00:02:49,720 --> 00:02:51,510
sigma minus for the atom.

51
00:02:51,510 --> 00:02:53,500
And then you have
four combinations,

52
00:02:53,500 --> 00:02:56,450
two are co-rotating, two
are counter-rotating.

53
00:02:56,450 --> 00:02:57,100
OK.

54
00:02:57,100 --> 00:02:59,990
What we are doing here is
second order perturbation theory

55
00:02:59,990 --> 00:03:02,330
with two optical fields.

56
00:03:02,330 --> 00:03:05,380
If you would not do any
rotating wave approximation,

57
00:03:05,380 --> 00:03:06,610
how many terms would we get?

58
00:03:20,595 --> 00:03:21,445
AUDIENCE: Eight.

59
00:03:21,445 --> 00:03:22,295
PROFESSOR: Eight?

60
00:03:22,295 --> 00:03:24,420
I think it's
multiplicative because we

61
00:03:24,420 --> 00:03:30,544
have four processes
involving one-- oops.

62
00:03:30,544 --> 00:03:32,490
Now I'm getting confused.

63
00:03:32,490 --> 00:03:35,390
I wanted to say 16, 4 times 4.

64
00:03:35,390 --> 00:03:38,590
But now I would say in
the first step, which

65
00:03:38,590 --> 00:03:41,930
is to the intermediate level,
we have four at frequency one,

66
00:03:41,930 --> 00:03:44,750
four at frequency,
which makes eight.

67
00:03:44,750 --> 00:03:47,480
But then I think in the second
step you get eight more.

68
00:03:47,480 --> 00:03:51,570
So if we don't make an
approximation, we get 64 terms.

69
00:03:51,570 --> 00:03:55,756
But they're just
all combinations,

70
00:03:55,756 --> 00:03:59,020
all combinations of frequencies.

71
00:03:59,020 --> 00:04:01,130
Anyway, therefore, I
hope you appreciate

72
00:04:01,130 --> 00:04:03,320
that I did the rotating
wave approximation.

73
00:04:03,320 --> 00:04:05,970
I said I'm only interested
in the near-resonant terms.

74
00:04:05,970 --> 00:04:10,940
And then when we say we want to
go up in energy in two steps.

75
00:04:10,940 --> 00:04:13,090
We absorb two photos.

76
00:04:13,090 --> 00:04:14,870
We don't have any
emission of photons.

77
00:04:14,870 --> 00:04:16,620
These are all the
counter-rotating terms.

78
00:04:16,620 --> 00:04:18,999
You have only the
absorption of photons,

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00:04:18,999 --> 00:04:20,499
and then we have
four possibilities.

80
00:04:23,900 --> 00:04:26,620
Yes, I mean, you
have a term where

81
00:04:26,620 --> 00:04:28,910
two photons are emitted
from the ground state.

82
00:04:28,910 --> 00:04:31,804
This is sort of now
doubly counter-rotatable.

83
00:04:31,804 --> 00:04:32,720
We're not going there.

84
00:04:32,720 --> 00:04:34,750
It's not adding
anything new to it.

85
00:04:34,750 --> 00:04:37,290
You know what
counter-rotating terms mean.

86
00:04:37,290 --> 00:04:39,360
In this chapter on
two photons, I'm

87
00:04:39,360 --> 00:04:43,090
completely focused on
the resonant terms.

88
00:04:46,560 --> 00:04:49,800
But since four terms
is still too many,

89
00:04:49,800 --> 00:04:54,390
I want to just tell you what
is special about two photons.

90
00:04:54,390 --> 00:04:56,620
I focus now on a
situation, and that's

91
00:04:56,620 --> 00:05:00,220
the most common situation in
the laboratory, where there

92
00:05:00,220 --> 00:05:03,880
is a near-resonant
intermediate state

93
00:05:03,880 --> 00:05:07,640
and that is sort of now
filtering out one of the terms.

94
00:05:07,640 --> 00:05:10,890
If this intermediate state
is resonant with omega 1,

95
00:05:10,890 --> 00:05:13,260
then we only want
to consider now

96
00:05:13,260 --> 00:05:16,810
the process where the first
step to the intermediate step

97
00:05:16,810 --> 00:05:20,850
is driven by the field
e1 and the second step

98
00:05:20,850 --> 00:05:24,720
to the final state b is
driven by the field e2.

99
00:05:24,720 --> 00:05:27,340
So therefore, we
have only one term,

100
00:05:27,340 --> 00:05:30,290
which dominates
out of those four,

101
00:05:30,290 --> 00:05:33,150
or dominates out
of those 64, which

102
00:05:33,150 --> 00:05:35,390
we would have gotten
without any approximation.

103
00:05:35,390 --> 00:05:37,510
It's one term-- the
near-resonant term--

104
00:05:37,510 --> 00:05:39,127
which dominates.

105
00:05:39,127 --> 00:05:40,710
So that's what we
want to discuss now.

106
00:05:52,140 --> 00:05:55,550
So this is the term
we want to consider.

107
00:05:55,550 --> 00:06:00,070
And it's the same we've
always done in lowest order

108
00:06:00,070 --> 00:06:02,940
perturbation, second
order perturbation theory,

109
00:06:02,940 --> 00:06:04,120
in two steps.

110
00:06:04,120 --> 00:06:05,260
It's now in two steps.

111
00:06:05,260 --> 00:06:08,390
But if you're asking, what is
the transition probability?

112
00:06:08,390 --> 00:06:10,680
The transition
probability, we have

113
00:06:10,680 --> 00:06:14,220
to get the probability to
be in the excited state.

114
00:06:14,220 --> 00:06:16,580
And then we have
the usual situation

115
00:06:16,580 --> 00:06:19,920
that this term can be written
as the sine squared divided

116
00:06:19,920 --> 00:06:25,020
by this, and it turns into
a delta function times t.

117
00:06:25,020 --> 00:06:28,130
And when we divide the
probability, the amplitude

118
00:06:28,130 --> 00:06:31,590
squared by t, we get a rate.

119
00:06:31,590 --> 00:06:33,280
And this is Fermi's Golden Rule.

120
00:06:33,280 --> 00:06:38,140
It is exactly the same you have
seen probably more than 100

121
00:06:38,140 --> 00:06:38,640
times.

122
00:06:42,090 --> 00:06:48,490
So therefore, we have
now a transition rate,

123
00:06:48,490 --> 00:06:50,960
which is Fermi's Golden Rule.

124
00:06:50,960 --> 00:06:52,620
This is the delta function.

125
00:06:52,620 --> 00:06:54,360
I called it the
function f because I

126
00:06:54,360 --> 00:06:57,770
want to discuss the
spectral profile a minute.

127
00:06:57,770 --> 00:07:00,840
But then-- and this
is the only difference

128
00:07:00,840 --> 00:07:03,430
to Fermi's Golden Rule
with a signal photon--

129
00:07:03,430 --> 00:07:07,900
the relevant matrix element
is, because we have two steps,

130
00:07:07,900 --> 00:07:12,310
is the product of two matrix
elements squared for step one

131
00:07:12,310 --> 00:07:13,780
and for step two.

132
00:07:13,780 --> 00:07:16,080
And because we have
an intermediate step,

133
00:07:16,080 --> 00:07:19,510
we have to divide by
the energy mismatch

134
00:07:19,510 --> 00:07:24,300
by the detuning in the
intermediate state.

135
00:07:24,300 --> 00:07:27,180
But remember, in the
one photon picture,

136
00:07:27,180 --> 00:07:31,260
Fermi's Golden Rule
is the matrix elements

137
00:07:31,260 --> 00:07:34,890
squared times the
spectral function.

138
00:07:34,890 --> 00:07:39,170
So in this matrix element
squared in frequency units,

139
00:07:39,170 --> 00:07:42,800
neglecting factors of 2,
was the Rabi frequency.

140
00:07:42,800 --> 00:07:44,970
So therefore, very
naturally, we want

141
00:07:44,970 --> 00:07:49,010
to define this as a
two-photon Rabi frequency

142
00:07:49,010 --> 00:07:52,690
whereas each matrix element
here divided by h bar

143
00:07:52,690 --> 00:07:55,670
was the single photon
Rabi frequency.

144
00:07:55,670 --> 00:07:59,070
So therefore, what we obtained
for the two photon processes,

145
00:07:59,070 --> 00:08:02,120
we have a two-photon
Rabi frequency,

146
00:08:02,120 --> 00:08:04,670
which is the product of
the single photon Rabi

147
00:08:04,670 --> 00:08:10,480
frequency for each step,
divided by the energy detuning

148
00:08:10,480 --> 00:08:11,655
from the immediate state.

149
00:08:20,180 --> 00:08:33,409
So therefore, our
result looks almost

150
00:08:33,409 --> 00:08:36,030
indistinguishable from the
result built on single photons.

151
00:08:36,030 --> 00:08:39,240
We've just cleverly
defined our quantities.

152
00:08:39,240 --> 00:08:43,169
The rate to go from a
to b is Rabi frequency

153
00:08:43,169 --> 00:08:46,550
squared, but it is the
two photon Rabi frequency.

154
00:08:46,550 --> 00:08:48,790
And the delta
function is the delta

155
00:08:48,790 --> 00:08:52,430
function for the resonance-- the
energy difference between state

156
00:08:52,430 --> 00:08:55,800
a and b-- but now not
just minus omega 1.

157
00:08:55,800 --> 00:08:58,780
It is minus the sum of
omega 1 plus omega 2

158
00:08:58,780 --> 00:09:01,290
because we have stacked up the
two photons in the two photon

159
00:09:01,290 --> 00:09:01,790
process.

160
00:09:05,420 --> 00:09:08,440
So let me just write that down.

161
00:09:08,440 --> 00:09:22,990
It looks like the
one photon excitation

162
00:09:22,990 --> 00:09:32,190
but with suitably
defined Rabi frequencies.

163
00:09:32,190 --> 00:09:34,230
So in other words, if
you were interested just

164
00:09:34,230 --> 00:09:37,340
in the physics of two
levels-- Rabi oscillation,

165
00:09:37,340 --> 00:09:40,900
you name it-- you can just
say the same thing happens.

166
00:09:40,900 --> 00:09:43,920
The only difference is that
instead of having a coupling

167
00:09:43,920 --> 00:09:46,820
directly by a matrix
element, we are now

168
00:09:46,820 --> 00:09:49,910
coupled by this two
photon Rabi frequency.

169
00:09:49,910 --> 00:09:52,940
And all your equation,
you know, everything--

170
00:09:52,940 --> 00:09:57,000
you can consider line
shape, spontaneous emission,

171
00:09:57,000 --> 00:10:00,110
saturation-- all the
phenomena we have discussed

172
00:10:00,110 --> 00:10:02,440
for single photon are analogous.

173
00:10:02,440 --> 00:10:07,100
You just have to use the density
of state calculated for the two

174
00:10:07,100 --> 00:10:10,230
photons and you have to use
the two photon Rabi frequency.

175
00:10:18,770 --> 00:10:19,300
OK.

176
00:10:19,300 --> 00:10:23,990
So I started out by telling you
about two photon processes, two

177
00:10:23,990 --> 00:10:26,140
photon absorption.

178
00:10:26,140 --> 00:10:35,050
But what is maybe even
more important in the way

179
00:10:35,050 --> 00:10:41,480
how it is used in experiments
are Raman processes.

180
00:10:41,480 --> 00:10:47,090
So let me just show what I mean.

181
00:10:47,090 --> 00:10:51,220
If you have a
state a and b, this

182
00:10:51,220 --> 00:10:54,380
can be two different vibrational
states of a molecule.

183
00:10:54,380 --> 00:10:57,130
It can be two hyperfine
states of an atom.

184
00:10:57,130 --> 00:11:00,660
Or if you think about
a break process,

185
00:11:00,660 --> 00:11:02,900
it could be the
same internal state

186
00:11:02,900 --> 00:11:05,270
of the atom-- the same
hyperfine state-- but with two

187
00:11:05,270 --> 00:11:07,240
different momenta.

188
00:11:07,240 --> 00:11:11,050
Then it is a Raman process
only in the external degree

189
00:11:11,050 --> 00:11:13,610
of freedom, in the
motion away function.

190
00:11:13,610 --> 00:11:17,490
The only change is you
change the momentum.

191
00:11:17,490 --> 00:11:22,060
We need our intermediate
state, which is often

192
00:11:22,060 --> 00:11:25,340
an electronically excited state.

193
00:11:25,340 --> 00:11:27,040
We are detuned.

194
00:11:27,040 --> 00:11:32,710
And now we have one photon going
up and one photon going down.

195
00:11:36,050 --> 00:11:42,110
Historically, people distinguish
between the situation

196
00:11:42,110 --> 00:11:49,550
where the final state is
lower or higher in energy.

197
00:11:49,550 --> 00:11:54,070
One is called the
Stokes process.

198
00:11:54,070 --> 00:11:57,620
The other one is
called anti-Stokes.

199
00:11:57,620 --> 00:12:00,530
But as long as you use
laser beams to stimulate it,

200
00:12:00,530 --> 00:12:04,760
you don't even care which state
is higher or lower in energy.

201
00:12:04,760 --> 00:12:09,970
But if you work in molecules
with a thermal ensemble,

202
00:12:09,970 --> 00:12:13,000
then you have certain
states thermally populated

203
00:12:13,000 --> 00:12:14,080
and others not.

