1
00:00:00,070 --> 00:00:01,770
The following
content is provided

2
00:00:01,770 --> 00:00:04,010
under a Creative
Commons license.

3
00:00:04,010 --> 00:00:06,860
Your support will help MIT
OpenCourseWare continue

4
00:00:06,860 --> 00:00:10,720
to offer high quality
educational resources for free.

5
00:00:10,720 --> 00:00:13,330
To make a donation or
view additional materials

6
00:00:13,330 --> 00:00:17,202
from hundreds of MIT courses,
visit MIT OpenCourseWare

7
00:00:17,202 --> 00:00:17,827
at ocw.mit.edu.

8
00:00:21,300 --> 00:00:24,430
PROFESSOR: Good afternoon.

9
00:00:24,430 --> 00:00:26,310
Let's get started.

10
00:00:28,920 --> 00:00:32,189
So we continue our
discussion today

11
00:00:32,189 --> 00:00:35,320
about light atom interaction.

12
00:00:35,320 --> 00:00:38,952
And just to sort of
remind you where we are,

13
00:00:38,952 --> 00:00:43,410
we started last week,
even before spring

14
00:00:43,410 --> 00:00:47,620
break, to talk about
the matrix element which

15
00:00:47,620 --> 00:00:49,410
provides a coupling.

16
00:00:49,410 --> 00:00:52,090
So now we have a coupling
between the two states.

17
00:00:52,090 --> 00:00:54,150
And we want to
understand what is

18
00:00:54,150 --> 00:00:56,990
the coupling doing
to the system.

19
00:00:56,990 --> 00:00:59,310
What is a dynamical
evolution of the system?

20
00:00:59,310 --> 00:01:01,100
How does the atomic
wave function

21
00:01:01,100 --> 00:01:09,020
evolve when we couple two
states using optical fields?

22
00:01:09,020 --> 00:01:15,230
And well, as usual, we start
with the basic phenomena.

23
00:01:15,230 --> 00:01:17,720
And this is we do
perturbation theory

24
00:01:17,720 --> 00:01:20,500
with a dipole Hamiltonian.

25
00:01:20,500 --> 00:01:25,170
And we have done that on Monday.

26
00:01:25,170 --> 00:01:27,190
We've done perturbation theory.

27
00:01:27,190 --> 00:01:29,830
And I made an
important distinction

28
00:01:29,830 --> 00:01:34,570
between a monochromatic
case and [INAUDIBLE] case.

29
00:01:34,570 --> 00:01:38,190
The monochromatic case gave
us in perturbation theory

30
00:01:38,190 --> 00:01:39,640
this result.

31
00:01:39,640 --> 00:01:41,450
And if you stare
at it for a while,

32
00:01:41,450 --> 00:01:43,940
you see, well, these are
just Rabi oscillations.

33
00:01:43,940 --> 00:01:46,720
It's not the formula
of Rabi oscillations

34
00:01:46,720 --> 00:01:48,670
with the generalized
Rabi frequency

35
00:01:48,670 --> 00:01:51,650
and power broadening because
it's perturbation theory.

36
00:01:51,650 --> 00:01:55,160
So these are Rabi oscillations
in the perturbative limit.

37
00:01:55,160 --> 00:01:59,690
But also on Monday I showed
you that for interactions

38
00:01:59,690 --> 00:02:03,920
with monochromatic light, we can
just rewrite the Hamiltonian.

39
00:02:03,920 --> 00:02:06,800
That it not just looks
like, it is exactly

40
00:02:06,800 --> 00:02:12,670
the Hamiltonian for spin
1/2 in a magnetic field.

41
00:02:12,670 --> 00:02:15,090
And we have already
discussed the solution.

42
00:02:15,090 --> 00:02:17,611
So, therefore, we know for
monochromatic radiation,

43
00:02:17,611 --> 00:02:19,360
we can just go beyond
perturbation theory.

44
00:02:19,360 --> 00:02:21,380
We can solve it exactly.

45
00:02:21,380 --> 00:02:21,970
OK.

46
00:02:21,970 --> 00:02:24,675
So monochromatic
radiation we understand.

47
00:02:24,675 --> 00:02:29,910
At least as long as we have
just the coupling of the atom

48
00:02:29,910 --> 00:02:33,270
to a single mode of the
electromagnetic field.

49
00:02:33,270 --> 00:02:36,174
And now we come back
to the broadband case.

50
00:02:36,174 --> 00:02:38,340
I give a little bit more
of a summary and an outline

51
00:02:38,340 --> 00:02:40,690
after I finish the
broadband case.

52
00:02:40,690 --> 00:02:45,770
So what we have right now is
we have Rabi oscillations,

53
00:02:45,770 --> 00:02:48,310
which we derived
for a single mode.

54
00:02:48,310 --> 00:02:52,670
And the broadband case
is such that we now

55
00:02:52,670 --> 00:02:59,940
assume that we have a
spectrum of frequencies.

56
00:02:59,940 --> 00:03:03,620
And we assume that to be
flat and broad in a moment.

57
00:03:03,620 --> 00:03:06,580
And what we have to
do now is we have

58
00:03:06,580 --> 00:03:09,470
to substitute in
this formula the Rabi

59
00:03:09,470 --> 00:03:12,100
frequency by the electric field.

60
00:03:12,100 --> 00:03:15,200
And the electric field
becomes using the connection

61
00:03:15,200 --> 00:03:23,070
between electric field
and energy density,

62
00:03:23,070 --> 00:03:25,280
eventually the Rabi
frequency squared

63
00:03:25,280 --> 00:03:30,150
gets replaced by an integral
over the spectral density.

64
00:03:30,150 --> 00:03:31,800
It looks mathematically exact.

65
00:03:31,800 --> 00:03:33,650
But, of course,
you all recognize

66
00:03:33,650 --> 00:03:37,170
that I've made a very
important assumption here.

67
00:03:37,170 --> 00:03:39,630
Namely, that there is no
correlation whatsoever

68
00:03:39,630 --> 00:03:41,550
between the different
frequencies.

69
00:03:41,550 --> 00:03:45,400
Because by integrating
here, I'm just

70
00:03:45,400 --> 00:03:47,410
summing up the e
squares assuming

71
00:03:47,410 --> 00:03:50,034
that there's no interference,
no coherence, no correlation

72
00:03:50,034 --> 00:03:51,450
between the different
frequencies.

73
00:03:54,390 --> 00:03:59,840
OK, so with those
assumptions I can now just

74
00:03:59,840 --> 00:04:01,970
formulate the
mathematics exactly.

75
00:04:01,970 --> 00:04:06,080
I calculate now the probability
to be in the excited state

76
00:04:06,080 --> 00:04:10,550
by assuming we have this kind of
pertrubative Rabi oscillations

77
00:04:10,550 --> 00:04:12,500
at every frequency component.

78
00:04:12,500 --> 00:04:16,209
But we integrate
over all frequencies.

79
00:04:16,209 --> 00:04:17,790
Any questions?

80
00:04:17,790 --> 00:04:24,465
So now-- and this is where
I ended the last lecture.

81
00:04:27,730 --> 00:04:31,257
And there are actually now, this
is really an interesting case.

82
00:04:31,257 --> 00:04:32,840
I know I sometimes
spend a lot of time

83
00:04:32,840 --> 00:04:35,230
on sort of where
the math is simple,

84
00:04:35,230 --> 00:04:37,280
but the physics is
really interesting.

85
00:04:37,280 --> 00:04:41,440
So the probability to find
an atom in the excited state

86
00:04:41,440 --> 00:04:45,510
is now this function
which replaces,

87
00:04:45,510 --> 00:04:47,610
which are the
Rabit oscillations,

88
00:04:47,610 --> 00:04:51,290
convoluted integrated
over the spectral density.

89
00:04:51,290 --> 00:04:53,005
And now we have to
consider two cases.

90
00:04:56,490 --> 00:04:59,870
If this function is flat and
is spectrally very broad,

91
00:04:59,870 --> 00:05:03,030
we can pull it out
of the integral

92
00:05:03,030 --> 00:05:07,040
However, this function
here, which are your Rabi

93
00:05:07,040 --> 00:05:10,240
oscillations-- sine squared
of an [? augument ?] divided

94
00:05:10,240 --> 00:05:13,630
by the [? augument. ?] I've
plotted it here for you.

95
00:05:13,630 --> 00:05:16,900
This function is
actually very peculiar.

96
00:05:16,900 --> 00:05:20,660
It has a height of t square.

97
00:05:20,660 --> 00:05:23,280
If you're doing Taylor
Expansion for short times t.

98
00:05:23,280 --> 00:05:26,380
But the width is
t to the minus 1.

99
00:05:26,380 --> 00:05:29,570
So as time goes by, it
gets narrower, and taller,

100
00:05:29,570 --> 00:05:30,910
and taller.

101
00:05:30,910 --> 00:05:34,510
And if you take t squared by
t to the minus 1, you get t.

102
00:05:34,510 --> 00:05:38,530
So the integral
over it is just t.

103
00:05:38,530 --> 00:05:43,780
So in other words,
we have a function

104
00:05:43,780 --> 00:05:47,350
which goes as t square.

105
00:05:47,350 --> 00:05:49,430
But the integral is t.

106
00:05:49,430 --> 00:05:52,280
And if the width is so narrow
that the width doesn't really

107
00:05:52,280 --> 00:05:56,430
matter, we can say this function
has become a delta function.

108
00:05:56,430 --> 00:05:58,900
But the integral over
the function is t.

109
00:05:58,900 --> 00:06:01,530
So it becomes t times
the delta function.

110
00:06:01,530 --> 00:06:03,400
And now we the two
limiting cases.

111
00:06:07,960 --> 00:06:14,250
If very short times t-- this
Rabi oscillation function

112
00:06:14,250 --> 00:06:16,420
is extremely broad.

113
00:06:16,420 --> 00:06:19,250
And at infinitesimal
times t, it is

114
00:06:19,250 --> 00:06:22,120
broader than any
spectral bandwidths.

115
00:06:22,120 --> 00:06:23,870
So actually what we
are doing is when

116
00:06:23,870 --> 00:06:27,780
we solve this integral at very
short time, this is broader.

117
00:06:27,780 --> 00:06:30,000
And we pull this
out of the integral.

118
00:06:30,000 --> 00:06:32,940
And this gives us a
prefect of t squared.

119
00:06:32,940 --> 00:06:35,053
So, therefore, the
excitation probability

120
00:06:35,053 --> 00:06:38,995
is t squared times the integral
of the spectral function which

121
00:06:38,995 --> 00:06:40,245
is just the totally intensity.

122
00:06:42,750 --> 00:06:46,810
However, if you
wait a little time,

123
00:06:46,810 --> 00:06:49,050
and eventually if
the time is such

124
00:06:49,050 --> 00:06:53,530
that the time is longer
than the inverse bandwidths

125
00:06:53,530 --> 00:06:58,290
of the spectral radiation,
then the spectral radiation

126
00:06:58,290 --> 00:07:01,470
is broader than
this function, f.

127
00:07:01,470 --> 00:07:04,150
And we pull this
out of the integral.

128
00:07:04,150 --> 00:07:07,270
And then what we have is we
have simply the integral here

129
00:07:07,270 --> 00:07:11,250
over the delta function
which gives us a factor of t.

130
00:07:11,250 --> 00:07:12,770
So we have two functions.

131
00:07:12,770 --> 00:07:15,910
And whichever is broader can
be pulled out of the integral.

132
00:07:15,910 --> 00:07:18,010
And that means in one
case, we have a probability

133
00:07:18,010 --> 00:07:21,090
which is t squared times
the total intensity.

134
00:07:21,090 --> 00:07:23,310
And in the other case,
we have a behavior

135
00:07:23,310 --> 00:07:27,270
which is the time times the
spectral density at 0 detuning.

136
00:07:33,440 --> 00:07:34,360
OK.

137
00:07:34,360 --> 00:07:38,180
So now let's-- just a side
remark before I interpret this

138
00:07:38,180 --> 00:07:42,590
result, this function you
would recover actually the Rabi

139
00:07:42,590 --> 00:07:43,330
oscillation.

140
00:07:43,330 --> 00:07:45,050
I plotted it in
that way emphasizing

141
00:07:45,050 --> 00:07:46,680
the amplitude and the width.

142
00:07:46,680 --> 00:07:51,150
But if you look for fixed
detuning and you vary the time,

143
00:07:51,150 --> 00:07:53,390
this function is
sort of spreading out

144
00:07:53,390 --> 00:07:56,580
and at a fixed detuning,
you'll go up and down.

145
00:07:56,580 --> 00:07:58,100
And these are the
Rabi oscillations

146
00:07:58,100 --> 00:08:02,720
we have discussed for
monochromatic radiation before.

147
00:08:02,720 --> 00:08:06,070
So these two limiting cases
are actually very important.

148
00:08:06,070 --> 00:08:09,830
At very short times, the
behavior is, you know,

149
00:08:09,830 --> 00:08:12,210
we have broad
spectral radiation.

150
00:08:12,210 --> 00:08:14,600
But if the time is shorter
than the inverse bandwidth

151
00:08:14,600 --> 00:08:17,630
of the radiation, even
the broadband radiation

152
00:08:17,630 --> 00:08:20,070
is like monochromatic wave.

153
00:08:20,070 --> 00:08:24,680
And the short time behavior is
t squared times the intensity.

154
00:08:24,680 --> 00:08:27,670
That's exactly what we got
for monochromatic light.

