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PROFESSOR: Well, we
are in the middle

9
00:00:26,290 --> 00:00:28,850
of something really interesting.

10
00:00:28,850 --> 00:00:32,200
We are talking about squeezing.

11
00:00:32,200 --> 00:00:35,200
We are talking about
non classical light.

12
00:00:35,200 --> 00:00:37,540
And today, I sort of
want to wrap it up.

13
00:00:37,540 --> 00:00:40,060
And I think it will really
be an exciting class.

14
00:00:40,060 --> 00:00:42,780
But before I continue
with the material,

15
00:00:42,780 --> 00:00:45,800
I want to address a question,
which actually came up

16
00:00:45,800 --> 00:00:48,410
in discussions with
several of the students.

17
00:00:48,410 --> 00:00:54,340
And this is, I realize
that some people said, OK.

18
00:00:54,340 --> 00:00:55,890
Everything makes sense.

19
00:00:55,890 --> 00:00:57,510
But what are we plotting?

20
00:00:57,510 --> 00:00:59,600
What is really squeezed?

21
00:00:59,600 --> 00:01:02,170
Are we squeezing in
the spatial domain?

22
00:01:02,170 --> 00:01:04,780
Are we squeezing in
the temporal domain?

23
00:01:04,780 --> 00:01:07,840
So the plots look wonderful,
with these ellipses

24
00:01:07,840 --> 00:01:08,960
and the circles.

25
00:01:08,960 --> 00:01:11,870
But, what is it really
we are doing here?

26
00:01:11,870 --> 00:01:14,140
So let me address that.

27
00:01:14,140 --> 00:01:37,760
First of all, we are talking
about a single harmonic

28
00:01:37,760 --> 00:01:39,100
oscillator.

29
00:01:39,100 --> 00:01:43,140
We showed that the
[INAUDIBLE] equations

30
00:01:43,140 --> 00:01:46,890
can be reduced to a bunch of
harmonic oscillator equations.

31
00:01:46,890 --> 00:01:49,630
One for each mode.

32
00:01:49,630 --> 00:01:52,400
And now we are
talking, today and

33
00:01:52,400 --> 00:01:56,400
in the previous few classes,
about one single mode,

34
00:01:56,400 --> 00:01:58,710
about one harmonic oscillator.

35
00:01:58,710 --> 00:02:02,760
And the harmonic oscillator
has canonically variables

36
00:02:02,760 --> 00:02:05,040
of momentum and position.

37
00:02:05,040 --> 00:02:08,479
But this is just to make
a connection for you

38
00:02:08,479 --> 00:02:10,919
with something you have learned.

39
00:02:10,919 --> 00:02:15,000
What we are talking about is
a single harmonic oscillator,

40
00:02:15,000 --> 00:02:19,010
which is one single mode of
the electromagnetic field.

41
00:02:19,010 --> 00:02:21,650
So maybe let me draw
a cartoon for that.

42
00:02:24,880 --> 00:02:28,880
So let's assume
we have a cavity.

43
00:02:28,880 --> 00:02:32,950
We have an electromagnetic wave.

44
00:02:32,950 --> 00:02:36,700
There is propagation,
where it's e to the ikz.

45
00:02:36,700 --> 00:02:39,300
There is transverse confinement.

46
00:02:39,300 --> 00:02:41,580
Maybe there is a
Gaussian e to the minus

47
00:02:41,580 --> 00:02:47,350
x squared plus y squared over
sum [INAUDIBLE] parameter.

48
00:02:47,350 --> 00:02:53,430
All of that is simply
the spatial mode.

49
00:02:53,430 --> 00:02:55,980
And we just take that for
given because we're not

50
00:02:55,980 --> 00:02:58,260
solving the spatial
differential equation.

51
00:02:58,260 --> 00:03:03,400
All we are doing is we are
looking at this one mode.

52
00:03:03,400 --> 00:03:06,720
And the two degrees
of freedom is

53
00:03:06,720 --> 00:03:11,230
that this mode can have a
certain number of photons.

54
00:03:11,230 --> 00:03:12,520
It's the amplitude.