204
00:12:14,080 --> 00:12:15,850
And then it makes a
difference whether you

205
00:12:15,850 --> 00:12:19,160
start from the ground state
a or from an excited state b.

206
00:12:30,010 --> 00:12:30,690
OK.

207
00:12:30,690 --> 00:12:37,030
So actually, everything
for the Raman process

208
00:12:37,030 --> 00:12:42,330
is completely analogous-- is
completely covered, actually--

209
00:12:42,330 --> 00:12:45,920
by what I wrote down for
you in perturbation theory.

210
00:12:45,920 --> 00:12:50,710
It's just if you had kept
all the 64 states-- to go up

211
00:12:50,710 --> 00:12:53,810
with one photon and down with
one photon one was one of them,

212
00:12:53,810 --> 00:12:56,280
but we discarded
it because we were

213
00:12:56,280 --> 00:12:58,520
only interested in going up.

214
00:12:58,520 --> 00:13:01,740
So in other words,
what was previously

215
00:13:01,740 --> 00:13:05,410
one of the
counter-rotating terms,

216
00:13:05,410 --> 00:13:09,460
where omega had a plus sign and
omega two had a minus sign, now

217
00:13:09,460 --> 00:13:11,660
it becomes a resonant
term because we

218
00:13:11,660 --> 00:13:14,740
have arranged our two
levels a and b in such a way

219
00:13:14,740 --> 00:13:18,420
that the near-resonant
process is that one.

220
00:13:18,420 --> 00:13:20,550
So in other words,
I mention it to you

221
00:13:20,550 --> 00:13:23,060
and it's just getting too
messy to write it down,

222
00:13:23,060 --> 00:13:27,090
when we have e to the i omega
t and e to the minus i omega t,

223
00:13:27,090 --> 00:13:31,790
I mentioned once to you is
the sign plus or minus means

224
00:13:31,790 --> 00:13:35,340
whether we absorb a photon
or whether we emit a photon.

225
00:13:35,340 --> 00:13:38,680
If you use a fully quantized
picture with a and a degas,

226
00:13:38,680 --> 00:13:41,640
the a for the
quantized description

227
00:13:41,640 --> 00:13:43,900
becomes an e to the i omega t.

228
00:13:43,900 --> 00:13:45,880
In the semi-classical
description of the

229
00:13:45,880 --> 00:13:48,270
a dega has a minus sign.

230
00:13:48,270 --> 00:13:51,740
So therefore, if you look
at all the combinations

231
00:13:51,740 --> 00:13:57,090
between plus i omega t, minus i
omega t, for this Raman process

232
00:13:57,090 --> 00:14:02,240
we want to select e to the minus
omega i and e to the plus omega

233
00:14:02,240 --> 00:14:04,170
too.

234
00:14:04,170 --> 00:14:06,595
And that means we are
focusing on this process.

235
00:14:16,160 --> 00:14:21,460
Therefore, everything
for the Raman process

236
00:14:21,460 --> 00:14:25,290
is analogous to the
two-photon absorption process.

237
00:14:25,290 --> 00:14:27,420
The only thing we
have to do is we

238
00:14:27,420 --> 00:14:30,260
have to change the sign
on the second frequency

239
00:14:30,260 --> 00:14:33,110
because the second photon is
emitted in a stimulated way

240
00:14:33,110 --> 00:14:35,310
and not absorbed.

241
00:14:35,310 --> 00:14:38,610
And therefore, our
detuning is the detuning

242
00:14:38,610 --> 00:14:41,630
from the Raman resonance.

243
00:14:41,630 --> 00:14:47,970
And therefore, OK, back
to Fermi's Golden Rule--

244
00:14:47,970 --> 00:14:51,400
the rate in Fermi's Golden
Rule is the matrix element

245
00:14:51,400 --> 00:14:56,100
squared times the
spectral density indicated

246
00:14:56,100 --> 00:14:57,630
by the delta function.

247
00:14:57,630 --> 00:15:02,130
The delta function is
now at the frequency,

248
00:15:02,130 --> 00:15:07,280
which is given by the
two-photon detuning.

249
00:15:07,280 --> 00:15:09,210
And the two-photon
Rabi frequency

250
00:15:09,210 --> 00:15:12,180
is exactly the same what we
had for two-photon absorption,

251
00:15:12,180 --> 00:15:15,330
the product of single-photon
Rabi frequencies

252
00:15:15,330 --> 00:15:16,640
divided by the detuning.

253
00:15:21,050 --> 00:15:23,680
Any questions?

254
00:15:23,680 --> 00:15:25,233
Yes.

255
00:15:25,233 --> 00:15:26,108
AUDIENCE: [INAUDIBLE]

256
00:15:40,755 --> 00:15:42,380
PROFESSOR: Well, we
are usually talking

257
00:15:42,380 --> 00:15:46,290
about-- in an atomic system--
about very narrow resonances.

258
00:15:46,290 --> 00:15:48,620
And we are working
hard on our lasers

259
00:15:48,620 --> 00:15:50,340
to be close to one resonance.

260
00:15:50,340 --> 00:15:53,740
And it would be an amazing
coincidence if accidentally it

261
00:15:53,740 --> 00:15:56,050
would be in resonance
with another one.

262
00:15:56,050 --> 00:15:59,120
But I would say, if you
had a situation-- let's

263
00:15:59,120 --> 00:16:01,730
say in a molecule, which
is a high density of state,

264
00:16:01,730 --> 00:16:04,710
the Raman process would
be a rotation or vibration

265
00:16:04,710 --> 00:16:08,630
of Raman process involving
the ground state.

266
00:16:08,630 --> 00:16:12,860
And if you're unlucky and don't
choose your lasers wisely,

267
00:16:12,860 --> 00:16:14,940
the two laser photons
could get you high up

268
00:16:14,940 --> 00:16:17,140
into an electronically
excited state.

269
00:16:17,140 --> 00:16:19,450
And this may have some
detrimental effect,

270
00:16:19,450 --> 00:16:21,330
depending what you want to do.

271
00:16:21,330 --> 00:16:24,660
But in general, I would say if
you have more than one process,

272
00:16:24,660 --> 00:16:27,240
there is no interesting
interference term.

273
00:16:27,240 --> 00:16:29,670
You just get two
different rates.

274
00:16:29,670 --> 00:16:32,190
One is the two-photon Raman
rate and the other one

275
00:16:32,190 --> 00:16:34,120
is the two-photon
absorption rate.

276
00:16:34,120 --> 00:16:35,703
And you just have
both simultaneously.

277
00:16:38,330 --> 00:16:40,380
They're not leading to
the same final state.

278
00:16:40,380 --> 00:16:43,210
If something leads to
the same final state--

279
00:16:43,210 --> 00:16:46,060
this is more subtle because you
kind of cram the interference

280
00:16:46,060 --> 00:16:47,900
effects.

281
00:16:47,900 --> 00:16:51,434
But we'll discuss some of those
things in our next chapter

282
00:16:51,434 --> 00:16:51,975
on coherence.

283
00:16:55,542 --> 00:16:56,375
Any other questions?

284
00:17:01,890 --> 00:17:02,390
OK.

285
00:17:07,490 --> 00:17:13,910
I want to now take it one level
higher, where we talk still

286
00:17:13,910 --> 00:17:16,400
about two-photon
processes but we

287
00:17:16,400 --> 00:17:21,660
are allowing one of the photons
to be spontaneously emitted.

288
00:17:21,660 --> 00:17:24,579
Again, we don't have
to learn new things.

289
00:17:24,579 --> 00:17:29,200
We just have to map it to
knowledge we already have.

290
00:17:29,200 --> 00:17:35,810
And let me sort of do
it in the following way.

291
00:17:35,810 --> 00:17:38,760
I just want to sort of give
you a clear understanding what

292
00:17:38,760 --> 00:17:42,740
this expression for
the two-photon rate is.

293
00:17:42,740 --> 00:17:45,170
If we assume laser
one and laser two

294
00:17:45,170 --> 00:17:48,706
are near-resonants with
a transition a to k and k

295
00:17:48,706 --> 00:17:53,940
to b, respectively,
we can sort of

296
00:17:53,940 --> 00:17:58,790
look at the two-photon
process in the following way.

297
00:17:58,790 --> 00:18:02,480
We can say the photon
omega 2 cannot be absorbed

298
00:18:02,480 --> 00:18:04,240
by the initial state.

299
00:18:04,240 --> 00:18:07,710
It can only be absorbed
by the initial state

300
00:18:07,710 --> 00:18:15,040
because omega 1 mixes in with a
certain probability, the state

301
00:18:15,040 --> 00:18:16,869
k into the ground state.

302
00:18:16,869 --> 00:18:18,410
So if you would
first forget about it

303
00:18:18,410 --> 00:18:20,510
and you just do
perturbation theory,

304
00:18:20,510 --> 00:18:23,050
then you would say in
perturbation theory

305
00:18:23,050 --> 00:18:26,250
with the field one,
the state a has now

306
00:18:26,250 --> 00:18:29,300
a probability,
given by this term,

307
00:18:29,300 --> 00:18:32,290
that the state a has
now an admixture.

308
00:18:32,290 --> 00:18:35,860
And now, if we have sort of--
we have dressed up our state

309
00:18:35,860 --> 00:18:41,780
a with admixing for
the near-resonant field

310
00:18:41,780 --> 00:18:44,500
some probability
of state k into it.

311
00:18:44,500 --> 00:18:48,670
And this stressed state now has
sort of a stepping stone here.

312
00:18:48,670 --> 00:18:50,870
And from this stepping
stone, it can now

313
00:18:50,870 --> 00:18:53,180
absorb the photon omega 2.

314
00:18:53,180 --> 00:18:55,540
So that's how we
should think about it.

315
00:18:55,540 --> 00:19:00,030
I'm treating this dressing
up of the initial state

316
00:19:00,030 --> 00:19:02,360
just a perturbation theory.

317
00:19:02,360 --> 00:19:05,490
And that's why everything
was in one formula

318
00:19:05,490 --> 00:19:07,720
when I applied
perturbation theory.

319
00:19:07,720 --> 00:19:13,210
But as we especially cover
in the second semester

320
00:19:13,210 --> 00:19:17,370
of the course, you can also
say if omega 1 is very, very

321
00:19:17,370 --> 00:19:21,370
strong, you can exactly
diagonalize the Hilbert

322
00:19:21,370 --> 00:19:23,860
space of states k and a.

323
00:19:23,860 --> 00:19:26,730
And this is called the
crest atom picture.

324
00:19:26,730 --> 00:19:30,790
But again, what happens is
you mix those two states.

325
00:19:30,790 --> 00:19:35,120
And it is the admixture now
in a non-perturbative way

326
00:19:35,120 --> 00:19:38,320
of state k, which is sort
of the stepping stone.

327
00:19:38,320 --> 00:19:40,500
And from this
stepping stone on, you

328
00:19:40,500 --> 00:19:42,450
can absorb a photon omega 2.

329
00:20:02,660 --> 00:20:07,210
So we could actually--
let me just redraw this--

330
00:20:07,210 --> 00:20:14,410
that we want to go
to the final state b.

331
00:20:14,410 --> 00:20:21,110
But in this kind of
picture I just suggested,

332
00:20:21,110 --> 00:20:23,510
I start with a dress state a.

333
00:20:23,510 --> 00:20:28,060
But what is relevant is
only kind of this admixture.

334
00:20:28,060 --> 00:20:35,880
And from this admixture,
we can absorb omega 2.

335
00:20:35,880 --> 00:20:40,930
The real state k
is somewhere else.

336
00:20:40,930 --> 00:20:46,310
And so it looks like, actually,
now a two-level system,

337
00:20:46,310 --> 00:20:49,120
where we go from
the dashed line--

338
00:20:49,120 --> 00:20:54,090
called the virtual state--
to the final state b.

339
00:20:54,090 --> 00:20:58,960
And let me just point out
what this virtual state is.

340
00:21:01,790 --> 00:21:03,880
Well, you know already
everything about it

341
00:21:03,880 --> 00:21:07,260
because everything which
can be known about it

342
00:21:07,260 --> 00:21:10,120
is what we have
derived in our formula.

343
00:21:10,120 --> 00:21:13,650
I'm just interpreting
the perturbation theory

344
00:21:13,650 --> 00:21:15,040
I've written down to you.

345
00:21:15,040 --> 00:21:17,224
And if I now call
it a virtual state,

346
00:21:17,224 --> 00:21:18,640
there is nothing
more you can ever

347
00:21:18,640 --> 00:21:22,190
know about this state than
what was in this formula.

348
00:21:22,190 --> 00:21:27,320
But it's maybe helpful
to summarize it.

349
00:21:27,320 --> 00:21:31,410
Because we have a resonant
with frequency omega 2

350
00:21:31,410 --> 00:21:37,070
in this situation, it is clear
that the energy of this state

351
00:21:37,070 --> 00:21:40,540
is where the dashed line is.

352
00:21:40,540 --> 00:21:43,270
It's not the energy
of the real state k.

353
00:21:43,270 --> 00:21:46,180
The stepping stone is created
with the first photon.

354
00:21:46,180 --> 00:21:48,200
And the dashed line
is the energy level

355
00:21:48,200 --> 00:21:51,554
of the virtual state.

356
00:21:51,554 --> 00:21:52,470
What is its character?

357
00:21:55,030 --> 00:21:58,070
Spatial bay function and such?