155
00:08:27,670 --> 00:08:30,256
In other words, if
you're broadening,

156
00:08:30,256 --> 00:08:31,630
you have a broad
spectral source,

157
00:08:31,630 --> 00:08:33,980
but if your inverse
time is broader

158
00:08:33,980 --> 00:08:36,020
than the broadening of
the spectral source,

159
00:08:36,020 --> 00:08:38,409
you're back to the
monochromatic case.

160
00:08:38,409 --> 00:08:40,030
So a lot of people get confused.

161
00:08:40,030 --> 00:08:43,370
I mean, I see that often
in part 3 oral exams, that

162
00:08:43,370 --> 00:08:45,080
no matter what your
spectral bandwidth

163
00:08:45,080 --> 00:08:47,050
is-- unless it's infinite.

164
00:08:47,050 --> 00:08:49,140
But let's discuss a
pathological case.

165
00:08:49,140 --> 00:08:52,530
For any broadband
light, at short moments,

166
00:08:52,530 --> 00:08:55,080
the system evolves as t square.

167
00:08:55,080 --> 00:08:59,050
And t square is the hallmark
of coherent time evolution.

168
00:08:59,050 --> 00:09:00,890
The amplitude goes with t.

169
00:09:00,890 --> 00:09:03,350
The probability
goes with t square.

170
00:09:03,350 --> 00:09:07,400
And only when the bandwidth
of the spectral light

171
00:09:07,400 --> 00:09:11,200
dominates when we can replace
this by the delta function.

172
00:09:11,200 --> 00:09:13,390
Then we are in the
regime, which you all

173
00:09:13,390 --> 00:09:18,350
know is Fermi's Golden Rule,
where we [INAUDIBLE] equation.

174
00:09:18,350 --> 00:09:20,540
And it all comes
from this formula.

175
00:09:20,540 --> 00:09:23,540
It all comes from assuming
Rabi oscillations.

176
00:09:23,540 --> 00:09:26,820
But then performing
the spectral integral

177
00:09:26,820 --> 00:09:28,380
over the Rabi oscillation.

178
00:09:28,380 --> 00:09:30,960
This takes us from
the narrowband case

179
00:09:30,960 --> 00:09:33,471
to the broadband case.

180
00:09:33,471 --> 00:09:34,800
Any questions about that?

181
00:09:34,800 --> 00:09:35,300
Yes?

182
00:09:35,300 --> 00:09:37,920
AUDIENCE: So what says
the bandwidth here?

183
00:09:37,920 --> 00:09:41,097
Because we assume
its narrowness arises

184
00:09:41,097 --> 00:09:42,680
just because of the
short [INAUDIBLE].

185
00:09:42,680 --> 00:09:44,140
PROFESSOR: I will
use an LED which

186
00:09:44,140 --> 00:09:45,860
has a few nanometer bandwidths.

187
00:09:45,860 --> 00:09:49,540
Use sunlight which has a few
hundred nanometer bandwidths.

188
00:09:49,540 --> 00:09:50,850
Whatever your light source is.

189
00:09:55,850 --> 00:09:57,930
At this point, it's
a general discussion.

190
00:09:57,930 --> 00:10:00,996
And I'm not going beyond
the two limiting cases here.

191
00:10:00,996 --> 00:10:02,870
Of course, if you have
a complicated spectral

192
00:10:02,870 --> 00:10:05,170
distribution, well,
you're on your own

193
00:10:05,170 --> 00:10:07,700
to solve this integral.

194
00:10:07,700 --> 00:10:09,895
But at short times,
we have this behavior.

195
00:10:09,895 --> 00:10:13,560
At long times, we have this.

196
00:10:13,560 --> 00:10:16,920
And the two regimes
are separated

197
00:10:16,920 --> 00:10:21,410
by the time where I have
a crossover in one case.

198
00:10:21,410 --> 00:10:24,310
This is broader and I can
pull it out in the other case.

199
00:10:24,310 --> 00:10:27,370
This function is broader and
I pull it out of the integral.

200
00:10:27,370 --> 00:10:29,480
And just where
this happens, this

201
00:10:29,480 --> 00:10:33,710
is where we have the transition
from t squared to t behavior.

202
00:10:33,710 --> 00:10:35,895
From coherent evolution
to rate equation.

203
00:10:49,790 --> 00:10:50,290
OK.

204
00:10:57,595 --> 00:10:58,095
Yes.

205
00:11:07,700 --> 00:11:11,230
So let's now look at the
situation of broadband light.

206
00:11:11,230 --> 00:11:13,600
Because later today I
want to discuss with you

207
00:11:13,600 --> 00:11:15,380
Einstein's a and b coefficient.

208
00:11:15,380 --> 00:11:17,860
A very classical topic
of atomic physics.

209
00:11:17,860 --> 00:11:20,980
A very famous concept
introduced by Einstein.

210
00:11:20,980 --> 00:11:23,870
And actually I want to use
the perturbation theory

211
00:11:23,870 --> 00:11:26,710
for broadband light which
we have now formulated

212
00:11:26,710 --> 00:11:30,120
to derive for you the b
coefficient of Einstein's a

213
00:11:30,120 --> 00:11:32,780
and b coefficient theory.

214
00:11:32,780 --> 00:11:33,280
OK.

215
00:11:33,280 --> 00:11:44,375
So for large time, we
can now talk about rate.

216
00:11:49,450 --> 00:11:54,420
Which is the probability
which increases linearly

217
00:11:54,420 --> 00:11:58,180
with time per unit time.

218
00:11:58,180 --> 00:12:04,160
And this was the
matrix element squared.

219
00:12:04,160 --> 00:12:08,470
2 epsilon 0 h-bar squared, 2 pi.

220
00:12:08,470 --> 00:12:10,260
The 2 cancels.

221
00:12:10,260 --> 00:12:14,690
And then we have from the delta
function, the spectral density

222
00:12:14,690 --> 00:12:15,690
at 0 detuning.

223
00:12:18,330 --> 00:12:20,755
Which is the spectral density
at the resonance frequency.

224
00:12:27,570 --> 00:12:30,250
So we have a rate equation now.

225
00:12:30,250 --> 00:12:33,760
That the rate equation
is the b coefficient.

226
00:12:33,760 --> 00:12:39,240
Einstein's famous b coefficient
times the spectral density.

227
00:12:39,240 --> 00:12:44,530
And the b coefficient is
now the proportionality

228
00:12:44,530 --> 00:12:48,640
constant between in
the equation above.

229
00:12:48,640 --> 00:12:53,280
Which is pi d squared.

230
00:12:53,280 --> 00:12:57,500
But now in all the formula
for the b coefficient,

231
00:12:57,500 --> 00:12:59,790
there's a factor of 3.

232
00:12:59,790 --> 00:13:04,770
Because the assumption is made
that we have isotropy of space.

233
00:13:04,770 --> 00:13:07,160
The atom are randomly oriented.

234
00:13:07,160 --> 00:13:20,310
And, therefore, dx squared
for given polarization,

235
00:13:20,310 --> 00:13:23,230
the dipole moment projected on a
polarization of the light which

236
00:13:23,230 --> 00:13:27,462
is dx squared is just
1/3 of the absolute value

237
00:13:27,462 --> 00:13:28,670
of the dipole moment squared.

238
00:13:38,980 --> 00:13:41,640
So in other words,
just to remind you

239
00:13:41,640 --> 00:13:45,300
what I have actually
discussed is nothing else

240
00:13:45,300 --> 00:13:46,820
than Fermi's Golden Rule.

241
00:13:49,610 --> 00:13:53,030
And I could've reminded
you of Fermi's Golden Rule

242
00:13:53,030 --> 00:13:55,490
where the rate is given.

243
00:13:55,490 --> 00:13:58,330
I just use a standard
notation of textbooks.

244
00:13:58,330 --> 00:14:02,580
You take the matrix element
squared, you multiply by 2 pi,

245
00:14:02,580 --> 00:14:04,135
and then you have
a delta function.

246
00:14:06,780 --> 00:14:12,659
And the delta function implies
a delta function is always

247
00:14:12,659 --> 00:14:14,200
a reminder that it
needs integration.

248
00:14:18,740 --> 00:14:21,305
So whenever you have a delta
function in Fermi's Golden

249
00:14:21,305 --> 00:14:24,980
Rule, you have to integrate.

250
00:14:24,980 --> 00:14:26,760
And there are two possibilities.

251
00:14:26,760 --> 00:14:31,710
You have to integrate over the
spectrum of external fields.

252
00:14:31,710 --> 00:14:33,550
That's what we just did.

253
00:14:33,550 --> 00:14:38,610
The other possibility is, which
doesn't apply to what we just

254
00:14:38,610 --> 00:14:41,350
discussed, that you
have to integrate over

255
00:14:41,350 --> 00:14:43,120
a continuum of final states.

256
00:14:48,120 --> 00:14:50,790
This will be important when
we use a Fermi's Golden Rule

257
00:14:50,790 --> 00:14:53,350
expression to talk about
spontaneous emission

258
00:14:53,350 --> 00:14:56,574
where we have a continuum
of final states.

259
00:14:56,574 --> 00:14:57,990
So anyway, I could
have just said,

260
00:14:57,990 --> 00:14:59,700
let's start with
Fermi's Golden Rule

261
00:14:59,700 --> 00:15:02,030
and let's jump to
the final result.

262
00:15:02,030 --> 00:15:04,990
But I really wanted
to emphasize here

263
00:15:04,990 --> 00:15:08,880
the sort of intimate connection
between Rabi oscillation,

264
00:15:08,880 --> 00:15:11,090
the t square
dependence, and how this

265
00:15:11,090 --> 00:15:12,730
turns into a rate equation.

266
00:15:22,500 --> 00:15:25,030
OK.

267
00:15:25,030 --> 00:15:33,106
Let's just summarize what
we have done in a table.

268
00:15:36,600 --> 00:15:42,955
We have seen two
different regime.

269
00:15:47,880 --> 00:15:50,990
In one case with
the Rabi resonance,

270
00:15:50,990 --> 00:16:01,850
we are discussing a single
final state of the atom.

271
00:16:01,850 --> 00:16:06,720
A single mode of the
electromagnetic field.

272
00:16:06,720 --> 00:16:11,070
All energy levels, all
states are discrete.

273
00:16:11,070 --> 00:16:18,650
We are talking about unitary
reversible time evolution.

274
00:16:18,650 --> 00:16:21,650
When we had rate
equations, we are

275
00:16:21,650 --> 00:16:25,750
talking about many final states.

276
00:16:25,750 --> 00:16:27,720
We integrate over them.

277
00:16:27,720 --> 00:16:36,870
Or and/or many modes
of the external field.

278
00:16:36,870 --> 00:16:39,750
We are naturally dealing
not with a discrete number,

279
00:16:39,750 --> 00:16:41,225
but with a continuum of states.

280
00:16:45,310 --> 00:16:49,360
The time evolution has
become irreversible.

281
00:16:49,360 --> 00:16:53,090
And is therefore no
longer unitary evolution,

282
00:16:53,090 --> 00:16:55,450
but it's a
dissipative evolution.

283
00:17:00,210 --> 00:17:07,520
And all this came
about not because we

284
00:17:07,520 --> 00:17:10,970
have spontaneous emission.

285
00:17:10,970 --> 00:17:12,919
I will tell you
throughout this course

286
00:17:12,919 --> 00:17:15,300
that spontaneous emission
is not as spontaneous

287
00:17:15,300 --> 00:17:17,200
as everybody assumes.

288
00:17:17,200 --> 00:17:20,349
Spontaneous emission is
actually unitary time evolution.

289
00:17:20,349 --> 00:17:22,770
Unless you discard information.

290
00:17:22,770 --> 00:17:25,480
But a lot of people think
rate equation irreversibility

291
00:17:25,480 --> 00:17:29,370
comes from something
which is genuinely

292
00:17:29,370 --> 00:17:31,737
spontaneous and irreproducable.

293
00:17:31,737 --> 00:17:33,320
I don't know anything
in physics which

294
00:17:33,320 --> 00:17:34,736
is spontaneous and
irreproducible.

295
00:17:34,736 --> 00:17:35,890
But we come to that later.

296
00:17:35,890 --> 00:17:38,740
And this is an example where
we obtain rate equation

297
00:17:38,740 --> 00:17:40,570
by simply driving a system.

298
00:17:40,570 --> 00:17:43,292
And the irreversibility comes
by performing the integral

299
00:17:43,292 --> 00:17:44,375
over the spectral density.

300
00:17:47,410 --> 00:17:50,290
So let me just write that down.

301
00:17:50,290 --> 00:17:55,150
Due to integration.

302
00:17:57,810 --> 00:18:02,980
Since we integrate
over an infinite number

303
00:18:02,980 --> 00:18:05,735
of modes or states.

304
00:18:14,050 --> 00:18:14,640
Any questions?

305
00:18:19,630 --> 00:18:20,310
OK, great.

306
00:18:20,310 --> 00:18:25,250
I wanted to make sure
that this is very clear.

307
00:18:25,250 --> 00:18:29,130
OK, at this point,
let me just summarize

308
00:18:29,130 --> 00:18:33,650
where we are in our discussion
of atom light interaction.

309
00:18:33,650 --> 00:18:36,110
We've actually made
a lot of progress.

310
00:18:36,110 --> 00:18:38,855
We have discussed
matrix elements.

311
00:18:43,010 --> 00:18:48,450
We have discussed the coupling
of atoms to an external field

312
00:18:48,450 --> 00:18:51,349
at the level of the
Schroedinger equation.

