1
00:00:00,070 --> 00:00:01,780
The following
content is provided

2
00:00:01,780 --> 00:00:04,030
under a Creative
Commons license.

3
00:00:04,030 --> 00:00:06,880
Your support will help MIT
OpenCourseWare continue

4
00:00:06,880 --> 00:00:10,740
to offer high quality
educational resources for free.

5
00:00:10,740 --> 00:00:13,360
To make a donation or
view additional materials

6
00:00:13,360 --> 00:00:17,258
from hundreds of MIT courses,
visit MIT OpenCourseWare

7
00:00:17,258 --> 00:00:17,883
at ocw.mit.edu.

8
00:00:21,190 --> 00:00:22,610
PROFESSOR: So good afternoon.

9
00:00:22,610 --> 00:00:24,160
Let's have an on-time departure.

10
00:00:27,510 --> 00:00:34,080
Last week, we talked about
quantum states of light.

11
00:00:34,080 --> 00:00:37,960
So we're talking about the
photon, the quantum description

12
00:00:37,960 --> 00:00:40,590
of the electromagnetic field.

13
00:00:40,590 --> 00:00:45,490
And I introduced for you the
most classical quantum state

14
00:00:45,490 --> 00:00:49,620
of the electromagnetic field,
namely the coherent state.

15
00:00:49,620 --> 00:00:51,990
I don't want to go through
the formalism again,

16
00:00:51,990 --> 00:00:55,700
but the coherent state
has a simple definition,

17
00:00:55,700 --> 00:00:57,920
simple but subtle.

18
00:00:57,920 --> 00:01:01,170
It's an eigenstate of the
annihilation operator,

19
00:01:01,170 --> 00:01:04,800
and it has a complex
eigenvalue alpha.

20
00:01:04,800 --> 00:01:06,480
And from that, you
can pretty much

21
00:01:06,480 --> 00:01:08,930
derive everything you want to
know about coherent states.

22
00:01:08,930 --> 00:01:09,779
Colin.

23
00:01:09,779 --> 00:01:12,174
AUDIENCE: What was the
purpose of-- whoever

24
00:01:12,174 --> 00:01:15,207
discovered eigenstates-- what
was the purpose of trying

25
00:01:15,207 --> 00:01:17,292
to find the eigenvalues
of destruction?

26
00:01:17,292 --> 00:01:18,750
PROFESSOR: Yeah,
isn't that subtle?

27
00:01:18,750 --> 00:01:20,100
Isn't that weird?

28
00:01:20,100 --> 00:01:24,330
How does somebody want to
find the eigenfunctions

29
00:01:24,330 --> 00:01:27,050
of an annihilation operator,
because the annihilation

30
00:01:27,050 --> 00:01:28,790
operator is non-Hermitian?

31
00:01:28,790 --> 00:01:30,510
And why the
annihilation operator?

32
00:01:30,510 --> 00:01:32,520
Why not the creation operator?

33
00:01:32,520 --> 00:01:34,880
Well, if you try to find the
eigenstates of the creation

34
00:01:34,880 --> 00:01:37,030
operator, you run into problems.

35
00:01:37,030 --> 00:01:39,070
The eigenstates are
non-normalizable.

36
00:01:39,070 --> 00:01:40,630
It doesn't make any sense.

37
00:01:40,630 --> 00:01:44,140
So here, creation and
annihilation are not symmetric.

38
00:01:44,140 --> 00:01:47,230
You have to pick the
annihilation operator.

39
00:01:47,230 --> 00:01:49,680
Colin, my gut feeling is
that somebody found it

40
00:01:49,680 --> 00:01:50,970
in a different way.

41
00:01:50,970 --> 00:01:53,540
The person who popularized
those states was [INAUDIBLE],

42
00:01:53,540 --> 00:01:57,510
and he got amply
rewarded for that.

43
00:01:57,510 --> 00:02:00,760
And I think you
define coherent state

44
00:02:00,760 --> 00:02:02,700
as some form of
superposition state.

45
00:02:02,700 --> 00:02:05,640
Maybe people have figured out
if you take a superposition

46
00:02:05,640 --> 00:02:08,620
state of number states that
this has special properties,

47
00:02:08,620 --> 00:02:12,500
that this is a minimum
uncertainty state, which

48
00:02:12,500 --> 00:02:15,530
is as close as we can
go in quantum physics

49
00:02:15,530 --> 00:02:19,540
to obtain a classical
electric field.

50
00:02:19,540 --> 00:02:21,470
So I think maybe people
found it like that.

51
00:02:21,470 --> 00:02:24,040
And then they say, hey, but
there is another operator

52
00:02:24,040 --> 00:02:27,330
definition eigenstate of
the annihilation operator.

53
00:02:27,330 --> 00:02:29,896
I'm 99 percent certain.

54
00:02:29,896 --> 00:02:31,270
But maybe you can
ask [INAUDIBLE]

55
00:02:31,270 --> 00:02:35,720
when he comes to a CUA seminar
if he discovered coherent state

56
00:02:35,720 --> 00:02:37,890
as eigenstates of the
annihilation operator.

57
00:02:37,890 --> 00:02:40,234
I don't think so.

58
00:02:40,234 --> 00:02:40,900
Other questions?

59
00:02:46,170 --> 00:02:48,150
So what I want to do
is-- what the theme

60
00:02:48,150 --> 00:02:50,510
of the end of last
lecture and today

61
00:02:50,510 --> 00:02:55,460
is that the coherent
state is a complex number.

62
00:02:55,460 --> 00:02:59,190
The coherent state
has a time evolution.

63
00:02:59,190 --> 00:03:02,150
It moves in a circle.

64
00:03:02,150 --> 00:03:07,000
And this is really the phasor
of the electric field associated

65
00:03:07,000 --> 00:03:07,620
with it.

66
00:03:07,620 --> 00:03:11,150
So coherent state,
the alpha value

67
00:03:11,150 --> 00:03:14,160
is directly related
to an electric field.

68
00:03:14,160 --> 00:03:19,240
That's why this state
is closely related

69
00:03:19,240 --> 00:03:23,820
to the classical limit of
the electromagnetic field.

70
00:03:33,260 --> 00:03:36,070
We discussed some properties
of the coherent state.

71
00:03:36,070 --> 00:03:39,130
We looked at the
fluctuations and showed

72
00:03:39,130 --> 00:03:42,295
that it's a
Poissonian statistic.

73
00:03:44,850 --> 00:03:48,530
And then we use now--
once you define something,

74
00:03:48,530 --> 00:03:50,360
you can use it as a tool.

75
00:03:50,360 --> 00:03:53,570
We are now using
the coherent states

76
00:03:53,570 --> 00:03:56,880
to look at any
other quantum state

77
00:03:56,880 --> 00:03:59,930
of the electromagnetic field,
any statistical operator which

78
00:03:59,930 --> 00:04:03,830
describes photons by
forming the diagram matrix

79
00:04:03,830 --> 00:04:09,680
element of the statistical
operator with alpha.

80
00:04:09,680 --> 00:04:13,200
If you just want to sort of
think about it intuitively,

81
00:04:13,200 --> 00:04:15,525
alpha is like the
best description

82
00:04:15,525 --> 00:04:17,550
of an electromagnetic field.

83
00:04:17,550 --> 00:04:20,110
So if you write
down this, we ask,

84
00:04:20,110 --> 00:04:23,870
What is the probability for
an arbitrary quantum state

85
00:04:23,870 --> 00:04:26,770
described by an arbitrary
statistical operator-- what

86
00:04:26,770 --> 00:04:29,180
is the probability that
the electric field is

87
00:04:29,180 --> 00:04:32,560
alpha in the
complex description?

88
00:04:32,560 --> 00:04:34,650
So that's what those
quasi-probabilities

89
00:04:34,650 --> 00:04:38,830
are-- Q of alpha.

90
00:04:38,830 --> 00:04:44,460
And we immediately looked
at some examples, of course,

91
00:04:44,460 --> 00:04:46,430
of probabilities.

92
00:04:46,430 --> 00:04:54,290
We showed that the vacuum
state is sort of an area.

93
00:04:54,290 --> 00:04:55,300
It's a Gaussian.

94
00:04:55,300 --> 00:05:01,850
And the area is-- it's an area.

95
00:05:01,850 --> 00:05:02,900
I don't know.

96
00:05:02,900 --> 00:05:07,400
I forgot the [? prefect. ?]
It's on the order of 1 or 1/2.

97
00:05:07,400 --> 00:05:10,775
We realize that
thermal states are also

98
00:05:10,775 --> 00:05:13,890
a Gaussian centered
at the origin.

99
00:05:13,890 --> 00:05:18,000
But it is a much
wider distribution.

100
00:05:18,000 --> 00:05:21,480
So thermal states have
much more uncertainty

101
00:05:21,480 --> 00:05:24,910
in the electric field
than the vacuum state.

102
00:05:24,910 --> 00:05:27,410
And then we looked at
the coherent state.

103
00:05:27,410 --> 00:05:30,580
And now, of course, when we
looked at the coherent state,

104
00:05:30,580 --> 00:05:34,660
we realize that coherent
states are maybe not

105
00:05:34,660 --> 00:05:37,520
as wonderful as I tried
to make you believe.

106
00:05:37,520 --> 00:05:39,560
They have some nasty properties.

107
00:05:39,560 --> 00:05:43,540
And that is when you ask,
What is the probability?

108
00:05:43,540 --> 00:05:47,580
The quasi-probability of the
coherent state would be beta.

109
00:05:47,580 --> 00:05:49,410
Now, if you would
expect a delta function,

110
00:05:49,410 --> 00:05:52,200
because when you ask, What is
the probability of a momentum

111
00:05:52,200 --> 00:05:53,277
state to be p naught?

112
00:05:53,277 --> 00:05:54,860
If you have a momentum
state p naught,

113
00:05:54,860 --> 00:05:57,180
it has a delta
function at p naught.

114
00:05:57,180 --> 00:05:59,230
But here, it's not
a delta function.

115
00:05:59,230 --> 00:06:04,510
It's a Gaussian of the
standard deviation.

116
00:06:04,510 --> 00:06:07,210
I think the standard deviation
here is-- what is it,

117
00:06:07,210 --> 00:06:08,820
square root 1 over
square root 2?

118
00:06:08,820 --> 00:06:12,090
Because the Gaussian has
usually 2 times sigma squared.

119
00:06:12,090 --> 00:06:14,150
And since the
denominator is unity--

120
00:06:14,150 --> 00:06:16,770
so the standard deviation
of 1 over square root 2.

121
00:06:16,770 --> 00:06:21,950
And that showed us that
the coherent states are not

122
00:06:21,950 --> 00:06:22,540
orthogonal.

123
00:06:22,540 --> 00:06:24,040
You have to be a
little bit careful.

124
00:06:24,040 --> 00:06:26,340
It's not your
standard basis set.

125
00:06:26,340 --> 00:06:27,820
It's an overcomplete basis.

126
00:06:31,800 --> 00:06:37,840
So therefore the
coherent state is not

127
00:06:37,840 --> 00:06:39,750
a delta function in
the causal probability.

128
00:06:39,750 --> 00:06:43,280
It's a little bit blurred
circle with an area

129
00:06:43,280 --> 00:06:46,730
on the order of unity.

130
00:06:46,730 --> 00:06:53,380
And you have some homework
assignment to look at it.

131
00:06:53,380 --> 00:06:55,610
So in some way, how
you should look at

132
00:06:55,610 --> 00:07:02,320
it is that if you see some
quasi-probability distribution,

133
00:07:02,320 --> 00:07:04,040
the distance from
the origin-- this

134
00:07:04,040 --> 00:07:06,820
is the absolute value
of the electric field.

135
00:07:06,820 --> 00:07:10,020
The phase here is
almost 45 degrees.

136
00:07:10,020 --> 00:07:11,605
But if you look at
this uncertainty,

137
00:07:11,605 --> 00:07:13,785
you would also say,
Well, the phase

138
00:07:13,785 --> 00:07:16,990
is uncertain to
within this angle.

139
00:07:16,990 --> 00:07:19,540
So you can pretty much
read the uncertainty

140
00:07:19,540 --> 00:07:23,690
in the total electric field,
the uncertainty in the phase.

141
00:07:23,690 --> 00:07:25,150
You can read pretty
much everything

142
00:07:25,150 --> 00:07:26,420
you want from this diagram.

143
00:07:30,970 --> 00:07:34,800
Now, when we look at
a number state-- Well,

144
00:07:34,800 --> 00:07:36,440
you often know in
quantum mechanics,

145
00:07:36,440 --> 00:07:39,000
number and phase
are complimentary.

146
00:07:39,000 --> 00:07:41,040
If the number of
photons is fixed,

147
00:07:41,040 --> 00:07:42,860
you know nothing
about the phase.

148
00:07:42,860 --> 00:07:46,580
And indeed the quasi-probability
of a number state is a ring.

149
00:07:46,580 --> 00:07:47,530
It has no phase.

150
00:07:47,530 --> 00:07:51,268
It has completely random
phase over the 2 pi circle.

151
00:07:54,140 --> 00:07:57,370
The energy is sharp
of a number state,

152
00:07:57,370 --> 00:08:01,120
since the energy is e squared.

153
00:08:01,120 --> 00:08:05,150
You may have expected
a delta function

154
00:08:05,150 --> 00:08:06,730
in the radial coordinate.

155
00:08:06,730 --> 00:08:08,320
But what you get
is also something

156
00:08:08,320 --> 00:08:09,630
blurred on the order of unity.

157
00:08:09,630 --> 00:08:13,460
And I want to say something
about that in a second.

158
00:08:13,460 --> 00:08:17,190
Finally, we discuss
the time dependence.

159
00:08:17,190 --> 00:08:19,580
And the time dependence
is very easy.

160
00:08:19,580 --> 00:08:22,380
After all, we're dealing
with an harmonic oscillator.

161
00:08:22,380 --> 00:08:24,500
And in an harmonic
oscillator, if you

162
00:08:24,500 --> 00:08:30,760
have a plane of x and p,
symmetric and anti-symmetric

163
00:08:30,760 --> 00:08:33,500
combination of a and a
dagger, in this plane

164
00:08:33,500 --> 00:08:36,289
in an harmonic oscillator,
the quantum state

165
00:08:36,289 --> 00:08:39,860
is just rotating circle,
a rotation with omega.

