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PROFESSOR: We're
going to continue

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00:00:21,935 --> 00:00:26,520
our discussion of correlation
functions, and in particular,

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00:00:26,520 --> 00:00:28,980
g2 of 0.

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00:00:28,980 --> 00:00:35,780
If you think I'm focusing a
lot on one quantity, g2 of 0,

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00:00:35,780 --> 00:00:37,290
that's correct.

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00:00:37,290 --> 00:00:39,230
But I have to tell
you, I have been

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involved in so many
discussions with people

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00:00:42,400 --> 00:00:46,670
who looked at g2 of 0 with
both Einstein condensates

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00:00:46,670 --> 00:00:49,890
when Bose-Einstein condensates
in [INAUDIBLE] were discovered

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00:00:49,890 --> 00:00:53,680
here at MIT by [INAUDIBLE]
and collaborators.

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00:00:53,680 --> 00:01:00,230
There was some mystery about, in
essence, the g2 of 0 function.

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00:01:00,230 --> 00:01:04,250
So a lot of physics
actually goes into it.

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00:01:04,250 --> 00:01:06,610
And I've learned a lot
from my own research

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00:01:06,610 --> 00:01:08,030
and through discussions.

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00:01:08,030 --> 00:01:11,830
And in this unit, I
want to summarize it.

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00:01:11,830 --> 00:01:16,240
So the confusion sometimes
comes because people

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00:01:16,240 --> 00:01:18,920
use classical pictures,
quantum pictures,

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00:01:18,920 --> 00:01:21,010
and they may not
see the relation.

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00:01:21,010 --> 00:01:23,435
So let's continue
the discussion.

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00:01:29,990 --> 00:01:35,300
I want to go through
four different views

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00:01:35,300 --> 00:01:37,550
of the same physics
that you will

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00:01:37,550 --> 00:01:40,120
recognize that
they're interrelated.

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00:01:40,120 --> 00:01:41,910
The first one, just
to repeat that,

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00:01:41,910 --> 00:01:45,460
is if you have light which
is Gaussian intensity

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00:01:45,460 --> 00:01:48,690
fluctuations, light
from random sources.

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00:01:48,690 --> 00:01:51,330
I think if you would look
at the twinkle of starlight,

34
00:01:51,330 --> 00:01:55,120
if you look at the
light from a light bulb,

35
00:01:55,120 --> 00:01:56,970
because of the
central limit theorem,

36
00:01:56,970 --> 00:02:00,240
you will find that the intensity
that the electric field

37
00:02:00,240 --> 00:02:02,575
has a Gaussian distribution,
and the intensity

38
00:02:02,575 --> 00:02:04,870
in exponential distribution.

39
00:02:04,870 --> 00:02:12,540
And if you then ask, what is
g2 of 0, you'll find it's 2.

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00:02:12,540 --> 00:02:15,480
Because for an
exponential distribution,

41
00:02:15,480 --> 00:02:17,240
the average of the
intensity squared

42
00:02:17,240 --> 00:02:20,660
is two times the intensity,
the average of the-- what

43
00:02:20,660 --> 00:02:21,160
did I say.

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00:02:21,160 --> 00:02:23,470
This average of the
squared intensity

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00:02:23,470 --> 00:02:28,695
is two times the average
of the intensity squared.

46
00:02:33,470 --> 00:02:36,560
Before I forget it, one
student asked me after class,

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00:02:36,560 --> 00:02:41,170
I'm discussing g2 of 0,
but what about g2 of tau?

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00:02:41,170 --> 00:02:44,260
Well, my understanding is
that almost all of the physics

49
00:02:44,260 --> 00:02:46,045
is in g2 of 0.

50
00:02:46,045 --> 00:02:51,640
And there is a simple number,
1 or 2, in those special cases.

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00:02:51,640 --> 00:02:55,340
What happens is, if you have
completely independent sources,

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00:02:55,340 --> 00:02:57,600
if you completely
independent statistics,

53
00:02:57,600 --> 00:03:02,335
then the probability of
finding a second particle,

54
00:03:02,335 --> 00:03:04,700
a second photon, is
independent of the first.

55
00:03:04,700 --> 00:03:08,670
So it means g2 of
tau would be unity.

56
00:03:08,670 --> 00:03:11,730
This is the case for an
uncoordinated system.

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00:03:11,730 --> 00:03:17,210
But if you have g2 of 0 which is
higher, this means correlation.

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00:03:17,210 --> 00:03:19,100
This means some
form of coherence.

59
00:03:19,100 --> 00:03:22,520
And usually, the
correlated value simply

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00:03:22,520 --> 00:03:26,840
decays to the uncorrelated value
with a characteristic coherence

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00:03:26,840 --> 00:03:27,730
time.

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00:03:27,730 --> 00:03:29,830
And this characteristic
coherence time

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00:03:29,830 --> 00:03:32,880
can be inhomogeneous, technical.

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00:03:32,880 --> 00:03:37,400
It can be limited by the Fourier
transform of a light pulse,

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00:03:37,400 --> 00:03:39,610
the monochromaticity,
the bend which

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00:03:39,610 --> 00:03:41,990
delta omega is 1 over tau.

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00:03:41,990 --> 00:03:45,450
So it has all of
the usual suspects

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00:03:45,450 --> 00:03:48,580
for what limits
the coherence time.

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00:03:48,580 --> 00:03:51,230
So therefore in
general, we're not

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00:03:51,230 --> 00:03:54,740
learning something fundamental
in looking at g2 of tau.

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00:03:54,740 --> 00:03:58,400
We're just learning the usual
spiel about correlation times

72
00:03:58,400 --> 00:03:59,260
and coherence times.

73
00:04:06,580 --> 00:04:12,560
The second angle I gave you
with the brief interference

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00:04:12,560 --> 00:04:15,720
is something which
is deeply insightful.

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00:04:15,720 --> 00:04:17,430
It shows you the following.

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00:04:17,430 --> 00:04:22,029
If you have a plane wave,
the intensity is constant.

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00:04:22,029 --> 00:04:25,640
If you have two plane
waves which interfere,

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00:04:25,640 --> 00:04:28,540
you create density fluctuations,
because the interference

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00:04:28,540 --> 00:04:31,660
pattern can be constructive
and can be destructive.

80
00:04:31,660 --> 00:04:34,630
So therefore, if the
intensity fluctuates,

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00:04:34,630 --> 00:04:41,360
then i square average is
no longer i average square.

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00:04:41,360 --> 00:04:45,060
And what happens is because
of the nature of interference,

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00:04:45,060 --> 00:04:53,240
because the average of
cosine square kx is-- well,

84
00:04:53,240 --> 00:04:55,570
there are factors of 1/2
and 2 which just come out

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00:04:55,570 --> 00:04:57,580
of cosine and sine
square function.

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00:04:57,580 --> 00:05:01,150
You get exactly g2 of 2.

87
00:05:01,150 --> 00:05:03,780
But that should already
tell you something,

88
00:05:03,780 --> 00:05:08,860
which will be important for both
quantum particles, and light,

89
00:05:08,860 --> 00:05:10,600
and the classic limit.

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00:05:10,600 --> 00:05:16,230
The moment if you have light or
particles in one single mode,

91
00:05:16,230 --> 00:05:18,410
you can't have any
form of interference.

92
00:05:18,410 --> 00:05:20,150
Only two modes can interfere.

93
00:05:20,150 --> 00:05:23,940
And you will always
get a g2 function of 1.

94
00:05:23,940 --> 00:05:26,490
But we'll come to
that in a second.

95
00:05:26,490 --> 00:05:29,328
Any questions at that point?

96
00:05:29,328 --> 00:05:29,828
Yes.

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00:05:29,828 --> 00:05:32,369
AUDIENCE: I thought that when
we talked about the g2 function

98
00:05:32,369 --> 00:05:36,946
several weeks ago, you said
that if we had a single mode

99
00:05:36,946 --> 00:05:39,850
thermal light, it would
have the g2 function of 2.

100
00:05:45,520 --> 00:05:48,000
And that's therefore,
all we were

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00:05:48,000 --> 00:05:51,370
using for lasers when we
used them in experiments

102
00:05:51,370 --> 00:05:53,700
is the fact that it's single
mode, not the coherence

103
00:05:53,700 --> 00:05:54,885
in the g2 function.

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00:06:00,110 --> 00:06:04,090
PROFESSOR: You have a good--
that's a good point, Cory.

105
00:06:04,090 --> 00:06:07,400
Let's just try to
connect the two pictures.

106
00:06:20,750 --> 00:06:22,430
Slowly, slowly.

107
00:06:22,430 --> 00:06:24,660
What did we discuss?

108
00:06:24,660 --> 00:06:27,110
This question is great for
me to repeat something.

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00:06:27,110 --> 00:06:30,070
When we looked at
black body radiation,

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00:06:30,070 --> 00:06:32,050
I wrote down the
partition function.

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00:06:32,050 --> 00:06:35,650
And I said I'm only
considering a single mode.

112
00:06:35,650 --> 00:06:43,670
And then, we have of course
in thermal equilibrium

113
00:06:43,670 --> 00:06:46,680
a Boltzmann type, or
Bose-Einstein type,

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00:06:46,680 --> 00:06:50,890
distribution of finding
certain numbers,

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00:06:50,890 --> 00:06:55,800
number of photons
populating the single mode.

116
00:06:55,800 --> 00:07:00,915
And that resulted in
a g2 function of 2.

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00:07:11,792 --> 00:07:13,958
AUDIENCE: It's not really
single mode though, right?

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00:07:13,958 --> 00:07:16,025
Because you're summing
over many frequencies.

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00:07:16,025 --> 00:07:17,650
PROFESSOR: No, it
was single frequency.

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00:07:24,710 --> 00:07:26,190
I have to think about it.

121
00:07:26,190 --> 00:07:28,090
That's a really good question.

122
00:07:28,090 --> 00:07:32,562
Right now, I would quickly
say, in thermal equilibrium--

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00:07:32,562 --> 00:07:34,895
I'll give you partial answer,
and when I think about it,

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00:07:34,895 --> 00:07:38,960
I'll give you full
answer on Friday

125
00:07:38,960 --> 00:07:41,430
at the beginning
of the next class--

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00:07:41,430 --> 00:07:43,080
I think would be
calculated where

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00:07:43,080 --> 00:07:46,150
the thermal fluctuations
in the intensity.

128
00:07:46,150 --> 00:07:48,515
And the intensity fluctuations,
and those intensity

129
00:07:48,515 --> 00:07:49,056
fluctuations.

130
00:07:53,960 --> 00:07:57,585
Those intensity fluctuations
result in a g2 function of 2.

131
00:07:57,585 --> 00:07:58,210
That's correct.

132
00:08:01,500 --> 00:08:04,780
Maybe what I want
you to say here

133
00:08:04,780 --> 00:08:09,470
is, if you have two modes
which have all maybe

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00:08:09,470 --> 00:08:11,720
single occupation or
constant occupation,

135
00:08:11,720 --> 00:08:15,340
and they interfere, then they
create a spatial interference

136
00:08:15,340 --> 00:08:18,930
pattern if the phase between
the two modes is random,

137
00:08:18,930 --> 00:08:22,340
which gives rise to
additional fluctuation.

138
00:08:22,340 --> 00:08:27,890
So most likely, the two
fluctuations are independent.

139
00:08:27,890 --> 00:08:30,470
One is the fluctuations
because of the interference

140
00:08:30,470 --> 00:08:32,630
of two modes.

141
00:08:32,630 --> 00:08:37,000
But when I say the single
mode has a g2 function of 1,

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00:08:37,000 --> 00:08:40,565
I meant actually a single
mode with constant amplitude.

143
00:08:43,260 --> 00:08:45,964
I'll double check and give you a
more complete answer on Friday.

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00:08:45,964 --> 00:08:46,630
Other questions?

145
00:08:50,770 --> 00:08:51,270
OK.

146
00:08:58,100 --> 00:09:07,255
The third view is classical
versus quantum statistics.

147
00:09:24,610 --> 00:09:28,200
And I think this really
shows that we need at least

148
00:09:28,200 --> 00:09:31,515
some quantumness to
find correlations.

149
00:09:35,850 --> 00:09:39,070
Let's assume we
have n particles.

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00:09:39,070 --> 00:09:42,105
And we have N possible states.

151
00:09:49,180 --> 00:09:57,060
And if it's a
classical system, let's

152
00:09:57,060 --> 00:10:04,270
assume we want to find one
particle in a certain state.