313
00:18:51,349 --> 00:18:52,890
And we have done
perturbation theory.

314
00:18:56,880 --> 00:19:01,310
And in perturbation
theory, we found

315
00:19:01,310 --> 00:19:05,665
Rabi oscillations and
we found rate equations.

316
00:19:09,830 --> 00:19:12,340
That's where we are right now.

317
00:19:12,340 --> 00:19:16,720
So the feature now
which is missing

318
00:19:16,720 --> 00:19:19,500
is, of course, damping
spontaneous emission

319
00:19:19,500 --> 00:19:22,790
irreversability-- another
form of irreversibility.

320
00:19:22,790 --> 00:19:27,290
Right now our Rabi
oscillations are undamped.

321
00:19:27,290 --> 00:19:29,290
Whether we obtain them
in perturbation theory

322
00:19:29,290 --> 00:19:34,440
or whether we use the
spin formalism to get them

323
00:19:34,440 --> 00:19:38,600
in the resonant in the
strong coupling case.

324
00:19:38,600 --> 00:19:43,780
And here, for the rate equation,
the way how we have solved it,

325
00:19:43,780 --> 00:19:46,250
the probability to be
in the excited state

326
00:19:46,250 --> 00:19:48,164
just increases forever.

327
00:19:48,164 --> 00:19:49,830
The system will never
reach equilibrium.

328
00:19:55,960 --> 00:20:00,726
But that means in both cases,
we have a missing element.

329
00:20:00,726 --> 00:20:02,100
And this is
spontaneous emission.

330
00:20:14,860 --> 00:20:16,720
So for the next
hour or two, we'll

331
00:20:16,720 --> 00:20:19,140
talk about aspects of
spontaneous emission.

332
00:20:19,140 --> 00:20:23,190
Spontaneous emission
will actually eventually

333
00:20:23,190 --> 00:20:25,560
lead to damping of
Rabi oscillation.

334
00:20:25,560 --> 00:20:29,070
And to a saturation
of the excitation.

335
00:20:33,310 --> 00:20:36,330
OK, so we're now discussing
spontaneous emission.

336
00:20:36,330 --> 00:20:39,490
And we will discuss it
in actually three levels.

337
00:20:39,490 --> 00:20:48,180
One is I will discuss
Einstein's a and b coefficients.

338
00:20:48,180 --> 00:20:50,390
I sometimes hesitate.

339
00:20:50,390 --> 00:20:52,489
Should I really discuss
Einstein's a and b

340
00:20:52,489 --> 00:20:53,030
coefficients?

341
00:20:53,030 --> 00:20:54,890
It's sort of old fashioned.

342
00:20:54,890 --> 00:20:57,680
And I have already in
perturbation theory

343
00:20:57,680 --> 00:20:59,660
given you a
microscopic derivation

344
00:20:59,660 --> 00:21:01,620
of Einstein's b coefficient.

345
00:21:01,620 --> 00:21:05,960
But everybody who is
an atomic physicist

346
00:21:05,960 --> 00:21:08,060
knows about Einstein's
a and b coefficient.

347
00:21:08,060 --> 00:21:10,490
It was really a stroke
of a genius to do it.

348
00:21:10,490 --> 00:21:12,920
And it becomes sort
of our language.

349
00:21:12,920 --> 00:21:15,801
So what I'm doing here is
I'm not beating it to death,

350
00:21:15,801 --> 00:21:17,425
but I give you sort
of a short summary.

351
00:21:19,970 --> 00:21:22,450
It's also sort of I make a
few comments which is actually

352
00:21:22,450 --> 00:21:24,040
amazing.

353
00:21:24,040 --> 00:21:27,280
How Einstein actually got
results from the a and b

354
00:21:27,280 --> 00:21:31,280
coefficient which you
can only get otherwise

355
00:21:31,280 --> 00:21:33,590
if you quantize
electromagnetic field.

356
00:21:33,590 --> 00:21:35,560
So it's also sort of
historically interesting

357
00:21:35,560 --> 00:21:40,480
that Einstein actually developed
the theory of the a and b

358
00:21:40,480 --> 00:21:42,810
coefficient before the
Schroedinger equation.

359
00:21:42,810 --> 00:21:44,970
Before quantum
mechanics was developed.

360
00:21:44,970 --> 00:21:47,820
And often you call Schroedinger
equation the first quantization

361
00:21:47,820 --> 00:21:51,180
and the field quantization,
the second quantization.

362
00:21:51,180 --> 00:21:55,000
So in some sense,
Einstein actually

363
00:21:55,000 --> 00:21:59,490
preempted or had already the
results of second quantization

364
00:21:59,490 --> 00:22:02,450
before first quantization
was developed.

365
00:22:02,450 --> 00:22:05,780
Anyway, it's a landmark
paper, how Einstein did it.

366
00:22:05,780 --> 00:22:08,170
And that's why I
want to discuss it.

367
00:22:08,170 --> 00:22:11,150
But it's partially also
in order to give you

368
00:22:11,150 --> 00:22:13,180
the historical context.

369
00:22:13,180 --> 00:22:19,000
But then, of course, we want to
use the modern formalism of use

370
00:22:19,000 --> 00:22:23,680
of quantization of the
electromagnetic field.

371
00:22:23,680 --> 00:22:27,295
And we have already
obtained just now the result

372
00:22:27,295 --> 00:22:31,650
for Einstein's b coefficient
by just looking at the induced

373
00:22:31,650 --> 00:22:34,440
by the absorption rate
or the stimulated rate.

374
00:22:34,440 --> 00:22:38,340
But then eventually having
a microscopic quantization.

375
00:22:38,340 --> 00:22:41,260
By having a quantization of
the electromagnetic field,

376
00:22:41,260 --> 00:22:49,060
we can also do now microscopic
fully quantum first principle

377
00:22:49,060 --> 00:22:51,760
calculation of
the a coefficient.

378
00:22:51,760 --> 00:22:53,510
So then we have already
the b coefficient,

379
00:22:53,510 --> 00:22:56,730
we get the a coefficient out
of microscopic calculation.

380
00:22:56,730 --> 00:22:59,187
So we don't really need
Einstein's treatment of a and b

381
00:22:59,187 --> 00:22:59,770
at this point.

382
00:22:59,770 --> 00:23:03,840
But it's nice to
see the connections.

383
00:23:03,840 --> 00:23:04,340
So anyway.

384
00:23:04,340 --> 00:23:05,740
So this is the agenda.

385
00:23:05,740 --> 00:23:07,870
Einstein's a and b
coefficient to pull out

386
00:23:07,870 --> 00:23:11,350
spontaneous emission
without even putting it in.

387
00:23:11,350 --> 00:23:14,000
Then we'll talk about
field quantization which

388
00:23:14,000 --> 00:23:15,960
automatically leads
us to a treatment

389
00:23:15,960 --> 00:23:19,070
of spontaneous emission.

390
00:23:19,070 --> 00:23:19,830
Any questions?

391
00:23:23,470 --> 00:23:28,935
So how was Einstein
able to show that there

392
00:23:28,935 --> 00:23:33,540
is spontaneous
emission without sort

393
00:23:33,540 --> 00:23:35,975
of knowing the quantum
character of fields?

394
00:23:39,690 --> 00:23:43,714
Well, the point was
he knew and understood

395
00:23:43,714 --> 00:23:45,380
that there would be
thermal equilibrium.

396
00:23:48,540 --> 00:23:52,060
He said, I know what
thermal equilibrium is.

397
00:23:52,060 --> 00:23:55,330
Thermal equilibrium is
the Boltzmann coefficient.

398
00:23:55,330 --> 00:23:58,246
A Boltzmann
probability for an atom

399
00:23:58,246 --> 00:23:59,370
to be in the excited state.

400
00:23:59,370 --> 00:24:00,830
The probability to be
in the excited state

401
00:24:00,830 --> 00:24:02,780
is just the Boltzmann
factor and depends

402
00:24:02,780 --> 00:24:06,360
on temperature in the usual way.

403
00:24:06,360 --> 00:24:09,710
And he also knew that
the spectrum of light

404
00:24:09,710 --> 00:24:12,210
would follow a
plank distribution.

405
00:24:12,210 --> 00:24:15,500
And if you put those things
together, you go beyond that.

406
00:24:15,500 --> 00:24:17,750
Because you are in
thermal equilibrium.

407
00:24:17,750 --> 00:24:20,470
This here what we
derived so far does not

408
00:24:20,470 --> 00:24:21,580
have thermal equilibrium.

409
00:24:21,580 --> 00:24:23,950
And thermal
equilibrium only comes

410
00:24:23,950 --> 00:24:26,460
through the damping of
spontaneous emission.

411
00:24:26,460 --> 00:24:31,010
So, therefore, by Einstein just
using Boltzmann distribution

412
00:24:31,010 --> 00:24:34,880
and Planck law, he got
spontaneous emission.

413
00:24:34,880 --> 00:24:37,456
And this is what I
just want to show you.

414
00:24:37,456 --> 00:24:38,830
For most of you,
it's a reminder.

415
00:24:43,510 --> 00:24:44,010
OK.

416
00:24:44,010 --> 00:24:45,680
Einstein's a and b coefficients.

417
00:25:04,820 --> 00:25:08,660
I will post one of Einstein's
papers on the website.

418
00:25:08,660 --> 00:25:11,450
He was also the first
to actually discuss

419
00:25:11,450 --> 00:25:13,750
mechanical forces of light.

420
00:25:13,750 --> 00:25:18,710
He realized that if you have
a gas at a temperature which

421
00:25:18,710 --> 00:25:21,830
is different from the
temperature in the womb,

422
00:25:21,830 --> 00:25:24,040
the gas has to equilibrate.

423
00:25:24,040 --> 00:25:29,270
And the gas can only equilibrate
loose, excess velocity

424
00:25:29,270 --> 00:25:34,480
by transferring its
momentum to the photons.

425
00:25:34,480 --> 00:25:37,360
So some equations of
laser cooling, the fact

426
00:25:37,360 --> 00:25:40,240
that light can exchange
momentum with a particle.

427
00:25:40,240 --> 00:25:43,610
And this is eventually
what leads to equilibrium,

428
00:25:43,610 --> 00:25:48,232
was already in papers at the
beginning of the 20th century.

429
00:25:48,232 --> 00:25:50,190
And it's just amazing if
you read those papers.

430
00:25:50,190 --> 00:25:53,530
How modern the language is
and how clear the language is.

431
00:25:53,530 --> 00:25:56,520
But here, I'm not talking
about the mechanical effects.

432
00:25:56,520 --> 00:25:58,540
But the mechanical
effects of light

433
00:25:58,540 --> 00:26:05,590
which many people in this
class use for a living, this

434
00:26:05,590 --> 00:26:07,340
is actually part
of this equation.

435
00:26:07,340 --> 00:26:10,410
Because the equilibrium
between-- again,

436
00:26:10,410 --> 00:26:12,970
I discuss here Einstein's
a and b coefficient--

437
00:26:12,970 --> 00:26:17,410
the equilibrium between
the electronic structure,

438
00:26:17,410 --> 00:26:20,060
the ground and excited
state with a photon field.

439
00:26:20,060 --> 00:26:22,740
But Einstein also
considered the equilibrium

440
00:26:22,740 --> 00:26:25,360
between the motional
degree of the atom.

441
00:26:25,360 --> 00:26:27,390
And equilibrium between
the motional degree

442
00:26:27,390 --> 00:26:29,800
of the atom and
the radiation field

443
00:26:29,800 --> 00:26:33,140
requires the spontaneous force.

444
00:26:33,140 --> 00:26:35,830
The spontaneous radiation force.

445
00:26:35,830 --> 00:26:38,120
I'm not discussing it here.

446
00:26:38,120 --> 00:26:42,580
But I'm discussing here is now
an equilibrium between ground

447
00:26:42,580 --> 00:26:44,250
state and excited state.

448
00:26:56,450 --> 00:27:03,520
So the probability to find
an atom in the excited state

449
00:27:03,520 --> 00:27:10,390
is simply described by
the Boltzmann factor.

450
00:27:13,470 --> 00:27:15,450
Now it's traditional
in the discussion

451
00:27:15,450 --> 00:27:16,960
of Einstein's a
and b coefficient

452
00:27:16,960 --> 00:27:24,260
to allow for degeneracy factors
at ground and excited state.

453
00:27:24,260 --> 00:27:27,620
I have to say I
usually hate that I

454
00:27:27,620 --> 00:27:28,960
try not to talk about levels.

455
00:27:28,960 --> 00:27:30,900
I just talk about
quantum states.

456
00:27:30,900 --> 00:27:33,800
Non-degenerate individual
quantum states.

457
00:27:33,800 --> 00:27:36,720
So in that sense, I try
to characterize population

458
00:27:36,720 --> 00:27:39,070
in a quantum state,
not in a level.

459
00:27:39,070 --> 00:27:42,230
But it is standard to
follow Einstein's concept

460
00:27:42,230 --> 00:27:44,270
where we have degeneracies.

461
00:27:44,270 --> 00:27:47,920
I'm not emphasizing them
here, but I will just

462
00:27:47,920 --> 00:27:50,610
write them down
where they belong.

463
00:27:50,610 --> 00:27:52,430
OK, so this takes care.

464
00:27:52,430 --> 00:27:58,240
We know what is a fraction of
atoms in the excited state.

465
00:27:58,240 --> 00:28:00,890
So this is the equilibrium.

466
00:28:00,890 --> 00:28:10,720
The next thing we
need is the light.