166
00:08:39,860 --> 00:08:43,870
And indeed we showed that when
we apply the time evolution

167
00:08:43,870 --> 00:08:46,990
operator-- and some of
you were right, of course,

168
00:08:46,990 --> 00:08:51,890
with a minus sign-- it moves
with e to the minus i omega t.

169
00:08:51,890 --> 00:08:59,780
And therefore everything
rotates in a clockwise way.

170
00:09:02,310 --> 00:09:11,540
Now, we discussed the operator
of the electric field.

171
00:09:11,540 --> 00:09:17,920
And in this
quasi-probability-- Sorry,

172
00:09:17,920 --> 00:09:20,930
we discussed the operator
of the electric field.

173
00:09:20,930 --> 00:09:23,750
And I hope you remember
that in the analogy

174
00:09:23,750 --> 00:09:26,670
with the harmonic oscillator,
the electric field was

175
00:09:26,670 --> 00:09:28,099
a minus a dagger.

176
00:09:28,099 --> 00:09:29,515
And this is the
momentum operator.

177
00:09:32,700 --> 00:09:36,910
In those quasi-probabilities--
and we will see more about it--

178
00:09:36,910 --> 00:09:39,530
something which is
sharp in momentum

179
00:09:39,530 --> 00:09:43,640
is a sliver parallel
to the x-axis.

180
00:09:43,640 --> 00:09:48,320
So therefore, you can
regard the vertical axis,

181
00:09:48,320 --> 00:09:51,630
which is the imaginary part of
alpha, as the momentum axis.

182
00:09:51,630 --> 00:09:54,490
And you can regard this
as-- the horizontal axis--

183
00:09:54,490 --> 00:09:56,270
as the x-axis.

184
00:09:56,270 --> 00:09:59,500
So therefore, since
momentum is electric field,

185
00:09:59,500 --> 00:10:01,180
you always get
the electric field

186
00:10:01,180 --> 00:10:03,890
by projecting onto
the vertical axis.

187
00:10:03,890 --> 00:10:06,510
And if you project
this fuzzy ball,

188
00:10:06,510 --> 00:10:10,810
you get a value 0
with some uncertainty.

189
00:10:10,810 --> 00:10:12,960
And if this
quasi-probability starts

190
00:10:12,960 --> 00:10:15,800
to rotate due to
the time evolution,

191
00:10:15,800 --> 00:10:20,790
we get an oscillating
electromagnetic field, almost

192
00:10:20,790 --> 00:10:23,890
classical except
for that fuzziness.

193
00:10:32,870 --> 00:10:37,720
So that's where we
want to continue.

194
00:10:37,720 --> 00:10:39,760
Any questions about that?

195
00:10:39,760 --> 00:10:40,559
Yes?

196
00:10:40,559 --> 00:10:43,912
AUDIENCE: Why is there no
phase fuzziness in the E t?

197
00:10:46,790 --> 00:10:50,940
PROFESSOR: No phase fuzziness.

198
00:10:50,940 --> 00:10:52,820
There is a phrase fuzziness.

199
00:10:52,820 --> 00:10:55,830
For instance, if you would
say the phase is determined

200
00:10:55,830 --> 00:10:58,650
by the 0 crossing, you don't
know exactly when the 0

201
00:10:58,650 --> 00:11:00,480
crossing happened,
and that corresponds

202
00:11:00,480 --> 00:11:03,870
to an uncertainty in the phase.

203
00:11:03,870 --> 00:11:07,220
Trust me, everything
is in this diagram.

204
00:11:07,220 --> 00:11:10,490
Now, there are two things
we want to continue.

205
00:11:10,490 --> 00:11:16,700
One is I want to show you
that the coherent state is

206
00:11:16,700 --> 00:11:18,570
a minimum uncertainty state.

207
00:11:18,570 --> 00:11:27,372
The product of delta
x delta p is just--

208
00:11:27,372 --> 00:11:29,380
is it h-bar or h-bar over two?

209
00:11:29,380 --> 00:11:31,370
One of the two.

210
00:11:31,370 --> 00:11:33,370
So it's a minimum
uncertainty state.

211
00:11:33,370 --> 00:11:36,500
And therefore, you can
never have a quantum state

212
00:11:36,500 --> 00:11:39,260
which is less fuzzy
than the coherent state.

213
00:11:39,260 --> 00:11:42,630
So this fuzziness here is
the intrinsic uncertainty

214
00:11:42,630 --> 00:11:44,110
of quantum physics.

215
00:11:44,110 --> 00:11:47,460
So that's what we
want to discuss today.

216
00:11:47,460 --> 00:11:51,150
But then we will
immediately start

217
00:11:51,150 --> 00:11:53,000
with non-classical states.

218
00:11:53,000 --> 00:11:56,140
And that is, well, if
this area is determined

219
00:11:56,140 --> 00:11:58,970
by Heisenberg's
uncertainty relation, what

220
00:11:58,970 --> 00:12:02,650
can be maybe deform the
circle into an ellipse,

221
00:12:02,650 --> 00:12:04,910
and these are three
states of light.

222
00:12:04,910 --> 00:12:05,785
So that's an outlook.

223
00:12:05,785 --> 00:12:07,826
That's what you're going
to do in the second half

224
00:12:07,826 --> 00:12:08,410
of the class.

225
00:12:14,940 --> 00:12:23,720
But before I do that, I
want to be a little bit more

226
00:12:23,720 --> 00:12:25,690
accurate about
quasi-probabilities.

227
00:12:28,210 --> 00:12:37,830
And this is almost
like a disclaimer now.

228
00:12:46,640 --> 00:12:50,980
Just to give you
the bigger picture.

229
00:12:50,980 --> 00:12:53,900
What we want to achieve with
those quasi-probabilities,

230
00:12:53,900 --> 00:12:57,300
we want to do what phase space
densities in classical physics.

231
00:12:57,300 --> 00:13:01,150
We have a coordinate which is
x, a coordinate which is p.

232
00:13:01,150 --> 00:13:04,510
And you can often describe
a classical system

233
00:13:04,510 --> 00:13:07,070
if you know the phase
space distribution,

234
00:13:07,070 --> 00:13:09,650
if you know the probability
that a particle has

235
00:13:09,650 --> 00:13:12,280
position x and momentum p.

236
00:13:12,280 --> 00:13:20,070
So all this is about
phase space densities.

237
00:13:26,740 --> 00:13:32,902
Probability of x and p.

238
00:13:32,902 --> 00:13:36,730
Of course, writing it down, you
immediately see the problem.

239
00:13:36,730 --> 00:13:40,490
In quantum mechanics, you cannot
measure x and p simultaneously.

240
00:13:40,490 --> 00:13:43,440
These are non-commuting
variables.

241
00:13:43,440 --> 00:13:50,860
So therefore, what happens is,
if you now define a phase space

242
00:13:50,860 --> 00:13:54,370
function, which is done in
quantum mechanics textbooks,

243
00:13:54,370 --> 00:13:58,050
you can actually do it
in three different ways.

244
00:13:58,050 --> 00:14:05,300
And the three different
ways are Q, P, and W.

245
00:14:05,300 --> 00:14:07,740
The definition--
I don't want to go

246
00:14:07,740 --> 00:14:10,190
through the mathematical
subtleties-- the definition

247
00:14:10,190 --> 00:14:15,210
of those functions involves
the operator definition, a

248
00:14:15,210 --> 00:14:19,970
and a dagger, or,
which is equivalent,

249
00:14:19,970 --> 00:14:21,930
the x and p operator.

250
00:14:25,080 --> 00:14:30,120
If you define something in units
of x and p or a and a dagger,

251
00:14:30,120 --> 00:14:33,120
you can have a product
which is fully symmetric i

252
00:14:33,120 --> 00:14:39,220
in the ordering of x p, which
is anti-normal or normal.

253
00:14:39,220 --> 00:14:41,990
So in other words,
the order matters.

254
00:14:45,070 --> 00:14:47,850
And you have three choices.

255
00:14:47,850 --> 00:14:54,080
In an operator product, you can
have symmetric ordering, which

256
00:14:54,080 --> 00:14:58,310
means not x p, but x p
plus p x, then it's over 2.

257
00:14:58,310 --> 00:15:00,900
That's symmetric.

258
00:15:00,900 --> 00:15:04,660
You can have an ordering
which is called normal

259
00:15:04,660 --> 00:15:06,315
and one which is
called anti-normal.

260
00:15:14,430 --> 00:15:22,695
Ordering of operators in the
operator base definition.

261
00:15:33,930 --> 00:15:37,430
Of course, if you
have three choices,

262
00:15:37,430 --> 00:15:39,240
you would say, Which
one is the best?

263
00:15:39,240 --> 00:15:40,840
Which one is the winner?

264
00:15:40,840 --> 00:15:44,490
But the fact is all three
have their advantages

265
00:15:44,490 --> 00:15:45,240
and disadvantages.

266
00:15:55,520 --> 00:16:01,410
So they all have
pluses and minuses.

267
00:16:01,410 --> 00:16:06,630
The reason why I picked
for the course Q of alpha

268
00:16:06,630 --> 00:16:09,900
is that it's a real probability,
it's always positive.

269
00:16:15,060 --> 00:16:19,520
It is a diagonal matrix element
of a statistical operator,

270
00:16:19,520 --> 00:16:21,160
and this has to be positive.

271
00:16:21,160 --> 00:16:23,260
So it's a real probability.

272
00:16:23,260 --> 00:16:30,100
The other guys, P of alpha,
can be positive or negative.

273
00:16:30,100 --> 00:16:33,230
And also, W of alpha can
be positive or negative.

274
00:16:41,420 --> 00:16:44,540
So if you use a P
alpha distribution

275
00:16:44,540 --> 00:16:56,340
statistical operator, it
can be written like this.

276
00:16:56,340 --> 00:17:07,680
And as a result, the coherent
state is now not this Gaussian.

277
00:17:07,680 --> 00:17:09,579
It doesn't have this
Gaussian distribution

278
00:17:09,579 --> 00:17:11,099
as a course of probability.

279
00:17:11,099 --> 00:17:13,180
It's what you want--
what maybe some of you

280
00:17:13,180 --> 00:17:16,000
wanted to see-- oh, by the
way, it's a delta function.

281
00:17:16,000 --> 00:17:18,680
The probability of the
coherent state alpha

282
00:17:18,680 --> 00:17:24,520
has a delta function peak at
alpha, which is sort of nice.

283
00:17:24,520 --> 00:17:30,350
And the number state is not
a ring of a finite radius.

284
00:17:30,350 --> 00:17:32,600
I just mentioned to
you the number state.

285
00:17:32,600 --> 00:17:35,800
You would naively expect
the energy is sharp.

286
00:17:35,800 --> 00:17:38,080
The square root of the
energy's electric field,

287
00:17:38,080 --> 00:17:39,970
shouldn't it be sharp?

288
00:17:39,970 --> 00:17:41,980
And indeed, it is sharp.

289
00:17:41,980 --> 00:17:44,390
It's actually worse
than a delta function.

290
00:17:44,390 --> 00:17:46,125
It's a derivative
of a delta function.

291
00:17:51,390 --> 00:17:55,690
But at least here,
in the probability P,

292
00:17:55,690 --> 00:17:59,470
which is also called
the Glauber-Sudarshan P

293
00:17:59,470 --> 00:18:05,890
representation, you
get the delta function,

294
00:18:05,890 --> 00:18:09,205
which may be very natural
for certain purposes.

295
00:18:09,205 --> 00:18:12,189
AUDIENCE: Since the [INAUDIBLE]
and you can express one

296
00:18:12,189 --> 00:18:15,300
in terms of the
other, is that unique?

297
00:18:15,300 --> 00:18:19,810
PROFESSOR: It is unique by
some symmetry choice here.

298
00:18:19,810 --> 00:18:30,240
So that's the advantage
of it, that by-- I'm not

299
00:18:30,240 --> 00:18:31,460
going into the mathematics.

300
00:18:31,460 --> 00:18:32,410
I'm not giving
you the definition

301
00:18:32,410 --> 00:18:33,618
but the way how it's defined.

302
00:18:33,618 --> 00:18:34,180
It's unique.

303
00:18:34,180 --> 00:18:35,760
It can be written in such a way.

304
00:18:35,760 --> 00:18:38,220
And the way how the
quasi-probabilities

305
00:18:38,220 --> 00:18:42,100
P-- the P representation--
is defined,

306
00:18:42,100 --> 00:18:44,110
you get a delta function
at the coherent state.

307
00:18:46,770 --> 00:18:50,860
And finally, W stands
for Wigner distribution.

308
00:18:55,480 --> 00:19:01,570
And the Wigner distribution
is something you actually

309
00:19:01,570 --> 00:19:03,980
find in most quantum
mechanics textbooks.

310
00:19:03,980 --> 00:19:07,300
The Q and P distribution are
more common in quantum optics.

311
00:19:07,300 --> 00:19:10,490
But the Wigner distribution
has the advantage

312
00:19:10,490 --> 00:19:17,380
that the projection
on the x- and y-axes

313
00:19:17,380 --> 00:19:26,950
are indeed psi of x
squared, psi of p squared.

314
00:19:26,950 --> 00:19:30,760
So you get actually the x
wave function and the p wave

315
00:19:30,760 --> 00:19:32,580
function.

316
00:19:32,580 --> 00:19:35,950
So now, as a full
disclaimer, if you

317
00:19:35,950 --> 00:19:40,600
want the electric field, which
is the momentum of the harmonic

318
00:19:40,600 --> 00:19:44,680
oscillator, which is the
electromagnetic field, what you

319
00:19:44,680 --> 00:19:49,030
really want to project is the
W function, because for the W

320
00:19:49,030 --> 00:19:51,250
function, the Wigner
function, the projection

321
00:19:51,250 --> 00:19:53,060
is exactly the
momentum distribution

322
00:19:53,060 --> 00:19:56,240
in the electric field.

323
00:19:56,240 --> 00:20:06,500
So the Wigner distribution is
closest to the classic phase

324
00:20:06,500 --> 00:20:09,560
space distribution,
as close as you

325
00:20:09,560 --> 00:20:13,030
come without
violating commutators.

326
00:20:18,300 --> 00:20:20,600
But of course it
has a disadvantage

327
00:20:20,600 --> 00:20:22,850
that it has negative values.

328
00:20:22,850 --> 00:20:25,710
And try to explain to
your next neighbor what

329
00:20:25,710 --> 00:20:28,990
is a negative probability.