358
00:21:58,070 --> 00:22:00,690
Well, it is exactly the
intermediate state k.

359
00:22:04,080 --> 00:22:06,870
And what is the population?

360
00:22:06,870 --> 00:22:08,710
If you had a
two-level system, we

361
00:22:08,710 --> 00:22:13,460
sort of start with 100%
amplitude in state 1.

362
00:22:13,460 --> 00:22:18,910
But here, our
population is diminished

363
00:22:18,910 --> 00:22:23,560
by the probability at which
we have it admixed the state.

364
00:22:23,560 --> 00:22:26,960
So in other words,
if I really wanted

365
00:22:26,960 --> 00:22:28,960
and I want to use
this concept, I

366
00:22:28,960 --> 00:22:30,600
could make a
simplified description

367
00:22:30,600 --> 00:22:32,010
of two-photon process.

368
00:22:32,010 --> 00:22:34,870
I would just say the
two-photon process is just

369
00:22:34,870 --> 00:22:36,970
a similar photon
process starting

370
00:22:36,970 --> 00:22:39,179
from the virtual state,
and the virtual state

371
00:22:39,179 --> 00:22:40,470
is created by the first photon.

372
00:22:45,780 --> 00:22:50,720
Well, you would say,
well, why do you do that?

373
00:22:50,720 --> 00:22:53,530
I think the picture
I've just drawn

374
00:22:53,530 --> 00:22:59,880
for you is sort of
helpful when we discuss

375
00:22:59,880 --> 00:23:05,030
now two-photon emission--
spontaneous emission.

376
00:23:05,030 --> 00:23:07,090
With two lasers,
it's sort of simple.

377
00:23:07,090 --> 00:23:11,830
But with two-photon emission,
we have the situation

378
00:23:11,830 --> 00:23:14,730
that we start in
an excited state b.

379
00:23:14,730 --> 00:23:19,140
We have one laser,
omega 1, and that's it.

380
00:23:19,140 --> 00:23:23,600
But we will find out that,
eventually, the system

381
00:23:23,600 --> 00:23:25,850
populates the ground state.

382
00:23:25,850 --> 00:23:27,990
And one possible
process-- and that's

383
00:23:27,990 --> 00:23:31,160
the one we are focusing
on is-- that it was first

384
00:23:31,160 --> 00:23:33,710
emitting a photon
in a stimulated way,

385
00:23:33,710 --> 00:23:35,740
but the second photon,
since we're not

386
00:23:35,740 --> 00:23:37,970
offering any extra
stimulation, had

387
00:23:37,970 --> 00:23:40,890
to be emitted spontaneously.

388
00:23:40,890 --> 00:23:42,920
Now, you say, well,
how do I calculate it?

389
00:23:42,920 --> 00:23:45,020
I can write out long equations.

390
00:23:45,020 --> 00:23:47,290
But with the concept
I've given to you,

391
00:23:47,290 --> 00:23:50,520
it should be clear that what
we actually have is here,

392
00:23:50,520 --> 00:23:54,310
we have nothing-- let's
just look at the first part

393
00:23:54,310 --> 00:23:56,800
as a dressed atom,
an atom in state

394
00:23:56,800 --> 00:24:00,060
b with some
admixture of state k.

395
00:24:00,060 --> 00:24:04,720
And this admixutre can now decay
by a single-photon process.

396
00:24:04,720 --> 00:24:07,990
So in other words, you don't
need to re-derive anything.

397
00:24:07,990 --> 00:24:12,220
You can just sort of use
analogy to write down

398
00:24:12,220 --> 00:24:15,900
what is the spontaneous emission
rate out of the state k.

399
00:24:19,080 --> 00:24:23,590
So what you would
write down now is

400
00:24:23,590 --> 00:24:28,970
the rate for this
two-photon emission-- one

401
00:24:28,970 --> 00:24:32,770
photon stimulated, the
second photon spontaneous--

402
00:24:32,770 --> 00:24:36,980
is simply the Einstein
a coefficient,

403
00:24:36,980 --> 00:24:40,720
or the spontaneous emission rate
out of this intermediate state

404
00:24:40,720 --> 00:24:42,910
gamma ka.

405
00:24:42,910 --> 00:24:47,410
But then we have to multiply
with the probability

406
00:24:47,410 --> 00:24:49,560
that this state is
present in the state

407
00:24:49,560 --> 00:24:52,480
b because b-- that's
what we assumed

408
00:24:52,480 --> 00:24:54,710
has no direct matrix
element to emit to state a.

409
00:24:58,340 --> 00:24:59,890
And now you should
sort of be amazed

410
00:24:59,890 --> 00:25:02,370
about the beauty of
concepts you have learned.

411
00:25:02,370 --> 00:25:04,800
We are talking about
something which

412
00:25:04,800 --> 00:25:07,950
maybe before this lecture--
wow, one photon stimulated,

413
00:25:07,950 --> 00:25:09,170
one photon spontaneous.

414
00:25:09,170 --> 00:25:10,690
That must be complicated.

415
00:25:10,690 --> 00:25:17,590
But it's just that,
except for one thing.

416
00:25:17,590 --> 00:25:20,730
And this is the
following-- remember,

417
00:25:20,730 --> 00:25:24,410
you should always remember
how we derived the formula

418
00:25:24,410 --> 00:25:26,550
for spontaneous emission.

419
00:25:26,550 --> 00:25:28,460
The physics of
spontaneous emission

420
00:25:28,460 --> 00:25:31,840
is that you can put one photon
in each of the empty modes

421
00:25:31,840 --> 00:25:33,850
and you have to
sum overall modes.

422
00:25:33,850 --> 00:25:35,680
And what was important
was the density

423
00:25:35,680 --> 00:25:37,850
of modes at the frequency.

424
00:25:37,850 --> 00:25:40,320
And now the frequency is omega.

425
00:25:40,320 --> 00:25:43,530
So when we calculated
the spontaneous rate--

426
00:25:43,530 --> 00:25:47,770
emission rate-- the decay
of the excited state k,

427
00:25:47,770 --> 00:25:52,570
we had an omega cubed dependence
at the resonance frequency--

428
00:25:52,570 --> 00:25:57,010
at the resonant frequency
for the transition ka.

429
00:25:57,010 --> 00:25:59,510
But now we are
interested in the density

430
00:25:59,510 --> 00:26:02,030
of states at frequency omega.

431
00:26:02,030 --> 00:26:04,720
Well, you'll remember,
two factors of omega

432
00:26:04,720 --> 00:26:06,640
are from the density of states.

433
00:26:06,640 --> 00:26:10,440
One came from translating
for momentum matrix

434
00:26:10,440 --> 00:26:15,310
element to-- no, one
was-- yes, it was set.

435
00:26:15,310 --> 00:26:18,840
But one was pretty much the
single-photon Rabi frequency.

436
00:26:18,840 --> 00:26:24,590
And this is also at the
frequency of the photon.

437
00:26:24,590 --> 00:26:28,560
So you just have to correct
for the omega cubed factor.

438
00:26:28,560 --> 00:26:31,560
And this is our result now
for two-photon emission.

439
00:26:42,795 --> 00:26:43,295
OK.

440
00:26:48,140 --> 00:27:00,780
This may be even more relevant
at least in the research which

441
00:27:00,780 --> 00:27:05,820
is done in my group and
in other groups at MIT,

442
00:27:05,820 --> 00:27:08,150
are again Raman processes.

443
00:27:08,150 --> 00:27:11,165
We often have Raman
processes-- you know,

444
00:27:11,165 --> 00:27:14,470
you need a reason why
you want a two-photon.

445
00:27:14,470 --> 00:27:18,280
If you can't reach the upper
state with a single photon like

446
00:27:18,280 --> 00:27:21,032
people in [INAUDIBLE]-- they
may just use two photons.

447
00:27:21,032 --> 00:27:22,490
But this is more
limit because they

448
00:27:22,490 --> 00:27:25,210
don't have the laser,
which can bridge the gap.

449
00:27:25,210 --> 00:27:28,140
In situations where you
work with alkali atoms,

450
00:27:28,140 --> 00:27:31,040
we are often very happy with--
we have one resonant line

451
00:27:31,040 --> 00:27:33,790
and we can do all the laser
cooling, everything we want.

452
00:27:33,790 --> 00:27:38,330
But often, we don't like that
the line width of the excited

453
00:27:38,330 --> 00:27:40,180
state is very large.

454
00:27:40,180 --> 00:27:43,242
And therefore, certain
precision work,

455
00:27:43,242 --> 00:27:45,450
where we want to be very
accurate in what we're doing

456
00:27:45,450 --> 00:27:51,860
with the atoms, cannot be
done on the d1 or d2 line.

457
00:27:51,860 --> 00:27:56,630
And therefore, we often
use a two-level system,

458
00:27:56,630 --> 00:27:59,270
which consists of
two hyperfine states

459
00:27:59,270 --> 00:28:01,120
because then there
is no broadening

460
00:28:01,120 --> 00:28:03,150
to do spontaneous emission.

461
00:28:03,150 --> 00:28:07,620
So one motivation why we again
and again and again consider

462
00:28:07,620 --> 00:28:09,730
two-photon processes
in our laboratory

463
00:28:09,730 --> 00:28:14,410
is because it gives us excess
to very narrow resonances.

464
00:28:14,410 --> 00:28:16,220
But I think you
get the gist now,

465
00:28:16,220 --> 00:28:18,590
and it will become
even clearer later on,

466
00:28:18,590 --> 00:28:22,220
that often when you do a
transition between two ground

467
00:28:22,220 --> 00:28:26,060
state levels, a
lot of the physics

468
00:28:26,060 --> 00:28:28,190
is the same as of
a single photon.

469
00:28:28,190 --> 00:28:30,440
You just replace your
single photon Rabi frequency

470
00:28:30,440 --> 00:28:32,920
by two-photon Rabi
frequency, and suddenly you

471
00:28:32,920 --> 00:28:34,630
can do everything
you ever wanted

472
00:28:34,630 --> 00:28:36,240
to do with a single photon.

473
00:28:36,240 --> 00:28:40,350
But now with the benefit of
having a very narrow resonance.

474
00:28:40,350 --> 00:28:46,090
So anyway, therefore, I think
the most important aspect

475
00:28:46,090 --> 00:28:51,750
of two-photon processes
99% of the research

476
00:28:51,750 --> 00:28:54,180
people in our field are
involved is actually

477
00:28:54,180 --> 00:28:56,640
in the form of Raman processes.

478
00:28:56,640 --> 00:28:59,760
So therefore, but for
pedagogical reasons,

479
00:28:59,760 --> 00:29:02,870
I just like to start out
with two-photon process--

480
00:29:02,870 --> 00:29:06,070
one photon stimulated,
one photon spontaneous.

481
00:29:06,070 --> 00:29:10,270
But now let's just fold it over,
and we have the lambda type

482
00:29:10,270 --> 00:29:11,280
transition.

483
00:29:11,280 --> 00:29:17,775
The first photon is absorbed
and the second photon

484
00:29:17,775 --> 00:29:21,060
is spontaneously emitted.

485
00:29:21,060 --> 00:29:23,022
I think as you
realize when you go

486
00:29:23,022 --> 00:29:26,360
from the two-photon
absorption to Raman process,

487
00:29:26,360 --> 00:29:28,820
there's nothing you
have to re-learn.

488
00:29:28,820 --> 00:29:31,640
You just have to be careful with
the signs of omega 1 and omega

489
00:29:31,640 --> 00:29:33,540
2.

490
00:29:33,540 --> 00:29:51,000
So if I would ask you now
what is the rate-- what

491
00:29:51,000 --> 00:29:55,130
is the rate of the
spontaneous Raman process,

492
00:29:55,130 --> 00:30:02,690
well, it is the probability to
be in the intermediate state.

493
00:30:02,690 --> 00:30:08,220
And this probability
is just re-writing

494
00:30:08,220 --> 00:30:12,060
in a different way what
we have used, is the Rabi

495
00:30:12,060 --> 00:30:16,000
frequency of the first
step squared, divided

496
00:30:16,000 --> 00:30:18,270
by the detuning squared.

497
00:30:18,270 --> 00:30:22,010
So this is the probability
to be in the excited state.

498
00:30:22,010 --> 00:30:25,490
Now, the spontaneous
emission occurs

499
00:30:25,490 --> 00:30:29,660
with Einstein's a
coefficient connecting

500
00:30:29,660 --> 00:30:33,020
the intermediate state
to the ground state.

501
00:30:33,020 --> 00:30:36,370
And then, of course,
as we just learned,

502
00:30:36,370 --> 00:30:40,240
we have to correct the
spectral density and such,

503
00:30:40,240 --> 00:30:42,100
or we have to use
for the calculation

504
00:30:42,100 --> 00:30:44,905
of the density of modes.

505
00:30:49,350 --> 00:30:51,516
The correct omega
factors-- we have

506
00:30:51,516 --> 00:30:54,760
to calculate it at the
frequency of the emitted photon.

507
00:30:58,490 --> 00:31:00,540
This is actually
also-- this kind

508
00:31:00,540 --> 00:31:04,360
of spontaneous
Raman process-- has

509
00:31:04,360 --> 00:31:06,450
been very important
historically.

510
00:31:06,450 --> 00:31:09,990
Before the advent of
lasers, all you had

511
00:31:09,990 --> 00:31:15,210
is light bulbs, strong light
bulbs, maybe mercury lamp

512
00:31:15,210 --> 00:31:18,860
which put a lot of light
into the mercury light,

513
00:31:18,860 --> 00:31:22,310
and at least it was
somewhat spectrally narrow.