467
00:28:10,720 --> 00:28:14,080
And Einstein assumed that
it's a spectral density

468
00:28:14,080 --> 00:28:15,740
in a black-body cavity.

469
00:28:21,010 --> 00:28:27,710
So we need the energy density
per frequency interval.

470
00:28:27,710 --> 00:28:31,210
And this is nothing else
than the occupation number

471
00:28:31,210 --> 00:28:36,730
of the mode times the
energy of the photon.

472
00:28:36,730 --> 00:28:41,185
Times the density of states.

473
00:28:46,630 --> 00:28:50,360
The photon number
per mode is just

474
00:28:50,360 --> 00:28:57,650
given by the
Bose-Einstein factor.

475
00:28:57,650 --> 00:29:01,040
Bose-Einstein statistics factor.

476
00:29:01,040 --> 00:29:10,710
The mode density is, as you
know, in three dimension.

477
00:29:10,710 --> 00:29:14,720
Omega squared, pi
square over c cube.

478
00:29:19,950 --> 00:29:28,470
So, therefore, the spectral
density of black-body radiation

479
00:29:28,470 --> 00:29:33,430
has-- and we need that,
and omega cube dependence.

480
00:29:38,180 --> 00:29:47,380
And then it has this
Bose-Einstein denominator

481
00:29:47,380 --> 00:29:48,810
in the well-known form.

482
00:29:48,810 --> 00:29:52,460
So this is now Planck's
black-body spectrum

483
00:29:52,460 --> 00:29:54,190
in the units where we need it.

484
00:29:57,410 --> 00:30:07,110
So all we need is now to find
the famous Einstein a and b

485
00:30:07,110 --> 00:30:08,090
coefficient.

486
00:30:08,090 --> 00:30:11,270
We have to write down a
rate equation for the atoms.

487
00:30:17,660 --> 00:30:20,820
So the fact is we know
already what equilibrium is.

488
00:30:20,820 --> 00:30:23,050
Excited state versus
ground state population

489
00:30:23,050 --> 00:30:24,310
is the Boltzmann factor.

490
00:30:24,310 --> 00:30:26,100
But now we write
down a rate equation

491
00:30:26,100 --> 00:30:28,040
which involves a
black-body field.

492
00:30:28,040 --> 00:30:31,410
And then we compare the
solution of the rate equation

493
00:30:31,410 --> 00:30:33,520
to the solution we already know.

494
00:30:33,520 --> 00:30:37,170
And from that, we get
Einstein's a and b coefficient.

495
00:30:37,170 --> 00:30:47,860
OK, so the change in the
population of the excited state

496
00:30:47,860 --> 00:30:51,270
has three different terms.

497
00:30:51,270 --> 00:30:58,360
One is the energy density
of the black-body radiation

498
00:30:58,360 --> 00:31:04,080
can cause stimulated emission.

499
00:31:11,004 --> 00:31:12,920
So, therefore, it's
proportional to the number

500
00:31:12,920 --> 00:31:14,460
of atoms in the excited state.

501
00:31:20,140 --> 00:31:22,480
The energy density of
the black-body radiation

502
00:31:22,480 --> 00:31:25,464
can cause absorption.

503
00:31:25,464 --> 00:31:26,880
This is proportional
to the number

504
00:31:26,880 --> 00:31:28,280
of atoms in the ground state.

505
00:31:44,190 --> 00:31:49,800
And then this
equation as it stands

506
00:31:49,800 --> 00:31:51,700
would lead to contradiction
when I compare

507
00:31:51,700 --> 00:31:55,040
the solution of this equation to
the Boltzmann factor we already

508
00:31:55,040 --> 00:31:56,220
know.

509
00:31:56,220 --> 00:32:01,800
And the only way to fix it
is to add an extra term.

510
00:32:01,800 --> 00:32:03,450
Which is spontaneous emission.

511
00:32:06,710 --> 00:32:09,800
If spontaneous emission
were not necessary,

512
00:32:09,800 --> 00:32:13,550
this a coefficient could in
the end turn out to be 0.

513
00:32:13,550 --> 00:32:15,320
Or it can be undetermined.

514
00:32:15,320 --> 00:32:18,355
But as we see, it is
necessary for consistency.

515
00:32:22,760 --> 00:32:25,955
So this is pretty much
the famous rate equation.

516
00:32:34,790 --> 00:32:38,005
And we are interested in
the equilibrium solution.

517
00:32:41,850 --> 00:32:50,640
In equilibrium, all
derivatives vanish.

518
00:32:50,640 --> 00:33:02,120
And then by setting
the derivatives to 0,

519
00:33:02,120 --> 00:33:03,980
I have one equation.

520
00:33:03,980 --> 00:33:11,970
And I will rewrite the equation
by putting the spectral density

521
00:33:11,970 --> 00:33:13,740
of the light on one side.

522
00:33:17,450 --> 00:33:19,280
And everything else
on the other side.

523
00:33:19,280 --> 00:33:22,976
And what we have here
is the a coefficient.

524
00:33:26,650 --> 00:33:30,660
The ground state population,
the excited state population.

525
00:33:30,660 --> 00:33:31,423
The b coefficient.

526
00:33:40,910 --> 00:33:43,570
So this is the spectral
density of-- it's

527
00:33:43,570 --> 00:33:46,340
just an expression for
the spectral density.

528
00:33:46,340 --> 00:33:52,570
We want to put in now that
the excited state fraction is

529
00:33:52,570 --> 00:33:55,640
given by a Boltzmann factor.

530
00:33:55,640 --> 00:34:04,650
So, therefore, Ne over Ng
becomes the Boltzmann factor.

531
00:34:04,650 --> 00:34:07,310
And, yes, there are
these degeneracy factors.

532
00:34:12,320 --> 00:34:15,060
So I've pretty much divided the
denominator and the numerator

533
00:34:15,060 --> 00:34:18,120
by the population in
the excited state.

534
00:34:18,120 --> 00:34:21,350
And here I get Beg

535
00:34:21,350 --> 00:34:26,170
OK, so this is the result
for the spectral density.

536
00:34:26,170 --> 00:34:31,409
But we know already that
the spectral density

537
00:34:31,409 --> 00:34:33,570
has to be of the Planck form.

538
00:34:33,570 --> 00:34:35,870
So now we can
simply compare what

539
00:34:35,870 --> 00:34:39,170
we know to what we obtained
from the rate equation

540
00:34:39,170 --> 00:34:41,435
and make sure that it matches.

541
00:34:44,139 --> 00:34:53,580
So it's good we have
the exponential factor.

542
00:34:53,580 --> 00:34:57,580
And by bringing this
expression to the form

543
00:34:57,580 --> 00:34:59,720
of the other
expression, we actually

544
00:34:59,720 --> 00:35:02,730
have to fulfill two conditions.

545
00:35:02,730 --> 00:35:07,060
One is in order to
make sure that kind

546
00:35:07,060 --> 00:35:10,440
of the total expression is
OK, it gives us a ratio.

547
00:35:10,440 --> 00:35:12,290
The Planck body
spectrum is normalized.

548
00:35:12,290 --> 00:35:14,080
There is no unknown prefactor.

549
00:35:14,080 --> 00:35:17,250
So this determines
the ratio of a and b.

550
00:35:17,250 --> 00:35:21,910
And also since we have
this functional form

551
00:35:21,910 --> 00:35:23,730
of the Bose-Einstein
statistics which

552
00:35:23,730 --> 00:35:25,930
has this exponential
factor minus 1,

553
00:35:25,930 --> 00:35:29,640
it gives us also a relation
between the 2B coefficient.

554
00:35:29,640 --> 00:35:32,697
That one is the b coefficient
for stimulated emission.

555
00:35:32,697 --> 00:35:34,280
And the other one
is the b coefficient

556
00:35:34,280 --> 00:35:35,155
for force absorption.

557
00:35:40,791 --> 00:35:41,290
OK.

558
00:35:41,290 --> 00:35:45,880
So with that, we
have the relation

559
00:35:45,880 --> 00:35:48,460
between the a coefficient
and the b coefficient.

560
00:35:55,940 --> 00:36:05,950
And we find that the B
coefficient for absorption

561
00:36:05,950 --> 00:36:08,537
and emission are the same.

562
00:36:08,537 --> 00:36:10,370
Well, we know it's the
same coupling matrix.

563
00:36:10,370 --> 00:36:12,340
I mean, the Hamiltonian
which connects ground

564
00:36:12,340 --> 00:36:14,840
to excited state, excited
state to ground state.

565
00:36:14,840 --> 00:36:17,650
But if you really want to
deal with degenerate states

566
00:36:17,650 --> 00:36:22,720
and not formulated for states,
you have degeneracy factors.

567
00:36:22,720 --> 00:36:26,820
OK So I could stop here.

568
00:36:26,820 --> 00:36:28,960
This is sort of the
textbook result.

569
00:36:28,960 --> 00:36:34,150
But I want to rewrite the
result that we recognize

570
00:36:34,150 --> 00:36:37,120
the quantization of the
electromagnetic field.

571
00:36:37,120 --> 00:36:41,300
So instead of just looking at
the power in Planck spectrum,

572
00:36:41,300 --> 00:36:42,990
spectral density, and such.

573
00:36:42,990 --> 00:36:49,790
I want to bring in
the photon number.

574
00:36:49,790 --> 00:36:52,140
I've already given you the
Bose-Einstein distribution

575
00:36:52,140 --> 00:36:54,140
for the photon
number in the mode.

576
00:36:54,140 --> 00:36:59,795
So I take now equation
a and multiply it

577
00:36:59,795 --> 00:37:02,640
with the average photon
number in a mode of omega.

578
00:37:11,010 --> 00:37:23,420
This gives me on
the left hand side--

579
00:37:23,420 --> 00:37:25,230
I'm multiplying this
with a photon number.

580
00:37:25,230 --> 00:37:28,650
So on the left hand side, I
have a times the photon number.

581
00:37:28,650 --> 00:37:32,990
On the right hand side, when
I put in the photon number,

582
00:37:32,990 --> 00:37:34,800
the photon number
with this expression

583
00:37:34,800 --> 00:37:37,800
just give me the
Planck distribution,

584
00:37:37,800 --> 00:37:40,950
the spectral energy.

585
00:37:40,950 --> 00:37:47,550
So yes.

586
00:37:47,550 --> 00:37:53,840
This gives me the
spectral energy density

587
00:37:53,840 --> 00:37:54,850
times the b coefficient.

588
00:38:05,245 --> 00:38:05,745
Yes.

589
00:38:11,130 --> 00:38:14,590
And this is nothing else
than stimulated emission.

590
00:38:14,590 --> 00:38:18,340
So we realize that
stimulated emission

591
00:38:18,340 --> 00:38:21,660
is nothing else than n
times the photon number.

592
00:38:21,660 --> 00:38:28,510
The photon number n times
spontaneous emission.

593
00:38:28,510 --> 00:38:39,700
Similarly, we know that
the rate for absorption

594
00:38:39,700 --> 00:38:46,040
becomes now, well,
the same unless we

595
00:38:46,040 --> 00:38:49,340
have degeneracy factors.

596
00:38:49,340 --> 00:38:54,030
But just for the
fundamental discussion,

597
00:38:54,030 --> 00:38:56,130
let's avoid the [? p ?]
counting how many

598
00:38:56,130 --> 00:38:57,760
degenerate levels a level have.

599
00:38:57,760 --> 00:38:59,970
Let's just assume we have
a situation that we just

600
00:38:59,970 --> 00:39:01,320
count every state individually.

601
00:39:19,420 --> 00:39:22,220
Then I can summarize this
result saw in the following.

602
00:39:25,000 --> 00:39:32,010
That the total rate
for emission was

603
00:39:32,010 --> 00:39:35,700
proportional to n for
stimulated emission.

604
00:39:35,700 --> 00:39:40,350
And then we have the extra
1 for spontaneous emission.

605
00:39:40,350 --> 00:39:45,180
Whereas the rate
for absorption was

606
00:39:45,180 --> 00:39:46,970
n times the
spontaneous emission.

607
00:39:51,900 --> 00:39:57,510
So we find that this important
formula that emission

608
00:39:57,510 --> 00:40:00,140
has an n plus 1 factor.

609
00:40:00,140 --> 00:40:03,560
Absorption has an n factor.

610
00:40:03,560 --> 00:40:06,030
And it is, of course,
this extra plus 1

611
00:40:06,030 --> 00:40:09,340
which was absolutely crucial to
establish thermal equilibrium.

612
00:40:09,340 --> 00:40:11,850
If a had been 0, no
thermal equilibrium

613
00:40:11,850 --> 00:40:12,980
would have been reached.

614
00:40:17,150 --> 00:40:20,480
So in other words,
what is already

615
00:40:20,480 --> 00:40:23,350
in Einstein's treatment
of the a and b coefficient

616
00:40:23,350 --> 00:40:26,600
is that if you understand
absorption, which

617
00:40:26,600 --> 00:40:29,780
you can understand with
the Schroedinger equation,

618
00:40:29,780 --> 00:40:32,260
and you understand
and you write it

619
00:40:32,260 --> 00:40:36,610
in the fundamental
way in photon numbers,

620
00:40:36,610 --> 00:40:39,780
then spontaneous
emission is just

621
00:40:39,780 --> 00:40:42,180
the rate of absorption
divided by n.

622
00:40:44,710 --> 00:40:49,010
Spontaneous emission is like
induced emission in its rate.

623
00:40:49,010 --> 00:40:50,680
But by just one single photon.