330
00:20:28,990 --> 00:20:34,500
Some people actually see
negative probabilities spring

331
00:20:34,500 --> 00:20:36,420
out the non-classical character.

332
00:20:36,420 --> 00:20:38,990
And I know in our
field, in AMO physics,

333
00:20:38,990 --> 00:20:41,380
a few years ago, in
my lifetime, there

334
00:20:41,380 --> 00:20:45,010
were really attempts to use
quantum state homography

335
00:20:45,010 --> 00:20:48,950
and measure for the first time
a negative Wigner distribution

336
00:20:48,950 --> 00:20:50,530
and show that to the world.

337
00:20:50,530 --> 00:20:53,520
So that meant something
to a lot of people.

338
00:20:53,520 --> 00:21:00,300
Anyway, you can read about it
in quantum physics textbooks.

339
00:21:00,300 --> 00:21:02,690
All I want you to know
is to know about it,

340
00:21:02,690 --> 00:21:05,947
but then also just relax.

341
00:21:05,947 --> 00:21:08,030
In the bigger picture, all
the three distributions

342
00:21:08,030 --> 00:21:09,340
are the same.

343
00:21:09,340 --> 00:21:12,270
It's more sort of on the
level of whether something

344
00:21:12,270 --> 00:21:14,660
is a delta function
or has widths unity.

345
00:21:14,660 --> 00:21:17,630
So on the small
scale, it matters.

346
00:21:17,630 --> 00:21:20,450
But if you map out
something on a bigger scale,

347
00:21:20,450 --> 00:21:24,390
they are all related
to each other.

348
00:21:24,390 --> 00:21:26,830
And for the rest off
today and the next class,

349
00:21:26,830 --> 00:21:29,520
when I show you those phase
space distribution, when

350
00:21:29,520 --> 00:21:33,240
I say I project onto
the vertical axis

351
00:21:33,240 --> 00:21:36,190
to get the electric
field, I'm not

352
00:21:36,190 --> 00:21:39,340
completely rigorous which
of the three functions

353
00:21:39,340 --> 00:21:40,280
I've really chosen.

354
00:21:43,687 --> 00:21:44,270
Any questions?

355
00:21:57,260 --> 00:21:57,760
OK.

356
00:22:00,670 --> 00:22:03,070
So I promised you--
and this is what

357
00:22:03,070 --> 00:22:07,300
we're going to do now is
we're want to understand

358
00:22:07,300 --> 00:22:10,723
in more depth the fluctuations.

359
00:22:14,640 --> 00:22:19,550
And in particular,
I want to show you

360
00:22:19,550 --> 00:22:24,610
that coherent states are
minimum uncertainty states.

361
00:22:37,620 --> 00:22:41,250
So by identifying the
vertical axis with p,

362
00:22:41,250 --> 00:22:46,230
the horizontal axis with
x, we immediately expect

363
00:22:46,230 --> 00:22:50,260
that we find a result related
the Heisenberg uncertainty

364
00:22:50,260 --> 00:22:55,480
principle, which says sit the
widths in P and the widths in Q

365
00:22:55,480 --> 00:22:59,780
has to be larger
than h-bar over 2.

366
00:22:59,780 --> 00:23:10,930
And we are mainly talking with
light about the photon creation

367
00:23:10,930 --> 00:23:14,210
and annihilation operator.

368
00:23:14,210 --> 00:23:21,640
But, just as a reminder, the
momentum and position operator

369
00:23:21,640 --> 00:23:28,906
are symmetric and
anti-symmetric combinations.

370
00:23:36,280 --> 00:23:47,540
So let me just call
those [? prefecters ?]

371
00:23:47,540 --> 00:23:51,843
Q naught and this one P naught.

372
00:23:59,100 --> 00:24:06,230
Well, we can then
immediately, by just

373
00:24:06,230 --> 00:24:11,190
using elementary
commutator, calculate

374
00:24:11,190 --> 00:24:17,276
what are the expectation values
in a coherent state for P,

375
00:24:17,276 --> 00:24:20,330
P squared, Q, and Q squared.

376
00:24:24,380 --> 00:24:38,550
For P, it is-- the P
operator is a dagger minus a.

377
00:24:38,550 --> 00:24:42,270
If we act with a on
alpha, we get alpha,

378
00:24:42,270 --> 00:24:43,880
because alpha-- the
coherent state--

379
00:24:43,880 --> 00:24:47,060
is an eigenstate of alpha.

380
00:24:47,060 --> 00:24:49,850
Now, with a dagger we act
on the left hand side.

381
00:24:49,850 --> 00:24:51,460
And we get alpha star.

382
00:24:59,450 --> 00:25:01,330
Well, with P squared
and Q squared,

383
00:25:01,330 --> 00:25:04,750
you have to use one
or two more steps

384
00:25:04,750 --> 00:25:06,550
to get rid of the products.

385
00:25:06,550 --> 00:25:10,350
But ultimately, you can
express space all that just

386
00:25:10,350 --> 00:25:17,400
in powers of alpha, alpha
squared, alpha star, and such.

387
00:25:17,400 --> 00:25:24,040
So what we want is
we want to find out

388
00:25:24,040 --> 00:25:32,210
what are the fluctuations delta
Q, delta P. Just as a reminder,

389
00:25:32,210 --> 00:25:40,190
delta Q squared is of course Q
squared average minus Q average

390
00:25:40,190 --> 00:25:41,010
squared.

391
00:25:41,010 --> 00:25:43,772
And Q squared
average and Q squared

392
00:25:43,772 --> 00:25:45,105
is what we have just calculated.

393
00:25:55,330 --> 00:25:58,370
Actually, let me leave it
in as a reminder for you--

394
00:25:58,370 --> 00:26:03,890
Q squared minus Q average
squared, the square root of it.

395
00:26:03,890 --> 00:26:11,990
And what we find is that it's
square root h-bar over 2 omega

396
00:26:11,990 --> 00:26:17,710
and, for delta P, it is square
root h-bar over omega 2.

397
00:26:17,710 --> 00:26:23,190
And what we verify, that
indeed, for a coherent state,

398
00:26:23,190 --> 00:26:26,635
the product of the 2 is
Heisenberg uncertainty limit.

399
00:26:31,110 --> 00:26:44,855
So therefore, coherent states
are minimum uncertainty states.

400
00:27:00,360 --> 00:27:02,560
And maybe, Colin, to
address your question,

401
00:27:02,560 --> 00:27:04,590
I could imagine that
coherent states were maybe

402
00:27:04,590 --> 00:27:07,650
invented by simply saying, We
have an harmonic oscillator

403
00:27:07,650 --> 00:27:08,200
data.

404
00:27:08,200 --> 00:27:12,480
We want to find the
minimum uncertainty

405
00:27:12,480 --> 00:27:17,130
states for which the
uncertainty in x and p,

406
00:27:17,130 --> 00:27:21,660
when expressed in natural
units, is the same.

407
00:27:21,660 --> 00:27:23,740
In other words,
the coherent state

408
00:27:23,740 --> 00:27:27,780
is the solution to the
following question.

409
00:27:27,780 --> 00:27:36,470
If you plot the
quasi-probability distribution,

410
00:27:36,470 --> 00:27:40,980
you give yourself
an uncertainty area.

411
00:27:40,980 --> 00:27:49,060
One horizontal
uncertainty is delta Q.

412
00:27:49,060 --> 00:27:55,500
The vertical
uncertainty is delta P.

413
00:27:55,500 --> 00:28:00,570
And the minimum
uncertainty state

414
00:28:00,570 --> 00:28:05,080
means that the area of the
2 uncertainties, the area

415
00:28:05,080 --> 00:28:18,040
of the shaded region, delta P
times delta Q is h-bar over 2.

416
00:28:18,040 --> 00:28:31,094
Or if I want to
rewrite that in-- Yes,

417
00:28:31,094 --> 00:28:37,210
in the real part of alpha
times the uncertainty

418
00:28:37,210 --> 00:28:41,810
in the imaginary part of alpha
for the quasi-probability,

419
00:28:41,810 --> 00:28:43,100
then this is 1/4.

420
00:28:57,650 --> 00:28:58,150
OK.

421
00:28:58,150 --> 00:29:04,460
So what we have learned
is that-- just one second.

422
00:29:04,460 --> 00:29:12,480
We have learned about one way
to characterize uncertainties--

423
00:29:12,480 --> 00:29:16,660
the quantumness of the
electromagnetic field--

424
00:29:16,660 --> 00:29:22,950
by looking at the uncertainties
in the x and p variable.

425
00:29:22,950 --> 00:29:26,230
And that led us to minimum
uncertainty states.

426
00:29:26,230 --> 00:29:28,830
Now, I want to now
introduce to you

427
00:29:28,830 --> 00:29:35,930
two other ways of characterizing
the uncertainty of quantum

428
00:29:35,930 --> 00:29:37,300
states of light.

429
00:29:37,300 --> 00:29:43,370
One is we can ask, What are
the fluctuations in the photon

430
00:29:43,370 --> 00:29:44,490
number?

431
00:29:44,490 --> 00:29:45,950
Right now, we have
asked, What are

432
00:29:45,950 --> 00:29:48,274
the fluctuations in
the electric field?

433
00:29:48,274 --> 00:29:49,690
But the next
question is, What are

434
00:29:49,690 --> 00:29:51,400
the fluctuations in
the photon number?

435
00:29:54,200 --> 00:29:56,490
Or we can ask, what
are fluctuations

436
00:29:56,490 --> 00:29:59,950
in the intensity when
we measure the intensity

437
00:29:59,950 --> 00:30:02,732
of the electromagnetic field?

438
00:30:02,732 --> 00:30:03,440
So let's do that.

439
00:30:08,890 --> 00:30:22,265
The fluctuations-- It
seems I've lost my lines.

440
00:30:34,260 --> 00:30:35,730
OK.

441
00:30:35,730 --> 00:30:41,950
So the fluctuations
of the intensity

442
00:30:41,950 --> 00:30:45,380
are usually expressed by the
second order temporal coherence

443
00:30:45,380 --> 00:30:47,039
function.

444
00:30:47,039 --> 00:30:48,580
That's what we want
to introduce now.

445
00:31:09,810 --> 00:31:21,550
Yes, this is the second
order temporal correlation

446
00:31:21,550 --> 00:31:23,570
or coherence function.

447
00:31:28,880 --> 00:31:30,950
What I'm always
encountering in this course

448
00:31:30,950 --> 00:31:33,720
is I would just like
to immediately tell you

449
00:31:33,720 --> 00:31:37,110
how it is defined in terms
of a's and a daggers.

450
00:31:37,110 --> 00:31:37,950
It's simple.

451
00:31:37,950 --> 00:31:39,130
It's quantum mechanical.

452
00:31:39,130 --> 00:31:40,320
It's exact.

453
00:31:40,320 --> 00:31:46,190
But I always feel that if
you want to really appreciate

454
00:31:46,190 --> 00:31:49,560
the quantum character, you have
to know the classic description

455
00:31:49,560 --> 00:31:50,710
first.

456
00:31:50,710 --> 00:31:55,270
So I want to first tell you what
is the second order coherence

457
00:31:55,270 --> 00:31:57,540
function for
classical light, which

458
00:31:57,540 --> 00:31:59,690
has a classic description.

459
00:31:59,690 --> 00:32:03,394
And then you'll see
what is the difference

460
00:32:03,394 --> 00:32:04,560
for quantum states of light.

461
00:32:07,840 --> 00:32:12,040
So the classical
description is you

462
00:32:12,040 --> 00:32:14,785
measure the intensity of light.

463
00:32:14,785 --> 00:32:16,020
You're just sitting here.

464
00:32:16,020 --> 00:32:18,870
You receive light
from a light bulb.

465
00:32:18,870 --> 00:32:21,180
You measure the
intensity at time t.

466
00:32:21,180 --> 00:32:27,620
You measure the intensity
at time tau later.

467
00:32:27,620 --> 00:32:28,965
And then you form the product.

468
00:32:32,380 --> 00:32:37,730
And you normalize it by the
average intensity squared.

469
00:32:37,730 --> 00:32:41,580
So if tau equals 0,
it's nothing else

470
00:32:41,580 --> 00:32:45,830
than the intensity
squared average divided

471
00:32:45,830 --> 00:32:47,910
by the average of the
intensity squared.

472
00:33:00,100 --> 00:33:07,360
So this is the classical
definition, g2 of tau.

473
00:33:07,360 --> 00:33:11,770
And I've left the
proof to the homework

474
00:33:11,770 --> 00:33:24,560
to show that the classical g2
of tau is always larger than 1.

475
00:33:24,560 --> 00:33:29,000
But it's pretty much what
you show is for tau equal 0,

476
00:33:29,000 --> 00:33:32,680
it means I squared
average is larger

477
00:33:32,680 --> 00:33:35,770
than I average squared, which
is, of course, always the case.

478
00:33:35,770 --> 00:33:40,230
But you can show that this is
also the case for finite tau.

479
00:33:40,230 --> 00:33:48,150
So quantum mechanically, we will
see that the g2 function is not

480
00:33:48,150 --> 00:33:52,060
necessarily larger than 1,
it can be smaller than 1.

481
00:33:52,060 --> 00:33:53,700
And that's actually
an interesting--

482
00:33:53,700 --> 00:33:57,070
you can see-- litmus
test for the quantumness.

483
00:33:57,070 --> 00:34:02,770
If you generate states of the
electromagnetic field-- Fock

484
00:34:02,770 --> 00:34:05,560
states, photon number states--
we see that in a moment.

485
00:34:05,560 --> 00:34:09,350
And they have a g2 of tau
which is smaller than 1.

486
00:34:09,350 --> 00:34:12,749
You know it is not
possible in any way

487
00:34:12,749 --> 00:34:18,900
to associate an intensity
of the electromagnetic field

488
00:34:18,900 --> 00:34:21,230
with that photon
state, because whenever

489
00:34:21,230 --> 00:34:24,600
you can associate a
classical intensity with it

490
00:34:24,600 --> 00:34:28,409
and use a classical intensity
to calculate the second order

491
00:34:28,409 --> 00:34:30,210
correlation function,
you get something

492
00:34:30,210 --> 00:34:32,030
which is larger than 1.