514
00:31:22,310 --> 00:31:23,870
But still very, very broad.

515
00:31:23,870 --> 00:31:27,000
And you couldn't really
resonantly, you know,

516
00:31:27,000 --> 00:31:30,610
stack up two light bulbs and
have enough spectral power

517
00:31:30,610 --> 00:31:33,170
to excite to a certain state.

518
00:31:33,170 --> 00:31:37,340
But look here-- you could
still, with a strong light bulb,

519
00:31:37,340 --> 00:31:39,920
create an admixture
of the excited state.

520
00:31:39,920 --> 00:31:43,110
This virtual line
was terribly broad

521
00:31:43,110 --> 00:31:46,930
because of the width of the
light bulb-- spectral width--

522
00:31:46,930 --> 00:31:50,910
but then this spontaneous photon
was just compensating for it.

523
00:31:50,910 --> 00:31:55,030
So in other words, it should
be obvious-- this process only

524
00:31:55,030 --> 00:31:58,810
depends on the power
for the first step.

525
00:31:58,810 --> 00:32:00,680
And it doesn't really
matter if the power

526
00:32:00,680 --> 00:32:03,560
is delivered by a
laser or a light bulb.

527
00:32:03,560 --> 00:32:06,210
The rate for this
process is the same.

528
00:32:06,210 --> 00:32:08,010
And this actually
was the discovery

529
00:32:08,010 --> 00:32:11,340
by Raman, which was rewarded
with the Nobel Prize,

530
00:32:11,340 --> 00:32:14,800
for suddenly observing
when you excited molecules

531
00:32:14,800 --> 00:32:18,720
with a very strong
light bulb, you suddenly

532
00:32:18,720 --> 00:32:23,660
saw very different frequencies
of photons coming out.

533
00:32:23,660 --> 00:32:26,040
And this was a
landmark discovery.

534
00:32:35,720 --> 00:32:38,360
OK, what is next?

535
00:32:38,360 --> 00:32:40,600
I've already written it out.

536
00:32:40,600 --> 00:32:50,680
So that's also important for a
lot of research within the CUA.

537
00:32:50,680 --> 00:32:54,820
When we simply want to change
the momentum state of an atom,

538
00:32:54,820 --> 00:32:56,750
we have two lasers.

539
00:32:56,750 --> 00:32:57,700
We go up and down.

540
00:33:00,610 --> 00:33:04,130
We are not changing
the internal state.

541
00:33:04,130 --> 00:33:08,180
But it is still a Raman
process because state a and b

542
00:33:08,180 --> 00:33:09,990
differ by the photon recoil.

543
00:33:09,990 --> 00:33:11,810
So we're not going
back to the same state.

544
00:33:11,810 --> 00:33:14,770
We are going back to the
same internal state a,

545
00:33:14,770 --> 00:33:18,840
but it may have 2h [INAUDIBLE],
two-photon recoil different.

546
00:33:18,840 --> 00:33:22,060
And therefore, as long
as, in quantum physics,

547
00:33:22,060 --> 00:33:26,510
one quantum number is different,
it is a different state.

548
00:33:26,510 --> 00:33:29,070
And therefore, if you
just think about it

549
00:33:29,070 --> 00:33:32,620
Rayleigh scattering resonance
fluorescence, that you go up

550
00:33:32,620 --> 00:33:35,390
and you go down
to the next state.

551
00:33:35,390 --> 00:33:38,110
You may think you have
your favorite atom,

552
00:33:38,110 --> 00:33:41,280
and you go up on a
cycling transition.

553
00:33:41,280 --> 00:33:46,945
Well, when the atom goes
up and then emits a photon

554
00:33:46,945 --> 00:33:50,560
on a cycling transition,
there is recoil involved.

555
00:33:50,560 --> 00:33:53,220
So actually what you're doing
is on the cycling transition,

556
00:33:53,220 --> 00:33:58,540
you cycle it through many, many
spontaneous Raman processes.

557
00:33:58,540 --> 00:34:00,220
So this is the
correct description

558
00:34:00,220 --> 00:34:03,160
of resonant fluorescence
and Rayleigh scattering.

559
00:34:09,927 --> 00:34:10,510
Any questions?

560
00:34:19,370 --> 00:34:19,870
Good.

561
00:34:23,050 --> 00:34:29,989
I've just mentioned that when
we do Reyleigh scattering,

562
00:34:29,989 --> 00:34:33,560
we have to consider that
the photons have momentum,

563
00:34:33,560 --> 00:34:36,219
and this takes us
to another state.

564
00:34:36,219 --> 00:34:38,230
Let's now be a little
bit more careful

565
00:34:38,230 --> 00:34:43,010
and consider what is
the role of the momentum

566
00:34:43,010 --> 00:34:46,719
in the transition-- in
the two-photon process.

567
00:34:46,719 --> 00:34:49,060
And in particular,
I want to come back

568
00:34:49,060 --> 00:34:51,290
to this Fermi's
Golden Rule formula

569
00:34:51,290 --> 00:34:57,800
and include in this spectral
profile, which in the simplest

570
00:34:57,800 --> 00:34:59,690
case is always the Lorentzian.

571
00:34:59,690 --> 00:35:02,880
But I want to include
now Doppler shifts.

572
00:35:02,880 --> 00:35:04,960
So in other words,
what I want to do now

573
00:35:04,960 --> 00:35:08,370
is I want to talk about
recoil and Doppler shifts

574
00:35:08,370 --> 00:35:11,950
in a two-photon
process and see how

575
00:35:11,950 --> 00:35:14,740
it will affect the line shape.

576
00:35:20,910 --> 00:35:25,880
All we have to do is--
or maybe let me back up.

577
00:35:25,880 --> 00:35:28,890
You should maybe consider
what I've discussed so far,

578
00:35:28,890 --> 00:35:32,190
is the situation of an
atom which has no motion.

579
00:35:32,190 --> 00:35:36,020
We could just fully focus on
the internal degree of freedom.

580
00:35:36,020 --> 00:35:38,545
And just to remind
you, we have discussed

581
00:35:38,545 --> 00:35:43,230
two ways how you can eliminate
motion out of the picture.

582
00:35:43,230 --> 00:35:46,920
One is assume the atom
has infinite mass.

583
00:35:46,920 --> 00:35:48,550
That's one possibility.

584
00:35:48,550 --> 00:35:52,210
The second one is assume
the atom is tightly

585
00:35:52,210 --> 00:35:57,900
localized in an ion trap,
deep in the [INAUDIBLE],

586
00:35:57,900 --> 00:36:01,570
that it is localized to
less than the wavelength.

587
00:36:01,570 --> 00:36:09,460
And actually, the two ways of
how you can eliminate recoil

588
00:36:09,460 --> 00:36:12,040
in Doppler shift are
actually the same.

589
00:36:12,040 --> 00:36:14,910
When I said, give the
atom an infinite mass,

590
00:36:14,910 --> 00:36:17,390
well, you give the
atom an infinite mass

591
00:36:17,390 --> 00:36:19,880
by tightly connecting
it to the laboratory.

592
00:36:19,880 --> 00:36:22,470
And this is what tight
confinement in an ion or atom

593
00:36:22,470 --> 00:36:24,020
trap does.

594
00:36:24,020 --> 00:36:28,220
Because then the recoil is
absorbed no longer by the atom,

595
00:36:28,220 --> 00:36:30,730
but by your
experimental structure,

596
00:36:30,730 --> 00:36:33,020
or by your whole laboratory,
or even by the building,

597
00:36:33,020 --> 00:36:34,560
if you want.

598
00:36:34,560 --> 00:36:40,260
But now, we are kind of going
beyond this restriction.

599
00:36:40,260 --> 00:36:44,510
We are now saying, OK, now
we allow the atom to move.

600
00:36:44,510 --> 00:36:49,040
And we can deal with that by
simply saying when the atom has

601
00:36:49,040 --> 00:36:55,030
a velocity v, we can transform--
we can just use the Galilean

602
00:36:55,030 --> 00:36:58,600
transformation and say OK,
the physics is the same.

603
00:36:58,600 --> 00:37:01,790
However, the atom,
due to its velocity,

604
00:37:01,790 --> 00:37:05,040
sees a slightly
different frequency.

605
00:37:05,040 --> 00:37:07,250
So therefore, we
have our Lorentzian.

606
00:37:07,250 --> 00:37:10,020
But now, we calculate
our Lorentzian

607
00:37:10,020 --> 00:37:13,720
by using the frequencies
perceived by the atom.

608
00:37:22,400 --> 00:37:25,160
The different
signs-- plus minus--

609
00:37:25,160 --> 00:37:26,960
are, of course,
distinguishing whether we

610
00:37:26,960 --> 00:37:29,110
have two-photon absorption
or Raman process.

611
00:37:33,490 --> 00:37:39,850
And the frequency shift, going
into the frame of the atom,

612
00:37:39,850 --> 00:37:45,275
gives us Doppler ts k1v and k2v.

613
00:37:45,275 --> 00:37:47,900
And now you'll see that there is
something which is potentially

614
00:37:47,900 --> 00:37:50,420
interesting, and I
want to discuss that.

615
00:37:50,420 --> 00:37:56,740
If k1 and k2 end up
here with a minus sign,

616
00:37:56,740 --> 00:38:01,170
it may eliminate Doppler
shifts, maybe even completely.

617
00:38:01,170 --> 00:38:02,940
So this is something new.

618
00:38:02,940 --> 00:38:05,830
If you have a single
photon, you always

619
00:38:05,830 --> 00:38:07,920
transfer momentum to the atom.

620
00:38:07,920 --> 00:38:12,560
But if you have two photons,
the two momenta can cancel.

621
00:38:12,560 --> 00:38:14,870
And this is actually
a powerful method

622
00:38:14,870 --> 00:38:18,820
to avoid Doppler
broadening in spectroscopy.

623
00:38:18,820 --> 00:38:21,660
So let me elaborate on that.

624
00:38:29,120 --> 00:38:31,760
The message you get
from this formula--

625
00:38:31,760 --> 00:38:34,640
it's actually much
easier to say it in words

626
00:38:34,640 --> 00:38:37,290
than to write it
mathematically because it's all

627
00:38:37,290 --> 00:38:39,190
hidden in the plus minus sign.

628
00:38:39,190 --> 00:38:48,130
But the gist is that two-photon
absorption or Raman processes--

629
00:38:48,130 --> 00:39:00,260
so any kind of
two-photon process--

630
00:39:00,260 --> 00:39:06,490
they are like
single-photon transitions.

631
00:39:15,538 --> 00:39:17,390
And we've already
discussed it, that we

632
00:39:17,390 --> 00:39:21,320
said we have Fermi's Golden
Rule like for single photon.

633
00:39:21,320 --> 00:39:23,500
It's just we have to use
the suitable definition

634
00:39:23,500 --> 00:39:26,140
of the two-photon
Rabi frequency.

635
00:39:26,140 --> 00:39:30,390
And we had also seen that
we can treat, actually,

636
00:39:30,390 --> 00:39:32,370
the two photons at one.

637
00:39:32,370 --> 00:39:35,140
But the frequencies,
sort of, of this photon--

638
00:39:35,140 --> 00:39:37,300
if you want to regard
the two-photon process

639
00:39:37,300 --> 00:39:41,880
as a super photon, the
frequency of the super photon

640
00:39:41,880 --> 00:39:45,200
is now either the
sum or the difference

641
00:39:45,200 --> 00:39:47,480
off the two
frequencies, depending

642
00:39:47,480 --> 00:39:51,490
whether we look at two-photon
absorption or Raman process.

643
00:39:51,490 --> 00:39:55,470
But what we see now from
the Doppler shift formula,

644
00:39:55,470 --> 00:39:59,360
that we can apply the
same also to the momentum.

645
00:39:59,360 --> 00:40:03,560
It is as if we had a
super photon, which

646
00:40:03,560 --> 00:40:08,130
drives one transition, but the
momentum of this super photon

647
00:40:08,130 --> 00:40:12,760
is now the sum or the
difference of the two

648
00:40:12,760 --> 00:40:15,250
momenta of the photon.

649
00:40:15,250 --> 00:40:18,910
And if you wonder, how
do you sum them up?

650
00:40:18,910 --> 00:40:23,980
What really matters is what is
after the two photons have been

651
00:40:23,980 --> 00:40:26,750
exchanged, what is
the total momentum

652
00:40:26,750 --> 00:40:28,470
transferred to the atom.

653
00:40:28,470 --> 00:40:32,660
So what appears here--
k1 plus minus k2--

654
00:40:32,660 --> 00:40:44,740
is the total momentum
transfer to the atom.

655
00:40:44,740 --> 00:40:48,730
And you see that if you
have two-photon absorption,

656
00:40:48,730 --> 00:40:52,140
if the two laser beams
are counter-propagating,

657
00:40:52,140 --> 00:40:54,630
the total momentum
transfer is zero.

658
00:40:54,630 --> 00:40:57,660
But if you have a Raman process
where you absorb one photon

659
00:40:57,660 --> 00:41:00,020
and then emit it,
the momentum transfer

660
00:41:00,020 --> 00:41:04,480
is zero, assuming similar
frequencies, when the two Raman

661
00:41:04,480 --> 00:41:06,360
beams are parallel.