153
00:10:04,270 --> 00:10:07,220
We have a big box and want
to-- we have a big box,

154
00:10:07,220 --> 00:10:08,990
have a small
sub-volume, and say what

155
00:10:08,990 --> 00:10:11,440
is the probability of
finding one particle here?

156
00:10:15,550 --> 00:10:21,960
So this probability
to find one particle

157
00:10:21,960 --> 00:10:31,190
is P1 is small n over big
N. The probability to find

158
00:10:31,190 --> 00:10:31,900
two particles.

159
00:10:34,550 --> 00:10:41,770
If you have non-interacting
classical particles,

160
00:10:41,770 --> 00:10:43,670
well, it's simply P1 squared.

161
00:10:46,880 --> 00:10:49,000
Independent classical
particles, they

162
00:10:49,000 --> 00:10:50,560
don't care what they are doing.

163
00:10:50,560 --> 00:10:52,810
You grab into a
sub-volume, which

164
00:10:52,810 --> 00:10:54,220
is maybe phase space cell.

165
00:10:54,220 --> 00:10:55,770
And therefore,
your quantum state,

166
00:10:55,770 --> 00:10:58,360
you have probability
P of finding one.

167
00:10:58,360 --> 00:11:01,820
But since each particle
moves around independently,

168
00:11:01,820 --> 00:11:06,180
the probability to find two
particles is just P squared.

169
00:11:06,180 --> 00:11:08,566
It's a little bit like if
you toss two coins, one

170
00:11:08,566 --> 00:11:10,190
with your left and
one with your right,

171
00:11:10,190 --> 00:11:11,690
and you ask what
is the probability

172
00:11:11,690 --> 00:11:15,285
to find a head or
tail, and you simply

173
00:11:15,285 --> 00:11:16,410
multiply the probabilities.

174
00:11:18,940 --> 00:11:24,880
But now, following reasonings
which Bose and Einstein

175
00:11:24,880 --> 00:11:31,180
introduced, you want to use
counting statistics, which

176
00:11:31,180 --> 00:11:33,860
takes into account the
indistinguishability

177
00:11:33,860 --> 00:11:34,595
of particles.

178
00:11:39,870 --> 00:11:45,490
So if we go from the
classical distribution

179
00:11:45,490 --> 00:11:53,360
to the distribution of
indistinguishable particles,

180
00:11:53,360 --> 00:12:08,190
then the classical probability
P 1 is reduced by n factorial,

181
00:12:08,190 --> 00:12:19,040
because we are not--
let me put it this way.

182
00:12:19,040 --> 00:12:20,805
The number-- I don't
want to go through

183
00:12:20,805 --> 00:12:23,040
a mathematical argument.

184
00:12:23,040 --> 00:12:33,240
If you grab one particle,
it can be the probability

185
00:12:33,240 --> 00:12:38,880
to find one particle
is represented

186
00:12:38,880 --> 00:12:42,090
by a microconfiguration
classically.

187
00:12:42,090 --> 00:12:45,630
And each configuration
when you permute particles

188
00:12:45,630 --> 00:12:47,500
counts as independent.

189
00:12:47,500 --> 00:12:50,180
But if you have quantum
indistinguishability,

190
00:12:50,180 --> 00:12:53,750
you're not counting
permutations as

191
00:12:53,750 --> 00:12:55,580
an independent configuration.

192
00:12:55,580 --> 00:12:58,325
And therefore, you have a
reduction by n factorial.

193
00:13:02,280 --> 00:13:05,960
If you look at the probability
of finding two particles,

194
00:13:05,960 --> 00:13:09,300
the reduction is-- this is
just the counting statistics

195
00:13:09,300 --> 00:13:12,100
n factorial by 2.

196
00:13:12,100 --> 00:13:20,500
So therefore, you will find
that classically, that quantum

197
00:13:20,500 --> 00:13:33,240
mechanically for
bosons, is two times.

198
00:13:46,990 --> 00:13:50,930
The probability to find two
bosons in one quantum state

199
00:13:50,930 --> 00:13:56,260
is two times the probability
squared to find one boson.

200
00:13:56,260 --> 00:13:57,845
So this is just
counting statistics.

201
00:14:03,990 --> 00:14:09,870
This can be applied
to conditions

202
00:14:09,870 --> 00:14:12,350
in a Bose-Einstein
in a Bose gas.

203
00:14:23,000 --> 00:14:27,730
When you have inelastic
collisions, spin relaxation,

204
00:14:27,730 --> 00:14:34,620
dipolar collisions,
two body collisions,

205
00:14:34,620 --> 00:14:37,560
two-body collisions
have a rate gamma 2,

206
00:14:37,560 --> 00:14:41,280
which is proportional to
the probability of finding

207
00:14:41,280 --> 00:14:45,297
two particles at the same
time at the same location.

208
00:14:45,297 --> 00:14:46,755
And that brings in
the g2 function.

209
00:14:49,300 --> 00:14:58,350
Whereas three-body
collisions, gamma 3,

210
00:14:58,350 --> 00:15:01,300
reflects the third
order correlation

211
00:15:01,300 --> 00:15:06,890
function g3, which is
defined in an analogous way.

212
00:15:06,890 --> 00:15:18,730
So if you now compare
at the same density

213
00:15:18,730 --> 00:15:25,060
the two-body rate coefficient,
like the same density

214
00:15:25,060 --> 00:15:29,340
means we are looking at the
rate of two-body collisions.

215
00:15:29,340 --> 00:15:35,630
And we compare between
two-body collision

216
00:15:35,630 --> 00:15:38,500
in a thermal cloud, which
is a g2 factor of two,

217
00:15:38,500 --> 00:15:41,780
a thermal cloud of bosons.

218
00:15:41,780 --> 00:15:43,660
Whereas a Bose-Einstein
condensate

219
00:15:43,660 --> 00:15:47,330
is a constant occupation
of particles in one mode

220
00:15:47,330 --> 00:15:51,670
and has therefore a g2 factor
of 1, exactly as [INAUDIBLE]

221
00:15:51,670 --> 00:15:53,500
of the Bose-Einstein
condensate is

222
00:15:53,500 --> 00:15:56,960
for particles for meta waves,
but the laser is for light.

223
00:15:59,560 --> 00:16:03,040
Whereas, if you look at
three-body collisions,

224
00:16:03,040 --> 00:16:05,730
those who do experiments with
Bose-Einstein condensates

225
00:16:05,730 --> 00:16:07,930
will know that
usually the lifetime

226
00:16:07,930 --> 00:16:09,589
of Bose-Einstein
condensates is limited

227
00:16:09,589 --> 00:16:10,630
by three-body collisions.

228
00:16:13,670 --> 00:16:18,290
Well, I didn't derive
it, but actually

229
00:16:18,290 --> 00:16:20,880
yes, we have the Gaussian
statistics for randomness.

230
00:16:20,880 --> 00:16:26,380
For Gaussian statistics, the
n-body collision function

231
00:16:26,380 --> 00:16:29,580
has a factor of n factorial.

232
00:16:29,580 --> 00:16:31,700
So therefore,
three-body collisions

233
00:16:31,700 --> 00:16:36,300
scale with 3 factorial, and
two-body collisions with 2

234
00:16:36,300 --> 00:16:37,930
factorial.

235
00:16:37,930 --> 00:16:40,090
So in other words, you
have a thermal cloud,

236
00:16:40,090 --> 00:16:43,180
you have a Bose-Einstein
condensate at the same density.

237
00:16:43,180 --> 00:16:45,470
But what matters for
two-body collision

238
00:16:45,470 --> 00:16:47,790
is not the average density.

239
00:16:47,790 --> 00:16:50,240
It is the average of n square.

240
00:16:50,240 --> 00:16:52,750
And the fluctuations
in the thermal cloud

241
00:16:52,750 --> 00:16:56,110
because of the g2 function
are two times enhanced

242
00:16:56,110 --> 00:16:59,290
compared to
Bose-Einstein condensate.

243
00:16:59,290 --> 00:17:02,130
And therefore, you find
that at the same density,

244
00:17:02,130 --> 00:17:03,047
you have more loss.

245
00:17:03,047 --> 00:17:05,380
You have a higher rate of two
and three-body collisions.

246
00:17:22,690 --> 00:17:30,280
Some of this was not--
became better understood

247
00:17:30,280 --> 00:17:42,530
soon after Bose-Einstein
condensates were realized.

248
00:17:42,530 --> 00:17:45,950
The Boulder group studied
three-body collisions.

249
00:17:45,950 --> 00:17:54,350
And for two-body
collisions, myself

250
00:17:54,350 --> 00:17:57,210
in the post-doc
clarified the situation

251
00:17:57,210 --> 00:18:01,960
that even the mean field energy,
which everybody had measured

252
00:18:01,960 --> 00:18:06,490
before, requires two particles
to interact with short range

253
00:18:06,490 --> 00:18:09,930
interaction is therefore
proportional to g2.

254
00:18:09,930 --> 00:18:12,320
And therefore, when people
had determined the mean field

255
00:18:12,320 --> 00:18:14,870
energy of a condensate
without really knowing

256
00:18:14,870 --> 00:18:16,752
they had already
determined the g2 function

257
00:18:16,752 --> 00:18:17,960
of Bose-Einstein condensates.

258
00:18:31,040 --> 00:18:34,880
OK, so this is pretty much
the counting statistic

259
00:18:34,880 --> 00:18:38,220
which you do in
undergraduate class

260
00:18:38,220 --> 00:18:42,520
when you derive the three
different statistics--

261
00:18:42,520 --> 00:18:45,150
Boltzmann statistics,
Bose-Einstein statistics, Fermi

262
00:18:45,150 --> 00:18:46,410
statistics.

263
00:18:46,410 --> 00:18:54,000
And it gives you the result
of that is also for bosons,

264
00:18:54,000 --> 00:18:55,750
you have the factor of 2.

265
00:18:55,750 --> 00:18:57,830
For fermions, of
course, you get 0.

266
00:18:57,830 --> 00:19:02,630
The average between bosons
2 and fermions 0 is 1.

267
00:19:02,630 --> 00:19:06,330
And this is the
classical statistics.

268
00:19:06,330 --> 00:19:06,935
Any questions?

269
00:19:20,960 --> 00:19:25,370
Then let me finish
this discussion

270
00:19:25,370 --> 00:19:30,625
with a quantum
mechanical description.

271
00:19:39,010 --> 00:19:47,430
If you describe the
detection or collision of two

272
00:19:47,430 --> 00:19:54,985
particles in two modes.

273
00:19:58,260 --> 00:19:59,960
So you have mode.

274
00:19:59,960 --> 00:20:04,390
A mode operator, annihilation
operator a1 you and a2.

275
00:20:04,390 --> 00:20:06,920
But if you are
asking what happens

276
00:20:06,920 --> 00:20:10,580
if I measure two particles,
you annihilate two particles

277
00:20:10,580 --> 00:20:13,300
with an operator a1 and a2.

278
00:20:13,300 --> 00:20:16,130
But then, because of
the indistinguishability

279
00:20:16,130 --> 00:20:22,350
of particles, you
have to consider

280
00:20:22,350 --> 00:20:27,600
an exchange term, which is a2a1.

281
00:20:27,600 --> 00:20:29,440
So whenever you
detect two particles

282
00:20:29,440 --> 00:20:31,520
in two different
modes, your signal

283
00:20:31,520 --> 00:20:36,370
is proportional to something
related to a1a2 plus a2a1.

284
00:20:39,120 --> 00:20:50,050
This extra exchange term
gives you an extra factor of 2

285
00:20:50,050 --> 00:20:55,140
for bosons, which is
exactly what appears

286
00:20:55,140 --> 00:20:58,760
in the g2 function.

287
00:20:58,760 --> 00:21:06,960
And, of course, if you have
an exchange term for fermions,

288
00:21:06,960 --> 00:21:09,930
you get 0.

289
00:21:09,930 --> 00:21:13,640
And this, of course,
leads to the antibunching,

290
00:21:13,640 --> 00:21:19,610
or the g2 value
of 0 for fermions.

291
00:21:23,610 --> 00:21:36,450
However, and this is obvious,
if you have a single mode,

292
00:21:36,450 --> 00:21:42,470
then the only operator to detect
two particles is a1 times a1.

293
00:21:42,470 --> 00:21:44,910
And this has no exchange term.

294
00:21:51,120 --> 00:21:55,560
And this is the
situation of the laser

295
00:21:55,560 --> 00:21:58,440
and the Bose-Einstein
condensate.