493
00:34:32,030 --> 00:34:34,440
So often therefore
again, when people

494
00:34:34,440 --> 00:34:38,929
want to show we really have now
non-classical photon states,

495
00:34:38,929 --> 00:34:44,020
they show g2 of tau
is smaller than 1.

496
00:34:44,020 --> 00:34:46,469
This is similar to
what I said before when

497
00:34:46,469 --> 00:34:51,040
you want to show that you
have non-classical light,

498
00:34:51,040 --> 00:34:53,780
you do quantum state
homography, measure the Wigner

499
00:34:53,780 --> 00:34:56,540
distribution, and show that
the Wigner distribution has

500
00:34:56,540 --> 00:34:58,360
negative quasi-probabilities.

501
00:34:58,360 --> 00:35:01,590
That's again something which
is classically not possible.

502
00:35:01,590 --> 00:35:06,407
It's only possible if you have
a truly non-classical state.

503
00:35:06,407 --> 00:35:08,365
AUDIENCE: So in the
definition of g2 classical,

504
00:35:08,365 --> 00:35:11,732
the expectation values would
need averaging over time, not

505
00:35:11,732 --> 00:35:15,797
over ensembles of I of t?

506
00:35:15,797 --> 00:35:16,380
PROFESSOR: OK.

507
00:35:19,280 --> 00:35:21,240
You can average over ensembles.

508
00:35:21,240 --> 00:35:25,660
So you can have 1,000 light
bulbs, switch them all on,

509
00:35:25,660 --> 00:35:28,010
and then measure at
a certain time z.

510
00:35:28,010 --> 00:35:31,190
But what we assume here
is that the light bulb

511
00:35:31,190 --> 00:35:32,580
is on continuously.

512
00:35:32,580 --> 00:35:34,490
So things don't
really depend on t.

513
00:35:38,620 --> 00:35:41,990
I've implicitly taken care of
it by defining the g2 function

514
00:35:41,990 --> 00:35:44,520
only of tau and
not of t and tau.

515
00:35:44,520 --> 00:35:48,560
You assume any time is the same,
because it's a distribution,

516
00:35:48,560 --> 00:35:51,530
it's this ensemble
in steady state.

517
00:35:51,530 --> 00:35:54,920
But it's sort of the same
story again and again.

518
00:35:54,920 --> 00:35:58,650
In classical physics, you can
determine an ensemble average

519
00:35:58,650 --> 00:36:01,640
by taking an ergotic
system and observing it

520
00:36:01,640 --> 00:36:03,250
at many, many times.

521
00:36:03,250 --> 00:36:06,410
And then the idea is
that one system as time

522
00:36:06,410 --> 00:36:09,880
goes by will sample
all possible states.

523
00:36:09,880 --> 00:36:13,350
Or you can prepare many,
many identical systems

524
00:36:13,350 --> 00:36:17,239
and do more of what is
an ensemble average.

525
00:36:17,239 --> 00:36:18,780
So in other words,
you would actually

526
00:36:18,780 --> 00:36:21,740
think, if you switch on a
light bulb with a stable power

527
00:36:21,740 --> 00:36:24,310
supply, that the light
emitted by the light bulb

528
00:36:24,310 --> 00:36:29,570
will go through all possible
quantum states as time evolves.

529
00:36:29,570 --> 00:36:31,580
And therefore, the
temporal average

530
00:36:31,580 --> 00:36:33,080
is equal to the
ensemble average.

531
00:36:55,380 --> 00:37:01,073
So how do we generalize
that to quantum mechanics?

532
00:37:04,360 --> 00:37:11,610
Well, one possibility
would be that-- OK.

533
00:37:11,610 --> 00:37:13,299
First of all, for
quantum mechanics,

534
00:37:13,299 --> 00:37:14,340
we have to use operators.

535
00:37:23,470 --> 00:37:31,910
And one choice would
be that, well, you

536
00:37:31,910 --> 00:37:37,420
say the intensity of
the electromagnetic wave

537
00:37:37,420 --> 00:37:40,600
is proportional to the
electric field squared.

538
00:37:40,600 --> 00:37:44,770
And you want to use the operator
for the electric field squared.

539
00:37:44,770 --> 00:37:48,090
Now, there is a little
bit of a problem,

540
00:37:48,090 --> 00:37:51,340
because what we really
mean quantum mechanically

541
00:37:51,340 --> 00:37:55,790
by g2 of tau is we
measure the intensity now

542
00:37:55,790 --> 00:37:57,210
and a little bit later.

543
00:37:57,210 --> 00:38:01,520
But measuring the intensity
really means absorbing photons,

544
00:38:01,520 --> 00:38:05,040
because the only way how you can
measure the intensity of light

545
00:38:05,040 --> 00:38:07,280
is with a photomultiplier.

546
00:38:07,280 --> 00:38:09,790
It makes click, the
photon is absorbed.

547
00:38:09,790 --> 00:38:14,280
And this is not fully described
by the electric field.

548
00:38:14,280 --> 00:38:17,240
Just assume you
have no photon, you

549
00:38:17,240 --> 00:38:20,600
are in the zero-point state
of your harmonic oscillator,

550
00:38:20,600 --> 00:38:24,760
and your E squared has
zero-point fluctuations.

551
00:38:24,760 --> 00:38:30,640
So what is more closely
related to an experiment how

552
00:38:30,640 --> 00:38:34,210
you measure the
correlation function is

553
00:38:34,210 --> 00:38:39,640
you want to look
at something else,

554
00:38:39,640 --> 00:38:55,180
namely at the probability
of absorbing 2 photons.

555
00:39:01,940 --> 00:39:03,690
So you start with
an initial state.

556
00:39:07,140 --> 00:39:09,750
You annihilate 2 photons.

557
00:39:09,750 --> 00:39:12,570
And then you have a final state.

558
00:39:12,570 --> 00:39:15,220
But if you're only
interested in what

559
00:39:15,220 --> 00:39:17,080
is the probability you
want to characterize

560
00:39:17,080 --> 00:39:22,140
your initial state, you may want
to sum over all final state.

561
00:39:22,140 --> 00:39:23,890
And this is your
total probability

562
00:39:23,890 --> 00:39:28,250
that you can absorb 2 photons
out of an initial state.

563
00:39:28,250 --> 00:39:33,110
But since you're in sum
now over all final state,

564
00:39:33,110 --> 00:39:40,290
this turns into--
Now, I could put

565
00:39:40,290 --> 00:39:45,970
for you as in the final
state is a complete basis.

566
00:39:45,970 --> 00:39:47,405
But I can also take it out.

567
00:39:53,650 --> 00:39:56,520
So this is what we get.

568
00:39:56,520 --> 00:40:02,200
So this suggests
that experiments

569
00:40:02,200 --> 00:40:07,630
where we look at two subsequent
clicks of a photomultiplier

570
00:40:07,630 --> 00:40:10,960
where we determine the
photon correlation,

571
00:40:10,960 --> 00:40:17,090
that this is measuring a
correlation function, which,

572
00:40:17,090 --> 00:40:21,060
for quantum states of
light, should be defined

573
00:40:21,060 --> 00:40:27,000
as the expectation value of
a dagger, a dagger, a, a.

574
00:40:29,770 --> 00:40:35,870
And now we have to normalize
by the probability squared

575
00:40:35,870 --> 00:40:43,735
of absorbing 1 photon, which
is a dagger a expectation

576
00:40:43,735 --> 00:40:44,485
value squared.

577
00:40:48,940 --> 00:40:51,820
I just tried to be a
little bit too motivated.

578
00:40:51,820 --> 00:40:53,620
In many, many textbooks,
the discussion

579
00:40:53,620 --> 00:40:54,995
would start with
this expression.

580
00:40:54,995 --> 00:40:57,090
You ask, Where
does it come from?

581
00:40:57,090 --> 00:41:01,460
And you realize yes,
for some measurement

582
00:41:01,460 --> 00:41:03,980
with photomultipliers,
that's what you measure.

583
00:41:03,980 --> 00:41:11,060
But I wanted to show you how
it is related to the intensity

584
00:41:11,060 --> 00:41:13,440
correlation function
defined classically.

585
00:41:13,440 --> 00:41:14,246
Cody.

586
00:41:14,246 --> 00:41:15,662
AUDIENCE: But this
looks like it's

587
00:41:15,662 --> 00:41:18,750
constant in tau,
which doesn't really

588
00:41:18,750 --> 00:41:22,410
make sense intuitively to me,
because at least classically,

589
00:41:22,410 --> 00:41:25,790
it shouldn't be constant in tau.

590
00:41:25,790 --> 00:41:27,120
PROFESSOR: OK.

591
00:41:27,120 --> 00:41:28,750
That's the next
thing I wanted to say

592
00:41:28,750 --> 00:41:33,580
is I've swept here-- I
just wanted to give you

593
00:41:33,580 --> 00:41:36,720
the structure of the
operators and not

594
00:41:36,720 --> 00:41:39,250
get distracted by
discussing time.

595
00:41:39,250 --> 00:41:41,510
I've dropped the
time argument here.

596
00:41:41,510 --> 00:41:52,550
But the fact is that as
long as we limit ourselves

597
00:41:52,550 --> 00:41:55,070
to a single mode of the
electromagnetic field,

598
00:41:55,070 --> 00:41:59,910
a single harmonic oscillator,
things are independent of tau.

599
00:41:59,910 --> 00:42:05,110
In other words, g2 of
tau equals g2 of 0,

600
00:42:05,110 --> 00:42:07,790
as long as we're dealing
with single mode light.

601
00:42:07,790 --> 00:42:11,480
You can actually say--
and that sort of tells you

602
00:42:11,480 --> 00:42:15,750
where the fluctuations
come-- if g2 of tau changes,

603
00:42:15,750 --> 00:42:18,530
it comes because you
have several modes

604
00:42:18,530 --> 00:42:21,990
of the electromagnetic field,
which, as a function of time,

605
00:42:21,990 --> 00:42:24,820
constructively and
destructively interfere.

606
00:42:24,820 --> 00:42:27,790
But if you have a single
mode, in a single mode--

607
00:42:27,790 --> 00:42:29,590
and what is a single mode?

608
00:42:29,590 --> 00:42:31,190
It's just a sine wave.

609
00:42:31,190 --> 00:42:33,330
And nothing happens
as a function of time.

610
00:42:33,330 --> 00:42:35,200
It's constant.

611
00:42:35,200 --> 00:42:37,334
AUDIENCE: So both
modes go inside.

612
00:42:44,280 --> 00:42:47,030
PROFESSOR: I'm squeezing a
textbook of quantum optics

613
00:42:47,030 --> 00:42:48,550
into two classes.

614
00:42:48,550 --> 00:42:52,170
And I want to give you the
ideas and the concepts.

615
00:42:52,170 --> 00:42:56,550
What I've sort of mixed
up here deliberately

616
00:42:56,550 --> 00:42:59,830
is I've given you the classic
definition of the intensity

617
00:42:59,830 --> 00:43:04,150
correlation function, which is
the famous Hanbury Brown Twiss

618
00:43:04,150 --> 00:43:06,640
experiment, and
used correlations

619
00:43:06,640 --> 00:43:08,530
as a function of time.

620
00:43:08,530 --> 00:43:10,580
Then I've motivated
how this should

621
00:43:10,580 --> 00:43:13,520
be defined for quantum
states of light.

622
00:43:13,520 --> 00:43:16,730
But when I transitioned to
quantum states of light,

623
00:43:16,730 --> 00:43:21,290
I decided to deal with
only one mode of the light.

624
00:43:21,290 --> 00:43:24,840
We should now sum
over-- I should

625
00:43:24,840 --> 00:43:28,600
put double or triple
indices on all the alphas

626
00:43:28,600 --> 00:43:31,090
for polarization,
for spatial modes,

627
00:43:31,090 --> 00:43:34,600
for different frequencies,
and we sum over all of them.

628
00:43:34,600 --> 00:43:38,470
But instead what I did is
I wanted to just show you

629
00:43:38,470 --> 00:43:41,704
the simple case-- and I
think you will be thankful

630
00:43:41,704 --> 00:43:43,370
for that in your
homework, that you only

631
00:43:43,370 --> 00:43:45,010
have to deal with
the simple case--

632
00:43:45,010 --> 00:43:47,410
that you only have to
look at a's and a daggers

633
00:43:47,410 --> 00:43:50,400
in the operator algebra
for single mode,

634
00:43:50,400 --> 00:43:52,700
for single harmonic oscillator.

635
00:43:52,700 --> 00:43:54,680
But what we lose,
so to speak here,

636
00:43:54,680 --> 00:43:57,530
is there is nothing
interesting going on in time.

637
00:43:57,530 --> 00:44:01,170
I've already told you that,
for a single mode, all

638
00:44:01,170 --> 00:44:04,630
of these quasi-probabilities,
they just rotate in a circle.

639
00:44:04,630 --> 00:44:07,160
So the time evolution
of the system

640
00:44:07,160 --> 00:44:09,730
you're describing right
now is completely boring.

641
00:44:09,730 --> 00:44:11,340
It's really a rotation.

642
00:44:11,340 --> 00:44:13,577
And if you would rotate
your head at omega,

643
00:44:13,577 --> 00:44:14,410
nothing will happen.

644
00:44:14,410 --> 00:44:16,840
And this is exactly
what you see here.

645
00:44:16,840 --> 00:44:22,200
So you find everything you
want to for coherence time.

646
00:44:22,200 --> 00:44:26,010
Coherence time is the time for
2 modes to get out of phase.

647
00:44:26,010 --> 00:44:29,560
But if you have 1 mode,
there is no coherence time.

648
00:44:29,560 --> 00:44:32,420
And when you find,
for classical light,

649
00:44:32,420 --> 00:44:37,880
that the g2 function has a peak
which decays with the time,

650
00:44:37,880 --> 00:44:41,030
it is the time for modes
to get out of phase.

651
00:44:41,030 --> 00:44:44,690
But in a single mode
picture, this is absent.

652
00:44:44,690 --> 00:44:48,600
Anyway, what is important now
for the discussion of quantum

653
00:44:48,600 --> 00:44:51,460
character of light
is really, we find

654
00:44:51,460 --> 00:44:52,820
that in a single mode picture.

655
00:44:52,820 --> 00:44:56,400
So I want to show you
now-- or at least give you

656
00:44:56,400 --> 00:44:57,910
the summary of
the results, which

657
00:44:57,910 --> 00:45:00,240
can be very easily derived,
because the math is very

658
00:45:00,240 --> 00:45:02,860
simple of those
operators, that we are now

659
00:45:02,860 --> 00:45:04,260
with that definition.