662
00:41:06,360 --> 00:41:11,326
So in other words, the situation
without momentum transfer--

663
00:41:11,326 --> 00:41:13,200
and you will see in a
moment, or I've already

664
00:41:13,200 --> 00:41:15,570
said this is a situation
where you are Doppler

665
00:41:15,570 --> 00:41:18,880
free-- for two-photon
absorption the geometry is

666
00:41:18,880 --> 00:41:20,480
counter-propagating.

667
00:41:20,480 --> 00:41:23,800
For Raman processes,
it is co-propagating.

668
00:41:23,800 --> 00:41:27,440
Then you have no or minimal
momentum transfer to the atom.

669
00:41:31,110 --> 00:41:31,660
OK.

670
00:41:31,660 --> 00:41:34,080
So this total
momentum transfer is

671
00:41:34,080 --> 00:41:47,340
minimized for k1 equals minus
k2 for two-photon absorption,

672
00:41:47,340 --> 00:41:55,890
or for k1 equals k2
co-propagating laser beams

673
00:41:55,890 --> 00:41:58,970
in the case of
the Raman process.

674
00:41:58,970 --> 00:42:00,329
Oops.

675
00:42:00,329 --> 00:42:01,245
Now you have it twice.

676
00:42:05,170 --> 00:42:08,070
Let me just focus, because
there is very special interest

677
00:42:08,070 --> 00:42:12,560
in that, for precision
spectroscopy of hydrogen.

678
00:42:12,560 --> 00:42:20,540
Let me assume we have just
one laser, which produces--

679
00:42:20,540 --> 00:42:23,030
but we want to dive a
two-photon process with it.

680
00:42:23,030 --> 00:42:26,010
In this situation,
omega 1 equals omega 2.

681
00:42:26,010 --> 00:42:28,550
The momentum transfers
are the same.

682
00:42:28,550 --> 00:42:30,890
And if we arrange
for the two photons

683
00:42:30,890 --> 00:42:33,530
to be absorbed from
opposite directions,

684
00:42:33,530 --> 00:42:38,260
we reach the situation where the
Doppler shift is really zero.

685
00:42:41,600 --> 00:42:46,820
So this is the way where we
do Doppler free spectroscopy.

686
00:42:49,960 --> 00:42:53,790
And two-photon
spectroscopy is one

687
00:42:53,790 --> 00:42:59,840
of the handful of methods
of practical importance

688
00:42:59,840 --> 00:43:02,310
for eliminating the first
order Doppler shift.

689
00:43:05,230 --> 00:43:11,780
So if you take an atom-- and
let me just quote Dan Kleppner

690
00:43:11,780 --> 00:43:14,570
research, where this
is the hydrogen atom--

691
00:43:14,570 --> 00:43:21,440
and you have two laser beams
from opposite direction.

692
00:43:21,440 --> 00:43:23,120
How will the spectrum look like?

693
00:43:26,190 --> 00:43:30,220
Well, we have the feature,
which I just emphasized,

694
00:43:30,220 --> 00:43:32,500
that the two photons-- one
is absorbed from the left

695
00:43:32,500 --> 00:43:34,850
and from the right,
and therefore you

696
00:43:34,850 --> 00:43:38,370
get a very, very sharp line.

697
00:43:38,370 --> 00:43:38,980
Really sharp.

698
00:43:38,980 --> 00:43:41,170
I will talk about
it in a second.

699
00:43:41,170 --> 00:43:45,650
But you cannot, of course,
suppress the process where you

700
00:43:45,650 --> 00:43:49,420
absorb two photons from the left
or two photons from the right.

701
00:43:49,420 --> 00:43:53,960
And therefore, you have
sort of a broad pedestal.

702
00:43:53,960 --> 00:43:59,190
So the pedestal is where you
take one photon-- both photons

703
00:43:59,190 --> 00:44:03,180
from the same side, whereas the
Doppler free peak is where you

704
00:44:03,180 --> 00:44:05,745
have photons from
counter-propagating directions.

705
00:44:18,560 --> 00:44:23,500
Since hydrogen is of
methological importance,

706
00:44:23,500 --> 00:44:26,120
measurements of-- fundamental
measurements of-- the Lamb

707
00:44:26,120 --> 00:44:29,060
shift, comparisons
with QED calculations,

708
00:44:29,060 --> 00:44:32,500
measurement of the
Rydberg constant-- these

709
00:44:32,500 --> 00:44:36,160
are all done by
hydrogen spectroscopy.

710
00:44:36,160 --> 00:44:38,270
So therefore, it
is very important

711
00:44:38,270 --> 00:44:43,610
to have precision method which
suppresses the Doppler effect.

712
00:44:43,610 --> 00:44:48,960
However, let me point out
that once you have completely

713
00:44:48,960 --> 00:44:52,480
eliminated the first
order Doppler broadening,

714
00:44:52,480 --> 00:44:58,150
you are then limited by the
second order Doppler effect.

715
00:44:58,150 --> 00:45:01,110
And as I pointed out,
there is no geometry-- no

716
00:45:01,110 --> 00:45:02,950
arrangements of
beams, no tricks you

717
00:45:02,950 --> 00:45:06,520
can play-- because
one contribution

718
00:45:06,520 --> 00:45:08,290
to the second order
Doppler effect

719
00:45:08,290 --> 00:45:13,030
simply comes from time dilation,
that in the frame of the atom,

720
00:45:13,030 --> 00:45:16,380
relativistically speaking,
the clock ticks differently.

721
00:45:16,380 --> 00:45:18,550
And therefore, the
spectral line is different.

722
00:45:27,630 --> 00:45:31,570
So this is what we had discussed
already for the second order

723
00:45:31,570 --> 00:45:35,530
Doppler effect, and this is
not suppressed by two-photon.

724
00:45:35,530 --> 00:45:40,490
If you now estimate what is
the relative line width-- so

725
00:45:40,490 --> 00:45:43,300
what is the delta, the
line broadening you

726
00:45:43,300 --> 00:45:45,780
to the second order
Doppler effect, in relation

727
00:45:45,780 --> 00:45:47,398
to the transition frequency?

728
00:45:58,970 --> 00:46:01,940
So just give me a second.

729
00:46:01,940 --> 00:46:02,510
Yeah.

730
00:46:02,510 --> 00:46:05,620
Then this omega cancels
and we have an expression

731
00:46:05,620 --> 00:46:07,120
which is sort of interesting.

732
00:46:07,120 --> 00:46:10,320
It is mv squared.

733
00:46:10,320 --> 00:46:13,730
It's the energy--
the thermal energy--

734
00:46:13,730 --> 00:46:16,590
but since we have normalized
it to the transition frequency,

735
00:46:16,590 --> 00:46:19,180
it becomes now
the thermal energy

736
00:46:19,180 --> 00:46:24,170
relative to the rest
mass of the atom.

737
00:46:24,170 --> 00:46:26,680
You would think, well,
this must be really tiny.

738
00:46:26,680 --> 00:46:28,560
Well, it is tiny.

739
00:46:28,560 --> 00:46:32,760
If you take room temperature, it
is 2 times 10 to the minus 11.

740
00:46:32,760 --> 00:46:34,860
But people are now
looking for precision

741
00:46:34,860 --> 00:46:39,200
in optical clocks, which is in
the 10 to the minus 15 range.

742
00:46:39,200 --> 00:46:44,040
So you really have to be
able to know the temperature,

743
00:46:44,040 --> 00:46:46,880
know the kinetic energy
distribution of the atom,

744
00:46:46,880 --> 00:46:48,790
and be able to correct for it.

745
00:46:48,790 --> 00:46:51,432
Or ultimately, if you
can't correct for it,

746
00:46:51,432 --> 00:46:53,890
you would not be able to correct
for it at room temperature

747
00:46:53,890 --> 00:46:57,100
because you can't know for
certain that the velocity

748
00:46:57,100 --> 00:46:59,260
distribution of the
atoms in your laser beam

749
00:46:59,260 --> 00:47:01,120
is exactly at room temperature.

750
00:47:01,120 --> 00:47:03,220
You really have to go
to low temperature,

751
00:47:03,220 --> 00:47:05,190
go to cryogenic temperatures.

752
00:47:05,190 --> 00:47:11,620
And in the famous experiments
in Munich and [INAUDIBLE] group,

753
00:47:11,620 --> 00:47:16,800
the typical situation is
for the 1s to 2s transition,

754
00:47:16,800 --> 00:47:20,240
they observed residual
line broadenings

755
00:47:20,240 --> 00:47:23,910
on the order of a
few hundred Hertz.

756
00:47:23,910 --> 00:47:29,790
This is not limited at all
by the natural lifetime

757
00:47:29,790 --> 00:47:31,520
of the 2s transition.

758
00:47:31,520 --> 00:47:34,450
As you calculate in your
last homework assignment,

759
00:47:34,450 --> 00:47:38,140
the lifetime of the
2s state is actually

760
00:47:38,140 --> 00:47:41,030
due to two-photon emission.

761
00:47:41,030 --> 00:47:42,030
Amazing.

762
00:47:42,030 --> 00:47:43,860
Two photons in series.

763
00:47:43,860 --> 00:47:46,740
This is the way how
the 2s state decays.

764
00:47:46,740 --> 00:47:50,630
And you will, actually, with
this rather simple description,

765
00:47:50,630 --> 00:47:53,310
get a fairly accurate
estimate for the lifetime,

766
00:47:53,310 --> 00:47:55,080
which is a fraction of a second.

767
00:47:57,680 --> 00:48:01,830
And [INAUDIBLE] group uses
a cryogenic experiment.

768
00:48:01,830 --> 00:48:05,310
They cool the hydrogen by
collisions with liquid helium,

769
00:48:05,310 --> 00:48:08,910
cooled vaults, to maybe a Kelvin
or so, three hundred times

770
00:48:08,910 --> 00:48:10,230
below room temperature.

771
00:48:10,230 --> 00:48:12,790
And this has been important
to reach this precision.

772
00:48:12,790 --> 00:48:16,060
And still, I think, even
at Kelvin temperature,

773
00:48:16,060 --> 00:48:25,470
the second order Doppler shift
is one of the important--

774
00:48:25,470 --> 00:48:27,350
is one of the
important systematics.

775
00:48:47,090 --> 00:48:47,590
OK.

776
00:48:47,590 --> 00:48:51,840
This is what I wanted to discuss
with you about two photons.

777
00:48:51,840 --> 00:48:53,105
Any questions?

778
00:48:55,920 --> 00:48:57,590
Yes.

779
00:48:57,590 --> 00:48:59,000
AUDIENCE: Just--
very lightly-- I

780
00:48:59,000 --> 00:49:00,835
would have thought [INAUDIBLE].

781
00:49:11,510 --> 00:49:13,230
PROFESSOR: That's correct.

782
00:49:13,230 --> 00:49:17,080
It really depends what you want.

783
00:49:17,080 --> 00:49:20,610
If you simply want an
atomic clock, all you want

784
00:49:20,610 --> 00:49:24,750
is a very, very, very
stable reference point.

785
00:49:24,750 --> 00:49:27,840
And people use for
that [INAUDIBLE]

786
00:49:27,840 --> 00:49:30,930
and strontium, or
the people who do

787
00:49:30,930 --> 00:49:33,370
ion traps use the aluminum ion.

788
00:49:33,370 --> 00:49:37,100
All they want is
a spectral line,

789
00:49:37,100 --> 00:49:41,330
which is sufficiently sharp,
sufficiently narrow, and also

790
00:49:41,330 --> 00:49:46,280
insensitive to magnetic
fields and electric fields.

791
00:49:46,280 --> 00:49:49,210
And so they select all over
from the whole periodic table

792
00:49:49,210 --> 00:49:50,870
what they want.

793
00:49:50,870 --> 00:49:54,430
But people who want to measure
fundamental constants--

794
00:49:54,430 --> 00:49:58,320
the Rydberg constant--
want to compare lame shift

795
00:49:58,320 --> 00:50:02,690
with first principle
QED calculations,

796
00:50:02,690 --> 00:50:05,410
sort of test the precision
of quantum electrodynamics,

797
00:50:05,410 --> 00:50:07,570
verify that quantum
electrodynamics is

798
00:50:07,570 --> 00:50:10,110
a complete description of
atom photon interaction,

799
00:50:10,110 --> 00:50:12,880
they can only deal
with simple systems

800
00:50:12,880 --> 00:50:15,300
where all the
calculations are possible.

801
00:50:15,300 --> 00:50:17,450
This is the case for hydrogen.

802
00:50:17,450 --> 00:50:21,475
And, actually, with some advance
in the numeric calculation

803
00:50:21,475 --> 00:50:24,540
of wave functions
and all that, it

804
00:50:24,540 --> 00:50:26,890
may also be possible
to do it with helium.

805
00:50:26,890 --> 00:50:28,640
I know in the literature
people have often

806
00:50:28,640 --> 00:50:31,230
suggested helium have
pushed the precision

807
00:50:31,230 --> 00:50:34,960
of two-electron calculations
further and further.

808
00:50:34,960 --> 00:50:38,077
I think so far it
hasn't kind of-- helium

809
00:50:38,077 --> 00:50:39,160
has not replaced hydrogen.

810
00:50:39,160 --> 00:50:40,670
It's still hydrogen.

811
00:50:40,670 --> 00:50:43,600
But this sort of tells
you what the choices

812
00:50:43,600 --> 00:50:46,520
if you want to test
fundamental physics

813
00:50:46,520 --> 00:50:50,680
or determine
fundamental constants.