624
00:40:55,190 --> 00:40:57,636
So as I pointed out, this
is a result which is usually

625
00:40:57,636 --> 00:40:59,010
obtained with
second quantization

626
00:40:59,010 --> 00:41:02,440
and it is already included in
Einstein's a and b coefficient.

627
00:41:05,600 --> 00:41:07,270
So we could stop here.

628
00:41:07,270 --> 00:41:09,420
We have already a major
result which is usually

629
00:41:09,420 --> 00:41:11,000
obtained in field quantization.

630
00:41:11,000 --> 00:41:13,639
But there is one deficiency
and we want clearly

631
00:41:13,639 --> 00:41:15,680
fix it and move on to the
microscopic derivation.

632
00:41:15,680 --> 00:41:17,300
And this is the following.

633
00:41:17,300 --> 00:41:20,350
Right now, we really assume
black-body radiation.

634
00:41:20,350 --> 00:41:23,170
And this ratio n plus
1 over n was only

635
00:41:23,170 --> 00:41:25,920
derived for average
photon numbers

636
00:41:25,920 --> 00:41:28,330
in a spectrally broad field.

637
00:41:28,330 --> 00:41:32,070
And what is left for
microscopic treatment which

638
00:41:32,070 --> 00:41:36,530
I want to present now is even
if you have just a single mode,

639
00:41:36,530 --> 00:41:39,170
the atom can only interact
with a single mode.

640
00:41:39,170 --> 00:41:41,670
We find that stimulated
emission and absorption

641
00:41:41,670 --> 00:41:45,330
is proportional to n, the number
of photons already present.

642
00:41:45,330 --> 00:41:49,110
And then there is plus 1
for spontaneous emission.

643
00:41:49,110 --> 00:41:53,700
So in other words, we do it now
sort of microscopically again.

644
00:41:53,700 --> 00:41:56,480
And what we get out of
it is that everything

645
00:41:56,480 --> 00:41:58,470
we learned from Einstein's
a and b coefficient

646
00:41:58,470 --> 00:42:00,400
is not just valid in
thermal equilibrium.

647
00:42:00,400 --> 00:42:02,560
It's not just valid
for average numbers.

648
00:42:02,560 --> 00:42:05,101
It's really valid for
single mode physics.

649
00:42:10,880 --> 00:42:13,110
OK, so the agenda
is what is next.

650
00:42:26,910 --> 00:42:27,410
Is valid.

651
00:42:32,680 --> 00:42:33,180
4n.

652
00:42:36,640 --> 00:42:39,590
So this expression
is valid not only

653
00:42:39,590 --> 00:42:46,280
for an average over many modes,
but for each single mode.

654
00:42:57,750 --> 00:43:01,644
Questions about Einstein's
a and b coefficient?

655
00:43:10,550 --> 00:43:11,180
OK.

656
00:43:11,180 --> 00:43:14,680
So we spend now the
rest of today and parts

657
00:43:14,680 --> 00:43:17,880
of next Monday in a
microscopic derivation

658
00:43:17,880 --> 00:43:21,260
of spontaneous emission
using field quantization.

659
00:43:21,260 --> 00:43:22,825
But I just want
to make you aware

660
00:43:22,825 --> 00:43:25,040
that we know already what it is.

661
00:43:25,040 --> 00:43:26,615
We have a semi-classical
derivation

662
00:43:26,615 --> 00:43:28,550
of the b coefficient.

663
00:43:28,550 --> 00:43:31,600
And Einstein's treatment
gives us the ratio of a and b.

664
00:43:31,600 --> 00:43:33,610
So we know already
at this point what

665
00:43:33,610 --> 00:43:35,720
the rate of spontaneous
emission is.

666
00:43:35,720 --> 00:43:38,150
But it is nice.

667
00:43:38,150 --> 00:43:40,900
I think also important
for our education

668
00:43:40,900 --> 00:43:43,200
to obtain it in
a microscopic way

669
00:43:43,200 --> 00:43:54,110
where we really show how we
have to-- sum overall modes

670
00:43:54,110 --> 00:43:56,866
and such to obtain
the expression.

671
00:43:56,866 --> 00:43:58,240
Also I want to
ask you questions.

672
00:43:58,240 --> 00:43:59,775
I want to ask you clicker
questions afterwards.

673
00:43:59,775 --> 00:44:01,630
And one clicker
question for you is

674
00:44:01,630 --> 00:44:03,180
what happens to
spontaneous emission

675
00:44:03,180 --> 00:44:05,210
in one and two dimension?

676
00:44:05,210 --> 00:44:07,130
Certain things will change.

677
00:44:07,130 --> 00:44:11,200
And it's much clearer
what will change

678
00:44:11,200 --> 00:44:14,300
if you have a clear
understanding how we sum up

679
00:44:14,300 --> 00:44:15,060
all of the modes.

680
00:44:15,060 --> 00:44:17,630
How all the possible
modes contribute

681
00:44:17,630 --> 00:44:20,530
to spontaneous emission.

682
00:44:20,530 --> 00:44:22,930
And, of course, in two
dimension and one dimension,

683
00:44:22,930 --> 00:44:26,270
you have a different
density of modes.

684
00:44:26,270 --> 00:44:31,150
So with that motivation, we
need a quantized electromagnetic

685
00:44:31,150 --> 00:44:32,666
field.

686
00:44:32,666 --> 00:44:35,480
Where we quantize the
field for each mode.

687
00:44:35,480 --> 00:44:38,456
And then we go back, we do
the summation of all modes.

688
00:44:38,456 --> 00:44:40,080
And we've really
understood in the most

689
00:44:40,080 --> 00:44:42,855
fundamental and microscopic way
how photons and light interact.

690
00:44:47,380 --> 00:44:50,990
OK, so our next chapter
is the quantization

691
00:44:50,990 --> 00:44:52,280
of the radiation field.

692
00:45:05,350 --> 00:45:06,785
We do-- yes, [INAUDIBLE]?

693
00:45:06,785 --> 00:45:09,260
AUDIENCE: I just
have a question.

694
00:45:09,260 --> 00:45:12,725
So when we compare the rate
equation and the distribution

695
00:45:12,725 --> 00:45:17,675
of the photons, so there is a
parameter, t, in both of them.

696
00:45:17,675 --> 00:45:20,645
So we just assume two
t's are the same because

697
00:45:20,645 --> 00:45:22,625
of their reaching
thermal equilibrium.

698
00:45:22,625 --> 00:45:23,730
PROFESSOR: Oh, yeah.

699
00:45:23,730 --> 00:45:24,230
Absolutely.

700
00:45:24,230 --> 00:45:28,430
I mean this is, of course,
what Einstein assumed

701
00:45:28,430 --> 00:45:31,770
that the thermal equilibrium
for atoms with the Boltzmann

702
00:45:31,770 --> 00:45:32,850
factor.

703
00:45:32,850 --> 00:45:36,690
And the thermal equilibrium for
photons described at the Planck

704
00:45:36,690 --> 00:45:39,305
distribution have to be reached
at the same temperature.

705
00:45:43,130 --> 00:45:45,860
It was a
thermodynamical argument

706
00:45:45,860 --> 00:45:48,240
assuming which
is, of course, one

707
00:45:48,240 --> 00:45:52,490
of the tenets of statistical
physics of thermodynamics.

708
00:45:52,490 --> 00:45:56,200
If you have two systems and
they interact with each other,

709
00:45:56,200 --> 00:45:58,817
they equilibrate at
the same temperature.

710
00:46:02,230 --> 00:46:03,540
Yes, this is very important.

711
00:46:03,540 --> 00:46:05,081
This was a very
important assumption.

712
00:46:10,650 --> 00:46:16,100
Of course, as we know, when we
have ultra cold atoms in a room

713
00:46:16,100 --> 00:46:18,690
temperature vacuum chamber,
the atoms do not equilibrate.

714
00:46:21,490 --> 00:46:25,850
But if you would trap them for
an infinite amount of time,

715
00:46:25,850 --> 00:46:28,180
they would equilibrate.

716
00:46:28,180 --> 00:46:31,512
It's just that we lose
the atoms from our trap.

717
00:46:31,512 --> 00:46:33,595
They're knocked out by
[INAUDIBLE] gas collisions.

718
00:46:33,595 --> 00:46:35,650
A lot of other things happen.

719
00:46:35,650 --> 00:46:39,750
But if you could isolate just
ultra cold atoms in a trap,

720
00:46:39,750 --> 00:46:42,350
they would stay in
this trap forever.

721
00:46:42,350 --> 00:46:46,060
There would be no other effect
shortening our observation

722
00:46:46,060 --> 00:46:46,660
time.

723
00:46:46,660 --> 00:46:50,800
Eventually, the atoms would
just boil out of your trap

724
00:46:50,800 --> 00:46:52,590
because black-body radiation.

725
00:46:52,590 --> 00:46:55,390
Momentum transfer from
black-body photons

726
00:46:55,390 --> 00:46:58,815
heats the atoms up
to room temperature.

727
00:47:01,680 --> 00:47:06,210
And this is, of course, one of
the things which really amaze

728
00:47:06,210 --> 00:47:08,520
people when laser
cooling came along.

729
00:47:08,520 --> 00:47:11,610
You know, everything at low
temperature was cryogenic.

730
00:47:11,610 --> 00:47:13,110
If you want to
keep a sample cold,

731
00:47:13,110 --> 00:47:17,640
you had to put liquid nitrogen
shields, helium shields.

732
00:47:17,640 --> 00:47:20,934
You had to put multiple
shields around--

733
00:47:20,934 --> 00:47:22,350
if you had an
optical [INAUDIBLE],

734
00:47:22,350 --> 00:47:24,830
you had a window of liquid
nitrogen temperature.

735
00:47:24,830 --> 00:47:26,900
One window at
helium temperature.

736
00:47:26,900 --> 00:47:29,690
Just to make sure that
the black-body radiation

737
00:47:29,690 --> 00:47:31,410
is absorbed and blocked.

738
00:47:31,410 --> 00:47:34,090
Because it would've been
absolutely detrimental

739
00:47:34,090 --> 00:47:36,670
if you had a sample at
very low temperature

740
00:47:36,670 --> 00:47:40,520
and it would've been exposed
to black-body radiation.

741
00:47:40,520 --> 00:47:43,720
So it's really a unique
feature of the atoms

742
00:47:43,720 --> 00:47:48,255
that they are, and this
is what you will calculate

743
00:47:48,255 --> 00:47:52,380
in this week's problem set, that
the atoms are almost completely

744
00:47:52,380 --> 00:47:54,720
transparent to the
black-body radiation.

745
00:47:54,720 --> 00:47:58,970
They only react if the hyperfine
frequency or they react far,

746
00:47:58,970 --> 00:48:01,300
far, far, far, far
off in the tail

747
00:48:01,300 --> 00:48:04,810
of the black-body radiation
with an electronic transition.

748
00:48:04,810 --> 00:48:07,850
But nevertheless, as
Einstein has taught us,

749
00:48:07,850 --> 00:48:10,840
and as we know from
general principles,

750
00:48:10,840 --> 00:48:14,020
this will not mean
that the atoms

751
00:48:14,020 --> 00:48:17,940
stay cold and are decoupled.

752
00:48:17,940 --> 00:48:23,260
It just means that it takes
maybe the age of the universe.

753
00:48:23,260 --> 00:48:24,990
I've never calculated
the number.

754
00:48:24,990 --> 00:48:28,415
It would really take forever
until the atoms in the atom

755
00:48:28,415 --> 00:48:35,320
trap reach the
ambient temperature.

756
00:48:35,320 --> 00:48:38,180
So Einstein's argument
was an idealized argument

757
00:48:38,180 --> 00:48:39,740
which in practice
would never happen.

758
00:48:39,740 --> 00:48:42,310
But if you exclude
all other processes,

759
00:48:42,310 --> 00:48:44,490
you have a consistent
system by saying

760
00:48:44,490 --> 00:48:47,100
I only have atoms with
their kinetic energy.

761
00:48:47,100 --> 00:48:48,740
I have black-body radiation.

762
00:48:48,740 --> 00:48:51,320
And everything has
to equilibrate.

763
00:48:51,320 --> 00:48:53,130
And as I said
before, the argument

764
00:48:53,130 --> 00:48:55,160
for Einstein's a
and b coefficient

765
00:48:55,160 --> 00:48:58,770
simply assumes that the ground
and excited state population

766
00:48:58,770 --> 00:48:59,850
equilibrium.

767
00:48:59,850 --> 00:49:02,010
But you can carry the
argument even further

768
00:49:02,010 --> 00:49:05,230
and say even the
Maxwell-Boltzmann velocity

769
00:49:05,230 --> 00:49:08,090
distribution of the
atoms has to equilibrate

770
00:49:08,090 --> 00:49:10,120
at the ambient temperature.

771
00:49:10,120 --> 00:49:11,200
Beautiful argument.

772
00:49:14,120 --> 00:49:15,700
And what you find
from this argument

773
00:49:15,700 --> 00:49:16,700
is it's really amazing.

774
00:49:16,700 --> 00:49:19,560
You find the photon
recall is h-bar k.

775
00:49:19,560 --> 00:49:24,300
Einstein pulled it out simply
by making this assumption.

776
00:49:24,300 --> 00:49:27,240
I will post the paper on that.

777
00:49:27,240 --> 00:49:28,960
OK, field quantization.

778
00:49:33,200 --> 00:49:36,190
We discuss the quantization
of the electromagnetic field

779
00:49:36,190 --> 00:49:38,200
really from first principles.