660
00:45:04,260 --> 00:45:08,340
We have g2 functions, which
are no longer following

661
00:45:08,340 --> 00:45:11,490
the classic constraint that
g2 has to be larger than 1.

662
00:45:11,490 --> 00:45:14,370
We find g2 functions
which are smaller than 1.

663
00:45:14,370 --> 00:45:17,110
And this sort of
tells us now where

664
00:45:17,110 --> 00:45:21,890
do we find the most
non-classical behavior,

665
00:45:21,890 --> 00:45:24,750
namely when g2 of tau
is as small as possible.

666
00:45:33,180 --> 00:45:34,810
OK.

667
00:45:34,810 --> 00:45:39,420
In your homework, you
will show immediately

668
00:45:39,420 --> 00:45:46,935
that the g2 function is
related to number fluctuations.

669
00:45:51,690 --> 00:45:55,745
It's related to an average
and an n-squared average.

670
00:46:00,570 --> 00:46:02,810
Let me just write that down.

671
00:46:02,810 --> 00:46:06,290
It's independent of tau.

672
00:46:06,290 --> 00:46:11,180
And the reason is we have
now limited ourselves

673
00:46:11,180 --> 00:46:14,460
to just one mode of the
electromagnetic field.

674
00:46:27,771 --> 00:46:28,757
Fano factor.

675
00:46:36,190 --> 00:46:37,305
Just give me one--

676
00:46:41,190 --> 00:46:45,660
OK So we are back
to-- We started

677
00:46:45,660 --> 00:46:47,670
with intensity fluctuations.

678
00:46:47,670 --> 00:46:50,170
But for a single mode of
the electromagnetic field,

679
00:46:50,170 --> 00:46:52,110
we are back to photon numbers.

680
00:46:52,110 --> 00:46:56,400
So what we are now expressing
with the g2 function

681
00:46:56,400 --> 00:47:00,805
are, in other words, just
fluctuations of photon numbers.

682
00:47:05,700 --> 00:47:08,760
I want to in a minute
draw you a table.

683
00:47:08,760 --> 00:47:11,070
What are the fluctuations
in the photon number

684
00:47:11,070 --> 00:47:13,610
for the states we have
encountered-- the number

685
00:47:13,610 --> 00:47:16,430
state, the Fock state,
the coherent state,

686
00:47:16,430 --> 00:47:18,080
and the thermal state?

687
00:47:18,080 --> 00:47:25,710
And we want to characterize
those quantum states of light

688
00:47:25,710 --> 00:47:28,490
by the g2 function.

689
00:47:28,490 --> 00:47:31,660
And actually, I can drop to
time argument or set it to 0.

690
00:47:36,500 --> 00:47:38,460
But then we first
make a reference

691
00:47:38,460 --> 00:47:43,460
that g2 can now
be smaller than 1.

692
00:47:43,460 --> 00:47:48,600
So g2 for a single mode light
is nothing else than a function.

693
00:47:48,600 --> 00:47:51,700
When you know what is an
average and n squared average,

694
00:47:51,700 --> 00:47:54,300
you know your g2 function.

695
00:47:54,300 --> 00:47:57,950
There is another
quantity, which we often

696
00:47:57,950 --> 00:48:02,700
use to characterize the
fluctuations in the photon

697
00:48:02,700 --> 00:48:03,230
number.

698
00:48:03,230 --> 00:48:06,400
And this is called
the Fano factor.

699
00:48:06,400 --> 00:48:13,940
The Fano factor is-- wants
to compare the fluctuations.

700
00:48:13,940 --> 00:48:19,712
n square average minus
n average squared,

701
00:48:19,712 --> 00:48:20,836
these are the fluctuations.

702
00:48:24,280 --> 00:48:26,850
The classical
fluctuation-- well,

703
00:48:26,850 --> 00:48:31,750
I say classical, classical
in the simplest case--

704
00:48:31,750 --> 00:48:33,533
are Poissonian fluctuations.

705
00:48:36,260 --> 00:48:39,880
So maybe we want to
normalize the fluctuations

706
00:48:39,880 --> 00:48:42,100
by Poissonian fluctuation.

707
00:48:42,100 --> 00:48:48,610
So for Poissonian statistics,
what I just wrote down

708
00:48:48,610 --> 00:48:49,450
would be 1.

709
00:48:58,570 --> 00:49:04,870
And well, if you
now subtract 1, we

710
00:49:04,870 --> 00:49:07,680
have the situation that
the final factor, which

711
00:49:07,680 --> 00:49:10,470
is positive, is
super-Poissonian--

712
00:49:10,470 --> 00:49:12,430
more fluctuation
than Poissonian--

713
00:49:12,430 --> 00:49:15,150
and the negative final
factor is sub-Poissonian.

714
00:49:19,030 --> 00:49:24,670
So with those
definitions, we can now

715
00:49:24,670 --> 00:49:35,810
compare the different states
of light we have introduced.

716
00:49:39,480 --> 00:49:44,770
We started out with black-body
radiation, thermal radiation.

717
00:49:44,770 --> 00:49:49,320
We defined coherent states.

718
00:49:49,320 --> 00:49:53,150
And the harmonic oscillator
description naturally

719
00:49:53,150 --> 00:49:55,560
gave us harmonic
oscillator eigenstates,

720
00:49:55,560 --> 00:49:59,540
the number of states,
or Fock states.

721
00:49:59,540 --> 00:50:03,050
So there are three ways
to characterize it.

722
00:50:03,050 --> 00:50:07,790
They are all useful.

723
00:50:07,790 --> 00:50:10,550
One is we can look at n squared.

724
00:50:10,550 --> 00:50:13,250
We can calculate
the Fano factor.

725
00:50:13,250 --> 00:50:15,730
Or we can calculate g2 of 0.

726
00:50:18,830 --> 00:50:26,030
For the coherent
state-- remember,

727
00:50:26,030 --> 00:50:28,600
the coherent state
is as close as we

728
00:50:28,600 --> 00:50:31,180
can come quantum
mechanically to the ideal

729
00:50:31,180 --> 00:50:34,090
of a pure electromagnetic wave.

730
00:50:34,090 --> 00:50:37,930
It has a Fano factor of 0.

731
00:50:37,930 --> 00:50:42,280
This means it's Poissonian.

732
00:50:42,280 --> 00:50:46,970
The g2 function
is simply 1, which

733
00:50:46,970 --> 00:50:51,230
is the lowest classical limit.

734
00:50:51,230 --> 00:50:55,340
So those two tell you that a
coherent state is sometimes

735
00:50:55,340 --> 00:50:57,880
what you think-- what
comes out of a laser is

736
00:50:57,880 --> 00:51:01,060
an ideal electromagnetic
wave, which

737
00:51:01,060 --> 00:51:04,160
has no temporal fluctuations
in the intensity.

738
00:51:04,160 --> 00:51:06,660
Therefore, g2 is 1.

739
00:51:06,660 --> 00:51:10,930
And the photon number is
Poissonian distributed.

740
00:51:10,930 --> 00:51:16,440
And that means that n
squared is an average times 1

741
00:51:16,440 --> 00:51:18,060
plus an average.

742
00:51:18,060 --> 00:51:19,720
We actually
discussed it earlier.

743
00:51:22,450 --> 00:51:25,890
The thermal state
is quite different.

744
00:51:25,890 --> 00:51:29,330
If you use-- kind
of pluck together

745
00:51:29,330 --> 00:51:33,715
the results we have obtained,
it has a Fano factor of n-bar.

746
00:51:38,310 --> 00:51:40,560
So this is super-Poissonian.

747
00:51:43,210 --> 00:51:45,440
If the occupation
number n is large,

748
00:51:45,440 --> 00:51:47,460
you have fluctuations
which are much,

749
00:51:47,460 --> 00:51:50,500
much larger than
Poissonian fluctuation.

750
00:51:50,500 --> 00:51:53,630
The g2 function is 2.

751
00:51:53,630 --> 00:51:56,040
It's sometimes
called thermal light.

752
00:51:56,040 --> 00:51:59,500
Chaotic light has
a g2 function of 2.

753
00:51:59,500 --> 00:52:03,790
Laser light or coherent state
has a g2 function of one.

754
00:52:03,790 --> 00:52:10,010
And n square average is
n-bar 1 plus 2 n-bar.

755
00:52:14,050 --> 00:52:14,550
OK.

756
00:52:14,550 --> 00:52:17,780
But now finally, maybe
the most interesting state

757
00:52:17,780 --> 00:52:20,620
from the perspective of
non-classical light--

758
00:52:20,620 --> 00:52:24,110
of quantum light-- is
the photon number state.

759
00:52:24,110 --> 00:52:28,890
Well, for the photon number
state, the number of photons

760
00:52:28,890 --> 00:52:31,520
is an eigenvalue.

761
00:52:31,520 --> 00:52:37,980
Therefore, n squared average
is n average squared.

762
00:52:37,980 --> 00:52:42,050
The Fano factor is minus 1.

763
00:52:42,050 --> 00:52:45,910
Sub-Poissonian distribution.

764
00:52:45,910 --> 00:52:50,990
And the g2 function, which
classically cannot go below 1,

765
00:52:50,990 --> 00:52:55,280
is now n minus 1 over n.

766
00:52:55,280 --> 00:52:57,600
It is smaller than 1.

767
00:52:57,600 --> 00:53:03,650
And you see immediately that
the biggest violation for g2

768
00:53:03,650 --> 00:53:13,780
is to go to minus 1 for the
case of a single photon state.

769
00:53:19,670 --> 00:53:22,388
Any questions?

770
00:53:22,388 --> 00:53:24,868
AUDIENCE: Shouldn't it
be 0 for [INAUDIBLE]?

771
00:53:28,840 --> 00:53:30,240
PROFESSOR: The g2 function?

772
00:53:30,240 --> 00:53:34,230
No, if you put in-- wait.

773
00:53:46,226 --> 00:53:46,725
Yes.

774
00:53:49,545 --> 00:53:50,045
Gosh.

775
00:54:02,931 --> 00:54:03,680
I'll double-check.

776
00:54:03,680 --> 00:54:04,763
I'm a little bit confused.

777
00:54:04,763 --> 00:54:17,197
But I-- At least
it's consistent now.

778
00:54:17,197 --> 00:54:19,196
I'm not sure if it's
right, but it's consistent.

779
00:54:22,130 --> 00:54:25,120
By the way, if
there is a question,

780
00:54:25,120 --> 00:54:28,390
I sometimes make a
question mark in my notes.

781
00:54:28,390 --> 00:54:32,605
And when I post the
notes-- maybe not when

782
00:54:32,605 --> 00:54:33,980
I post the notes
on the next day,

783
00:54:33,980 --> 00:54:37,900
but a few days later-- the
question marks are eliminated.

784
00:54:37,900 --> 00:54:43,540
Just as an example,
last class, there

785
00:54:43,540 --> 00:54:48,080
was some relation
for which I didn't

786
00:54:48,080 --> 00:54:50,860
remember how the
derivation was done.

787
00:54:50,860 --> 00:54:53,710
When I post those notes
today, the derivation

788
00:54:53,710 --> 00:54:55,720
is now in the notes.

789
00:54:55,720 --> 00:54:57,310
I'm not announcing
it separately.

790
00:54:57,310 --> 00:54:59,910
But whenever I have a
question mark in class

791
00:54:59,910 --> 00:55:02,660
and I don't think it's
worth an extra announcement,

792
00:55:02,660 --> 00:55:04,775
it is fixed in the
posted notes and you

793
00:55:04,775 --> 00:55:05,900
find the information there.

794
00:55:08,720 --> 00:55:09,920
OK.

795
00:55:09,920 --> 00:55:14,230
So this is now drawing our
attention to the single photon.

796
00:55:20,490 --> 00:55:22,050
And this is our next subsection.

797
00:55:26,529 --> 00:55:27,820
This shouldn't come unexpected.

798
00:55:36,770 --> 00:55:40,200
If you want to emphasize the
difference between quantum

799
00:55:40,200 --> 00:55:43,950
light and classic light,
where does it come from?

800
00:55:43,950 --> 00:55:46,600
Well, it comes from the
quantization of light.

801
00:55:46,600 --> 00:55:48,640
It comes from the
photon character.

802
00:55:48,640 --> 00:55:51,480
It comes from the
effect that light is not

803
00:55:51,480 --> 00:55:56,390
a continuous stream of energy,
it comes quantized in photons.

804
00:55:56,390 --> 00:56:00,240
So the granularity of light
due to the photon character

805
00:56:00,240 --> 00:56:05,044
is, of course, most pronounced
for a single photon.

806
00:56:05,044 --> 00:56:08,150
For instance, when we
define the g2 function

807
00:56:08,150 --> 00:56:10,750
as a correlation of
detecting a photon

808
00:56:10,750 --> 00:56:13,110
and detecting
another photon, well,

809
00:56:13,110 --> 00:56:16,110
if you've only one photon,
you find one photon,

810
00:56:16,110 --> 00:56:19,460
and for the next photon, there
is no photon to be detected.

811
00:56:19,460 --> 00:56:23,220
So the probability of
detecting two photons is 0.

812
00:56:23,220 --> 00:56:26,410
And that only happens when you
go down to similar photons.

813
00:56:26,410 --> 00:56:31,210
So this is when
certain fluctuations

814
00:56:31,210 --> 00:56:37,470
are most pronounced,
because the energy is

815
00:56:37,470 --> 00:56:38,860
dependant on a singular photon.

816
00:56:41,370 --> 00:56:43,216
Let me first address
one misconception.

817
00:56:48,250 --> 00:56:53,260
You can say, Well, let's
just use a coherent state.

818
00:56:53,260 --> 00:56:57,980
And we talk about
the attenuator--

819
00:56:57,980 --> 00:57:01,040
the quantum attenuator
with all its operator

820
00:57:01,040 --> 00:57:05,090
beauty-- probably not today
but it the next lecture.

821
00:57:05,090 --> 00:57:08,752
But let me already sort
of prepare you for that.

822
00:57:08,752 --> 00:57:11,210
When you have a coherent state,
when you have a laser beam,

823
00:57:11,210 --> 00:57:12,790
you can put an attenuator.

824
00:57:12,790 --> 00:57:16,110
And your laser beam gets
weaker and weaker and weaker.