814
00:50:50,680 --> 00:50:53,720
In all atoms other than
helium and hydrogen,

815
00:50:53,720 --> 00:50:57,914
you would be limited by the
infeasibility of many electron

816
00:50:57,914 --> 00:50:58,455
calculations.

817
00:51:06,270 --> 00:51:07,222
Yes?

818
00:51:07,222 --> 00:51:11,801
AUDIENCE: Going back to the
two-photon Raman process where

819
00:51:11,801 --> 00:51:13,970
you had the second
spontaneous emission,

820
00:51:13,970 --> 00:51:15,430
I just want to clarify.

821
00:51:15,430 --> 00:51:19,770
So in the past, all this time
when you were talking about

822
00:51:19,770 --> 00:51:24,674
off-resonant single photon
scattering, was this actually,

823
00:51:24,674 --> 00:51:28,212
really, the more descriptive
picture or was there actually

824
00:51:28,212 --> 00:51:29,128
a different physical--

825
00:51:29,128 --> 00:51:35,540
PROFESSOR: No, this is
off resonance scattering.

826
00:51:35,540 --> 00:51:40,870
And if you ask me, when do
you have a situation where you

827
00:51:40,870 --> 00:51:43,690
first absorb the photon and
then emit it, I would say,

828
00:51:43,690 --> 00:51:45,000
I would like to know that.

829
00:51:45,000 --> 00:51:47,170
I don't think there
is a situation where

830
00:51:47,170 --> 00:51:48,380
this is possible.

831
00:51:48,380 --> 00:51:51,380
You always-- you should always
use a two-photon picture.

832
00:51:51,380 --> 00:51:54,100
Or the only situation, if you
press a little bit harder,

833
00:51:54,100 --> 00:51:56,400
where I think you can
think in first absorption

834
00:51:56,400 --> 00:51:58,800
and then emission,
is if you have a gas

835
00:51:58,800 --> 00:52:00,700
with lots and lots
of collisions,

836
00:52:00,700 --> 00:52:03,540
the first photon may
not be fully resonant.

837
00:52:03,540 --> 00:52:05,710
But then you have a
collision and the collision

838
00:52:05,710 --> 00:52:08,110
stabilised, provides
the missing energy

839
00:52:08,110 --> 00:52:09,930
or takes away the extra energy.

840
00:52:09,930 --> 00:52:12,560
And then with something which is
much more complicated than you

841
00:52:12,560 --> 00:52:15,555
ever wanted, you have
now truly an atom

842
00:52:15,555 --> 00:52:18,200
in an excited state,
which has completely

843
00:52:18,200 --> 00:52:21,430
lost its memory from
how it was excited.

844
00:52:21,430 --> 00:52:25,210
And yes, for this atom,
it will now simply

845
00:52:25,210 --> 00:52:27,210
spontaneously emit a photon.

846
00:52:27,210 --> 00:52:29,030
But in all other
situations where

847
00:52:29,030 --> 00:52:33,470
you don't have any loss
of coherence or something

848
00:52:33,470 --> 00:52:36,550
in between, you're never
really going to the real state.

849
00:52:36,550 --> 00:52:38,200
You're always going
to virtual state.

850
00:52:38,200 --> 00:52:40,530
It's all two-photon.

851
00:52:40,530 --> 00:52:42,340
And, I mean, we discussed that.

852
00:52:42,340 --> 00:52:45,760
Remember the clicker question
where most of the class

853
00:52:45,760 --> 00:52:48,740
was confused when
I said we go up--

854
00:52:48,740 --> 00:52:51,530
if we're exciting the atom?

855
00:52:51,530 --> 00:52:53,510
But now we are asking,
what is the spectrum

856
00:52:53,510 --> 00:52:55,360
of the emitted photon?

857
00:52:55,360 --> 00:52:58,070
It is a delta function
at the drive field,

858
00:52:58,070 --> 00:53:00,190
and we discussed it at length.

859
00:53:00,190 --> 00:53:04,880
If you allow in your head any
picture of first absorption

860
00:53:04,880 --> 00:53:09,720
then emission, that's where all
the wrong answers come from.

861
00:53:09,720 --> 00:53:11,640
This is really the
picture behind it.

862
00:53:11,640 --> 00:53:12,650
This is what happens.

863
00:53:12,650 --> 00:53:16,390
Photon in, photon out--
should be described together,

864
00:53:16,390 --> 00:53:19,050
unless you have-- and I'm
just repeating myself--

865
00:53:19,050 --> 00:53:21,020
something in between,
which is sort

866
00:53:21,020 --> 00:53:22,814
of some form of
de-coherence which

867
00:53:22,814 --> 00:53:23,980
decouples the two processes.

868
00:53:40,220 --> 00:53:41,190
OK.

869
00:53:41,190 --> 00:53:45,590
But that's a wonderful opening
to our next chapter, namely

870
00:53:45,590 --> 00:53:47,780
coherence.

871
00:53:47,780 --> 00:53:52,810
To what extent-- we just
discussed one aspect of it

872
00:53:52,810 --> 00:53:55,200
and we come back to this
in this chapter-- to

873
00:53:55,200 --> 00:53:59,640
what extent is the
photon which is scattered

874
00:53:59,640 --> 00:54:01,160
coherent with the
incoming photon?

875
00:54:03,730 --> 00:54:10,210
So I want to feature,
in this last big chapter

876
00:54:10,210 --> 00:54:17,000
of this course, coherence in all
its different manifestations.

877
00:54:17,000 --> 00:54:18,540
I think this is rather unusual.

878
00:54:18,540 --> 00:54:21,120
I don't know of any
textbook or any other course

879
00:54:21,120 --> 00:54:22,270
where this is done.

880
00:54:22,270 --> 00:54:25,940
But this is similar in spirit to
what we did on line broadening.

881
00:54:25,940 --> 00:54:28,580
I felt I could create
spatial connections

882
00:54:28,580 --> 00:54:31,790
by discussing all possibly line
shifts and line broadenings

883
00:54:31,790 --> 00:54:32,870
together.

884
00:54:32,870 --> 00:54:37,070
And now I hope you will also
see certain common traits

885
00:54:37,070 --> 00:54:41,050
if I discuss together all
the different manifestations

886
00:54:41,050 --> 00:54:43,410
of coherence.

887
00:54:43,410 --> 00:54:57,830
So what we want to
discuss is-- we start out

888
00:54:57,830 --> 00:55:02,010
by talking about coherence
in a single atom.

889
00:55:05,890 --> 00:55:09,450
We can have coherence
between two levels.

890
00:55:12,970 --> 00:55:15,760
Usually, I try to
stop at the simplest

891
00:55:15,760 --> 00:55:17,520
possible manifestation.

892
00:55:17,520 --> 00:55:20,000
But when we talk
about coherence,

893
00:55:20,000 --> 00:55:23,700
I cannot stop at two levels
because there are many new

894
00:55:23,700 --> 00:55:27,770
qualitative features which come
into play when we have three

895
00:55:27,770 --> 00:55:30,520
levels-- like lasing
without inversion,

896
00:55:30,520 --> 00:55:32,600
like electromagnetically-induced
transparencies,

897
00:55:32,600 --> 00:55:35,090
for those of you who
have heard about it.

898
00:55:35,090 --> 00:55:37,190
On the other hand,
I can reassure you,

899
00:55:37,190 --> 00:55:40,100
I don't think there's anything
fundamental to be learned

900
00:55:40,100 --> 00:55:42,060
by going to four,
five, and six levels,

901
00:55:42,060 --> 00:55:43,990
so we will stop at three levels.

902
00:55:49,380 --> 00:55:53,140
We can have-- so this
is a single atom.

903
00:55:53,140 --> 00:55:58,980
But we can also have coherence
between different atoms.

904
00:56:03,390 --> 00:56:14,590
And phenomena we want to
discuss is superradiance,

905
00:56:14,590 --> 00:56:20,480
which is very, very much related
to the phenomenon of phase

906
00:56:20,480 --> 00:56:21,350
matching.

907
00:56:21,350 --> 00:56:23,360
Everybody who frequency
doubles a laser

908
00:56:23,360 --> 00:56:26,047
knows about phase matching
and phase matching conditions.

909
00:56:26,047 --> 00:56:28,130
You have to rotate the
crystal or heat the crustal

910
00:56:28,130 --> 00:56:31,220
to the temperature where
the whole crystal--

911
00:56:31,220 --> 00:56:35,160
all the atoms in the
crystal act coherently.

912
00:56:35,160 --> 00:56:38,590
But it's very related
to superradiance.

913
00:56:38,590 --> 00:56:41,990
There is a third aspect of
coherence between atoms which

914
00:56:41,990 --> 00:56:44,880
I will not discuss
this semester,

915
00:56:44,880 --> 00:56:48,419
and this is the situation
of Bose-Einstein condensates

916
00:56:48,419 --> 00:56:49,710
and macroscopic wave functions.

917
00:56:53,000 --> 00:56:55,510
This is covered together
with quantum gases

918
00:56:55,510 --> 00:57:06,240
in-- this is discussed in
the context of quantum gases

919
00:57:06,240 --> 00:57:08,570
in the second part
of the course.

920
00:57:08,570 --> 00:57:14,870
So having these very different
manifestations of coherence,

921
00:57:14,870 --> 00:57:18,300
I want to try now to give you
a definition of coherence.

922
00:57:18,300 --> 00:57:19,860
But it's a bit
difficult because I

923
00:57:19,860 --> 00:57:24,890
want to cover with my
definition all the cases I know.

924
00:57:24,890 --> 00:57:32,120
But with those examples in
mind, for me coherence--

925
00:57:32,120 --> 00:57:33,960
we have the phenomenon
of coherence.

926
00:57:33,960 --> 00:57:41,610
Coherence exists if there
is a well-defined phase.

927
00:57:51,620 --> 00:57:55,850
Well, if we have a phase--
a well-defined phase--

928
00:57:55,850 --> 00:58:01,530
it's always a phase between
quantum mechanical amplitudes.

929
00:58:01,530 --> 00:58:09,490
So we need two or
more amplitudes.

930
00:58:09,490 --> 00:58:12,070
So coherence exists if there
is a well-defined phase

931
00:58:12,070 --> 00:58:21,170
between two or more
amplitudes, but we can only

932
00:58:21,170 --> 00:58:25,980
observe it if those
amplitudes interfere.

933
00:58:25,980 --> 00:58:29,790
And it can be two amplitudes
describing two different atoms,

934
00:58:29,790 --> 00:58:32,720
or it can be two
amplitudes of two

935
00:58:32,720 --> 00:58:35,210
states within the same atom.

936
00:58:35,210 --> 00:58:37,110
But I will point
out to you-- what

937
00:58:37,110 --> 00:58:39,010
is really relevant is
an indistinguishable

938
00:58:39,010 --> 00:58:42,590
ability that those two
amplitudes are involved

939
00:58:42,590 --> 00:58:46,730
in two branches of a process
which has the same final state.

940
00:58:46,730 --> 00:58:48,960
And like in Feynman's
double-slit experiment,

941
00:58:48,960 --> 00:58:52,237
you don't know which
intermediate state was taken.

942
00:58:52,237 --> 00:58:54,070
And that's where coherence
manifests itself.

943
00:58:57,570 --> 00:59:01,770
So that means when we
observe an interference, that

944
00:59:01,770 --> 00:59:06,170
means we obesrve-- and
this is how we read out

945
00:59:06,170 --> 00:59:17,170
of a coherence-- one
observes a physical quantity,

946
00:59:17,170 --> 00:59:20,030
the population in a
certain quantum state,

947
00:59:20,030 --> 00:59:25,710
total electric field
emitted, but this quantity

948
00:59:25,710 --> 00:59:38,210
is usually proportional to the
square of the total amplitude.

949
00:59:42,840 --> 00:59:44,921
And that means we get
an interference term.

950
00:59:51,310 --> 00:59:52,915
So coherence is important.

951
00:59:57,130 --> 01:00:02,120
Let me provide one
additional motivation

952
01:00:02,120 --> 01:00:06,410
that coherence is an
important technique

953
01:00:06,410 --> 01:00:08,300
and important tool
for measurements.

954
01:00:17,600 --> 01:00:27,960
In a way, it's subtle but
trivial at the same time.

955
01:00:27,960 --> 01:00:33,820
Whenever we do spectroscopy,
we are actually

956
01:00:33,820 --> 01:00:39,710
interested in doing a
measurement of energy.

957
01:00:39,710 --> 01:00:42,500
We want to measure
energy levels.

958
01:00:42,500 --> 01:00:47,630
And those energy
levels can tell us

959
01:00:47,630 --> 01:00:54,540
something about magnetic
fields through Zeeman shifts.

960
01:00:54,540 --> 01:00:57,100
If you're addressing
energy levels

961
01:00:57,100 --> 01:01:02,270
in the gravitational field
for atom interferometry,

962
01:01:02,270 --> 01:01:07,080
the energy levels reflect
gravitational fields.

963
01:01:07,080 --> 01:01:10,570
Or if we are not
interested in-- if we

964
01:01:10,570 --> 01:01:14,520
try to eliminate or shield
the atoms from magnetic fields

965
01:01:14,520 --> 01:01:18,210
and we just want to
get the most precision

966
01:01:18,210 --> 01:01:20,380
in a reproducible
energy level, this

967
01:01:20,380 --> 01:01:23,940
is the situational
of atomic clocks.