296
00:21:58,440 --> 00:22:02,870
And that means they
have a g2 function of 1.

297
00:22:07,290 --> 00:22:09,250
So Cory, this is
another argument

298
00:22:09,250 --> 00:22:14,310
why single mode occupation
has a g2 function of 1.

299
00:22:14,310 --> 00:22:17,600
But I have to reconcile
it with your question.

300
00:22:17,600 --> 00:22:19,540
Right now, I think
there may be difference

301
00:22:19,540 --> 00:22:23,420
between a canonical ensemble
and grand canonical ensemble.

302
00:22:23,420 --> 00:22:31,730
In a canonical ensemble,
we have [INAUDIBLE]

303
00:22:31,730 --> 00:22:33,370
the total particle
number is fixed.

304
00:22:33,370 --> 00:22:35,230
But in a grand
canonical it fluctuates.

305
00:22:35,230 --> 00:22:37,380
And maybe this leads to
additional fluctuations

306
00:22:37,380 --> 00:22:41,850
for the case of a single
mode chaotic light.

307
00:22:41,850 --> 00:22:44,710
But I will have to think
about it, your question.

308
00:22:44,710 --> 00:22:47,860
But here is the
same argument why

309
00:22:47,860 --> 00:22:50,132
if you occupy only a
single mode-- if you look,

310
00:22:50,132 --> 00:22:51,965
for instance, at a
Bose-Einstein condensate,

311
00:22:51,965 --> 00:22:55,940
you do not have the
second exchange term.

312
00:22:55,940 --> 00:22:58,604
And the missing
exchange term just

313
00:22:58,604 --> 00:23:00,520
propagates through the
equations and gives you

314
00:23:00,520 --> 00:23:03,840
the g2 factor, the
g2 function of 1.

315
00:23:11,930 --> 00:23:14,510
OK so let me sort of wrap it up.

316
00:23:14,510 --> 00:23:17,920
I've given you different
angles at the g2 function.

317
00:23:17,920 --> 00:23:22,180
Some are simply classical
intensity distribution.

318
00:23:22,180 --> 00:23:26,460
Some is simply interference.

319
00:23:26,460 --> 00:23:31,030
Another aspect is Poissonian
counting statistics.

320
00:23:31,030 --> 00:23:33,120
My understanding and
my interpretation

321
00:23:33,120 --> 00:23:37,200
is they look as different
as they can possibly look,

322
00:23:37,200 --> 00:23:40,300
but they reflect
the same physics.

323
00:23:40,300 --> 00:23:45,540
Because interference of to
light only happens because light

324
00:23:45,540 --> 00:23:46,535
consists of photons.

325
00:23:46,535 --> 00:23:48,210
And photons are bosons.

326
00:23:48,210 --> 00:23:50,330
So the classical
interference and

327
00:23:50,330 --> 00:23:57,350
the quantum mechanical
counting of bosons

328
00:23:57,350 --> 00:24:03,190
lead not for random reasons,
but lead for profound reasons

329
00:24:03,190 --> 00:24:05,100
to the same result.

330
00:24:05,100 --> 00:24:08,850
And chaotic light, which
seems to be determined

331
00:24:08,850 --> 00:24:11,480
by just random
fluctuation, well,

332
00:24:11,480 --> 00:24:15,690
the random fluctuations
if you have single mode

333
00:24:15,690 --> 00:24:18,250
light, a thermal distribution
of single mode photons,

334
00:24:18,250 --> 00:24:19,980
comes from random phases.

335
00:24:19,980 --> 00:24:22,917
And these random phases
lead to random interference.

336
00:24:22,917 --> 00:24:24,375
And so we are back
to interference.

337
00:24:28,630 --> 00:24:41,830
Finally, let me give you my
view on the measurements of g2.

338
00:24:46,190 --> 00:24:47,830
When you look through
the literature,

339
00:24:47,830 --> 00:24:51,170
you find the famous Henry
[INAUDIBLE] experiment.

340
00:24:51,170 --> 00:24:53,960
In one of your
homework assignments,

341
00:24:53,960 --> 00:24:58,030
you looked at a
seminal experiment

342
00:24:58,030 --> 00:25:01,700
where people dropped atoms
out of cold atom clouds

343
00:25:01,700 --> 00:25:04,470
and measured the g2
function for cold atoms.

344
00:25:08,250 --> 00:25:11,330
Sometimes it's confusing
when you directly

345
00:25:11,330 --> 00:25:12,530
compare the two experiments.

346
00:25:15,740 --> 00:25:20,460
But let me try to give
you a common description,

347
00:25:20,460 --> 00:25:28,160
or the common denominator,
between all these experiments.

348
00:25:28,160 --> 00:25:40,000
You can say that all experiments
to measure the two particle

349
00:25:40,000 --> 00:25:43,610
correlation function
is about comparing

350
00:25:43,610 --> 00:25:47,910
the probability of finding two
particles with a probability

351
00:25:47,910 --> 00:25:49,170
of finding one particle.

352
00:25:52,100 --> 00:25:58,290
And well, you restrict your
measurement, your detection

353
00:25:58,290 --> 00:26:06,020
of particles, to
either one quantum

354
00:26:06,020 --> 00:26:20,950
state, one mode of the
electromagnetic field, one

355
00:26:20,950 --> 00:26:26,420
coherence length, or
coherence volume, of light.

356
00:26:26,420 --> 00:26:30,760
Or if you use the semi-classical
argument for the description

357
00:26:30,760 --> 00:26:36,660
of particles, you take
one phase space cell.

358
00:26:36,660 --> 00:26:40,125
This is one phase space cell,
one mode, one coherence volume,

359
00:26:40,125 --> 00:26:49,400
one quantum state, is sort
of the different definitions

360
00:26:49,400 --> 00:26:52,740
of what a quantum state
is, or what a wave packet

361
00:26:52,740 --> 00:26:56,610
is with Heisenberg
limited uncertainty.

362
00:26:56,610 --> 00:27:01,820
You cannot define a particle
in phase space to within better

363
00:27:01,820 --> 00:27:05,660
than the phase space
volume of a quantum state,

364
00:27:05,660 --> 00:27:09,686
which is h square.

365
00:27:09,686 --> 00:27:10,310
Or is it h bar?

366
00:27:10,310 --> 00:27:14,430
I don't think there
is a bar-- h square.

367
00:27:14,430 --> 00:27:17,020
And so this is how you
can relate wave packets

368
00:27:17,020 --> 00:27:19,480
to quantum states to
phase space volume.

369
00:27:19,480 --> 00:27:24,800
It's pretty much mathematically
I could say in different basis

370
00:27:24,800 --> 00:27:29,810
sets, in different basis
sets, it is one quantum state.

371
00:27:29,810 --> 00:27:31,230
But you can use wave packets.

372
00:27:31,230 --> 00:27:33,550
You can use time
dependent description.

373
00:27:33,550 --> 00:27:36,770
But what I said here--
one quantum state,

374
00:27:36,770 --> 00:27:38,960
one mode, one
coherence volume-- this

375
00:27:38,960 --> 00:27:42,420
is as well as you can
define a particle or photon

376
00:27:42,420 --> 00:27:47,220
to be by fundamental reasons
by Heisenberg's uncertainty

377
00:27:47,220 --> 00:27:48,372
relation.

378
00:27:48,372 --> 00:27:50,714
AUDIENCE: What do you mean
by one coherence volume?

379
00:27:50,714 --> 00:27:51,380
PROFESSOR: What?

380
00:27:51,380 --> 00:27:54,100
AUDIENCE: What do we mean
by one coherence volume?

381
00:27:54,100 --> 00:27:56,930
PROFESSOR: Well, if you have
a laser beam which propagates,

382
00:27:56,930 --> 00:28:01,770
and it is TEM00, you would say
the transverse coherence volume

383
00:28:01,770 --> 00:28:04,020
is the size of the
laser beam but then

384
00:28:04,020 --> 00:28:05,870
the longitudinal
coherence volume

385
00:28:05,870 --> 00:28:09,150
is given by the
coherence length.

386
00:28:09,150 --> 00:28:11,880
Just envision you have
a laser beam here.

387
00:28:11,880 --> 00:28:17,730
And the coherence volume where
all of the photons are coherent

388
00:28:17,730 --> 00:28:22,790
is the area of the laser beam
times the coherence length.

389
00:28:22,790 --> 00:28:26,930
And whenever you find a
photon in this kind of volume,

390
00:28:26,930 --> 00:28:30,860
then it is in a
Heisenberg Uncertainty

391
00:28:30,860 --> 00:28:33,830
limited or
Fourier-limited state.

392
00:28:33,830 --> 00:28:37,600
It is in, if it's a wave
packet and pulse laser,

393
00:28:37,600 --> 00:28:40,650
it's in a time
dependent quantum state.

394
00:28:40,650 --> 00:28:42,236
But it is as coherent.

395
00:28:42,236 --> 00:28:43,110
It is fully coherent.

396
00:28:45,990 --> 00:28:48,270
So you have to figure
out for your system what

397
00:28:48,270 --> 00:28:59,050
is this minimum-- what is the
coherence volume, what defines

398
00:28:59,050 --> 00:29:03,890
the fundamental phase
space cell of your system?

399
00:29:03,890 --> 00:29:06,710
And now you do the following.

400
00:29:06,710 --> 00:29:09,210
You are asking, what
is the probability

401
00:29:09,210 --> 00:29:10,950
to find two particles?

402
00:29:10,950 --> 00:29:15,110
And what is the probability
to find one particle?

403
00:29:15,110 --> 00:29:29,090
So if p is the probability
to find one particle,

404
00:29:29,090 --> 00:29:36,260
then you have three options.

405
00:29:36,260 --> 00:29:43,765
p square, 2 p square or
0 to find two particles.

406
00:29:49,200 --> 00:29:52,960
One is the classical case
of distinguishable particle.

407
00:29:52,960 --> 00:29:56,200
This is the case of bosons,
bosonic atoms or photons.

408
00:29:56,200 --> 00:29:59,970
Collins Therefore includes the
Hanbury Brown-Twiss experiment.

409
00:29:59,970 --> 00:30:01,050
And 0 is fermions.

410
00:30:06,450 --> 00:30:13,460
And the question is now
different experiments,

411
00:30:13,460 --> 00:30:17,090
how do you define,
with spatial resolution

412
00:30:17,090 --> 00:30:20,560
or temporal resolution,
the coherence volume?

413
00:30:20,560 --> 00:30:23,740
And that can involve
transverse coordination,

414
00:30:23,740 --> 00:30:25,860
temporal resolution,
and whatever

415
00:30:25,860 --> 00:30:34,690
you can use to define
the phase space

416
00:30:34,690 --> 00:30:38,500
cell or the coherence links,
the coherence volume of a laser

417
00:30:38,500 --> 00:30:39,050
beam.

418
00:30:39,050 --> 00:30:43,070
Let me give you one example,
which I think illustrates it.

419
00:30:43,070 --> 00:30:46,260
And this is the example
of an atom cloud.

420
00:30:46,260 --> 00:30:52,530
If you have an atom
cloud, all the particles

421
00:30:52,530 --> 00:30:55,950
which are in a volume of a
thermal de Broglie wavelengths

422
00:30:55,950 --> 00:30:56,510
are coherent.

423
00:31:04,970 --> 00:31:07,830
The momentum uncertainty of
particles in a thermal cloud

424
00:31:07,830 --> 00:31:11,590
is 1 over the thermal--
is the thermal momentum.

425
00:31:11,590 --> 00:31:14,810
And according to Heisenberg,
the position uncertainty

426
00:31:14,810 --> 00:31:16,770
related with this
momentum spread

427
00:31:16,770 --> 00:31:19,160
is just the thermal de
Broglie wavelengths.

428
00:31:19,160 --> 00:31:23,470
So you can say that all the
atoms in a cubic de Broglie

429
00:31:23,470 --> 00:31:27,460
wavelengths are coherent.

430
00:31:27,460 --> 00:31:33,130
All the atoms are in one
semi-classical quantum state.

431
00:31:33,130 --> 00:31:38,460
So therefore, this is sort
of atoms in one single mode.

432
00:31:38,460 --> 00:31:42,470
So the Hanbury Brown-Twiss
experiment with atoms,

433
00:31:42,470 --> 00:31:44,750
or the measurement
of the g2 function,

434
00:31:44,750 --> 00:31:46,720
could be defined as follows.