825
00:57:16,110 --> 00:57:17,840
But it stays a coherent state.

826
00:57:17,840 --> 00:57:20,490
And I will prove that
to you very soon.

827
00:57:20,490 --> 00:57:24,370
So you can now say that you
take your coherent state

828
00:57:24,370 --> 00:57:28,205
and you attenuate it down that
there is only one photon left.

829
00:57:28,205 --> 00:57:31,950
Is that a single photon state?

830
00:57:31,950 --> 00:57:33,480
The answer is no.

831
00:57:33,480 --> 00:57:35,990
It is an attenuated
coherent state.

832
00:57:35,990 --> 00:57:39,760
Coherent states, as I've just
shown you, are very classical.

833
00:57:39,760 --> 00:57:41,930
They've always
Poissonian distribution.

834
00:57:41,930 --> 00:57:43,980
They've always a
g2 function of 1.

835
00:57:43,980 --> 00:57:46,420
And attenuation is
not changing it.

836
00:57:46,420 --> 00:57:49,030
Attenuation is preserving that.

837
00:57:49,030 --> 00:57:52,730
So now I want to
show you explicitly

838
00:57:52,730 --> 00:57:58,100
why an attenuated coherent
state may have an average photon

839
00:57:58,100 --> 00:58:03,010
number of 1 but it shares
nothing else with a quantum

840
00:58:03,010 --> 00:58:06,270
state of 1 photon of
n equals 1 Fock state.

841
00:58:08,900 --> 00:58:16,770
So a coherent state with an
expectation value of 1 photon

842
00:58:16,770 --> 00:58:19,770
is not a single photon.

843
00:58:24,680 --> 00:58:33,640
And this can be, for
instance, expressed

844
00:58:33,640 --> 00:58:37,590
by looking what is
actually the probability,

845
00:58:37,590 --> 00:58:39,910
if you have such
a coherent state,

846
00:58:39,910 --> 00:58:44,270
to find 1, 2, or 3 photons.

847
00:58:44,270 --> 00:58:52,440
Well, the probability to find
0 photons-- no photon at all--

848
00:58:52,440 --> 00:58:54,550
is actually 1/3.

849
00:58:54,550 --> 00:59:00,740
So the probability to find 1
photon or to find 0 photons

850
00:59:00,740 --> 00:59:03,040
is the same.

851
00:59:03,040 --> 00:59:06,880
The probability to
find 2 photons is 0.18.

852
00:59:06,880 --> 00:59:09,360
So here you have a probability
of finding a photon

853
00:59:09,360 --> 00:59:12,490
and correlating it with
the next photon click.

854
00:59:12,490 --> 00:59:15,390
And you have even,
on first sight,

855
00:59:15,390 --> 00:59:25,690
a surprisingly large probability
to find 3 and 4 photons.

856
00:59:25,690 --> 00:59:30,810
So in 2 percent of the cases,
you will find 4 photons.

857
00:59:37,410 --> 00:59:43,930
Whereas in contrast, the Fock
state with quantum number n

858
00:59:43,930 --> 00:59:54,350
equals 1 is an eigenstate
of the number operator

859
00:59:54,350 --> 00:59:58,440
beta with eigenvalue n equals 1.

860
01:00:02,980 --> 01:00:08,580
So that tells you, if you
want to get n equals 1,

861
01:00:08,580 --> 01:00:11,000
if you want to get a
single photon state,

862
01:00:11,000 --> 01:00:18,060
you cannot just use a strong
laser beam or a strong light

863
01:00:18,060 --> 01:00:19,840
source and attenuate it.

864
01:00:19,840 --> 01:00:24,920
You have to work with something
which genuinely creates only 1

865
01:00:24,920 --> 01:00:30,090
photon without any ambiguity,
without any fluctuations,

866
01:00:30,090 --> 01:00:34,840
without any possibility
of creating 2 photons.

867
01:00:34,840 --> 01:00:38,920
And I would actually say
over the last 10 or 15 years,

868
01:00:38,920 --> 01:00:41,750
the creation of
single photons has

869
01:00:41,750 --> 01:00:44,790
been sort of a small
cottage industry,

870
01:00:44,790 --> 01:00:50,600
because single photons are often
needed for protocols in quantum

871
01:00:50,600 --> 01:00:57,530
computation-- for experiments
which really require accurate

872
01:00:57,530 --> 01:00:59,800
quantum state
preparation of light--

873
01:00:59,800 --> 01:01:03,050
and in particular,
non-classical light.

874
01:01:03,050 --> 01:01:09,970
But of course, there are ways
how you can get single photons.

875
01:01:09,970 --> 01:01:13,540
And this is, well, you
start with single atoms.

876
01:01:13,540 --> 01:01:17,910
If you have a single atom
in the excited state,

877
01:01:17,910 --> 01:01:19,590
it can emit only one photon.

878
01:01:34,930 --> 01:01:44,450
So in other words,
you cannot-- Usually,

879
01:01:44,450 --> 01:01:47,410
we don't have the tools to
prepare a single photon--

880
01:01:47,410 --> 01:01:50,530
to take a single photon
out of many, many photons--

881
01:01:50,530 --> 01:01:52,570
and store it separately.

882
01:01:52,570 --> 01:01:56,530
But what we can control
is single atoms.

883
01:01:56,530 --> 01:01:58,620
We can prepare single atoms.

884
01:01:58,620 --> 01:02:01,450
And then we can make
sure that single atoms

885
01:02:01,450 --> 01:02:02,900
create single photons.

886
01:02:02,900 --> 01:02:06,870
It's a little bit a way that
we cannot control the bullets

887
01:02:06,870 --> 01:02:07,820
which are fired.

888
01:02:07,820 --> 01:02:09,380
But we can control the guns.

889
01:02:09,380 --> 01:02:13,190
And we make sure that each gun
can emit exactly one bullet.

890
01:02:13,190 --> 01:02:14,910
So that's a way
how we can create

891
01:02:14,910 --> 01:02:16,916
non-classical states of light.

892
01:02:23,370 --> 01:02:24,690
Yes.

893
01:02:24,690 --> 01:02:33,738
So let's now look for
those single photons

894
01:02:33,738 --> 01:02:36,015
at the quasi-probability
distribution.

895
01:02:40,650 --> 01:02:45,580
So we obtain the
quasi-probability distribution

896
01:02:45,580 --> 01:02:50,880
by taking the
single photon state

897
01:02:50,880 --> 01:02:53,040
and projecting it
on a coherent state.

898
01:02:56,320 --> 01:02:59,790
You remember-- it's higher
up in the notes-- what

899
01:02:59,790 --> 01:03:03,540
the coherent state alpha
is in the Fock state basis,

900
01:03:03,540 --> 01:03:04,750
in a number basis.

901
01:03:04,750 --> 01:03:07,660
And so we're just
picking out n equals 1.

902
01:03:07,660 --> 01:03:11,365
And this was nothing
else than alpha

903
01:03:11,365 --> 01:03:15,790
squared times e to the
minus alpha squared.

904
01:03:15,790 --> 01:03:21,080
So therefore, in these
diagrams with the real part

905
01:03:21,080 --> 01:03:28,650
and imaginary part, if I
plot the quasi-probability

906
01:03:28,650 --> 01:03:35,700
distribution, we get a ring.

907
01:03:35,700 --> 01:03:40,650
The ring immediately tells us
that there is no phase defined.

908
01:03:40,650 --> 01:03:42,830
All the phases are
equally probable.

909
01:03:45,520 --> 01:03:47,800
And that also
means, if you don't

910
01:03:47,800 --> 01:03:53,550
have a phase, the average value
of the electric field this 0.

911
01:04:20,320 --> 01:04:27,020
Since the equation
of single photons

912
01:04:27,020 --> 01:04:30,630
is essential for studying
non-classical light,

913
01:04:30,630 --> 01:04:35,760
but also since it's a very
active frontier of our field--

914
01:04:35,760 --> 01:04:37,630
actually, one of the
leaders in this field

915
01:04:37,630 --> 01:04:40,320
is Professor Vladan
Vuletic here at MIT--

916
01:04:40,320 --> 01:04:43,160
let me at least give
you a taste of how

917
01:04:43,160 --> 01:04:46,990
to create single photons.

918
01:04:49,850 --> 01:04:54,500
I already gave you
the major ingredient.

919
01:04:54,500 --> 01:04:59,270
Namely, it's about-- it
involves single atoms.

920
01:04:59,270 --> 01:05:03,200
But it's a little bit
more demanding like that.

921
01:05:03,200 --> 01:05:06,630
So you want to take
1 atom home or 1 ion.

922
01:05:06,630 --> 01:05:11,310
But the problem is if the
atom or ion emits a light,

923
01:05:11,310 --> 01:05:14,690
it can emit the light
into all directions.

924
01:05:14,690 --> 01:05:18,310
And therefore, you have a
single photon afterwards.

925
01:05:18,310 --> 01:05:22,350
But you have many, many
different spacial modes.

926
01:05:22,350 --> 01:05:26,730
And in any given mode, it
will not have a single photon.

927
01:05:26,730 --> 01:05:33,420
So therefore, what you have
to add to the single atom

928
01:05:33,420 --> 01:05:36,790
or single ion is-- you
have to put it in a cavity.

929
01:05:49,030 --> 01:05:54,000
And then, in the limit of a
very, very high finesse cavity,

930
01:05:54,000 --> 01:05:57,560
the probability will
be very, very high

931
01:05:57,560 --> 01:06:00,990
that your photon is emitted
into the mode of the cavity

932
01:06:00,990 --> 01:06:05,030
and it is not emitted
perpendicular into other modes.

933
01:06:05,030 --> 01:06:08,380
But that itself is
not yet sufficient,

934
01:06:08,380 --> 01:06:10,560
because you have
a single photon,

935
01:06:10,560 --> 01:06:14,520
but you also want to know when
does the single photon arrive.

936
01:06:14,520 --> 01:06:16,070
You want to do experiments.

937
01:06:16,070 --> 01:06:17,870
You want single
photons on demand.

938
01:06:21,580 --> 01:06:34,140
And so one option is that
you prepare your atoms

939
01:06:34,140 --> 01:06:36,970
in the ground state.

940
01:06:36,970 --> 01:06:43,630
You take a pi pulse, which,
with 100 percent probability,

941
01:06:43,630 --> 01:06:46,700
excites the atom to
the excited state.

942
01:06:50,310 --> 01:06:53,790
And then within an
actual lifetime,

943
01:06:53,790 --> 01:06:57,330
or maybe even in
resonator-enhanced inverse

944
01:06:57,330 --> 01:07:02,000
natural line widths, the
single photon is emitted.

945
01:07:02,000 --> 01:07:04,790
So therefore, you
pump the system

946
01:07:04,790 --> 01:07:07,730
and then you know within
the next few nanoseconds

947
01:07:07,730 --> 01:07:10,720
your cavity mode will
have a single photon.

948
01:07:25,110 --> 01:07:37,010
The problem here is
that you're dealing--

949
01:07:37,010 --> 01:07:43,250
You have to prepare single
atoms, which is difficult.

950
01:07:43,250 --> 01:07:46,290
You have to couple them
to a high finesse cavity.

951
01:07:46,290 --> 01:07:50,410
There is technically
another approach,

952
01:07:50,410 --> 01:07:53,235
where you use many atoms.

953
01:07:53,235 --> 01:07:54,610
So you could say
that many atoms,

954
01:07:54,610 --> 01:07:56,380
you no longer have a single gun.

955
01:07:56,380 --> 01:07:59,720
So therefore, you can
get several photons.

956
01:07:59,720 --> 01:08:05,310
But the singleness in photons
comes because the photons

957
01:08:05,310 --> 01:08:08,390
are now heralded--
they are announced.

958
01:08:08,390 --> 01:08:10,890
And the idea is the following.

959
01:08:10,890 --> 01:08:16,240
If you start with the state 1,
and you have an excited state,

960
01:08:16,240 --> 01:08:17,344
and you have a pump pulse.

961
01:08:20,130 --> 01:08:24,000
Now, if you have many atoms, you
have a much higher efficiency

962
01:08:24,000 --> 01:08:25,930
of pumping atoms comes
to the excited state,

963
01:08:25,930 --> 01:08:29,490
because if you have n atoms,
it's n times more efficient.

964
01:08:29,490 --> 01:08:32,720
But now you take the
following situation.

965
01:08:32,720 --> 01:08:36,899
You wait until there
is a Raman transition--

966
01:08:36,899 --> 01:08:48,100
until you detect the photon
for Raman transition,

967
01:08:48,100 --> 01:08:53,939
where the excited state
decays to the state 2.

968
01:08:53,939 --> 01:09:00,810
At this moment, you know I
have now 1 atom in state 2.

969
01:09:03,510 --> 01:09:05,470
So in other words,
you're not starting

970
01:09:05,470 --> 01:09:08,840
with single atoms, which is
sometimes more demanding.

971
01:09:08,840 --> 01:09:10,840
How can you prepare the system?

972
01:09:10,840 --> 01:09:15,340
It has other disadvantages
that n atoms have.

973
01:09:18,149 --> 01:09:20,569
I'm not sure if I should
mention it, but with n atoms,

974
01:09:20,569 --> 01:09:23,740
they have a super-radiant
factor n with n atoms.

975
01:09:23,740 --> 01:09:27,099
You can get an n-times
enhancement of emission

976
01:09:27,099 --> 01:09:27,890
into a single mode.

977
01:09:27,890 --> 01:09:30,170
So there are real,
massive advantages

978
01:09:30,170 --> 01:09:32,220
in working with many atoms.

979
01:09:32,220 --> 01:09:36,220
But now you know that
one atom is in state 2

980
01:09:36,220 --> 01:09:38,990
when you detect
the first photon.

981
01:09:38,990 --> 01:09:44,790
And then you have the same
situation which we had above.

982
01:09:44,790 --> 01:09:50,210
You can now take your
system-- let me just

983
01:09:50,210 --> 01:09:51,990
redraw the level diagram.

984
01:09:51,990 --> 01:09:56,230
Excited state, state 1, state 2.

985
01:09:56,230 --> 01:10:01,810
You know now that you
have 1 single atom here.

986
01:10:01,810 --> 01:10:05,800
And by using a laser
pulse, you can excite it

987
01:10:05,800 --> 01:10:07,740
to the excited state.