968
01:01:23,940 --> 01:01:26,194
So pretty much when we
use atomic spectroscopy

969
01:01:26,194 --> 01:01:28,360
for any application, we are
interested in the energy

970
01:01:28,360 --> 01:01:29,680
levels.

971
01:01:29,680 --> 01:01:33,970
But this is very
deeply connected

972
01:01:33,970 --> 01:01:46,520
to coherence and the phase
because the relative phase

973
01:01:46,520 --> 01:01:54,300
between two states
is nothing else

974
01:01:54,300 --> 01:02:03,320
than the time integral
over the energy difference

975
01:02:03,320 --> 01:02:06,810
between two levels.

976
01:02:06,810 --> 01:02:09,360
So the phase evolved
between the two levels

977
01:02:09,360 --> 01:02:12,510
is the difference frequency
times time or time

978
01:02:12,510 --> 01:02:14,500
integrated dt.

979
01:02:14,500 --> 01:02:18,760
So therefore, when we are
talking about coherence,

980
01:02:18,760 --> 01:02:23,560
how can we maintain longer
coherence between energy levels

981
01:02:23,560 --> 01:02:26,940
or how can we create coherence
in three level systems?

982
01:02:26,940 --> 01:02:32,520
This is actually intricately
related to the fact

983
01:02:32,520 --> 01:02:35,671
that we can obtain more precise
information about the energy

984
01:02:35,671 --> 01:02:36,170
levels.

985
01:02:40,320 --> 01:02:48,350
Anyway, this is a very general
introduction to coherence.

986
01:02:48,350 --> 01:02:54,980
Before I talk about
manifestations of coherence,

987
01:02:54,980 --> 01:02:58,790
I have two clicker questions
because the first form

988
01:02:58,790 --> 01:03:01,160
of coherence I
want to discuss is

989
01:03:01,160 --> 01:03:05,400
the coherence of--
the coherence involved

990
01:03:05,400 --> 01:03:08,340
in exciting atoms and
the atom emitting light.

991
01:03:08,340 --> 01:03:11,230
It's related to the spontaneous
emission and scattering

992
01:03:11,230 --> 01:03:12,700
problem.

993
01:03:12,700 --> 01:03:20,900
So just to sort of figure
out what you know already,

994
01:03:20,900 --> 01:03:26,620
let me ask you something
about the nature

995
01:03:26,620 --> 01:03:27,750
of spontaneous emission.

996
01:03:42,870 --> 01:03:46,500
The first choice is
spontaneous emission,

997
01:03:46,500 --> 01:03:55,200
is nothing else than a
unitary transformation--

998
01:03:55,200 --> 01:04:07,560
unitary transformation or
unitary time evolution--

999
01:04:07,560 --> 01:04:13,710
of the wave function
of the total system.

1000
01:04:16,360 --> 01:04:24,340
And option B is
spontaneous emission

1001
01:04:24,340 --> 01:04:29,400
introduces-- well, maybe
through a master equations

1002
01:04:29,400 --> 01:04:39,270
or optical Bloch equation--
introduces a random phase

1003
01:04:39,270 --> 01:04:47,797
into the time evolution of
the quantum mechanical system.

1004
01:04:51,700 --> 01:04:54,150
So what is the picture you
have on spontaneous emission?

1005
01:05:17,820 --> 01:05:18,405
OK.

1006
01:05:18,405 --> 01:05:18,905
Stop.

1007
01:05:27,600 --> 01:05:28,480
OK.

1008
01:05:28,480 --> 01:05:32,900
So at least half of you
better pay attention now.

1009
01:05:32,900 --> 01:05:42,420
So the answer is whenever
you have a system

1010
01:05:42,420 --> 01:05:44,510
and it couples to the
electromagnetic field--

1011
01:05:44,510 --> 01:05:49,320
you just put a system in an
excited state and you wait.

1012
01:05:49,320 --> 01:05:54,080
The 100% unique and correct
answer is the system

1013
01:05:54,080 --> 01:06:02,160
involves with the
following operator,

1014
01:06:02,160 --> 01:06:15,820
and this is the operator we
have discussed many times.

1015
01:06:15,820 --> 01:06:19,780
This is the operator
which completely describes

1016
01:06:19,780 --> 01:06:24,340
the interaction of an atom
with the electromagnetic field.

1017
01:06:24,340 --> 01:06:29,740
And since the whole
system is completely

1018
01:06:29,740 --> 01:06:34,800
described by this
Hamiltonian, the whole system

1019
01:06:34,800 --> 01:06:37,164
undergoes a unitary
time evolution.

1020
01:06:42,600 --> 01:06:45,350
So if you talk about the
total system consisting

1021
01:06:45,350 --> 01:06:47,880
of the photons in all
electromagnetic-- in all

1022
01:06:47,880 --> 01:06:50,420
relevant modes-- and
the atomic system,

1023
01:06:50,420 --> 01:06:52,840
this total
quantomechanical system

1024
01:06:52,840 --> 01:06:55,850
has a unitary time evolution
with this operator.

1025
01:06:55,850 --> 01:06:57,900
And to the best
what our knowledge,

1026
01:06:57,900 --> 01:07:00,150
this is a complete
description which

1027
01:07:00,150 --> 01:07:02,508
covers all aspects
of the system.

1028
01:07:09,620 --> 01:07:10,930
OK.

1029
01:07:10,930 --> 01:07:19,070
But, and this is now
the next question,

1030
01:07:19,070 --> 01:07:23,750
there is a certain randomness
in spontaneous emission

1031
01:07:23,750 --> 01:07:25,450
when we go to the
laboratory and look

1032
01:07:25,450 --> 01:07:28,500
at the spontaneously
emitted photons.

1033
01:07:28,500 --> 01:07:31,080
And this is actually
what I want to work out

1034
01:07:31,080 --> 01:07:33,540
with you in--
maybe even today, I

1035
01:07:33,540 --> 01:07:36,370
think ten minutes
may be enough-- what

1036
01:07:36,370 --> 01:07:40,930
is really the information--
the phase information-- which

1037
01:07:40,930 --> 01:07:43,890
we have in a photon, which has
been spontaneously emitted.

1038
01:07:50,820 --> 01:07:53,430
I know how to phrase--
the randomness

1039
01:07:53,430 --> 01:07:54,670
of spontaneous emission.

1040
01:07:54,670 --> 01:07:56,030
Well, let me write it down.

1041
01:08:05,810 --> 01:08:07,660
First a very big disclaimer.

1042
01:08:07,660 --> 01:08:10,920
This question does not
contradict the first one.

1043
01:08:10,920 --> 01:08:14,290
The fact that we have a unitary
evolution with this operator

1044
01:08:14,290 --> 01:08:19,090
is 100% or 110% true.

1045
01:08:19,090 --> 01:08:23,510
But this operator
will actually lead

1046
01:08:23,510 --> 01:08:26,120
to final states of the
photon field, which

1047
01:08:26,120 --> 01:08:28,170
may not have a specific phase.

1048
01:08:28,170 --> 01:08:30,830
So if you say there is
some intuition that there

1049
01:08:30,830 --> 01:08:35,180
is something going on with
the phase, you may be correct.

1050
01:08:35,180 --> 01:08:39,460
But everything which is
going on with the phase

1051
01:08:39,460 --> 01:08:43,029
is the result of a
unitary time evolution.

1052
01:08:43,029 --> 01:08:46,450
The system itself is
described by an operator,

1053
01:08:46,450 --> 01:08:50,350
by a Schrodinger equation
for the total system.

1054
01:08:50,350 --> 01:08:54,680
But the question I have for
you now is if you detect,

1055
01:08:54,680 --> 01:08:58,279
let's say, the photon emitted
in spontaneous emission,

1056
01:08:58,279 --> 01:09:00,580
the randomness of
spontaneous emission,

1057
01:09:00,580 --> 01:09:06,760
the-- let me call
it loss of phase,

1058
01:09:06,760 --> 01:09:10,970
or at least the diminishment
of the read out.

1059
01:09:10,970 --> 01:09:13,470
There may be situations where
we have a laser beam which has

1060
01:09:13,470 --> 01:09:15,500
a well-defined phase,
photons are scattered,

1061
01:09:15,500 --> 01:09:17,920
and we just cannot retrieve
the phase of the laser beam

1062
01:09:17,920 --> 01:09:19,290
by looking at the photons.

1063
01:09:19,290 --> 01:09:21,270
So this is what I mean here.

1064
01:09:21,270 --> 01:09:24,550
Also, the photons can't come
out of a unitary time evolution.

1065
01:09:24,550 --> 01:09:27,670
My question now is, is the
randomness or this loss

1066
01:09:27,670 --> 01:09:41,430
of phase of spontaneously
emitted photons--

1067
01:09:41,430 --> 01:09:43,109
and now we should,
I want to know

1068
01:09:43,109 --> 01:09:48,969
your best guess--
what is it due to?

1069
01:09:52,300 --> 01:09:56,740
Is it only due to
the-- does it only

1070
01:09:56,740 --> 01:10:00,950
occur, is it only due
to the measurement

1071
01:10:00,950 --> 01:10:02,000
process of the photon?

1072
01:10:11,720 --> 01:10:22,730
Or is it due to
performing sort of

1073
01:10:22,730 --> 01:10:26,920
a partial trace ever-reaching
over certain states?

1074
01:10:33,707 --> 01:10:35,290
So if you're interested
in the photon,

1075
01:10:35,290 --> 01:10:37,890
maybe tracing out the
states of the atom.

1076
01:10:37,890 --> 01:10:43,810
Or ever-reaching over modes
of the electromagnetic field.

1077
01:10:43,810 --> 01:10:48,133
And question C is both
is actually possible.

1078
01:10:54,260 --> 01:10:57,210
So if you look at
spontaneously emitted photons

1079
01:10:57,210 --> 01:10:59,540
and they're not perfectly
phased coherent--

1080
01:10:59,540 --> 01:11:03,140
they're not reproducing
the phase of the laser who

1081
01:11:03,140 --> 01:11:08,460
has created them-- what
is the reason for that?

1082
01:11:08,460 --> 01:11:11,010
Is it always the
fundamental reason

1083
01:11:11,010 --> 01:11:13,430
or is there no
fundamental reason?

1084
01:11:13,430 --> 01:11:19,700
It's only a kind of reason of
ignoring information, taking

1085
01:11:19,700 --> 01:11:20,580
a partial trace?

1086
01:11:35,840 --> 01:11:36,340
OK.

1087
01:11:36,340 --> 01:11:39,951
That's pretty good.

1088
01:11:39,951 --> 01:11:40,450
Yes.

1089
01:11:45,010 --> 01:11:52,040
What I want to emphasize is
that A is very, very important,

1090
01:11:52,040 --> 01:11:53,500
and I want to discuss it now.

1091
01:11:53,500 --> 01:11:55,490
But B is always the case.

1092
01:11:55,490 --> 01:11:58,720
If you ignore the position
where the atom has scattered

1093
01:11:58,720 --> 01:12:00,810
the light-- if
atoms scatter light

1094
01:12:00,810 --> 01:12:03,600
and they are wavelengths
apart, then you

1095
01:12:03,600 --> 01:12:06,420
have maybe optical path
length differences.

1096
01:12:06,420 --> 01:12:08,860
The photon from the
laser hits an atom here.

1097
01:12:08,860 --> 01:12:10,300
It goes to your detector.

1098
01:12:10,300 --> 01:12:12,750
But from another
atom, the photon

1099
01:12:12,750 --> 01:12:16,610
has accumulated a different
spatial phase into the IKR.

1100
01:12:16,610 --> 01:12:19,730
And, of course, what you get
here is the random phase.

1101
01:12:19,730 --> 01:12:21,620
This is why quite
often when we scatter

1102
01:12:21,620 --> 01:12:23,480
light for many,
many atoms, we're

1103
01:12:23,480 --> 01:12:25,340
not even asking for the phase.

1104
01:12:25,340 --> 01:12:28,680
We say, one atom scatters
light-- really scatters light

1105
01:12:28,680 --> 01:12:30,610
at a certain intensity, i1.

1106
01:12:30,610 --> 01:12:33,320
And n atoms-- well, we
get n times the light.

1107
01:12:33,320 --> 01:12:35,800
You immediately perform
an incoherence sum

1108
01:12:35,800 --> 01:12:38,280
because you sort of
know deep in your heart

1109
01:12:38,280 --> 01:12:40,710
that there won't be
any interference.

1110
01:12:40,710 --> 01:12:42,645
So it's always
possible, of course,

1111
01:12:42,645 --> 01:12:46,220
to lose the phase
by not controlling

1112
01:12:46,220 --> 01:12:50,070
every aspect of your experiment.

1113
01:12:50,070 --> 01:12:51,780
So B is always possible.

1114
01:12:57,170 --> 01:13:00,450
But for so many reasons that
I don't want to discuss it,

1115
01:13:00,450 --> 01:13:03,920
but the measurement
process is very relevant.

1116
01:13:03,920 --> 01:13:07,810
And so, therefore, I would
have answered C here.

1117
01:13:11,320 --> 01:13:15,000
But let's now discuss the
most fundamental situation.

1118
01:13:15,000 --> 01:13:16,800
And the most
fundamental situation

1119
01:13:16,800 --> 01:13:21,105
is we take our favorite
atom with infinite mass--

1120
01:13:21,105 --> 01:13:23,490
no Doppler shift,
we just put it.