780
00:49:38,200 --> 00:49:41,990
From vector potential,
radiation field, Coulomb gauge,

781
00:49:41,990 --> 00:49:45,480
transverse vector
potential, in 8422.

782
00:49:45,480 --> 00:49:48,240
So we dedicate
one or two classes

783
00:49:48,240 --> 00:49:52,770
to just discuss all
the steps to have

784
00:49:52,770 --> 00:49:55,800
full quantization of the
electromagnetic field

785
00:49:55,800 --> 00:49:58,080
with all the bells and whistles.

786
00:49:58,080 --> 00:49:59,910
So sometimes when I
teach this course,

787
00:49:59,910 --> 00:50:02,647
I say, well, you've heard
about field quantization,

788
00:50:02,647 --> 00:50:03,480
I can refer to that.

789
00:50:03,480 --> 00:50:06,080
Or I can refer you to 8422.

790
00:50:06,080 --> 00:50:09,420
But in the end I thought,
why don't I just give you

791
00:50:09,420 --> 00:50:10,425
a 10 minute derivation.

792
00:50:10,425 --> 00:50:14,160
Just sort of focusing on
the essential because this

793
00:50:14,160 --> 00:50:16,005
makes this cause
more self-contained

794
00:50:16,005 --> 00:50:18,060
and more complete.

795
00:50:18,060 --> 00:50:21,280
So I give you now a
ten minute quantization

796
00:50:21,280 --> 00:50:22,820
of the electromagnetic field.

797
00:50:22,820 --> 00:50:26,100
Pretty much going
straight to showing you

798
00:50:26,100 --> 00:50:28,297
electromagnetic field
isn't harmonic oscillator.

799
00:50:28,297 --> 00:50:30,005
And now let's use the
quantum description

800
00:50:30,005 --> 00:50:31,280
of the harmonic oscillator.

801
00:50:31,280 --> 00:50:33,060
And then we have a
quantum description

802
00:50:33,060 --> 00:50:35,990
of the electromagnetic field.

803
00:50:35,990 --> 00:50:38,855
So this is not rigorous,
but it is logically compete.

804
00:50:42,650 --> 00:50:45,450
So we focus in the discussion
of the quantization

805
00:50:45,450 --> 00:50:48,040
of the electromagnetic
field, we focus just

806
00:50:48,040 --> 00:50:51,180
on a single mode of the
electromagnetic field.

807
00:50:54,560 --> 00:50:56,450
Each mode will be an
harmonic oscillator.

808
00:50:56,450 --> 00:50:59,549
And then we have many
harmonic oscillators

809
00:50:59,549 --> 00:51:01,090
comprising the
electromagnetic field.

810
00:51:05,020 --> 00:51:11,150
So a even single mode, we
assume that we have plain waves

811
00:51:11,150 --> 00:51:13,170
with a polarization,
with an amplitude.

812
00:51:26,980 --> 00:51:29,730
The electric field
is the derivative

813
00:51:29,730 --> 00:51:32,950
of the vector potential.

814
00:51:32,950 --> 00:51:37,520
And the shortest way
to show you an analogy

815
00:51:37,520 --> 00:51:40,940
with the harmonic
oscillator is to remind you

816
00:51:40,940 --> 00:51:45,570
that the total energy--
which is actually

817
00:51:45,570 --> 00:51:48,610
if you're wondering
about a factor of 2,

818
00:51:48,610 --> 00:51:52,870
the electric and the magnetic
part, the total energy

819
00:51:52,870 --> 00:52:02,090
is quadratic in the amplitude
of the vector potential.

820
00:52:08,340 --> 00:52:10,210
By the way, there
is a factor of 1/2

821
00:52:10,210 --> 00:52:12,250
because if you have a
sinusoidal variation,

822
00:52:12,250 --> 00:52:15,130
you take the time average,
cosine squared, which is 1/2.

823
00:52:18,070 --> 00:52:24,920
Well, if the total energy is
quadratic in the amplitude,

824
00:52:24,920 --> 00:52:31,160
this immediately allows
us to draw analogies

825
00:52:31,160 --> 00:52:33,095
to an harmonic oscillator.

826
00:52:35,620 --> 00:52:39,170
And we can use the
vector potential

827
00:52:39,170 --> 00:52:42,920
of the single mode of
the electromagnetic field

828
00:52:42,920 --> 00:52:46,090
to define two
quantities, q and p.

829
00:52:49,110 --> 00:52:51,490
Let me write it in that way.

830
00:52:51,490 --> 00:53:08,580
Omega q, plus ip, is related
to a in the following way.

831
00:53:13,870 --> 00:53:18,330
And yes, I was just
ranting about this.

832
00:53:18,330 --> 00:53:20,730
v is the volume.

833
00:53:20,730 --> 00:53:24,030
We assume everything happens
in a finite volume of space.

834
00:53:24,030 --> 00:53:30,650
Value would say I have two
new quantities, q and p.

835
00:53:30,650 --> 00:53:32,680
So I need two equations.

836
00:53:32,680 --> 00:53:37,885
And the two equations involved
a and a complex conjugate.

837
00:53:41,190 --> 00:53:45,600
So now we had an expression
for the total energy

838
00:53:45,600 --> 00:53:50,760
in terms of the amplitude
of the vector potential.

839
00:53:50,760 --> 00:53:53,830
So now I can rewrite it the
amplitude square of the vector

840
00:53:53,830 --> 00:53:56,910
potential is a times a star.

841
00:53:56,910 --> 00:54:02,700
And with that I get
the total energy

842
00:54:02,700 --> 00:54:07,500
to be proportional to
q square plus p square.

843
00:54:07,500 --> 00:54:10,350
And that should remind you,
and if everything was set up

844
00:54:10,350 --> 00:54:13,910
to remind you, that this looks
like a harmonic oscillator

845
00:54:13,910 --> 00:54:16,420
where q is the
position variable.

846
00:54:16,420 --> 00:54:17,930
And p is the momentum variable.

847
00:54:22,740 --> 00:54:24,730
So now, I mean all
this is classical.

848
00:54:24,730 --> 00:54:27,720
All this is just
clever definitions.

849
00:54:27,720 --> 00:54:30,330
But now we have to do a
leap to quantum physics.

850
00:54:30,330 --> 00:54:31,780
We cannot logically derive it.

851
00:54:31,780 --> 00:54:33,140
We have to make a leap.

852
00:54:33,140 --> 00:54:43,034
And the leap is that we
postulate that this should now

853
00:54:43,034 --> 00:54:44,950
be described as a quantum
harmonic oscillator.

854
00:54:47,710 --> 00:54:50,920
And this transition is
done by simply postulating

855
00:54:50,920 --> 00:54:55,210
that the two quantities
we have defined

856
00:54:55,210 --> 00:54:58,925
fulfill the canonical commutator
for position and momentum.

857
00:55:03,190 --> 00:55:06,910
So we've started with the vector
potential, expressed the energy

858
00:55:06,910 --> 00:55:08,890
as a vector
potential, and now we

859
00:55:08,890 --> 00:55:12,670
say we recognize through
those definitions

860
00:55:12,670 --> 00:55:15,470
that this is an harmonic
oscillator with variables q

861
00:55:15,470 --> 00:55:18,606
and p which are defined in
terms of the vector potential.

862
00:55:21,360 --> 00:55:25,520
So then you know if you have the
quantized harmonic oscillator,

863
00:55:25,520 --> 00:55:30,155
you immediately introduce
creation and annihilation

864
00:55:30,155 --> 00:55:30,655
operators.

865
00:55:38,570 --> 00:55:41,930
Which are linear
superpositions of q and p

866
00:55:41,930 --> 00:55:44,540
in the following form.

867
00:55:44,540 --> 00:55:48,650
And a dagger has
a minus sign here.

868
00:55:48,650 --> 00:55:52,780
And all the prefactors were
cleverly set up in such a way

869
00:55:52,780 --> 00:55:55,390
that the commutator of
a and a dagger is 1.

870
00:55:59,150 --> 00:56:01,510
And now we can do all
the substitutions.

871
00:56:01,510 --> 00:56:05,350
We can express p and
q by a and a dagger.

872
00:56:05,350 --> 00:56:09,440
But p and q were related
to the vector potential, a.

873
00:56:09,440 --> 00:56:12,970
And the vector potential a
square defined the energy.

874
00:56:12,970 --> 00:56:16,130
So now we have an expression
for the energy which

875
00:56:16,130 --> 00:56:18,610
is no longer involving
a or p and q.

876
00:56:18,610 --> 00:56:20,890
It involves a and a dagger.

877
00:56:20,890 --> 00:56:27,260
And surprise, surprise, we
find that our total energy

878
00:56:27,260 --> 00:56:31,040
because we have operators
now has become a Hamiltonian,

879
00:56:31,040 --> 00:56:35,190
Has this well-known result
with the photon number operator

880
00:56:35,190 --> 00:56:37,870
a dagger, a plus 1/2.

881
00:56:47,080 --> 00:56:52,520
So this is sort of the
quickest way which takes us

882
00:56:52,520 --> 00:56:54,860
in a few minutes to the
quantized electromagnetic

883
00:56:54,860 --> 00:56:57,180
field.

884
00:56:57,180 --> 00:56:59,930
Of course, all I
need is to come back

885
00:56:59,930 --> 00:57:02,480
to spontaneous
emission stimulated

886
00:57:02,480 --> 00:57:07,010
emission are the matrix
elements of this operator's

887
00:57:07,010 --> 00:57:09,380
a and a dagger.

888
00:57:09,380 --> 00:57:12,510
And this is where,
of course, stimulated

889
00:57:12,510 --> 00:57:15,540
and spontaneous emission--
all that comes in.

890
00:57:15,540 --> 00:57:26,720
The non-vanishing
matrix elements

891
00:57:26,720 --> 00:57:30,850
in this description of
the electromagnetic field

892
00:57:30,850 --> 00:57:33,690
are the ones where a
annihilates a photon

893
00:57:33,690 --> 00:57:36,820
and the matrix element
is square root n.

894
00:57:36,820 --> 00:57:40,410
Or where a creates a
photon, adds a photon

895
00:57:40,410 --> 00:57:43,560
to n photons already present.

896
00:57:43,560 --> 00:57:47,125
And then the matrix
element is n plus 1.

897
00:57:50,770 --> 00:57:54,250
OK, so we went from a to q.

898
00:57:54,250 --> 00:57:55,630
And p.

899
00:57:55,630 --> 00:57:59,050
And we went to a and a dagger.

900
00:57:59,050 --> 00:58:04,500
But a is also related
to the electric field

901
00:58:04,500 --> 00:58:08,350
by taking the time derivative
of the vector potential.

902
00:58:13,990 --> 00:58:17,480
So now, of course, we can
go from our expressions

903
00:58:17,480 --> 00:58:20,770
of a and a dagger
all the way back.

904
00:58:20,770 --> 00:58:22,940
Just substitute,
substitute, substitute.

905
00:58:22,940 --> 00:58:29,220
And find an expression
for the electric field

906
00:58:29,220 --> 00:58:30,590
in terms of a and a dagger.

907
00:58:38,740 --> 00:58:50,616
The result is that we have
a and here we have a dagger.

908
00:58:50,616 --> 00:58:53,750
We have a polarization vector.

909
00:58:53,750 --> 00:58:59,440
We have the plain wave vector.

910
00:58:59,440 --> 00:59:01,740
And the complex conjugate.

911
00:59:01,740 --> 00:59:05,325
And the complex conjugate
of a is a dagger

912
00:59:05,325 --> 00:59:07,650
so the electric field
is a superposition

913
00:59:07,650 --> 00:59:09,270
of a and a dagger.

914
00:59:09,270 --> 00:59:12,700
The electric field
becomes an operator

915
00:59:12,700 --> 00:59:18,250
which is the sum of
creation-annihilation operator.

916
00:59:18,250 --> 00:59:21,850
So with that, we can go
back to our Hamiltonian.

917
00:59:21,850 --> 00:59:24,750
Our Hamiltonian for the
interaction between light

918
00:59:24,750 --> 00:59:28,640
and atoms in the
simplest possible case

919
00:59:28,640 --> 00:59:31,460
was the dipole Hamiltonian.

920
00:59:34,890 --> 00:59:38,320
Which involves a
dipole matrix element.

921
00:59:38,320 --> 00:59:40,020
The charge of the
electron is negative.

922
00:59:40,020 --> 00:59:42,410
That's why the minus
sign has disappeared.

923
00:59:42,410 --> 00:59:45,490
And now all we do is
from our treatment

924
00:59:45,490 --> 00:59:47,990
before in the Schroedinger
equation where

925
00:59:47,990 --> 00:59:50,420
the electric field
was an external field.

926
00:59:50,420 --> 00:59:53,550
Now the electric field
becomes the operator

927
00:59:53,550 --> 00:59:55,960
acting on the quantum state
of the electromagnetic field.

928
01:00:06,150 --> 01:00:15,000
So by the way, this prefactor
here is because the rest of it

929
01:00:15,000 --> 01:00:16,560
is just dimensionless.

930
01:00:16,560 --> 01:00:24,660
This prefactor has, so we
have the matrix element here,

931
01:00:24,660 --> 01:00:27,370
this prefactor is
an electric field.

932
01:00:27,370 --> 01:00:30,150
And it's something you
should always know.

933
01:00:30,150 --> 01:00:33,090
This electric field is
actually the electric field

934
01:00:33,090 --> 01:00:34,490
of a single photon.