988
01:10:07,740 --> 01:10:16,070
And then you observe
the single photon 2.

989
01:10:20,290 --> 01:10:24,250
So in other words,
the observation

990
01:10:24,250 --> 01:10:31,770
off the first photon
tells you that your system

991
01:10:31,770 --> 01:10:35,120
is prepared with
1 atom in state 2.

992
01:10:35,120 --> 01:10:37,970
And then you can get
a single photon out

993
01:10:37,970 --> 01:10:41,530
of it, which is the photon
for the inverse Raman

994
01:10:41,530 --> 01:10:45,249
transition when you pump
the system back to state 1.

995
01:10:45,249 --> 01:10:46,665
AUDIENCE: Doesn't
this rely on you

996
01:10:46,665 --> 01:10:52,848
being able to reliably prepare
a single atom in another quantum

997
01:10:52,848 --> 01:10:55,318
state?

998
01:10:55,318 --> 01:10:57,294
Won't there always
be some uncertainty

999
01:10:57,294 --> 01:10:59,764
as to how many atoms you
can prepare in one sitting?

1000
01:11:06,674 --> 01:11:08,340
PROFESSOR: There are
some uncertainties.

1001
01:11:08,340 --> 01:11:10,850
And what you're
saying is correct.

1002
01:11:10,850 --> 01:11:14,070
We don't have perfect
single photon sources.

1003
01:11:14,070 --> 01:11:19,520
And people characterize the
fidelity of the single photon

1004
01:11:19,520 --> 01:11:20,150
source.

1005
01:11:20,150 --> 01:11:26,750
For instance, if you detect the
single photon, you would say,

1006
01:11:26,750 --> 01:11:30,460
Now my system is ready to
emit a single photon triggered

1007
01:11:30,460 --> 01:11:31,752
by this pump pulse.

1008
01:11:31,752 --> 01:11:33,335
And if you can get
now a single photon

1009
01:11:33,335 --> 01:11:37,821
in 90 percent of the cases,
you publish a wonderful paper,

1010
01:11:37,821 --> 01:11:39,820
because you've set a new
record for the fidelity

1011
01:11:39,820 --> 01:11:41,490
of a single photon source.

1012
01:11:41,490 --> 01:11:43,820
So people are really
struggling with some

1013
01:11:43,820 --> 01:11:45,460
of those uncertainties.

1014
01:11:45,460 --> 01:11:49,770
But to involve, for
instance, three levels

1015
01:11:49,770 --> 01:11:52,690
is sort of an advantage,
because you're not

1016
01:11:52,690 --> 01:11:56,290
limited by the preparation of
a single atom, for instance.

1017
01:11:56,290 --> 01:11:57,540
You can have many atoms.

1018
01:11:57,540 --> 01:11:59,260
The atoms are always there.

1019
01:11:59,260 --> 01:12:02,810
And the moment one atom
is prepared in state 2,

1020
01:12:02,810 --> 01:12:06,360
this atom announces itself
with a single photon.

1021
01:12:06,360 --> 01:12:11,360
So it takes a lot of uncertainty
out that the system says,

1022
01:12:11,360 --> 01:12:13,420
with the first
photon, I'm ready now.

1023
01:12:13,420 --> 01:12:15,120
I can emit a single photon.

1024
01:12:15,120 --> 01:12:17,270
And then you get
your single photon.

1025
01:12:17,270 --> 01:12:19,930
Then you can gate
your whole experiment

1026
01:12:19,930 --> 01:12:23,680
to a time following the
detection of the first photon.

1027
01:12:23,680 --> 01:12:26,594
And for your gated
time afterwards, you

1028
01:12:26,594 --> 01:12:28,010
have a very, very
high probability

1029
01:12:28,010 --> 01:12:31,140
of finding this photon.

1030
01:12:31,140 --> 01:12:34,590
Or if you want, you can now
do experiments with 2 photons.

1031
01:12:34,590 --> 01:12:38,720
If you keep this laser on and
pump the atom immediately back,

1032
01:12:38,720 --> 01:12:40,990
you have actually a
single photon here,

1033
01:12:40,990 --> 01:12:42,550
followed by a
single photon here.

1034
01:12:42,550 --> 01:12:44,370
It's sort of click clack.

1035
01:12:44,370 --> 01:12:46,790
And now you can do
correlations between two

1036
01:12:46,790 --> 01:12:49,720
different single
photon states and such.

1037
01:12:49,720 --> 01:12:52,021
It's a very rich
frontier of our field.

1038
01:12:52,021 --> 01:12:53,104
You have another question?

1039
01:12:53,104 --> 01:12:54,078
AUDIENCE: Yeah.

1040
01:12:54,078 --> 01:12:57,487
Is this also within a cavity
now, like the single photon

1041
01:12:57,487 --> 01:12:58,950
[? reflection ?]
in the first test?

1042
01:12:58,950 --> 01:12:59,685
PROFESSOR: Yes.

1043
01:12:59,685 --> 01:13:04,290
Actually, these photons--
the wavy lines--

1044
01:13:04,290 --> 01:13:06,290
are emitted spontaneously.

1045
01:13:06,290 --> 01:13:09,800
Spontaneous emission,
pretty much by definition,

1046
01:13:09,800 --> 01:13:12,170
goes into all possible
spatial modes.

1047
01:13:12,170 --> 01:13:15,245
And the only way to control the
spatial mode is with a cavity.

1048
01:13:18,520 --> 01:13:20,690
What I don't want
to go into details--

1049
01:13:20,690 --> 01:13:23,170
but those who have an
understanding of that--

1050
01:13:23,170 --> 01:13:24,250
is the following.

1051
01:13:24,250 --> 01:13:31,060
If you n atoms and you prepare
one atom here in state 2,

1052
01:13:31,060 --> 01:13:34,990
you do not know which of
your n atoms is prepared.

1053
01:13:34,990 --> 01:13:40,250
So you have n indistinguishable
possibilities.

1054
01:13:40,250 --> 01:13:42,740
And if you have n
indistinguishable probabilities

1055
01:13:42,740 --> 01:13:46,840
which atom you have prepared,
the emission of the photon

1056
01:13:46,840 --> 01:13:49,490
back is n times enhanced.

1057
01:13:49,490 --> 01:13:53,030
So therefore, you
have actually a system

1058
01:13:53,030 --> 01:13:57,690
which has an n times stronger
coupling to the cavity.

1059
01:13:57,690 --> 01:14:02,820
So having n atoms makes
it much, much easier

1060
01:14:02,820 --> 01:14:04,900
to construct a high
finesse cavity.

1061
01:14:04,900 --> 01:14:07,850
You get this super
radiance increase

1062
01:14:07,850 --> 01:14:11,270
of the strong coupling for free.

1063
01:14:11,270 --> 01:14:16,280
And this is why there are
many reasons why you do not

1064
01:14:16,280 --> 01:14:19,250
want to work with the
single ion or a single atom.

1065
01:14:19,250 --> 01:14:20,160
It's possible.

1066
01:14:20,160 --> 01:14:21,940
For very fundamental
experiments,

1067
01:14:21,940 --> 01:14:23,920
people have shown
that it can be done.

1068
01:14:23,920 --> 01:14:27,550
But technologically,
it's much easier and much

1069
01:14:27,550 --> 01:14:30,030
more robust to work
with many atoms.

1070
01:14:30,030 --> 01:14:32,370
But then you need
the many atoms have

1071
01:14:32,370 --> 01:14:38,034
to tell you when 1 atom
is prepared to emit now

1072
01:14:38,034 --> 01:14:38,700
a single photon.

1073
01:14:43,200 --> 01:14:46,510
If you're interested
in this subject,

1074
01:14:46,510 --> 01:14:50,060
talk to the students
in [INAUDIBLE] school.

1075
01:14:50,060 --> 01:14:52,467
They're really the
world experts in that.

1076
01:15:06,861 --> 01:15:07,360
OK.

1077
01:15:07,360 --> 01:15:11,510
Finally, we're not really
getting to squeeze light.

1078
01:15:11,510 --> 01:15:13,840
We start with squeezing
the light tomorrow.

1079
01:15:16,360 --> 01:15:20,620
The last thing in
this major section

1080
01:15:20,620 --> 01:15:33,860
is the famous Hanbury
Brown Twiss experiment.

1081
01:15:38,150 --> 01:15:43,390
This was a landmark
experiment done in the 1950s.

1082
01:15:43,390 --> 01:15:47,520
And it was the first
experiment which

1083
01:15:47,520 --> 01:15:52,640
really looked at g2 functions
correlations, which one could

1084
01:15:52,640 --> 01:15:55,490
say was the beginning
of quantum optics

1085
01:15:55,490 --> 01:16:00,820
and modern experiments
with light.

1086
01:16:00,820 --> 01:16:03,630
Probably until then, light was
just an electromagnetic wave

1087
01:16:03,630 --> 01:16:05,190
people regarded as boring.

1088
01:16:05,190 --> 01:16:07,890
But by measuring
now correlations,

1089
01:16:07,890 --> 01:16:12,080
people realize that it's an
interesting object to study.

1090
01:16:12,080 --> 01:16:18,990
I want to just go
over the basic scheme.

1091
01:16:18,990 --> 01:16:21,600
You will look at it
a little bit more

1092
01:16:21,600 --> 01:16:24,370
closely in your
homework assignment.

1093
01:16:24,370 --> 01:16:27,720
So the idea behind an Hanbury
Brown Twiss experiment

1094
01:16:27,720 --> 01:16:31,870
is that you have a
light source and you

1095
01:16:31,870 --> 01:16:35,230
want to characterize it.

1096
01:16:45,020 --> 01:16:48,635
So the light source emits light.

1097
01:16:51,240 --> 01:16:54,790
And now, this is important.

1098
01:16:54,790 --> 01:16:57,390
Whenever you want
to detect 2 photons,

1099
01:16:57,390 --> 01:16:59,710
you want to figure out
it's 1 photon followed

1100
01:16:59,710 --> 01:17:02,550
by another photon.

1101
01:17:02,550 --> 01:17:05,280
Technically, you cannot do
it with a photomultiplier,

1102
01:17:05,280 --> 01:17:09,610
because when a photomultiplier
clicks, it needs many,

1103
01:17:09,610 --> 01:17:12,460
many nanoseconds for the
photomultiplier to read out

1104
01:17:12,460 --> 01:17:16,070
the signal and recover to
recharge up its electrodes

1105
01:17:16,070 --> 01:17:18,400
and be ready to detect
the next photon.

1106
01:17:18,400 --> 01:17:21,560
So therefore, when you
want to find click click--

1107
01:17:21,560 --> 01:17:24,650
double clicks in the
stream of light--

1108
01:17:24,650 --> 01:17:27,170
you have to involve
a beam splitter

1109
01:17:27,170 --> 01:17:28,610
and involve two photodetectors.

1110
01:17:31,760 --> 01:17:34,150
Sure, you now need an
adequate description.

1111
01:17:34,150 --> 01:17:36,250
But in principle,
you can now find

1112
01:17:36,250 --> 01:17:39,980
2 photons which are only
a few picoseconds apart,

1113
01:17:39,980 --> 01:17:47,220
because the first photon is
observed by the first detector,

1114
01:17:47,220 --> 01:17:49,220
and the second
photon is detected

1115
01:17:49,220 --> 01:17:51,209
by the second detector.

1116
01:17:51,209 --> 01:17:53,500
Sure, you would say, What
happens if you're in the beam

1117
01:17:53,500 --> 01:17:55,760
splitter, both photons
go to detector one?

1118
01:17:55,760 --> 01:17:57,200
Well, that's tough luck.

1119
01:17:57,200 --> 01:18:00,820
You take 1 chance in
2 that the 2 photons

1120
01:18:00,820 --> 01:18:02,570
go to the 2 different detectors.

1121
01:18:02,570 --> 01:18:06,730
But the beam splitter has to
be part of your experiment,

1122
01:18:06,730 --> 01:18:08,800
has to become part
of your description.

1123
01:18:08,800 --> 01:18:12,860
And this is what you do
in homework number one.

1124
01:18:12,860 --> 01:18:20,400
So what we are asking here when
we measure the g2 function.

1125
01:18:20,400 --> 01:18:23,540
Let's assume that tau equals 0.

1126
01:18:23,540 --> 01:18:29,730
We are reading out a signal.

1127
01:18:29,730 --> 01:18:32,835
And what we are using is
a coincidence detector.

1128
01:18:37,560 --> 01:18:40,110
We are looking that in a
very small temporal window,

1129
01:18:40,110 --> 01:18:43,330
2 photons are detected
simultaneously.

1130
01:18:43,330 --> 01:18:45,360
Well, this is the
quantum version.

1131
01:18:45,360 --> 01:18:48,190
The classical version
is the g2 function

1132
01:18:48,190 --> 01:18:51,400
is the product of the
intensity at time t times

1133
01:18:51,400 --> 01:18:54,880
the product of the intensity
at time t plus tau.

1134
01:18:54,880 --> 01:18:57,780
So what you would do
here in this circuit,

1135
01:18:57,780 --> 01:19:00,280
you would take the signal
from this detector, the signal

1136
01:19:00,280 --> 01:19:02,450
from that, and multiply the two.

1137
01:19:02,450 --> 01:19:06,480
This is how you determine I
of t times I of t plus tau.

1138
01:19:11,820 --> 01:19:14,490
So whether you do it
in the classical domain

1139
01:19:14,490 --> 01:19:19,820
or whether you do it
in the quantum domain,

1140
01:19:19,820 --> 01:19:23,050
this is the way how
you experimentally

1141
01:19:23,050 --> 01:19:28,930
measure the second order
correlation function.

1142
01:19:35,140 --> 01:19:41,100
In the classic limit, you
don't have to worry about it.

1143
01:19:41,100 --> 01:19:44,450
The classical limit is always
a limit of high intensity.

1144
01:19:44,450 --> 01:19:48,020
So at any given time, you
have a ton of photons.

1145
01:19:48,020 --> 01:19:51,245
And then, at beam splitter,
half of the photons go left,

1146
01:19:51,245 --> 01:19:52,640
half of the photons go right.

1147
01:19:52,640 --> 01:19:54,370
You have an equal splitting.

1148
01:19:54,370 --> 01:19:56,540
You have exactly
half the intensity

1149
01:19:56,540 --> 01:20:00,750
in the left arm and the right
arm after the beam splitter.