1121
01:13:23,490 --> 01:13:26,370
And we put it in a
cavity that it can only

1122
01:13:26,370 --> 01:13:29,690
interact with a single mode
of the electromagnetic field.

1123
01:13:29,690 --> 01:13:31,610
So this is a
fundamental situation.

1124
01:13:31,610 --> 01:13:33,660
And a lot of the
general situations,

1125
01:13:33,660 --> 01:13:37,130
you get by just summing
up over many modes,

1126
01:13:37,130 --> 01:13:39,980
summing up over many
positions of the atom,

1127
01:13:39,980 --> 01:13:42,130
introducing Doppler
effects and all that.

1128
01:13:42,130 --> 01:13:43,740
It just messes things up.

1129
01:13:43,740 --> 01:13:47,360
And this is more in
the spirit of answer B,

1130
01:13:47,360 --> 01:13:51,170
that you perform partial trace
and average over many states.

1131
01:13:51,170 --> 01:13:52,940
But let's now
pinpoint what I think

1132
01:13:52,940 --> 01:13:55,870
is intellectually the
most important one,

1133
01:13:55,870 --> 01:14:00,310
the pure situation
where an atom is just

1134
01:14:00,310 --> 01:14:03,150
talking to a single mode.

1135
01:14:03,150 --> 01:14:22,650
So let us assume that we have an
atom which starts in the ground

1136
01:14:22,650 --> 01:14:26,488
state, but now we
excite the atom.

1137
01:14:31,080 --> 01:14:37,170
And we excite it
by a short pulse.

1138
01:14:37,170 --> 01:14:38,850
And this can be
a pulse which has

1139
01:14:38,850 --> 01:14:41,860
a pulse aimed
between zero and pi.

1140
01:14:41,860 --> 01:14:45,860
And depending what
the pulse angle is,

1141
01:14:45,860 --> 01:14:50,660
it will admix-- we
have the ground state

1142
01:14:50,660 --> 01:14:55,210
and it will admix something
of the excited state.

1143
01:14:55,210 --> 01:14:59,990
And in case of a pi pulse, we
have 100% in the excited state.

1144
01:14:59,990 --> 01:15:07,180
So our atomic wave
function is this.

1145
01:15:07,180 --> 01:15:09,470
The excited state--
let's just assume

1146
01:15:09,470 --> 01:15:10,990
the photons are in resonance.

1147
01:15:10,990 --> 01:15:14,500
We know how to deal with
off resonant lasers.

1148
01:15:14,500 --> 01:15:19,370
So therefore, the
phase evolution

1149
01:15:19,370 --> 01:15:22,500
is e to the i omega naught t.

1150
01:15:22,500 --> 01:15:25,150
But now, and this is
what coherence is about,

1151
01:15:25,150 --> 01:15:28,910
there is a very specific phase.

1152
01:15:28,910 --> 01:15:34,370
And this phase phi
comes from the laser.

1153
01:15:34,370 --> 01:15:38,320
If you excite the
atom with a laser beam

1154
01:15:38,320 --> 01:15:43,860
but it has a phase shift,
then the atomic wave function

1155
01:15:43,860 --> 01:15:48,080
is phase shifted because
every amplitude you admixed

1156
01:15:48,080 --> 01:15:50,520
into the ground state in
form of an excited state

1157
01:15:50,520 --> 01:15:53,190
was driven by the operator--
the dipole operator,

1158
01:15:53,190 --> 01:15:55,500
e-- and e, the
electric field, has

1159
01:15:55,500 --> 01:15:58,120
the phase of the laser beam.

1160
01:15:58,120 --> 01:16:02,230
Sure, there may be-- and kind
of all other trivial factors

1161
01:16:02,230 --> 01:16:03,280
I've set to one here.

1162
01:16:03,280 --> 01:16:06,730
But there is the phase of
the laser, which directly

1163
01:16:06,730 --> 01:16:11,850
is imprinted into the
phase of the wave function.

1164
01:16:11,850 --> 01:16:12,350
OK.

1165
01:16:12,350 --> 01:16:14,920
So this is what the
atom does-- what

1166
01:16:14,920 --> 01:16:16,310
the laser beam does to the atom.

1167
01:16:16,310 --> 01:16:18,770
So we have now an atom
which is partially excited.

1168
01:16:18,770 --> 01:16:21,420
And it carries is an imprint
of the phase of the laser.

1169
01:16:24,210 --> 01:16:29,910
And now after the
laser pulse is over,

1170
01:16:29,910 --> 01:16:33,390
the photonic part of the
wave function is a vacuum.

1171
01:16:33,390 --> 01:16:35,690
We have no photon in our cavity.

1172
01:16:39,840 --> 01:16:44,040
And now we wait and we
allow spontaneous emission.

1173
01:16:44,040 --> 01:16:46,320
And spontaneous
emission is nothing else

1174
01:16:46,320 --> 01:16:49,600
than the time evolution
with the operator I just

1175
01:16:49,600 --> 01:16:50,874
discussed with you.

1176
01:16:54,360 --> 01:17:01,580
So after spontaneous
emission, well, one

1177
01:17:01,580 --> 01:17:04,800
is we know for sure the
atom is in the ground state.

1178
01:17:20,730 --> 01:17:22,820
Let me write down the result.

1179
01:17:22,820 --> 01:17:25,130
If we apply the
operator which couples

1180
01:17:25,130 --> 01:17:31,330
to the electromagnetic
field, and we

1181
01:17:31,330 --> 01:17:33,880
assume only
co-rotating terms here.

1182
01:17:33,880 --> 01:17:36,070
Let's just neglect
counter-rotating terms,

1183
01:17:36,070 --> 01:17:39,750
which can be-- which in
near-resonants are irrelevant.

1184
01:17:39,750 --> 01:17:43,230
What happens is we are now
propagating this wave function.

1185
01:17:43,230 --> 01:17:47,610
And the ground state-- with
the ground state of the photon,

1186
01:17:47,610 --> 01:17:49,670
does nothing.

1187
01:17:49,670 --> 01:17:53,400
However, the excited
state with zero photon,

1188
01:17:53,400 --> 01:17:56,140
we discussed that excited
state with zero photon

1189
01:17:56,140 --> 01:17:57,800
will actually do
Rabi oscillation.

1190
01:17:57,800 --> 01:18:00,290
Ground state with one
photon, excited state

1191
01:18:00,290 --> 01:18:01,500
with zero photons.

1192
01:18:01,500 --> 01:18:04,470
And so we have now our
knowledge from the vacuum Rabi

1193
01:18:04,470 --> 01:18:08,050
oscillation that this part of
the wave function does nothing.

1194
01:18:08,050 --> 01:18:10,290
This part of the wave
function undergoes

1195
01:18:10,290 --> 01:18:14,500
single photon vacuum
Rabi oscillations.

1196
01:18:14,500 --> 01:18:18,860
And if we start out with the
superposition of ground zero

1197
01:18:18,860 --> 01:18:22,140
and excited zero, well, this
is a superposition principle

1198
01:18:22,140 --> 01:18:23,730
of quantum mechanics.

1199
01:18:23,730 --> 01:18:27,480
We can just propagate this part,
and we can propagate that part.

1200
01:18:27,480 --> 01:18:31,360
So what I suggest is when we,
really at the fundamental level

1201
01:18:31,360 --> 01:18:33,370
now, discuss
spontaneous emission,

1202
01:18:33,370 --> 01:18:39,000
we allow this
part-- excited atom,

1203
01:18:39,000 --> 01:18:44,500
empty cavity-- to undergo half
a vacuum Rabi oscillation.

1204
01:18:44,500 --> 01:18:47,450
And then the excited state
is in the ground state

1205
01:18:47,450 --> 01:18:52,710
and the photon state
has one photon.

1206
01:18:52,710 --> 01:18:55,100
It's a completely
coherent Rabi oscillation.

1207
01:18:55,100 --> 01:18:59,150
I just allow half
a cycle to evolve.

1208
01:18:59,150 --> 01:19:05,320
And the result of that is, well,
nothing happened to this part.

1209
01:19:05,320 --> 01:19:08,700
And the Rabi
oscillations have now

1210
01:19:08,700 --> 01:19:10,260
taken us to the
one photon state.

1211
01:19:18,690 --> 01:19:25,390
And it has just
swapped excited zero

1212
01:19:25,390 --> 01:19:26,730
to ground state one photon.

1213
01:19:32,880 --> 01:19:40,720
So let me write down something
which is really remarkable,

1214
01:19:40,720 --> 01:19:48,450
and then we discuss on
Monday-- next Monday--

1215
01:19:48,450 --> 01:19:51,130
we discuss how we would
really measure the phase.

1216
01:19:51,130 --> 01:19:55,090
But just look at the two
expressions I've underlined.

1217
01:19:55,090 --> 01:19:59,920
And this tells us
that the quantum state

1218
01:19:59,920 --> 01:20:02,650
of the atom as a two-level
system-- ground excited

1219
01:20:02,650 --> 01:20:04,510
state with all the
phase factors--

1220
01:20:04,510 --> 01:20:08,930
has been now exactly
matched on the quantum

1221
01:20:08,930 --> 01:20:11,070
state of the cavity.

1222
01:20:11,070 --> 01:20:13,140
So if you regard ground
and excited state

1223
01:20:13,140 --> 01:20:16,450
as a two-level system, every
quantum mechanical subtlety

1224
01:20:16,450 --> 01:20:21,230
of the atomic system
has now disappeared.

1225
01:20:21,230 --> 01:20:22,520
It's in the ground state.

1226
01:20:22,520 --> 01:20:26,160
But everything
which was coherent,

1227
01:20:26,160 --> 01:20:29,070
which was a phase which was
interesting about the atomic

1228
01:20:29,070 --> 01:20:32,220
system, has been transferred
to the photon field.

1229
01:20:32,220 --> 01:20:33,101
Yes?

1230
01:20:33,101 --> 01:20:35,887
AUDIENCE: Do you want to
have an alpha e there also?

1231
01:20:35,887 --> 01:20:36,970
PROFESSOR: Oh yes, please.

1232
01:20:36,970 --> 01:20:38,290
Thank you.

1233
01:20:38,290 --> 01:20:41,035
When I said everything,
I meant everything.

1234
01:20:45,770 --> 01:20:46,270
Yes.

1235
01:20:51,640 --> 01:20:52,260
Yes.

1236
01:20:52,260 --> 01:20:52,760
OK.

1237
01:20:52,760 --> 01:20:54,530
So let me write that
down and then we

1238
01:20:54,530 --> 01:20:56,275
have-- so this is what I meant.

1239
01:20:59,130 --> 01:21:02,870
This is the most fundamental
aspect of spontaneous emission,

1240
01:21:02,870 --> 01:21:11,020
that the quantum
state of the atom

1241
01:21:11,020 --> 01:21:15,140
has been perfectly matched.

1242
01:21:15,140 --> 01:21:22,079
Perfectly mapped onto
the photon field.

1243
01:21:29,750 --> 01:21:35,770
And the one thing we have
to discuss on Wednesday

1244
01:21:35,770 --> 01:21:37,820
is the whole of the phase phi.

1245
01:21:40,630 --> 01:21:46,520
Phi-- we started out by phi
being the phase of the laser.

1246
01:21:46,520 --> 01:21:48,290
And if the laser is
in a coherent state,

1247
01:21:48,290 --> 01:21:50,700
I will talk about on Wednesday,
in a homodyne measurement

1248
01:21:50,700 --> 01:21:53,100
we can measure phi to
any accuracy you want.

1249
01:21:53,100 --> 01:21:55,140
We can determine the
phase of the laser

1250
01:21:55,140 --> 01:21:57,160
in a homodyne or
heterodyne experiment.

1251
01:21:57,160 --> 01:22:02,220
This phase phi has been
now perfectly imprinted

1252
01:22:02,220 --> 01:22:04,640
into a two-level
system for the atom.

1253
01:22:04,640 --> 01:22:08,790
And now it appears mapped
into a two-level system

1254
01:22:08,790 --> 01:22:10,810
for the photons--
the two-level system

1255
01:22:10,810 --> 01:22:13,990
between zero photons
and one photons.

1256
01:22:13,990 --> 01:22:16,830
But if we are now
doing a measurement

1257
01:22:16,830 --> 01:22:19,910
either on the atomic system
or on the photonic system,

1258
01:22:19,910 --> 01:22:26,500
we are limited in the accuracy
at which we can retrieve phi.

1259
01:22:26,500 --> 01:22:28,730
And this is what we want
to discuss on Wednesday.

1260
01:22:28,730 --> 01:22:32,735
And this is what I referred
to as the fundamental limit

1261
01:22:32,735 --> 01:22:35,430
of spontaneous emission
because we have not

1262
01:22:35,430 --> 01:22:37,010
lost any coherence here.

1263
01:22:37,010 --> 01:22:39,790
It's just if the phase
is only imprinted

1264
01:22:39,790 --> 01:22:42,810
in one particle--
one particle quantum

1265
01:22:42,810 --> 01:22:45,320
physics sets us a limitation.

1266
01:22:45,320 --> 01:22:50,141
Oh well, we can read
out the phase phi.

1267
01:22:50,141 --> 01:22:50,640
OK.

1268
01:22:50,640 --> 01:22:53,710
Any question?

1269
01:22:53,710 --> 01:22:55,860
To be continued on Monday.