435
00:31:46,720 --> 00:31:48,160
You have an atom cloud.

436
00:31:48,160 --> 00:31:51,010
And if you had an
electron microscope

437
00:31:51,010 --> 00:31:55,470
or some high resolution device,
you grab into your cloud

438
00:31:55,470 --> 00:31:57,890
and ask, what is the
probability for one particle.

439
00:31:57,890 --> 00:31:59,910
What is the probability
for two particles?

440
00:31:59,910 --> 00:32:05,730
And you will find that
p2 is 2 times p1 squared.

441
00:32:05,730 --> 00:32:09,250
If your volume is too big,
you lose the factor of 2,

442
00:32:09,250 --> 00:32:11,455
because you average over
uncorrelated volumes.

443
00:32:14,720 --> 00:32:20,410
Now, in your homework, you
were looking at the question,

444
00:32:20,410 --> 00:32:23,060
how can I really
grab into a cloud

445
00:32:23,060 --> 00:32:25,752
and just pick out atoms
out of one phase space

446
00:32:25,752 --> 00:32:28,970
cell of effective size,
lambda de Broglie cubed?

447
00:32:32,340 --> 00:32:49,080
And the way how it was done
is that you take an atom cloud

448
00:32:49,080 --> 00:32:55,220
and you drop it and expand it.

449
00:32:55,220 --> 00:32:59,260
When the cloud expands, there
is a mapping from momentum space

450
00:32:59,260 --> 00:33:02,200
into position space.

451
00:33:02,200 --> 00:33:21,120
Then you use some
form of pinhole,

452
00:33:21,120 --> 00:33:23,695
which provides
transverse coordination.

453
00:33:28,910 --> 00:33:44,745
You use a detection laser, which
gives you temporal resolution.

454
00:33:49,940 --> 00:33:52,160
And well, this was
part of your homework,

455
00:33:52,160 --> 00:33:57,110
but I just wanted to give
you the bird's view on it.

456
00:33:57,110 --> 00:34:00,160
By controlling the
transverse correlation

457
00:34:00,160 --> 00:34:03,850
and the temporal resolution
of the detection,

458
00:34:03,850 --> 00:34:08,580
you create a situation that
what you count only atoms which

459
00:34:08,580 --> 00:34:12,940
originated from one phase
space cell in your cloud.

460
00:34:12,940 --> 00:34:15,010
So in other words,
let's say experiment

461
00:34:15,010 --> 00:34:16,949
where without
electron microscope,

462
00:34:16,949 --> 00:34:20,110
without submicron
spatial resolution,

463
00:34:20,110 --> 00:34:23,080
you can literally
grab into a cloud,

464
00:34:23,080 --> 00:34:26,170
capture a volume of the
thermal de Broglie wavelengths,

465
00:34:26,170 --> 00:34:28,420
open your hand,
and figure out how

466
00:34:28,420 --> 00:34:30,510
is the probability for
two particles related

467
00:34:30,510 --> 00:34:32,604
to the probability of
finding one particle.

468
00:34:36,690 --> 00:34:37,565
Any questions?

469
00:34:47,699 --> 00:34:48,199
All right.

470
00:35:17,560 --> 00:35:18,370
OK, good.

471
00:35:18,370 --> 00:35:28,740
So it's time for new chapter.

472
00:35:49,870 --> 00:35:52,320
So in this chapter,
we want to look

473
00:35:52,320 --> 00:35:55,370
at interactions
between light and atoms

474
00:35:55,370 --> 00:35:57,710
using Feynman diagrams.

475
00:35:57,710 --> 00:36:03,180
And I know this part of the
course is a little bit formal.

476
00:36:03,180 --> 00:36:09,110
We are using exact solutions
for the time evolution

477
00:36:09,110 --> 00:36:11,330
operator in quantum physics.

478
00:36:11,330 --> 00:36:17,470
But I'm not doing it to teach
you sophisticated formalism.

479
00:36:17,470 --> 00:36:20,450
Well, it's interesting
to learn it anyway.

480
00:36:20,450 --> 00:36:26,310
But it is this picture
of really writing down

481
00:36:26,310 --> 00:36:32,730
the exact solution of the
time evolution operator, which

482
00:36:32,730 --> 00:36:38,140
helped me to understand much
better what virtual states are,

483
00:36:38,140 --> 00:36:41,060
what certain
virtual photons are.

484
00:36:41,060 --> 00:36:43,870
So there are things we
talk about it all the time.

485
00:36:43,870 --> 00:36:46,730
And the question is, how do
you define a virtual state?

486
00:36:46,730 --> 00:36:49,690
What is the virtual
emission of a photon?

487
00:36:49,690 --> 00:36:53,420
And the only way how I can
really explain it to you

488
00:36:53,420 --> 00:36:56,090
is by showing you the
equation and say look,

489
00:36:56,090 --> 00:37:01,020
the virtual photon is just
a term in this equation.

490
00:37:01,020 --> 00:37:04,530
So that's the goal
of this chapter.

491
00:37:04,530 --> 00:37:08,300
I don't want to overemphasize
the mathematical rigor,

492
00:37:08,300 --> 00:37:10,210
but I want to
really show you what

493
00:37:10,210 --> 00:37:13,070
it means to have
virtual photons and what

494
00:37:13,070 --> 00:37:16,160
exactly virtual states are.

495
00:37:16,160 --> 00:37:35,460
So let me first motivate that by
reminding you of two diagrams.

496
00:37:35,460 --> 00:37:37,650
In physics, we always
like to draw something--

497
00:37:37,650 --> 00:37:39,450
a few lines, a few doodles.

498
00:37:39,450 --> 00:37:44,880
And when we have a
two level system,

499
00:37:44,880 --> 00:37:49,690
we have the two processes which
are emission and absorption.

500
00:37:58,170 --> 00:38:07,010
Well, so this is easy as long
as the light is in resonance.

501
00:38:07,010 --> 00:38:16,170
But if the light is
not in resonance,

502
00:38:16,170 --> 00:38:25,390
we may ask what
about this process?

503
00:38:29,500 --> 00:38:34,840
And since somehow the weekly
line of the photon ends here,

504
00:38:34,840 --> 00:38:37,700
I may even put in a
dashed line and say,

505
00:38:37,700 --> 00:38:39,610
this is a virtual state.

506
00:38:39,610 --> 00:38:43,030
What does it mean?

507
00:38:43,030 --> 00:38:48,020
Or we can even ask,
is it possible--

508
00:38:48,020 --> 00:38:51,720
I'm just playing with straight
lines which are quantum state

509
00:38:51,720 --> 00:38:55,110
and wiggly lines--
is it possible

510
00:38:55,110 --> 00:38:56,470
that this process happens.

511
00:38:59,000 --> 00:39:03,690
That would actually mean that
an atom in the ground state,

512
00:39:03,690 --> 00:39:06,720
just out of the
blue, emits a photon.

513
00:39:06,720 --> 00:39:08,230
Can that happen?

514
00:39:08,230 --> 00:39:10,060
And I can again
draw a dashed line

515
00:39:10,060 --> 00:39:11,590
and say, here's a virtual state.

516
00:39:14,190 --> 00:39:19,070
Or, we can say we start
in the excited state.

517
00:39:19,070 --> 00:39:24,560
The atom is in
the excited state.

518
00:39:24,560 --> 00:39:27,730
But can the atom-- its
a two level system,

519
00:39:27,730 --> 00:39:31,100
so we don't include higher
states-- but can a two level

520
00:39:31,100 --> 00:39:39,130
atom in the excited state
absorb another photon?

521
00:39:42,770 --> 00:39:53,540
So let me just change the
color of these photons

522
00:39:53,540 --> 00:40:09,656
to red, because I want to--
and this would remain black.

523
00:40:14,540 --> 00:40:17,211
All right.

524
00:40:17,211 --> 00:40:17,710
Good.

525
00:40:17,710 --> 00:40:19,630
So question is, are
those processes possible?

526
00:40:24,690 --> 00:40:25,660
Yes or no?

527
00:40:25,660 --> 00:40:26,540
AUDIENCE: Yes.

528
00:40:26,540 --> 00:40:28,710
PROFESSOR: Yes, they are.

529
00:40:28,710 --> 00:40:34,550
And they have experimental,
observable consequences.

530
00:40:34,550 --> 00:40:38,380
However, they look
funny because something

531
00:40:38,380 --> 00:40:41,580
seems to be strange with
energy conservation.

532
00:40:41,580 --> 00:40:45,360
And what we will see
is that in the end,

533
00:40:45,360 --> 00:40:48,050
at the end of the
day, that means

534
00:40:48,050 --> 00:40:55,220
if you let the system evolve for
a long time, after a long time,

535
00:40:55,220 --> 00:40:59,655
energy has to be conserved.

536
00:41:02,760 --> 00:41:15,930
And therefore, we will
always need a second photon

537
00:41:15,930 --> 00:41:17,205
to conserve energy.

538
00:41:22,640 --> 00:41:25,690
So those weirder
processes, where

539
00:41:25,690 --> 00:41:27,760
photons are emitted
by ground states out

540
00:41:27,760 --> 00:41:32,620
of the blue or excited
states absorb other photons

541
00:41:32,620 --> 00:41:38,692
without-- also, there is
no higher lying state.

542
00:41:38,692 --> 00:41:40,025
That looks a little bit strange.

543
00:41:42,630 --> 00:41:45,250
So what you will
see is-- and I want

544
00:41:45,250 --> 00:41:47,410
to show you that
mathematically--

545
00:41:47,410 --> 00:41:53,790
if an atom emits a photon
out of the ground state,

546
00:41:53,790 --> 00:41:56,050
you would say, where does
the energy come from.

547
00:41:56,050 --> 00:42:03,510
But quantum mechanic allows us
to violate energy conservation

548
00:42:03,510 --> 00:42:07,840
by an amount delta e for a time
which is h bar over delta e.

549
00:42:07,840 --> 00:42:11,290
So for short movement,
energy can be violated.

550
00:42:11,290 --> 00:42:13,015
But then, you need
the second photon

551
00:42:13,015 --> 00:42:15,330
to reconcile energy
conservation.

552
00:42:15,330 --> 00:42:18,242
You can violate energy
only for short times.

553
00:42:18,242 --> 00:42:19,950
And if you would say,
what does it really

554
00:42:19,950 --> 00:42:22,910
mean to violate energy
conservation for short times?

555
00:42:22,910 --> 00:42:25,180
Well, I want to show
you the equation,

556
00:42:25,180 --> 00:42:26,940
which exactly explains
what it means.

557
00:42:36,700 --> 00:42:39,150
Next question, just to see.

558
00:42:39,150 --> 00:42:42,490
If you have an excited
state, and you just

559
00:42:42,490 --> 00:42:46,580
say it's possible to
absorb another atom,

560
00:42:46,580 --> 00:42:51,000
you are in the excited state,
and you absorb another photon.

561
00:42:51,000 --> 00:42:53,470
And at least you said,
yes this is possible.

562
00:42:53,470 --> 00:42:56,080
My question is, when you
are in the excited state

563
00:42:56,080 --> 00:42:59,980
and absorb a photon, in which
quantum state is the atom

564
00:42:59,980 --> 00:43:02,926
after absorbing a photon?

565
00:43:02,926 --> 00:43:04,012
AUDIENCE: Ground state.

566
00:43:04,012 --> 00:43:04,762
PROFESSOR: Pardon?

567
00:43:04,762 --> 00:43:05,720
AUDIENCE: Ground state.

568
00:43:05,720 --> 00:43:08,460
PROFESSOR: In the ground state.

569
00:43:08,460 --> 00:43:09,090
Yes.

570
00:43:09,090 --> 00:43:10,710
There is no other state.

571
00:43:10,710 --> 00:43:13,610
And the operator--
and I want to show you

572
00:43:13,610 --> 00:43:16,930
that-- when photons
are exchanged,

573
00:43:16,930 --> 00:43:19,904
always transforms the
ground to the excited state,

574
00:43:19,904 --> 00:43:21,320
because a dipole
operator connects

575
00:43:21,320 --> 00:43:23,140
the ground to the excited state.

576
00:43:23,140 --> 00:43:27,660
So therefore-- but we'll see
that all in the time evolution

577
00:43:27,660 --> 00:43:32,600
operator in the formal solution
that this dash line is actually

578
00:43:32,600 --> 00:43:34,500
the ground state.

579
00:43:34,500 --> 00:43:36,650
And while it's not obvious,
what is this dash line?