935
01:00:34,490 --> 01:00:35,980
This is the correct
normalization.

936
01:00:35,980 --> 01:00:38,660
If you want to factor out the
volume, the frequency, and all

937
01:00:38,660 --> 01:00:41,390
that, you combine these
factors in such a way

938
01:00:41,390 --> 01:00:44,830
that it's electric field
of a single photon.

939
01:00:44,830 --> 01:00:46,850
Then we have the dipole moment.

940
01:00:46,850 --> 01:00:51,890
And then we have an expression
with creation and annihilation

941
01:00:51,890 --> 01:00:59,300
operator over here.

942
01:00:59,300 --> 01:01:02,985
I assume now that the
atom sits at r equals 0.

943
01:01:02,985 --> 01:01:06,165
So why should I carry
forward an e to the ikr term.

944
01:01:06,165 --> 01:01:09,760
We conveniently place
the atoms at r equals 0.

945
01:01:09,760 --> 01:01:14,870
But I have to say a word or
two about the e to the i omega

946
01:01:14,870 --> 01:01:16,320
t factor.

947
01:01:16,320 --> 01:01:19,230
I have been deliberately
cavalier about my formulation

948
01:01:19,230 --> 01:01:21,900
in quantum physics,
of quantum mechanics,

949
01:01:21,900 --> 01:01:24,920
whether I use the Schroedinger
or the Heisenberg picture.

950
01:01:24,920 --> 01:01:26,770
And you know in the
Schroedinger picture

951
01:01:26,770 --> 01:01:29,250
the wave function is time
dependent, not the operator.

952
01:01:29,250 --> 01:01:32,280
In the Heisenberg picture,
it's the other way around.

953
01:01:32,280 --> 01:01:34,590
And I have to tell you every
time I do a calculation

954
01:01:34,590 --> 01:01:36,760
and look at it, I'm getting
confused about the two

955
01:01:36,760 --> 01:01:38,110
pictures.

956
01:01:38,110 --> 01:01:41,900
So anyway, trust me
that in this case when

957
01:01:41,900 --> 01:01:44,260
I want to discuss the
Schroedinger picture,

958
01:01:44,260 --> 01:01:47,670
the time dependent factor
should not be present.

959
01:01:47,670 --> 01:01:50,610
But you really have to
look at the derivation

960
01:01:50,610 --> 01:01:53,870
and carefully realize
the two are connected

961
01:01:53,870 --> 01:01:55,120
with a unitary transformation.

962
01:01:55,120 --> 01:01:56,703
You really have to
figure out in which

963
01:01:56,703 --> 01:01:58,150
[INAUDIBLE]
presentation you are.

964
01:01:58,150 --> 01:02:00,670
But I want to not focus
on the formality here.

965
01:02:00,670 --> 01:02:02,780
But I'm not carrying
forward this factor

966
01:02:02,780 --> 01:02:07,070
because I want to discuss
the Schroedinger picture.

967
01:02:07,070 --> 01:02:08,830
OK.

968
01:02:08,830 --> 01:02:11,050
Yes.

969
01:02:11,050 --> 01:02:18,081
So now we can look at the matrix
elements of our interaction

970
01:02:18,081 --> 01:02:18,580
Hamiltonian.

971
01:02:21,860 --> 01:02:24,850
And just to be clear,
we have written down

972
01:02:24,850 --> 01:02:31,400
this Hamiltonian for just a
single mode of the radiation

973
01:02:31,400 --> 01:02:31,900
field.

974
01:02:35,060 --> 01:02:36,920
Depending on what we
are interested in,

975
01:02:36,920 --> 01:02:40,350
we may have to sum
over many, many modes.

976
01:02:40,350 --> 01:02:48,270
So we are looking at transitions
from an initial state

977
01:02:48,270 --> 01:02:50,830
which may be an excited state.

978
01:02:50,830 --> 01:02:54,660
To a final state which
may be a ground state.

979
01:02:54,660 --> 01:02:58,220
And since we have quantized
the magnetic fields,

980
01:02:58,220 --> 01:03:04,570
we also have to specify the
state of the quantum field.

981
01:03:04,570 --> 01:03:09,040
And we assume that the uncoupled
Hamiltonian, of course,

982
01:03:09,040 --> 01:03:12,340
has simply number states as
eigenstates [INAUDIBLE] photons

983
01:03:12,340 --> 01:03:13,310
and prime photons.

984
01:03:16,540 --> 01:03:26,420
So the only non-vanishing matrix
elements are the following.

985
01:03:33,480 --> 01:03:35,880
e is a charge.

986
01:03:35,880 --> 01:03:39,460
e hat is the polarization.

987
01:03:39,460 --> 01:03:43,160
Epsilon 1 is the electric
field of a single photon.

988
01:03:43,160 --> 01:03:45,380
And, of course, we
only have a coupling

989
01:03:45,380 --> 01:03:47,580
by the fully
quantized Hamiltonian

990
01:03:47,580 --> 01:03:50,870
when we have a dipole
matrix element connecting

991
01:03:50,870 --> 01:03:51,665
state a and b.

992
01:03:51,665 --> 01:03:53,790
I mean these are all sort
of things we have already

993
01:03:53,790 --> 01:03:56,580
discussed in another context.

994
01:03:56,580 --> 01:04:01,750
But now the a's and a daggers
which only act on the photon

995
01:04:01,750 --> 01:04:12,480
field, give rise value
to two couplings.

996
01:04:12,480 --> 01:04:15,075
One is absorption
and one is emission.

997
01:04:17,690 --> 01:04:22,320
Absorption takes place when
we look at the matrix element

998
01:04:22,320 --> 01:04:27,410
when the final state
has one more photon.

999
01:04:27,410 --> 01:04:34,010
And emission takes
the other way around.

1000
01:04:34,010 --> 01:04:38,090
When the final state has one
more, which way do we go?

1001
01:04:42,057 --> 01:04:43,140
Let me just write it down.

1002
01:04:43,140 --> 01:04:44,014
And then read it off.

1003
01:04:51,320 --> 01:04:54,390
I think I've inverted, but
anyway, initial and final state

1004
01:04:54,390 --> 01:04:58,840
can differ by plus one
photon or minus one photon.

1005
01:04:58,840 --> 01:05:01,600
In one case it's absorption,
the other case it's emission.

1006
01:05:01,600 --> 01:05:03,660
And the matrix element
is n or n plus 1.

1007
01:05:08,640 --> 01:05:11,810
So one is absorption.

1008
01:05:11,810 --> 01:05:12,840
And one is emission.

1009
01:05:17,120 --> 01:05:24,670
So finally if we ask,
what are the rates

1010
01:05:24,670 --> 01:05:35,130
of absorption and
emission when we

1011
01:05:35,130 --> 01:05:42,940
assume we have a situation
where-- and we have now

1012
01:05:42,940 --> 01:05:45,680
discussed the matrix element
and this matrix element could

1013
01:05:45,680 --> 01:05:47,930
become the basis of
Fermi's Golden Rule.

1014
01:05:47,930 --> 01:05:50,490
We just have to specify time
dependent perturbation theory.

1015
01:05:50,490 --> 01:05:54,630
But in any case, whatever we
do when we talk about the rate,

1016
01:05:54,630 --> 01:05:57,650
it will involve the
matrix elements squared.

1017
01:05:57,650 --> 01:06:04,080
So now we can ask
what happens when

1018
01:06:04,080 --> 01:06:10,060
we couple ground
and excited states.

1019
01:06:13,140 --> 01:06:16,190
And let's assume we
have an excited state.

1020
01:06:16,190 --> 01:06:20,280
And we sum over all possible
photon occupation numbers

1021
01:06:20,280 --> 01:06:21,502
of the ground state.

1022
01:06:21,502 --> 01:06:23,710
Well, when we go from the
excited state to the ground

1023
01:06:23,710 --> 01:06:29,300
state, there will be only one
term contributing to the sum.

1024
01:06:29,300 --> 01:06:33,560
Where we have one photon more
because it has been emitted.

1025
01:06:33,560 --> 01:06:37,330
So, therefore, because of
the square root n and n

1026
01:06:37,330 --> 01:06:39,710
plus 1 dependence of
the matrix element,

1027
01:06:39,710 --> 01:06:44,845
we find that for the processes
where photon is emitted, where

1028
01:06:44,845 --> 01:06:47,700
the atomic system
gives away a photon,

1029
01:06:47,700 --> 01:06:51,930
the sum of all the possible
rates becomes simply n plus 1.

1030
01:06:51,930 --> 01:06:54,161
And in the case of
absorption it becomes n.

1031
01:06:57,110 --> 01:07:02,380
So in other words, we have now
done the field quantization

1032
01:07:02,380 --> 01:07:05,740
what Einstein pulled out of
a thermodynamic equilibrium

1033
01:07:05,740 --> 01:07:06,620
argument.

1034
01:07:06,620 --> 01:07:12,020
Namely that if you have a
system that the rate of emission

1035
01:07:12,020 --> 01:07:18,150
versus the rate of absorption
is n plus 1 over n.

1036
01:07:18,150 --> 01:07:21,630
But we did not assume any
spectral distribution.

1037
01:07:21,630 --> 01:07:23,590
We know this n plus
1 over n applies

1038
01:07:23,590 --> 01:07:27,876
to every single mode of
t electromagnetic field.

1039
01:07:40,220 --> 01:07:41,512
Questions about that?

1040
01:07:58,380 --> 01:08:05,360
I also want to tell you, just as
a side remark, a lot of people

1041
01:08:05,360 --> 01:08:09,320
think that when
emission is n plus 1,

1042
01:08:09,320 --> 01:08:12,150
the plus 1 is different from n.

1043
01:08:12,150 --> 01:08:16,620
That this plus 1 is sort of a
spontaneously emitted photon

1044
01:08:16,620 --> 01:08:18,870
which has maybe
some random phase.

1045
01:08:18,870 --> 01:08:22,170
And the n which is
stimulated photons,

1046
01:08:22,170 --> 01:08:26,319
they go in the same
mode as they joined

1047
01:08:26,319 --> 01:08:29,880
sort of the identical to
the photons already present.

1048
01:08:29,880 --> 01:08:32,225
I don't see any of
that in that treatment.

1049
01:08:35,660 --> 01:08:39,630
So a spontaneously
emitted photon

1050
01:08:39,630 --> 01:08:42,170
is identical to the
photon which would

1051
01:08:42,170 --> 01:08:43,510
be emitted in a stimulated way.

1052
01:08:46,560 --> 01:08:48,649
You just have n plus 1.

1053
01:08:48,649 --> 01:08:51,370
This is the matrix element
for coupling to this mode.

1054
01:08:57,029 --> 01:08:59,990
At some point,
spontaneous emission

1055
01:08:59,990 --> 01:09:01,979
can happen in many modes.

1056
01:09:01,979 --> 01:09:04,410
And if it goes to
many modes, then there

1057
01:09:04,410 --> 01:09:08,859
is some integral or
some summation involved.

1058
01:09:08,859 --> 01:09:11,479
And this can cause a
certain randomness.

1059
01:09:11,479 --> 01:09:14,060
But at the level
of a single mode,

1060
01:09:14,060 --> 01:09:18,689
I do not see any difference
between the one photon

1061
01:09:18,689 --> 01:09:21,790
and the n photons at
this level of discussion.

1062
01:09:21,790 --> 01:09:25,040
Just keep that in mind.

1063
01:09:25,040 --> 01:09:27,560
And actually, we'll
discuss micromasers.

1064
01:09:27,560 --> 01:09:30,670
You can have put an
excited atom in the cavity.

1065
01:09:30,670 --> 01:09:32,750
And you have a fully
reversible exchange.

1066
01:09:32,750 --> 01:09:34,510
You spontaneously
emit, you absorb.

1067
01:09:34,510 --> 01:09:36,319
You spontaneously
emit, you absorb.

1068
01:09:36,319 --> 01:09:39,649
You have Rabi oscillations
which involve a single photon.

1069
01:09:39,649 --> 01:09:43,180
And they involve
spontaneous emission.

1070
01:09:43,180 --> 01:09:44,380
Fully reversible.

1071
01:09:44,380 --> 01:09:46,184
Completely
[INAUDIBLE] evolution.

1072
01:09:49,140 --> 01:09:51,753
So we have 10 minutes left.

1073
01:10:02,400 --> 01:10:06,780
Yes, I think this is just
enough to derive for you.

1074
01:10:09,480 --> 01:10:13,250
Now to derive for you using
the fully quantized picture.

1075
01:10:15,800 --> 01:10:17,940
To derive flow from
first principles

1076
01:10:17,940 --> 01:10:21,060
microscopically an expression
for Einstein's a coefficient.

1077
01:10:27,389 --> 01:10:28,930
So in other words,
what I'm doing now

1078
01:10:28,930 --> 01:10:31,630
is I really directly
calculate for you

1079
01:10:31,630 --> 01:10:33,017
the rate of
spontaneous emission.

1080
01:10:33,017 --> 01:10:34,850
And I'm not getting it
through the back door

1081
01:10:34,850 --> 01:10:38,580
by treating absorption
and then saying, well,

1082
01:10:38,580 --> 01:10:39,990
there's n and n plus 1.

1083
01:10:39,990 --> 01:10:43,302
Or borrowing some
argument from Einstein.

1084
01:10:43,302 --> 01:10:44,760
It's such an
important quantity, we

1085
01:10:44,760 --> 01:10:47,750
should just hit the
system with a Hamiltonian

1086
01:10:47,750 --> 01:10:49,710
and out comes a
spontaneous emission rate.