1150
01:20:00,750 --> 01:20:04,150
So there is no problem at
all with the quantization,

1151
01:20:04,150 --> 01:20:06,640
because this is the
classical limit.

1152
01:20:06,640 --> 01:20:10,340
So classically you
can say-- actually,

1153
01:20:10,340 --> 01:20:13,390
I shouldn't say photon, I
should actually say intensity.

1154
01:20:13,390 --> 01:20:16,240
The intensity splits equally.

1155
01:20:24,530 --> 01:20:28,100
And if you do the measurement
with the coincidence detector,

1156
01:20:28,100 --> 01:20:32,180
you find the difference
between the g2 function,

1157
01:20:32,180 --> 01:20:34,010
which I discussed earlier.

1158
01:20:34,010 --> 01:20:38,540
That the g2 function is 1 for
coherent light or laser light,

1159
01:20:38,540 --> 01:20:41,475
and the g2 function is
2 for thermal light.

1160
01:20:47,070 --> 01:20:50,110
That's actually the only
way how you can distinguish

1161
01:20:50,110 --> 01:20:52,900
a light bulb from a laser beam.

1162
01:20:52,900 --> 01:20:55,930
If you put a light bulb into
a cavity or couple the light

1163
01:20:55,930 --> 01:20:58,320
from a light bulb into
a fiber, the light

1164
01:20:58,320 --> 01:21:01,120
becomes spatially a single mode.

1165
01:21:01,120 --> 01:21:03,120
If you put a very
narrow spectral filter

1166
01:21:03,120 --> 01:21:06,410
or Fabry-Perot cavity,
the light becomes

1167
01:21:06,410 --> 01:21:09,000
single mode in frequency domain.

1168
01:21:09,000 --> 01:21:12,450
So when you take a light bulb
and spatially and spectrally

1169
01:21:12,450 --> 01:21:17,140
filter it that it is single
mode, in terms of the mode,

1170
01:21:17,140 --> 01:21:20,630
it is single mode as
a single mode laser.

1171
01:21:20,630 --> 01:21:22,305
But it is the
correlation experimental

1172
01:21:22,305 --> 01:21:25,900
which shows you that you
started with a thermal source.

1173
01:21:25,900 --> 01:21:29,470
It has a g2 function of 2, and
you can never get rid of it,

1174
01:21:29,470 --> 01:21:31,690
whereas the laser beam
has a g2 function of 1.

1175
01:21:34,200 --> 01:21:34,850
OK.

1176
01:21:34,850 --> 01:21:37,220
So this is the
classical version.

1177
01:21:37,220 --> 01:21:39,970
In the quantum version,
especially when

1178
01:21:39,970 --> 01:21:48,300
we have a single
photon, the photon

1179
01:21:48,300 --> 01:21:52,660
can go to only one detector.

1180
01:21:55,660 --> 01:21:59,800
And that means, for this
extreme case of a single photon,

1181
01:21:59,800 --> 01:22:03,380
the g2 function is 0.

1182
01:22:03,380 --> 01:22:07,810
Anyway, you will look at
those situations in more

1183
01:22:07,810 --> 01:22:11,670
detail in you
homework assignment.

1184
01:22:11,670 --> 01:22:12,374
Yes?

1185
01:22:12,374 --> 01:22:14,082
AUDIENCE: Sir, if you
know that different

1186
01:22:14,082 --> 01:22:17,040
for tau non-zero, then even
the single mode in the function

1187
01:22:17,040 --> 01:22:20,387
would have a final g2
[INAUDIBLE], because there

1188
01:22:20,387 --> 01:22:21,970
would be some
probability [INAUDIBLE].

1189
01:22:43,600 --> 01:22:46,300
PROFESSOR: We have to be
now a little bit careful.

1190
01:22:46,300 --> 01:22:52,930
If you have a single mode Fock
state, there's only 1 photon.

1191
01:22:52,930 --> 01:22:56,655
And once this photon is
detected, we have vacuum,

1192
01:22:56,655 --> 01:22:59,340
we have no photon left.

1193
01:22:59,340 --> 01:23:06,450
So what may be confusing
here is, on first sight,

1194
01:23:06,450 --> 01:23:09,770
is that the way how the
experiment is done and has been

1195
01:23:09,770 --> 01:23:13,880
historically done, it's
not done we Fock states.

1196
01:23:13,880 --> 01:23:16,780
It's done with the light
source in a continuous stream

1197
01:23:16,780 --> 01:23:18,220
of photons.

1198
01:23:18,220 --> 01:23:19,820
So Hanbury Brown
Twiss experiment,

1199
01:23:19,820 --> 01:23:24,030
he looked at in
light bulb, or he

1200
01:23:24,030 --> 01:23:26,900
looked at star light and such,
and determined the intensity

1201
01:23:26,900 --> 01:23:27,760
correlation.

1202
01:23:27,760 --> 01:23:29,800
He couldn't do the
experiment with lasers,

1203
01:23:29,800 --> 01:23:31,660
but now we would
with a laser beam.

1204
01:23:31,660 --> 01:23:34,000
And we find that g2 is 1.

1205
01:23:34,000 --> 01:23:37,200
But in those
situations, we actually

1206
01:23:37,200 --> 01:23:42,646
do not have the way-- we
have actually a beam, which

1207
01:23:42,646 --> 01:23:45,180
is a stream of photons.

1208
01:23:45,180 --> 01:23:48,190
And this requires a little
bit different description.

1209
01:23:48,190 --> 01:23:52,210
In other words, when I
have a coherent state,

1210
01:23:52,210 --> 01:23:55,790
a laser beam is
always replenishing

1211
01:23:55,790 --> 01:23:56,740
the coherent state.

1212
01:23:56,740 --> 01:23:59,570
The laser beam preserves
the electric field,

1213
01:23:59,570 --> 01:24:03,220
whereas, strictly speaking,
the way how I described it

1214
01:24:03,220 --> 01:24:06,370
for pedagogical reasons is
you have a cavity, which

1215
01:24:06,370 --> 01:24:09,520
is filled with a
coherent state alpha.

1216
01:24:09,520 --> 01:24:12,010
And then you start
analyzing that.

1217
01:24:12,010 --> 01:24:16,080
But in other words,
what we have focused

1218
01:24:16,080 --> 01:24:18,890
in the simple description
is that we have a quantum

1219
01:24:18,890 --> 01:24:21,630
state which is prepared--
it's a closed system.

1220
01:24:21,630 --> 01:24:24,090
And now we do our detection.

1221
01:24:24,090 --> 01:24:26,940
The experiment how
it is done is often

1222
01:24:26,940 --> 01:24:30,920
done as in an open system,
where you couple the system

1223
01:24:30,920 --> 01:24:33,580
to a light source,
which is always

1224
01:24:33,580 --> 01:24:36,080
replenishing your experiment.

1225
01:24:36,080 --> 01:24:39,250
So to be careful, especially
with a single photon.

1226
01:24:39,250 --> 01:24:41,490
I think, in essence,
the experiment

1227
01:24:41,490 --> 01:24:44,090
would be done with a
single photon light source.

1228
01:24:44,090 --> 01:24:45,890
But the single
photon light source

1229
01:24:45,890 --> 01:24:48,370
has a high repetition rate.

1230
01:24:48,370 --> 01:24:50,880
I showed you how to
generate single photons.

1231
01:24:50,880 --> 01:24:53,430
You have an heralded
single photon.

1232
01:24:53,430 --> 01:24:55,710
You know the single
photon comes now.

1233
01:24:55,710 --> 01:24:57,110
You do the experiment.

1234
01:24:57,110 --> 01:25:00,160
And then you repeat it again.

1235
01:25:00,160 --> 01:25:01,730
And then what you
have is you only

1236
01:25:01,730 --> 01:25:05,580
look at one single
photon bunch at a time.

1237
01:25:05,580 --> 01:25:09,040
And then you find indeed, that
during that temporal window,

1238
01:25:09,040 --> 01:25:11,210
you will never find
a second photon.

1239
01:25:11,210 --> 01:25:13,650
So this may happen
in a few nanoseconds.

1240
01:25:13,650 --> 01:25:18,100
Then you wait a microsecond,
and the next photon arrives.

1241
01:25:18,100 --> 01:25:22,930
Of course, if you would now
describe your light source,

1242
01:25:22,930 --> 01:25:26,980
what you is the g2 of tau,
and tau is a microsecond,

1243
01:25:26,980 --> 01:25:30,360
and the microsecond is the time
between the first single photon

1244
01:25:30,360 --> 01:25:33,000
burst and the next
single photon burst,

1245
01:25:33,000 --> 01:25:36,380
now you will find that
your g2 function is not 0.

1246
01:25:36,380 --> 01:25:39,230
But this is then related
to the repetition

1247
01:25:39,230 --> 01:25:42,560
rate of a single photon and not
to the single photon itself.

1248
01:25:42,560 --> 01:25:44,810
I think you've got the taste.

1249
01:25:44,810 --> 01:25:46,460
Your homework is
a really simple.

1250
01:25:46,460 --> 01:25:48,250
You just deal with
a closed system.

1251
01:25:48,250 --> 01:25:50,260
You look at a quantum
state at a time.

1252
01:25:50,260 --> 01:25:53,570
But if you map it onto
a beam experiment,

1253
01:25:53,570 --> 01:25:55,110
you have to think
about what does it

1254
01:25:55,110 --> 01:25:58,790
mean to have a replenishment
of the quantum state.

1255
01:25:58,790 --> 01:25:59,790
Any question?

1256
01:25:59,790 --> 01:26:03,620
I know I have to stop, but.

1257
01:26:03,620 --> 01:26:06,100
AUDIENCE: In the beam
splitter, the other port

1258
01:26:06,100 --> 01:26:10,068
of the beam splitter should
do some stuff, right?

1259
01:26:12,570 --> 01:26:13,630
PROFESSOR: Yes.

1260
01:26:13,630 --> 01:26:17,900
Actually, this will keep us busy
for a while, that even if you

1261
01:26:17,900 --> 01:26:21,990
do not put in any light,
we put in the vacuum state.

1262
01:26:21,990 --> 01:26:25,220
And you will find that the
description of the beam

1263
01:26:25,220 --> 01:26:29,020
splitter in the quantum
state is incomplete,

1264
01:26:29,020 --> 01:26:32,750
unless you specify that your
beam splitter is coupling

1265
01:26:32,750 --> 01:26:36,490
in from a different
mode the vacuum state.

1266
01:26:36,490 --> 01:26:39,190
Yes, this will be
very important.

1267
01:26:39,190 --> 01:26:41,991
And we will cover it
in its full glory.

1268
01:26:41,991 --> 01:26:43,615
AUDIENCE: I just have
a quick question.

1269
01:26:43,615 --> 01:26:46,750
The way we can distinguish
from a thermal state

1270
01:26:46,750 --> 01:26:49,100
and a coherent state
is through the g2.

1271
01:26:49,100 --> 01:26:50,600
So from an experimental
perspective,

1272
01:26:50,600 --> 01:26:53,050
could we do most of
experiments if we just

1273
01:26:53,050 --> 01:26:57,350
had thermal sources that were
very single mode, so to speak?

1274
01:26:57,350 --> 01:26:59,570
PROFESSOR: Well, yes.

1275
01:26:59,570 --> 01:27:03,150
All the laser cooling, all the
[INAUDIBLE], all the absorption

1276
01:27:03,150 --> 01:27:06,990
imaging-- all that would
work if you had a single mode

1277
01:27:06,990 --> 01:27:08,390
thermal source.

1278
01:27:08,390 --> 01:27:11,440
Let's face it, we say we
use lasers for everything

1279
01:27:11,440 --> 01:27:12,040
we're doing.

1280
01:27:12,040 --> 01:27:14,930
But the only property
which distinguishes

1281
01:27:14,930 --> 01:27:17,130
a laser from the thermal
light source-- we're

1282
01:27:17,130 --> 01:27:19,640
not taking advantage of it.

1283
01:27:19,640 --> 01:27:22,950
Of course practically, if
you take a thermal source

1284
01:27:22,950 --> 01:27:24,950
and filter it down
to a single mode,

1285
01:27:24,950 --> 01:27:27,520
you will be left with
only a few photons.

1286
01:27:27,520 --> 01:27:31,720
You cannot create an intense
enough single mode light source

1287
01:27:31,720 --> 01:27:35,160
unless you use stimulated
emission into a single mode.

1288
01:27:35,160 --> 01:27:36,431
And that's a laser.

1289
01:27:36,431 --> 01:27:36,930
Colin.

1290
01:27:36,930 --> 01:27:41,385
AUDIENCE: The correlation
function for, for example,

1291
01:27:41,385 --> 01:27:44,924
an LED light-- that's not
really a thermal source.

1292
01:27:44,924 --> 01:27:47,215
It's not described by a
Maxwell-Boltzmann distribution.

1293
01:27:47,215 --> 01:27:50,680
And that would be closer
to something for a laser,

1294
01:27:50,680 --> 01:27:52,165
even though there is no cavity?

1295
01:27:55,150 --> 01:27:58,420
PROFESSOR: Well, LEDs in some
limit are quantum objects.

1296
01:28:05,732 --> 01:28:07,940
Actually, do you know what
the g2 function of LED is?

1297
01:28:07,940 --> 01:28:09,500
Does it become
laser-like, or does it

1298
01:28:09,500 --> 01:28:11,430
become even antibunched?

1299
01:28:11,430 --> 01:28:15,970
Because if it's a relaxation
mechanism-- So anyway,

1300
01:28:15,970 --> 01:28:21,320
what Colin says is
there are actually

1301
01:28:21,320 --> 01:28:24,210
more different light
sources that just the laser

1302
01:28:24,210 --> 01:28:26,650
and the thermal light source.

1303
01:28:26,650 --> 01:28:32,080
There are LEDs or semiconductor
devices, which provide photons

1304
01:28:32,080 --> 01:28:36,010
with interesting
statistical properties.

1305
01:28:36,010 --> 01:28:38,731
I've heard about it-- that
LED light sources have

1306
01:28:38,731 --> 01:28:40,230
some special
properties, but I don't

1307
01:28:40,230 --> 01:28:42,020
remember which was the one.

1308
01:28:42,020 --> 01:28:42,520
OK.

1309
01:28:42,520 --> 01:28:43,380
We have to stop.

1310
01:28:43,380 --> 01:28:45,360
I'll see you on Wednesday.