580
00:43:36,650 --> 00:43:37,580
AUDIENCE: The excited state.

581
00:43:37,580 --> 00:43:39,038
PROFESSOR: It is
the excited state.

582
00:43:39,038 --> 00:43:42,050
The atom in the ground
state emits a photon.

583
00:43:42,050 --> 00:43:44,730
Therefore, the atomic
system has now lower energy,

584
00:43:44,730 --> 00:43:46,960
because a photon
has been emitted.

585
00:43:46,960 --> 00:43:50,480
But the character of this
state is now the excited state.

586
00:43:53,320 --> 00:43:58,810
So we will later see that this
means that in a perturbation

587
00:43:58,810 --> 00:44:01,280
analysis, we are
violating energy.

588
00:44:01,280 --> 00:44:04,390
We are violating energy,
because the real excited

589
00:44:04,390 --> 00:44:11,460
state is omega photon plus
omega atom, the resonant energy

590
00:44:11,460 --> 00:44:12,190
for the atom.

591
00:44:12,190 --> 00:44:17,200
The real excited state is omega
photon plus omega atom higher.

592
00:44:17,200 --> 00:44:21,560
So you will actually find that
in a perturbation analysis,

593
00:44:21,560 --> 00:44:25,590
this term has an energy
denominator, which

594
00:44:25,590 --> 00:44:31,490
is of resonant by
exactly this separation.

595
00:44:31,490 --> 00:44:34,720
And this term, because the
real count state is here,

596
00:44:34,720 --> 00:44:37,550
the virtual state here which
is the count state is here,

597
00:44:37,550 --> 00:44:42,210
has actually the same energy
denominator as that state.

598
00:44:42,210 --> 00:44:43,790
And for this state,
of course, this

599
00:44:43,790 --> 00:44:46,060
is something you've seen of
resonant light scattering.

600
00:44:46,060 --> 00:44:47,842
The energy
[? denominator ?] is there.

601
00:44:52,470 --> 00:45:04,050
So, OK.

602
00:45:04,050 --> 00:45:12,800
So I'll explain to you that
we involve virtual states.

603
00:45:12,800 --> 00:45:18,920
So when we have
virtual states, we

604
00:45:18,920 --> 00:45:26,370
want to clarify what
is their energy.

605
00:45:26,370 --> 00:45:28,890
But we discussed that
already in the discussion.

606
00:45:28,890 --> 00:45:33,560
And what is the wave function,
ground or excited state.

607
00:45:38,950 --> 00:45:39,450
Yes.

608
00:45:51,000 --> 00:45:54,700
So just to repeat it, when
we draw those diagrams,

609
00:45:54,700 --> 00:46:02,150
those diagrams show
the energy of the atom,

610
00:46:02,150 --> 00:46:09,080
but taking into account
the energy of the photon.

611
00:46:09,080 --> 00:46:14,790
In other words, if you
have a ground state,

612
00:46:14,790 --> 00:46:18,820
and we absorb a photon,
the photon has disappeared,

613
00:46:18,820 --> 00:46:21,070
and the energy of
the atom is now

614
00:46:21,070 --> 00:46:22,820
the ground state and
the photon energy.

615
00:46:22,820 --> 00:46:24,780
And this is a dashed line.

616
00:46:24,780 --> 00:46:29,360
Similarly, when an atom emits
a photon in the ground state,

617
00:46:29,360 --> 00:46:31,020
the photon is emitted.

618
00:46:31,020 --> 00:46:35,400
Therefore, the atomic energy
is lower than the ground state

619
00:46:35,400 --> 00:46:36,980
by the photon energy.

620
00:46:36,980 --> 00:46:39,490
And we draw the
dashed line here.

621
00:46:39,490 --> 00:46:43,070
This is what we mean
when we say that is

622
00:46:43,070 --> 00:46:45,009
the energy of the virtual state.

623
00:46:45,009 --> 00:46:46,800
And this is what we
draw in those diagrams.

624
00:46:50,170 --> 00:46:53,660
By the way, yesterday
we had the CUA seminar.

625
00:46:53,660 --> 00:46:59,880
And the speaker was actually
talking about trapping atoms

626
00:46:59,880 --> 00:47:01,750
with quantum fluctuations.

627
00:47:01,750 --> 00:47:06,380
And he conceded
explicitly Casimir forces,

628
00:47:06,380 --> 00:47:08,000
forces of the vacuum.

629
00:47:08,000 --> 00:47:10,780
And diagrammatically,
forces of the vacuum

630
00:47:10,780 --> 00:47:14,880
come, because the ground state
atom emits a virtual photon

631
00:47:14,880 --> 00:47:17,000
and reabsorbs.

632
00:47:17,000 --> 00:47:19,810
This is actually, I
will mention it later,

633
00:47:19,810 --> 00:47:21,270
the same diagram
is actually also

634
00:47:21,270 --> 00:47:23,060
the diagram which leads
to the Lamb shift.

635
00:47:23,060 --> 00:47:26,810
These are all sort of when you
have atoms in the lowest state,

636
00:47:26,810 --> 00:47:29,450
and they interact with
the electromagnetic field.

637
00:47:29,450 --> 00:47:31,740
Well, if they interact
with the vacuum,

638
00:47:31,740 --> 00:47:33,140
they cannot absorb photon.

639
00:47:33,140 --> 00:47:38,350
All we can do is emit a photon,
and this leads to this diagram.

640
00:47:38,350 --> 00:47:40,780
But just to tell you that
you have to be careful,

641
00:47:40,780 --> 00:47:45,020
yesterday's speakers
actually used this process.

642
00:47:45,020 --> 00:47:46,240
But he drew this diagram.

643
00:47:46,240 --> 00:47:48,930
He drew it in the
opposite way, which

644
00:47:48,930 --> 00:47:51,650
I think confused some
people in the audience.

645
00:47:51,650 --> 00:47:54,430
The correct way is
to draw it like this.

646
00:47:54,430 --> 00:47:59,440
The virtual state for
vacuum fluctuations

647
00:47:59,440 --> 00:48:01,880
for the Lamb shift
for the Casimir force

648
00:48:01,880 --> 00:48:03,970
is below the current state.

649
00:48:07,360 --> 00:48:17,610
OK, so we have-- let me just
redraw those three diagrams.

650
00:48:17,610 --> 00:48:27,840
I want to now introduce
the time evolution.

651
00:48:27,840 --> 00:48:31,165
So I want to use those.

652
00:48:38,380 --> 00:48:40,235
Seems I'm running out of space.

653
00:48:50,280 --> 00:48:57,000
So we want to
introduce the time axis

654
00:48:57,000 --> 00:49:04,830
for-- so we had this diagram.

655
00:49:04,830 --> 00:49:08,480
We had this diagram.

656
00:49:08,480 --> 00:49:21,100
And we had-- this
is a virtual state.

657
00:49:21,100 --> 00:49:22,290
OK.

658
00:49:22,290 --> 00:49:32,090
So those three
diagrams correspond

659
00:49:32,090 --> 00:49:33,340
to the following situation.

660
00:49:37,690 --> 00:49:42,860
If time evolves from
the bottom to the top,

661
00:49:42,860 --> 00:49:46,010
we can now draw
the first process

662
00:49:46,010 --> 00:49:48,300
when atom in the
ground state first

663
00:49:48,300 --> 00:49:49,790
absorbs and then emits a photon.

664
00:50:01,750 --> 00:50:08,250
And let's label the second
photon in the purple color.

665
00:50:08,250 --> 00:50:12,260
So an atom starts out
in the ground state.

666
00:50:12,260 --> 00:50:14,980
Then it reaches
the time where it

667
00:50:14,980 --> 00:50:17,860
interacts the
electromagnetic field.

668
00:50:17,860 --> 00:50:21,320
There is a photon
which propagates

669
00:50:21,320 --> 00:50:27,060
from earlier times to this
time, t equals t prime.

670
00:50:27,060 --> 00:50:30,940
At the time t prime,
the photon disappears.

671
00:50:30,940 --> 00:50:36,440
And the atom, which was
in the ground state,

672
00:50:36,440 --> 00:50:38,980
goes to the excited state.

673
00:50:38,980 --> 00:50:42,690
It may propagate in the
excited state for a while.

674
00:50:42,690 --> 00:50:48,470
And then it emits the photon and
is back into the ground state.

675
00:50:48,470 --> 00:50:52,730
So this is the temporal
diagram for this process.

676
00:50:52,730 --> 00:50:59,640
For the next situation,
we also start

677
00:50:59,640 --> 00:51:03,540
with an atom in
the ground state.

678
00:51:03,540 --> 00:51:06,980
But now we have the
situation that a photon

679
00:51:06,980 --> 00:51:09,980
is emitted by the ground state.

680
00:51:12,600 --> 00:51:17,220
As a result, the atom
switches to the excited state.

681
00:51:17,220 --> 00:51:23,780
Strong violation of energy,
but possible for short times.

682
00:51:23,780 --> 00:51:31,780
Then there is a real photon,
which is now absorbed,

683
00:51:31,780 --> 00:51:35,030
absorbed by an atom
in the excited state.

684
00:51:35,030 --> 00:51:39,520
And that takes us back
down to the ground state.

685
00:51:39,520 --> 00:51:43,560
And finally, in
the third scenario,

686
00:51:43,560 --> 00:51:45,850
we have an atom in
the excited state.

687
00:51:49,080 --> 00:51:52,650
There is a real photon
which is absorbed

688
00:51:52,650 --> 00:51:56,290
that switches the atom
into the ground state.

689
00:51:56,290 --> 00:52:03,210
And then the atom in the
ground state emits a photon.

690
00:52:03,210 --> 00:52:06,440
And as a result, it is
back in the excited state.

691
00:52:17,390 --> 00:52:32,160
So let me just make a note
that when a ground state

692
00:52:32,160 --> 00:52:47,290
atom absorbs a photon, this
is the co-rotating term

693
00:52:47,290 --> 00:52:53,390
in the quantum description,
which a lot of you have seen.

694
00:52:53,390 --> 00:53:01,460
The opposite process of
emitting a virtual photon-- I

695
00:53:01,460 --> 00:53:05,060
shouldn't say virtual photon.

696
00:53:05,060 --> 00:53:05,940
It's a photon.

697
00:53:05,940 --> 00:53:09,050
What makes it virtual
we will see later

698
00:53:09,050 --> 00:53:12,010
when we have the more accurate
mathematical formulation.

699
00:53:12,010 --> 00:53:14,660
So this is a
counter-rotating term.

700
00:53:22,160 --> 00:53:34,890
And we have used the rule
that every photon, which

701
00:53:34,890 --> 00:53:40,250
with a real or virtual-- every
photon which is interchanged

702
00:53:40,250 --> 00:53:42,950
with the atom
absorbs or emitted--

703
00:53:42,950 --> 00:53:48,910
changes the atomic state
from ground to excited.

704
00:53:48,910 --> 00:53:52,660
So an atom goes from the
ground to the excited state

705
00:53:52,660 --> 00:53:55,810
either by photon absorption
or by photon emission.

706
00:53:55,810 --> 00:53:56,800
Both is possible.

707
00:54:00,440 --> 00:54:04,130
So whenever photon
appears or disappears,

708
00:54:04,130 --> 00:54:07,000
it changes-- let's say we
start in the ground state.

709
00:54:07,000 --> 00:54:09,030
We go to the excited state.

710
00:54:09,030 --> 00:54:15,060
And this is possible
by photon absorption--

711
00:54:15,060 --> 00:54:18,485
this is a co-rotating
term-- or emission.

712
00:54:23,170 --> 00:54:27,490
What I've just said
reflects that when

713
00:54:27,490 --> 00:54:32,910
we derived the
dipole approximation,

714
00:54:32,910 --> 00:54:40,330
that the essential term of the
dipole operator is of diagonal.

715
00:54:40,330 --> 00:54:42,110
The dipole operator
is an operator

716
00:54:42,110 --> 00:54:44,770
between calm and excited state.

717
00:54:44,770 --> 00:54:51,180
Therefore, it has those
two matrix elements.

718
00:54:51,180 --> 00:54:56,700
Whereas electric field
operator is the equation

719
00:54:56,700 --> 00:55:00,550
and annihilation
operator a and a dagger.

720
00:55:00,550 --> 00:55:05,410
Or which is very elegant
if I use for the two level

721
00:55:05,410 --> 00:55:08,160
system a spin one
half description.

722
00:55:08,160 --> 00:55:11,030
And I use sigma plus
and sigma minus,

723
00:55:11,030 --> 00:55:12,890
raising and lowering operator.