1087
01:10:49,710 --> 01:10:50,927
And this is what we're doing.

1088
01:11:08,830 --> 01:11:12,260
So the starting point is
what we have discussed

1089
01:11:12,260 --> 01:11:13,750
at the beginning of the class.

1090
01:11:13,750 --> 01:11:22,470
We want to discuss
Fermi's Golden Rule.

1091
01:11:22,470 --> 01:11:25,520
We want to use the rate.

1092
01:11:29,070 --> 01:11:33,090
And to remind you
the rate for process

1093
01:11:33,090 --> 01:11:38,170
is the matrix element
squared by h plus square.

1094
01:11:38,170 --> 01:11:43,505
And then we have to multiply
with the density of states.

1095
01:11:56,990 --> 01:11:59,511
So this is the
density of states.

1096
01:11:59,511 --> 01:12:00,260
Pair polarization.

1097
01:12:00,260 --> 01:12:05,240
Actually, I made a few
corrections to my notes

1098
01:12:05,240 --> 01:12:07,770
because I realize I
have be very, very

1099
01:12:07,770 --> 01:12:10,740
careful in telling you
what the states are.

1100
01:12:10,740 --> 01:12:12,850
Because this is what
this exercise is about.

1101
01:12:12,850 --> 01:12:17,330
And we are writing it
down for one mode by mode.

1102
01:12:17,330 --> 01:12:20,490
So the density of state
is now pair polarization.

1103
01:12:20,490 --> 01:12:23,960
We take care of
polarizations later.

1104
01:12:23,960 --> 01:12:25,410
Per unit frequency interval.

1105
01:12:41,315 --> 01:12:41,815
Yes.

1106
01:12:57,920 --> 01:13:00,940
So this rate, but now I
have to add one caveat.

1107
01:13:00,940 --> 01:13:03,040
I was just thinking how
I should express it.

1108
01:13:03,040 --> 01:13:06,350
This rate, if it's all
of spontaneous emission,

1109
01:13:06,350 --> 01:13:08,740
is the Einstein a coefficient.

1110
01:13:08,740 --> 01:13:10,300
But there is one caveat.

1111
01:13:10,300 --> 01:13:12,990
And this is the emission
of an atom which

1112
01:13:12,990 --> 01:13:15,275
has a dipole moment
is not isotropic.

1113
01:13:15,275 --> 01:13:16,650
So I have to be
a little bit more

1114
01:13:16,650 --> 01:13:18,130
careful with the solid angle.

1115
01:13:18,130 --> 01:13:20,640
I cannot just calculate a
rate and assume everything is

1116
01:13:20,640 --> 01:13:21,460
isotopic.

1117
01:13:21,460 --> 01:13:24,400
If I would do that, I
would save a few minutes.

1118
01:13:24,400 --> 01:13:27,760
But I would have really swept
something under the rug.

1119
01:13:27,760 --> 01:13:33,280
So what I'm calculating first is
the rate in a given unit angle.

1120
01:13:33,280 --> 01:13:36,630
And then I do an integration
over the unit angle.

1121
01:13:36,630 --> 01:13:40,720
And eventually I will integrate
over the dipole [INAUDIBLE].

1122
01:13:40,720 --> 01:13:47,510
So, therefore, the density
of the photon states

1123
01:13:47,510 --> 01:13:51,190
is sort of photons with their
k vectors go into all space,

1124
01:13:51,190 --> 01:13:56,210
but I wanted to have the density
of states per unit angle.

1125
01:13:56,210 --> 01:14:08,430
And this quantity is omega
square, 8 pi cube, c cube,

1126
01:14:08,430 --> 01:14:13,910
times v. And, of course, if
you multiply this by 4 pi,

1127
01:14:13,910 --> 01:14:15,790
you get your normal
density of states.

1128
01:14:15,790 --> 01:14:19,500
Because the density of
states is isotropic.

1129
01:14:19,500 --> 01:14:21,630
But the rate which
we calculate will not

1130
01:14:21,630 --> 01:14:23,775
be isotopic because of
the dipole matrix element.

1131
01:14:23,775 --> 01:14:25,540
And the dipole pattern.

1132
01:14:25,540 --> 01:14:29,030
So, therefore, we start with
a differential formulation.

1133
01:14:29,030 --> 01:14:31,150
Spontaneous emission
per solid angle.

1134
01:14:31,150 --> 01:14:34,650
And then when we do, when we
integrate over the solid angle,

1135
01:14:34,650 --> 01:14:37,050
we have to take care of a
sine square factor because

1136
01:14:37,050 --> 01:14:40,480
of the dipole pattern.

1137
01:14:40,480 --> 01:14:43,320
Good.

1138
01:14:43,320 --> 01:14:48,015
Now Fermi's Golden Rule takes
us from an excited state

1139
01:14:48,015 --> 01:14:50,340
to ground state.

1140
01:14:50,340 --> 01:14:55,030
And since we use the
fully quantized treatment,

1141
01:14:55,030 --> 01:15:02,280
our product states of atomic
states and photon states.

1142
01:15:02,280 --> 01:15:07,760
And so we assume we start with
an atom in the excited states.

1143
01:15:07,760 --> 01:15:13,150
And all modes, mode 1,
mode 2, mode 3, are empty.

1144
01:15:16,630 --> 01:15:22,920
And the final state, well,
one photon is emitted

1145
01:15:22,920 --> 01:15:25,037
and it can appear
in any of the modes.

1146
01:15:25,037 --> 01:15:27,161
And we have to do an integral
of all possibilities.

1147
01:15:33,420 --> 01:15:33,960
Good.

1148
01:15:33,960 --> 01:15:38,400
So we did all the
work with quantizing

1149
01:15:38,400 --> 01:15:39,810
the electromagnetic field.

1150
01:15:39,810 --> 01:15:44,530
Because we want to calculate
those matrix elements.

1151
01:15:44,530 --> 01:15:47,340
Let me just carry over the
prefactors, the electric field

1152
01:15:47,340 --> 01:15:48,175
of a single photon.

1153
01:15:53,540 --> 01:15:57,360
Here we have the dipole moment
between the polarization

1154
01:15:57,360 --> 01:16:02,480
and the atomic dipole
matrix element.

1155
01:16:02,480 --> 01:16:07,170
And now since we are talking
about an emission problem,

1156
01:16:07,170 --> 01:16:10,470
we have from the matrix
elements squared as we just

1157
01:16:10,470 --> 01:16:13,070
discussed, an n plus 1 factor.

1158
01:16:13,070 --> 01:16:16,420
But the population we
start with 0 photons.

1159
01:16:16,420 --> 01:16:20,420
So, therefore, it's just one.

1160
01:16:20,420 --> 01:16:22,300
So all the work we
did on quantization

1161
01:16:22,300 --> 01:16:24,610
of the electromagnetic
field is that even

1162
01:16:24,610 --> 01:16:27,660
without any photon present,
we have a coupling.

1163
01:16:27,660 --> 01:16:31,570
You can say it's a coupling
caused by the vacuum which

1164
01:16:31,570 --> 01:16:34,310
is like the coupling
we would have gotten

1165
01:16:34,310 --> 01:16:36,386
if we have exactly
one photon per mode.

1166
01:16:42,190 --> 01:16:50,450
So let me just write that out.

1167
01:17:14,840 --> 01:17:15,557
Yes.

1168
01:17:15,557 --> 01:17:17,980
I think we can finish that.

1169
01:17:20,810 --> 01:17:26,150
OK, we have now taken care
of the matrix element.

1170
01:17:26,150 --> 01:17:30,260
So we insert the matrix element
now in our Fermi's Golden Rule

1171
01:17:30,260 --> 01:17:32,035
expression for the ad omega.

1172
01:17:35,250 --> 01:17:40,750
Let me just keep track
of all the factors.

1173
01:17:40,750 --> 01:17:42,695
[INAUDIBLE] omega in
the matrix element.

1174
01:17:45,750 --> 01:17:50,160
This comes from the electric
field of a single photon.

1175
01:17:58,550 --> 01:18:00,700
We have a matrix element.

1176
01:18:00,700 --> 01:18:04,592
Dipole matrix element
times polarization.

1177
01:18:04,592 --> 01:18:10,470
And the density of states
gives us omega square.

1178
01:18:10,470 --> 01:18:13,930
So see already we'll get a
spontaneous emission omega cube

1179
01:18:13,930 --> 01:18:15,110
expression.

1180
01:18:15,110 --> 01:18:19,270
One omega comes because
the electric field

1181
01:18:19,270 --> 01:18:22,140
of a single photon, the
electric field squared

1182
01:18:22,140 --> 01:18:24,450
of a single photon is
proportional to omega.

1183
01:18:24,450 --> 01:18:27,970
And an omega square factor comes
from the density of states.

1184
01:18:27,970 --> 01:18:31,220
It's really important
to keep that apart.

1185
01:18:31,220 --> 01:18:33,370
The omega 3 dependence
has two different sources.

1186
01:18:47,010 --> 01:18:50,180
It's always nice to see that we
assumed an [INAUDIBLE] volume

1187
01:18:50,180 --> 01:18:51,150
and it cancels out.

1188
01:18:53,930 --> 01:18:55,970
OK.

1189
01:18:55,970 --> 01:19:12,060
Let me just write it down and
then we do the final step.

1190
01:19:12,060 --> 01:19:14,350
So if everything
were isotropic, I

1191
01:19:14,350 --> 01:19:15,940
could just multiply with 4 pi.

1192
01:19:15,940 --> 01:19:18,830
And the last factor
would be dropped.

1193
01:19:18,830 --> 01:19:22,940
But if you want to go from
the spontaneous emission

1194
01:19:22,940 --> 01:19:27,260
per solid angle to the
total spontaneous emission,

1195
01:19:27,260 --> 01:19:29,010
we have to average it.

1196
01:19:29,010 --> 01:19:32,380
And what has angular
factors is actually

1197
01:19:32,380 --> 01:19:35,460
the projection between
the atomic dipole

1198
01:19:35,460 --> 01:19:37,290
moment and the polarization.

1199
01:19:45,570 --> 01:19:48,750
So this is the relevant term.

1200
01:19:48,750 --> 01:19:52,830
And now we have to distinguish.

1201
01:19:52,830 --> 01:19:56,210
There are two polarizations.

1202
01:19:56,210 --> 01:19:59,182
One polarization,
the polarization

1203
01:19:59,182 --> 01:20:00,390
when we have a dipole moment.

1204
01:20:00,390 --> 01:20:03,700
And you have light which
is polarized in such a way

1205
01:20:03,700 --> 01:20:05,810
that the light goes here.

1206
01:20:05,810 --> 01:20:08,620
This is a dipole moment,
the light goes here.

1207
01:20:08,620 --> 01:20:10,640
And now the light
which propagates here

1208
01:20:10,640 --> 01:20:13,590
can have a
polarization like this.

1209
01:20:13,590 --> 01:20:19,470
Which has a projection of sine
theta, with the dipole moment.

1210
01:20:19,470 --> 01:20:22,520
And if the light goes there and
the polarization is like this,

1211
01:20:22,520 --> 01:20:24,610
it's orthogonal to
the dipole moment.

1212
01:20:24,610 --> 01:20:27,860
So the scalar product is 0.

1213
01:20:27,860 --> 01:20:33,770
So for one polarization, we
have a sine theta factor.

1214
01:20:33,770 --> 01:20:39,610
For the second polarization,
the scalar product

1215
01:20:39,610 --> 01:20:43,010
for the dipole moment is 0.

1216
01:20:43,010 --> 01:20:49,535
So, therefore, and that's
the last conclusion

1217
01:20:49,535 --> 01:20:50,410
I want to draw today.

1218
01:20:55,990 --> 01:21:06,890
Is this integration
over the solid angle

1219
01:21:06,890 --> 01:21:12,340
boils down to that we can
pull the matrix element out

1220
01:21:12,340 --> 01:21:14,760
of the integral.

1221
01:21:14,760 --> 01:21:16,980
And what is left is
the projection factor,

1222
01:21:16,980 --> 01:21:19,960
sine square theta.

1223
01:21:19,960 --> 01:21:24,020
We have to integrate over
the whole solid angle.

1224
01:21:24,020 --> 01:21:26,517
And this gives us 2/3.

1225
01:21:30,000 --> 01:21:33,370
So, therefore, our
final result is

1226
01:21:33,370 --> 01:21:41,586
that the microscopic expression
for the a coefficient

1227
01:21:41,586 --> 01:21:43,210
has its factor of 3
in the denominator.

1228
01:21:43,210 --> 01:21:45,700
And this factor of 3
only comes because I

1229
01:21:45,700 --> 01:21:49,440
correct the average
over the dipole pattern.

1230
01:21:49,440 --> 01:21:53,440
Well, then we have
4 pi epsilon 0.

1231
01:21:53,440 --> 01:21:56,430
I mentioned the
important dependence

1232
01:21:56,430 --> 01:21:57,708
on the frequency cubed.

1233
01:22:01,100 --> 01:22:06,810
OK, so this is our
final result for today.

1234
01:22:06,810 --> 01:22:12,430
And I will discuss next
week what are its units?

1235
01:22:12,430 --> 01:22:14,110
How big is it?

1236
01:22:14,110 --> 01:22:17,060
What is the quality factor
of the atomic oscillator?

1237
01:22:17,060 --> 01:22:20,970
But we can start next
week with this result.

1238
01:22:20,970 --> 01:22:22,820
All right.