724
00:55:12,890 --> 00:55:15,980
Sigma plus takes the ground
state to the excited state.

725
00:55:15,980 --> 00:55:19,090
Sigma minus takes the exciting
state to the ground state.

726
00:55:19,090 --> 00:55:23,230
Then in the fundamental
atom light interaction,

727
00:55:23,230 --> 00:55:25,770
we have those four terms.

728
00:55:25,770 --> 00:55:30,030
And now you see that
sigma plus takes the atom

729
00:55:30,030 --> 00:55:33,720
from the calm to
the excited state.

730
00:55:33,720 --> 00:55:37,850
But sigma plus appears
both with photon absorption

731
00:55:37,850 --> 00:55:39,640
and with photon emission.

732
00:55:39,640 --> 00:55:41,866
So therefore, the
atom can-- this

733
00:55:41,866 --> 00:55:43,490
is what the quantum
mechanical operator

734
00:55:43,490 --> 00:55:46,430
tells us-- go from the
ground to the excited state

735
00:55:46,430 --> 00:55:51,170
either by photon absorption
or by photon emission.

736
00:55:54,992 --> 00:55:55,575
Any questions?

737
00:56:00,840 --> 00:56:01,834
Yes.

738
00:56:01,834 --> 00:56:04,567
AUDIENCE: So if we were
to look beyond the dipole

739
00:56:04,567 --> 00:56:09,289
approximation, would
we see transitions

740
00:56:09,289 --> 00:56:11,277
that leave the
[INAUDIBLE] state?

741
00:56:20,230 --> 00:56:22,700
PROFESSOR: Yes or no.

742
00:56:22,700 --> 00:56:25,270
What we need here is a
bilinear interaction,

743
00:56:25,270 --> 00:56:28,920
which has an raising and
lowering operator for the atom.

744
00:56:28,920 --> 00:56:31,100
And here it has a
raising and lowering atom

745
00:56:31,100 --> 00:56:33,490
operator for the photon number.

746
00:56:33,490 --> 00:56:37,930
And it has the co-rotating
term where you raise here

747
00:56:37,930 --> 00:56:39,350
and you lower there.

748
00:56:39,350 --> 00:56:41,040
You lower here and
you raise there.

749
00:56:41,040 --> 00:56:45,110
But it has also terms where you
raise the atomic excitation,

750
00:56:45,110 --> 00:56:47,470
and you raise the
photon number by one,

751
00:56:47,470 --> 00:56:49,480
which of course violates energy.

752
00:56:49,480 --> 00:56:52,870
But the message I want
to give you is this

753
00:56:52,870 --> 00:56:54,250
is possible for short times.

754
00:56:54,250 --> 00:56:58,190
And this is exactly what the
quantum mechanical equations

755
00:56:58,190 --> 00:57:00,460
tell us.

756
00:57:00,460 --> 00:57:02,540
So the short answer
to your question

757
00:57:02,540 --> 00:57:05,090
is, as long as we
describe the system

758
00:57:05,090 --> 00:57:09,100
by the dipole interaction,
or what we have even beyond--

759
00:57:09,100 --> 00:57:10,900
I'm not using a
perturbation argument.

760
00:57:10,900 --> 00:57:13,570
I'm writing down the operator.

761
00:57:13,570 --> 00:57:16,470
And this operator can
now effect the system

762
00:57:16,470 --> 00:57:18,700
to all possible orders.

763
00:57:18,700 --> 00:57:22,060
So even the non-perturbative
strong coupling limit

764
00:57:22,060 --> 00:57:25,850
will always have products
of atomic raising operators,

765
00:57:25,850 --> 00:57:28,660
lowering operators with photon
raising and lowering operators.

766
00:57:32,790 --> 00:57:36,590
The little bit longer answer
is, some of what I'm telling you

767
00:57:36,590 --> 00:57:39,090
may be gauge dependence.

768
00:57:39,090 --> 00:57:46,810
If you work in the-- if you
use, not the dipole description,

769
00:57:46,810 --> 00:57:50,450
but you use the p
dot a Hamiltonian,

770
00:57:50,450 --> 00:57:53,850
this Hamiltonian
has a e square term.

771
00:57:53,850 --> 00:58:00,570
The e square term
allows the atoms

772
00:58:00,570 --> 00:58:04,420
to exchange two photons
at the same time.

773
00:58:04,420 --> 00:58:08,640
So in other words, in
the a square interaction,

774
00:58:08,640 --> 00:58:16,140
we may have a
vertex point in time

775
00:58:16,140 --> 00:58:19,280
where the atom interact
with the photon field

776
00:58:19,280 --> 00:58:23,060
where two photons are exchanged.

777
00:58:23,060 --> 00:58:26,100
And then the rules
are different.

778
00:58:26,100 --> 00:58:28,770
So in that sense-- but
this is quite often

779
00:58:28,770 --> 00:58:30,950
when we give a quantum
mechanical pictures, which

780
00:58:30,950 --> 00:58:34,590
has terms, which nicely
shows the time evolution.

781
00:58:34,590 --> 00:58:36,670
And you think wow, yeah,
this is what happens.

782
00:58:36,670 --> 00:58:38,640
One photon at a
time, the atom goes

783
00:58:38,640 --> 00:58:40,370
ground, excited,
excited, ground state.

784
00:58:40,370 --> 00:58:41,590
And you have your picture.

785
00:58:41,590 --> 00:58:44,920
And it is fully consistent
with the exact quantum

786
00:58:44,920 --> 00:58:46,420
mechanical result.

787
00:58:46,420 --> 00:58:48,822
You may be able to get
the same result out

788
00:58:48,822 --> 00:58:49,780
of a different picture.

789
00:58:53,830 --> 00:58:57,202
That point will
actually occur again

790
00:58:57,202 --> 00:58:58,660
when we talk, for
instance, when we

791
00:58:58,660 --> 00:59:09,390
talk about density operators.

792
00:59:09,390 --> 00:59:12,960
A density operator is an
average of our quantum states.

793
00:59:12,960 --> 00:59:15,640
But you can get the
same density operator

794
00:59:15,640 --> 00:59:19,860
by averaging quantum states
in different basis sets.

795
00:59:19,860 --> 00:59:25,910
So if you get one picture out
of one specific basis set,

796
00:59:25,910 --> 00:59:27,642
it gives you the
correct intuition,

797
00:59:27,642 --> 00:59:29,350
it's a correct
description, but it is not

798
00:59:29,350 --> 00:59:32,168
the only possible description.

799
00:59:32,168 --> 00:59:33,102
[? Collin ?].

800
00:59:33,102 --> 00:59:35,390
AUDIENCE: Maybe the
thing, I don't know.

801
00:59:35,390 --> 00:59:36,890
AUDIENCE: I think
what you're trying

802
00:59:36,890 --> 00:59:40,490
to get at is that picture
works as long as you can limit

803
00:59:40,490 --> 00:59:42,716
yourself to a two level system.

804
00:59:42,716 --> 00:59:44,810
Because that's
correct on a whole.

805
00:59:44,810 --> 00:59:47,010
Or is it like strong coupling?

806
00:59:47,010 --> 00:59:50,535
But as long as you can
identify the two level system,

807
00:59:50,535 --> 00:59:55,525
or I think the dipole versus
particle approximations

808
00:59:55,525 --> 00:59:59,517
[INAUDIBLE] an atom may have
transition probabilities

809
00:59:59,517 --> 01:00:03,010
to others, like from
an [INAUDIBLE] state.

810
01:00:03,010 --> 01:00:04,507
But--

811
01:00:04,507 --> 01:00:06,503
AUDIENCE: Actually,
I was just wondering

812
01:00:06,503 --> 01:00:09,247
if it was possible to have
it like a transition matrix

813
01:00:09,247 --> 01:00:12,531
element to [INAUDIBLE] coupling
the ground state to the ground

814
01:00:12,531 --> 01:00:13,030
state.

815
01:00:13,030 --> 01:00:13,770
PROFESSOR: Yes.

816
01:00:13,770 --> 01:00:16,920
The e square operator
does that, because

817
01:00:16,920 --> 01:00:21,206
in the other gauge,
which is pa, pa.

818
01:00:21,206 --> 01:00:25,450
p is in dipole operator.
p is the momentum operator

819
01:00:25,450 --> 01:00:26,550
for the atoms.

820
01:00:26,550 --> 01:00:29,776
And the p operator will
switch the atom from ground

821
01:00:29,776 --> 01:00:31,010
to excited state.

822
01:00:31,010 --> 01:00:34,820
a, the vector potential,
involves a and a daggers.

823
01:00:34,820 --> 01:00:38,730
So the p dot a term does
pretty much the same

824
01:00:38,730 --> 01:00:41,100
as the d dot e
term, because it's

825
01:00:41,100 --> 01:00:43,790
a bilinear product
of an operator which

826
01:00:43,790 --> 01:00:47,150
has an effect on the atoms and
has an effect on the photons.

827
01:00:47,150 --> 01:00:50,990
And it means you always switch
the quantum state in your atom

828
01:00:50,990 --> 01:00:53,430
and create or
annihilate the photon.

829
01:00:53,430 --> 01:00:56,380
But the a squared term--
it's just a square,

830
01:00:56,380 --> 01:00:58,350
it doesn't have a
p in front of it--

831
01:00:58,350 --> 01:01:06,820
will allow an atom in without
changing the state to create

832
01:01:06,820 --> 01:01:08,320
two photons, emit two photons.

833
01:01:08,320 --> 01:01:10,030
And then there's
probably a cross term

834
01:01:10,030 --> 01:01:12,700
which means to absorb
and emit a photon.

835
01:01:12,700 --> 01:01:17,547
You will actually, in one of
your next homework assignments,

836
01:01:17,547 --> 01:01:19,380
you will actually look
at the a square term.

837
01:01:19,380 --> 01:01:21,650
And I think it's a very
educational problem.

838
01:01:21,650 --> 01:01:23,375
So you really see
that you have two

839
01:01:23,375 --> 01:01:28,040
very different descriptions,
but the results fully agree.

840
01:01:28,040 --> 01:01:30,860
But this is already
an expert discussion.

841
01:01:30,860 --> 01:01:34,660
Why don't we for now just hang
onto the dipole approximation.

842
01:01:34,660 --> 01:01:37,070
We have d dot e,
and we just learn

843
01:01:37,070 --> 01:01:39,790
what is inside the
dipole approximation,

844
01:01:39,790 --> 01:01:43,400
and what is an exact quantum
mechanical description using

845
01:01:43,400 --> 01:01:45,670
the dipole approximation.

846
01:01:45,670 --> 01:01:49,650
But and then we can come
back to the discussion,

847
01:01:49,650 --> 01:01:52,730
is everything we
describe really real.

848
01:01:52,730 --> 01:01:54,750
And while the answer
is, what is real

849
01:01:54,750 --> 01:01:57,350
in quantum physics
is the final result,

850
01:01:57,350 --> 01:01:59,570
the intermediate
results, you have

851
01:01:59,570 --> 01:02:03,380
to say that this is
only one possibility

852
01:02:03,380 --> 01:02:04,725
to go to the final result.

853
01:02:08,290 --> 01:02:23,765
OK, so we want to-- let me
just put in one more page.

854
01:02:39,740 --> 01:02:44,820
So what we want to do now is
we want to do a calculation.

855
01:02:44,820 --> 01:02:47,080
It's a perturbative
calculation, but we

856
01:02:47,080 --> 01:02:48,200
can take it to all orders.

857
01:02:48,200 --> 01:02:51,480
So therefore it's
exact and general.

858
01:02:51,480 --> 01:02:58,720
We want to do a perturbative
calculation of transition

859
01:02:58,720 --> 01:02:59,220
amplitudes.

860
01:03:07,660 --> 01:03:17,020
And what I am following here
the discussion in atom photon

861
01:03:17,020 --> 01:03:23,290
interaction on those pages.

862
01:03:23,290 --> 01:03:26,140
And the idea is the following.

863
01:03:26,140 --> 01:03:29,670
We have the Hamiltonian
for the atom.

864
01:03:29,670 --> 01:03:31,790
And then we have a
Hamiltonian which

865
01:03:31,790 --> 01:03:35,320
describes the transverse field.

866
01:03:44,380 --> 01:03:48,000
And the interaction of
the transverse field

867
01:03:48,000 --> 01:03:50,690
with the atoms.

868
01:03:50,690 --> 01:03:55,920
What we want to use is we
want to use the interaction

869
01:03:55,920 --> 01:04:02,980
picture, which is often
the nicest picture

870
01:04:02,980 --> 01:04:08,380
to describe the effect of
the interaction between two

871
01:04:08,380 --> 01:04:10,570
systems.

872
01:04:10,570 --> 01:04:17,750
In the interaction picture,
you are transforming

873
01:04:17,750 --> 01:04:22,480
from you Schrodinger
type e function

874
01:04:22,480 --> 01:04:27,390
to wave function psi tilde.

875
01:04:27,390 --> 01:04:32,580
And those wave function
psi tilde include already

876
01:04:32,580 --> 01:04:35,860
the dynamics of the
Hamiltonian H naught.

877
01:04:43,620 --> 01:04:50,760
So that means that all operators
in the normal Schrodinger

878
01:04:50,760 --> 01:04:56,270
picture become now operators
in the interaction picture

879
01:04:56,270 --> 01:04:59,430
by canonical
transformation, which

880
01:04:59,430 --> 01:05:05,195
involves the dynamics due to
the unperturbed Hamiltonian.

881
01:05:10,850 --> 01:05:14,490
And so that's just
a reminder what

882
01:05:14,490 --> 01:05:16,610
the interaction picture is.

883
01:05:16,610 --> 01:05:20,800
And we are now interested what
happens in the interaction

884
01:05:20,800 --> 01:05:26,780
picture to an initial
wave function psi of ti.

885
01:05:26,780 --> 01:05:29,860
This wave function already
includes the time evolution

886
01:05:29,860 --> 01:05:31,610
due to H naught.

887
01:05:31,610 --> 01:05:35,240
So the only change now comes
because we have interactions

888
01:05:35,240 --> 01:05:37,220
with the electromagnetic field.

889
01:05:37,220 --> 01:05:39,950
So this allows us
now to focus only

890
01:05:39,950 --> 01:05:42,419
on the relevant interactions,
the interactions we

891
01:05:42,419 --> 01:05:43,210
want to understand.

892
01:05:51,190 --> 01:05:55,950
And the formalism which is
used in atom photon interaction

893
01:05:55,950 --> 01:05:58,530
focuses on the time
evolution operator,

894
01:05:58,530 --> 01:06:01,960
which I'm sure all
of you have seen.

895
01:06:01,960 --> 01:06:07,330
So the time evolution
of our system,

896
01:06:07,330 --> 01:06:09,510
how the wave function
involves from the initial

897
01:06:09,510 --> 01:06:14,990
to the final time, is described
by the time evolution operator.

898
01:06:17,930 --> 01:06:20,370
I will come back to the
wave function later,

899
01:06:20,370 --> 01:06:26,020
but in order to derive the time
evolution of the wave function,

900
01:06:26,020 --> 01:06:31,360
it's at least convenient
to focus first

901
01:06:31,360 --> 01:06:33,120
on the time evolution operator.

902
01:06:33,120 --> 01:06:34,700
So in other words,
I want to show

903
01:06:34,700 --> 01:06:37,850
you a formal solution for
the time evolution operator.

904
01:06:37,850 --> 01:06:40,800
And once we know the
time evolution operator,

905
01:06:40,800 --> 01:06:42,370
we apply it to
the wave function.

906
01:06:42,370 --> 01:06:44,619
And then we are talking again
about the wave function.

907
01:06:52,350 --> 01:06:58,140
OK, since the derivation
involves many equations

908
01:06:58,140 --> 01:07:02,050
and they are all printed in
the full beauty in atom photon

909
01:07:02,050 --> 01:07:08,970
interaction, I decided to
use pre-written slides here.

910
01:07:08,970 --> 01:07:12,530
Also, most of it is
fairly elementary.

911
01:07:12,530 --> 01:07:17,360
The equations are complicated,
but the concepts behind them

912
01:07:17,360 --> 01:07:19,540
are very, very simple.

913
01:07:19,540 --> 01:07:20,920
It's actually the
beauty of using

914
01:07:20,920 --> 01:07:22,490
a kind of a time
evolution operator.

915
01:07:22,490 --> 01:07:24,690
It's a little bit
mathematically formal,

916
01:07:24,690 --> 01:07:28,990
but it allows us to
express what is really

917
01:07:28,990 --> 01:07:32,160
going on in the solution
in very, very simple terms.

918
01:07:41,730 --> 01:07:44,460
So the goal is to do a
perturbative expansion

919
01:07:44,460 --> 01:07:47,300
for the time evolution operator.

920
01:07:47,300 --> 01:07:51,890
If we have no interaction
potential v, the interaction

921
01:07:51,890 --> 01:07:54,630
picture, nothing happens
in the interaction picture,

922
01:07:54,630 --> 01:07:56,380
and therefore, the
time evolution operator

923
01:07:56,380 --> 01:07:57,910
is a unity matrix.

924
01:07:57,910 --> 01:08:02,840
But if you have a coupling term
v, there is time evolution.

925
01:08:02,840 --> 01:08:05,130
And the time evolution
can be treated

926
01:08:05,130 --> 01:08:09,110
in lowest or in higher
orders in the perturbation.

927
01:08:09,110 --> 01:08:12,495
And we want to use now
perturbation expansion, where

928
01:08:12,495 --> 01:08:14,640
we have correction
terms, which is

929
01:08:14,640 --> 01:08:17,149
the first order, second
order, and nth order

930
01:08:17,149 --> 01:08:20,689
correction for time
evolution operator.

931
01:08:20,689 --> 01:08:22,790
So what we simply
want to do is we

932
01:08:22,790 --> 01:08:25,580
want to do an
iterative solution.

933
01:08:25,580 --> 01:08:28,130
We want to find an iterative
solution for the time evolution

934
01:08:28,130 --> 01:08:28,630
operator.

935
01:08:31,700 --> 01:08:34,240
The next two lines
is just a reminder.

936
01:08:37,080 --> 01:08:40,510
What is the differential
equation for the time evolution

937
01:08:40,510 --> 01:08:42,446
operator?

938
01:08:42,446 --> 01:08:45,029
Well, we are really just talking
about Schrodinger's equation.

939
01:08:45,029 --> 01:08:47,350
Schrodinger's equation
for the wave function

940
01:08:47,350 --> 01:08:49,850
turns into differential
equation for the time evolution

941
01:08:49,850 --> 01:08:50,979
operator.

942
01:08:50,979 --> 01:08:53,630
Schrodinger's equation in
the interaction picture

943
01:08:53,630 --> 01:08:56,970
says that the time
derivative of the wave

944
01:08:56,970 --> 01:08:59,310
function in the
interaction picture

945
01:08:59,310 --> 01:09:00,819
has nothing to do H naught.

946
01:09:00,819 --> 01:09:02,689
H naught has already
been absorbed

947
01:09:02,689 --> 01:09:04,710
in the definition of
the wave function.

948
01:09:04,710 --> 01:09:06,939
So the time evolution in
the interaction picture

949
01:09:06,939 --> 01:09:10,890
only comes from the
interaction term.

950
01:09:10,890 --> 01:09:20,810
And now, if you write the
wave function at time tf

951
01:09:20,810 --> 01:09:24,420
as the time evolved wave
function from the initial time

952
01:09:24,420 --> 01:09:29,600
to the final time, then you
take the derivative here.

953
01:09:29,600 --> 01:09:33,220
You actually take a derivative
of the time evolution operator.

954
01:09:33,220 --> 01:09:38,130
And therefore, in one step,
from Schrodinger's equation,

955
01:09:38,130 --> 01:09:41,609
you find a differential
equation for the time evolution

956
01:09:41,609 --> 01:09:43,710
operator.

957
01:09:43,710 --> 01:09:47,010
In other words, this is
now the operator equation

958
01:09:47,010 --> 01:09:48,880
for the time evolution operator.

959
01:09:48,880 --> 01:09:53,640
And it's nothing else than a
100% rewrite of the Schrodinger

960
01:09:53,640 --> 01:09:54,140
equation.

961
01:10:02,180 --> 01:10:04,900
Interrupt me if
you have questions.

962
01:10:04,900 --> 01:10:08,580
This part should be--
we'll go through that

963
01:10:08,580 --> 01:10:10,400
for solely pedagogical reasons.

964
01:10:10,400 --> 01:10:13,920
And that means you should
really understand it.

965
01:10:13,920 --> 01:10:22,250
So this equation can
now be formally solved

966
01:10:22,250 --> 01:10:23,740
in the following way.

967
01:10:23,740 --> 01:10:27,310
Take this expression for
the time evolution operator

968
01:10:27,310 --> 01:10:31,790
and insert it in this equation,
and you find an identity.

969
01:10:31,790 --> 01:10:33,720
Of course, you haven't
really solved it,

970
01:10:33,720 --> 01:10:37,610
because you have the time
evolution operator expressed

971
01:10:37,610 --> 01:10:39,830
by the time evolution operator.

972
01:10:39,830 --> 01:10:42,150
This looks like a
circular conclusion,

973
01:10:42,150 --> 01:10:43,960
which is just nonsense.

974
01:10:43,960 --> 01:10:46,300
But if you inspect
it more closely,

975
01:10:46,300 --> 01:10:50,570
you observe that the time
evolution operator here

976
01:10:50,570 --> 01:10:54,260
is expressed by the time
evolution operator multiplied

977
01:10:54,260 --> 01:10:59,370
with v. And if you now think
in an iterative solution

978
01:10:59,370 --> 01:11:01,590
that we want to express.

979
01:11:01,590 --> 01:11:04,110
We want to expand the
time evolution operator

980
01:11:04,110 --> 01:11:09,320
in the different orders in the
interaction parameter v. Then

981
01:11:09,320 --> 01:11:15,250
you find actually
that if you are only

982
01:11:15,250 --> 01:11:19,210
interested in the first order
of the interaction operator,

983
01:11:19,210 --> 01:11:21,220
you can use the
[INAUDIBLE] order here,

984
01:11:21,220 --> 01:11:23,620
because you get one
more power of v.

985
01:11:23,620 --> 01:11:25,330
So that's how you
get the first order.

986
01:11:25,330 --> 01:11:28,610
If you want to know the second
order of the time evolution

987
01:11:28,610 --> 01:11:30,720
operator, you can
plug-in the first order

988
01:11:30,720 --> 01:11:32,150
on the right hand side.

989
01:11:32,150 --> 01:11:36,410
So therefore, you get
an iterative solution.

990
01:11:36,410 --> 01:11:38,710
The first order solution is one.

991
01:11:38,710 --> 01:11:41,120
The [INAUDIBLE] order solution
is the unity operator.

992
01:11:41,120 --> 01:11:45,120
The first order solution is by
just putting the unity operator

993
01:11:45,120 --> 01:11:45,890
here.

994
01:11:45,890 --> 01:11:49,220
The second order solution
is by taking the first order

995
01:11:49,220 --> 01:11:51,480
solution, plugging
it in in here.

996
01:11:51,480 --> 01:11:53,980
And then you get two
integrals over the operator

997
01:11:53,980 --> 01:12:00,100
v. So in that sense,
we have formally

998
01:12:00,100 --> 01:12:09,000
solved the time evolution
of the system in all orders.

999
01:12:09,000 --> 01:12:10,524
So this is what
we get out of it.

1000
01:12:18,120 --> 01:12:20,660
Yes, the equations
are getting longer.

1001
01:12:20,660 --> 01:12:23,520
But the structure
is fairly obvious.

1002
01:12:23,520 --> 01:12:30,970
So you saw that the--
let me just flashback.

1003
01:12:30,970 --> 01:12:34,130
The first order term
had a temporal integral

1004
01:12:34,130 --> 01:12:38,650
over v. The second order term
has two temporal integrals

1005
01:12:38,650 --> 01:12:45,130
over v. And there's a time order
in between tau 1 and tau 2.

1006
01:12:45,130 --> 01:12:52,370
And the nth order term now
involves n temporal integrals

1007
01:12:52,370 --> 01:12:54,987
where the times are
ordered in such a way.

1008
01:13:03,610 --> 01:13:07,940
Now we want to go back to
the Schrodinger equation.

1009
01:13:07,940 --> 01:13:12,410
So we want to use this already
fairly complicated expression

1010
01:13:12,410 --> 01:13:18,540
for the time evolution
operator and apply it

1011
01:13:18,540 --> 01:13:20,860
to the initial wave function.

1012
01:13:20,860 --> 01:13:22,880
And we want to calculate
matrix elements

1013
01:13:22,880 --> 01:13:26,090
between the initial wave
function and some other phi

1014
01:13:26,090 --> 01:13:32,670
f, some other wave
function in a given basis.

1015
01:13:32,670 --> 01:13:43,380
So we are also going back.

1016
01:13:43,380 --> 01:13:45,130
I don't want to go
through too many steps.

1017
01:13:45,130 --> 01:13:49,270
Here, we're also going back
to the original Schrodinger

1018
01:13:49,270 --> 01:13:50,010
picture.

1019
01:13:50,010 --> 01:13:53,150
So I've taken all
the tildes off.

1020
01:13:53,150 --> 01:13:56,140
But you know the step
form the Schrodinger

1021
01:13:56,140 --> 01:13:58,590
picture to the
interaction picture

1022
01:13:58,590 --> 01:14:01,740
was only taking out the
Hamiltonian H naught.

1023
01:14:01,740 --> 01:14:07,880
And the Hamiltonian H naught
for eigenstates phi i and phi f,

1024
01:14:07,880 --> 01:14:13,260
is simply a time evolution, e
to the e ei with a certain time.

1025
01:14:13,260 --> 01:14:17,110
So all what this transformation
to the interaction picture

1026
01:14:17,110 --> 01:14:21,080
gave us, it eliminated, at
least for a short while,

1027
01:14:21,080 --> 01:14:24,210
all of those phase
factors, which are simply

1028
01:14:24,210 --> 01:14:25,715
the evolution of
the eigenfunction

1029
01:14:25,715 --> 01:14:26,840
in the Schrodinger picture.

1030
01:14:47,460 --> 01:14:52,100
So let me look at a typical
term which we have right now.

1031
01:14:52,100 --> 01:14:54,850
And you can't expect
something simpler

1032
01:14:54,850 --> 01:14:57,325
than this, because this
is the general solution.

1033
01:14:57,325 --> 01:15:00,200
It's an exact solution
to all orders.

1034
01:15:00,200 --> 01:15:02,890
It's a summation over
terms to all orders.

1035
01:15:02,890 --> 01:15:06,410
And eventually the summation of
an infinite number of diagrams

1036
01:15:06,410 --> 01:15:09,270
is an exact solution.

1037
01:15:09,270 --> 01:15:15,090
So what we've got here is we
have an integral over n times.

1038
01:15:15,090 --> 01:15:17,570
And the n times
are time ordered.

1039
01:15:17,570 --> 01:15:22,650
This is the formal solution for
the time evolution operator.

1040
01:15:22,650 --> 01:15:30,950
And that means that we
integrate overall times,

1041
01:15:30,950 --> 01:15:34,420
but what it is under
the integral sign

1042
01:15:34,420 --> 01:15:39,290
is a product of the
operator v at times

1043
01:15:39,290 --> 01:15:42,510
tau 1, tau 2, tau 3, tau 4.

1044
01:15:45,230 --> 01:15:50,240
Since we got back to the
Schrodinger picture, what

1045
01:15:50,240 --> 01:15:55,180
happens is between
times tau 1, and tau 2,

1046
01:15:55,180 --> 01:15:59,370
the particle simply evolves
with the phase factor

1047
01:15:59,370 --> 01:16:03,530
given by the
Hamiltonian H naught.

1048
01:16:03,530 --> 01:16:05,540
So the picture is the following.

1049
01:16:05,540 --> 01:16:11,030
The time evolution operator
in the interaction picture

1050
01:16:11,030 --> 01:16:24,760
is a product over the operator
v tilde at different times.

1051
01:16:24,760 --> 01:16:28,100
If you simply plug it back
into the above equation--

1052
01:16:28,100 --> 01:16:30,110
and you have to look at
it for a little while

1053
01:16:30,110 --> 01:16:32,810
to see that everything
works out correctly.

1054
01:16:32,810 --> 01:16:38,785
But what that means is then that
if you go to the Schrodinger

1055
01:16:38,785 --> 01:16:42,210
picture and write down the
operator in the Schrodinger

1056
01:16:42,210 --> 01:16:44,900
in the basis of
unperturbed eigenfunction,

1057
01:16:44,900 --> 01:16:50,900
that you have products of
the operator v at times

1058
01:16:50,900 --> 01:16:54,510
t1, tau 2, tau 3,
they're time ordered.

1059
01:16:57,590 --> 01:17:00,400
And because we have left
the interaction picture,

1060
01:17:00,400 --> 01:17:04,400
you also get the propagation
with the phase factors

1061
01:17:04,400 --> 01:17:07,850
given by the eigenenergies of
the unperturbed Hamiltonian H

1062
01:17:07,850 --> 01:17:09,620
naught.

1063
01:17:09,620 --> 01:17:12,480
So therefore, you get
just an infinite number

1064
01:17:12,480 --> 01:17:16,840
of terms which have all
the following structure,

1065
01:17:16,840 --> 01:17:20,830
that the particle propagates
in its eigenstate.

1066
01:17:20,830 --> 01:17:24,770
It reaches a vertex where
the interaction switches

1067
01:17:24,770 --> 01:17:27,520
the particle from one
state to the next.

1068
01:17:27,520 --> 01:17:30,890
Then in the next state, which
may now be the excited state,

1069
01:17:30,890 --> 01:17:33,850
we have propagation
in the excited state.

1070
01:17:33,850 --> 01:17:37,560
Then the excited state is,
again, exchanging a photon,

1071
01:17:37,560 --> 01:17:41,090
has an interaction term,
and is switched to-- well,

1072
01:17:41,090 --> 01:17:43,880
for the two level system, it has
to go back to the ground state.

1073
01:17:43,880 --> 01:17:47,740
In a more general situation, it
goes to another quantum state.

1074
01:17:47,740 --> 01:18:02,160
So therefore, what we
have exactly derived

1075
01:18:02,160 --> 01:18:06,640
is that the propagation,
or the time evolution,

1076
01:18:06,640 --> 01:18:12,940
of the wave function
is nothing else then

1077
01:18:12,940 --> 01:18:14,590
many, many of those factors.

1078
01:18:17,490 --> 01:18:21,060
And each of those factors
can be represented

1079
01:18:21,060 --> 01:18:22,460
by a diagram like this.

1080
01:18:33,260 --> 01:18:42,410
So let me just write down
what this diagram here

1081
01:18:42,410 --> 01:18:45,010
means mathematically.

1082
01:18:45,010 --> 01:18:51,970
It means that initially, we
have a photon of energy epsilon,

1083
01:18:51,970 --> 01:18:56,200
of polarization epsilon, wave
factor k, and a certain energy.

1084
01:18:56,200 --> 01:18:59,080
And we have
particles in state a.

1085
01:18:59,080 --> 01:19:10,970
That means that we get from the
initial time to the time tau 1.

1086
01:19:10,970 --> 01:19:14,650
And tau 1 is the time
of the first vertex

1087
01:19:14,650 --> 01:19:20,970
of the first photon exchange,
that the wave function evolves.

1088
01:19:23,850 --> 01:19:28,800
And it evolves with an energy
of the unperturbed Hamiltonian,

1089
01:19:28,800 --> 01:19:31,450
which is the atomic energy.

1090
01:19:31,450 --> 01:19:33,510
And we have one
photon h bar omega.

1091
01:19:36,160 --> 01:19:39,860
At the time tau 1--
and this is when

1092
01:19:39,860 --> 01:19:41,560
the diagram on
the left hand side

1093
01:19:41,560 --> 01:19:47,070
has a vertex-- we are now
bringing in the interaction

1094
01:19:47,070 --> 01:19:50,880
operator, which
acts on the atom.

1095
01:19:50,880 --> 01:19:55,100
It takes the atom from
state A to state B.

1096
01:19:55,100 --> 01:19:58,640
And it changes the
state of the photon.

1097
01:19:58,640 --> 01:20:01,690
In this case, a photon
with a certain wave factor

1098
01:20:01,690 --> 01:20:06,500
and polarization
simply disappears.

1099
01:20:06,500 --> 01:20:14,140
So this is now the time tau 1.

1100
01:20:14,140 --> 01:20:18,610
The next Vertex is
reached at time tau 2.

1101
01:20:18,610 --> 01:20:23,580
And between the time
tau 2 and tau 1,

1102
01:20:23,580 --> 01:20:29,710
the system evolves according
to the Hamiltonian H naught.

1103
01:20:29,710 --> 01:20:34,710
And the energy is now,
because we have no photon,

1104
01:20:34,710 --> 01:20:41,440
is simply the energy
of the atom in state B.

1105
01:20:41,440 --> 01:20:44,910
Then we reach the next vertex.

1106
01:20:44,910 --> 01:20:48,350
We start with a state
B and no photon.

1107
01:20:48,350 --> 01:20:52,190
The interaction
switches, as now I

1108
01:20:52,190 --> 01:20:54,000
assumed we have more
than two levels.

1109
01:20:54,000 --> 01:20:57,620
Two are state C. And we
have now a photon, which

1110
01:20:57,620 --> 01:21:02,600
may be different, k
prime epsilon prime.

1111
01:21:02,600 --> 01:21:08,410
Since we are now, for this
given example, I've selected n

1112
01:21:08,410 --> 01:21:11,290
equals 2, something
which is second order

1113
01:21:11,290 --> 01:21:12,790
in the perturbation.

1114
01:21:12,790 --> 01:21:19,600
But that means now that
between the time tau

1115
01:21:19,600 --> 01:21:25,790
2 and the final tie-in for
the time propagation, that's

1116
01:21:25,790 --> 01:21:27,340
now everything is done.

1117
01:21:27,340 --> 01:21:29,300
Things propagate.

1118
01:21:29,300 --> 01:21:33,380
And the unperturbed
Hamiltonian H naught

1119
01:21:33,380 --> 01:21:36,010
gives us now a
phase factor, which

1120
01:21:36,010 --> 01:21:40,580
reflects the energy
of the atomic state,

1121
01:21:40,580 --> 01:21:44,770
and the energy of
the photon in mode k

1122
01:21:44,770 --> 01:21:49,510
prime epsilon prime with
energy omega h per omega prime.

1123
01:21:49,510 --> 01:21:53,200
So in other words, I hope you
see that at the end of this--

1124
01:21:53,200 --> 01:21:55,510
I mean, this is why people
have Feynman diagrams.

1125
01:21:55,510 --> 01:21:58,630
It's a complicated, involved
mathematical formulation

1126
01:21:58,630 --> 01:22:00,270
with multiple integrations.

1127
01:22:00,270 --> 01:22:03,330
But at the end of the
day, what we have derived

1128
01:22:03,330 --> 01:22:06,772
is that the most
general solutions

1129
01:22:06,772 --> 01:22:13,330
to all orders in the probation
is nothing else than a sum

1130
01:22:13,330 --> 01:22:16,610
and integral over
diagrams like this.

1131
01:22:16,610 --> 01:22:21,210
So when I depict it here, the
second order diagram, of course

1132
01:22:21,210 --> 01:22:25,860
this has an exact
mathematical meaning.

1133
01:22:25,860 --> 01:22:28,350
And what we have to
do is when we solve

1134
01:22:28,350 --> 01:22:30,540
a quantum mechanical
equation, we

1135
01:22:30,540 --> 01:22:33,330
have to now allow those
interactions, which

1136
01:22:33,330 --> 01:22:37,910
happen at time tau 1 and tau 2,
to happen at arbitrary times.

1137
01:22:37,910 --> 01:22:40,680
So therefore, we have
to sum over amplitudes

1138
01:22:40,680 --> 01:22:44,162
by integrating over
times tau 1 and tau 2.

1139
01:22:47,050 --> 01:22:49,890
So on Friday-- I
have to stop now.

1140
01:22:49,890 --> 01:22:51,980
I think there's a
seminar right now.

1141
01:22:51,980 --> 01:22:56,540
On Friday, I will show you
that with this description,

1142
01:22:56,540 --> 01:23:00,720
we have actually captured
everything I explained to you

1143
01:23:00,720 --> 01:23:04,410
earlier-- those virtual
states, the emission

1144
01:23:04,410 --> 01:23:06,890
of virtual or real photons.

1145
01:23:06,890 --> 01:23:08,840
So everything that
which was maybe

1146
01:23:08,840 --> 01:23:12,180
qualitative at the
beginning of this section

1147
01:23:12,180 --> 01:23:15,340
has now a precise,
mathematical meaning.

1148
01:23:15,340 --> 01:23:17,720
But that's what we do on Friday.