55
00:03:12,520 --> 00:03:15,850
And the second one is you
can see the temporal phase.

56
00:03:15,850 --> 00:03:18,930
It can be a cosine omega t.
it can be a sine omega t.

57
00:03:18,930 --> 00:03:21,040
It can be a superposition.

58
00:03:21,040 --> 00:03:25,080
But whatever we are talking
about is in this mode.

59
00:03:25,080 --> 00:03:27,730
There is nothing happening
in the spatial domain.

60
00:03:27,730 --> 00:03:32,230
They're just asking, what is
the oscillation in this mode?

61
00:03:32,230 --> 00:03:34,070
The whole mode does
what it should.

62
00:03:34,070 --> 00:03:37,100
It has a prefect, of
which is the amplitude.

63
00:03:37,100 --> 00:03:40,440
And it has a temporal effect
of which we factor out.

64
00:03:40,440 --> 00:03:42,710
And this is what we
are talking about.

65
00:03:48,210 --> 00:03:51,500
Let me be a little
bit more specific

66
00:03:51,500 --> 00:03:59,780
and say, that when we
are plotting things,

67
00:03:59,780 --> 00:04:05,240
we are plotting the Q
representation, the phase space

68
00:04:05,240 --> 00:04:09,220
representation of the
statistical operator,

69
00:04:09,220 --> 00:04:12,640
which is simply describing
this single mode

70
00:04:12,640 --> 00:04:14,740
of the harmonic oscillator.

71
00:04:14,740 --> 00:04:17,870
And by performing the
diagram matrix elements,

72
00:04:17,870 --> 00:04:22,580
we obtain the Q distribution.

73
00:04:22,580 --> 00:04:26,320
In that case, we
have the vacuum.

74
00:04:26,320 --> 00:04:31,620
We have a displaced vacuum,
which is in coherent state.

75
00:04:31,620 --> 00:04:37,700
And our x's are, from
the very definition,

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00:04:37,700 --> 00:04:39,990
in the complex plane
with a real part

77
00:04:39,990 --> 00:04:41,391
of the imaginary part of alpha.

78
00:04:47,340 --> 00:04:54,210
However, we can also
define [? the  Veetner ?]

79
00:04:54,210 --> 00:04:57,060
distribution, which is another
phase based distribution.

80
00:04:57,060 --> 00:04:59,930
It's almost the same
as the Q distribution.

81
00:04:59,930 --> 00:05:01,770
It's just this little
bit smeared out

82
00:05:01,770 --> 00:05:04,750
by h bar because of
some commutators.

83
00:05:04,750 --> 00:05:06,660
But nothing you
have to worry about.

84
00:05:06,660 --> 00:05:14,560
In that case, the
projection of the W function

85
00:05:14,560 --> 00:05:19,520
on the vertical
axis, on the y-axis,

86
00:05:19,520 --> 00:05:23,925
is the momentum wave
function squared.

87
00:05:29,735 --> 00:05:36,425
On the x-axis, it is the
spatial wave function

88
00:05:36,425 --> 00:05:38,020
squared of the
harmonic oscillator.

89
00:05:41,730 --> 00:05:44,590
So therefore, we
may sometimes think,

90
00:05:44,590 --> 00:05:48,440
when we have a distribution
here, we project it.

91
00:05:48,440 --> 00:05:52,490
And we see what is the
momentum distribution.

92
00:05:52,490 --> 00:05:55,510
Or what is the
spatial distribution

93
00:05:55,510 --> 00:05:58,430
of the mechanical
harmonic oscillator.

94
00:05:58,430 --> 00:06:01,170
Which is analogous,
which is equivalent,

95
00:06:01,170 --> 00:06:05,340
to the one mode of the
electromagnetic field

96
00:06:05,340 --> 00:06:08,080
we are using.

97
00:06:08,080 --> 00:06:10,520
I know it may help
you to some extent

98
00:06:10,520 --> 00:06:17,470
to think about the P and Q.
But it may also be misleading

99
00:06:17,470 --> 00:06:20,540
because it gives you the
impression something is moving

100
00:06:20,540 --> 00:06:23,240
with a momentum
P, in real space.

101
00:06:23,240 --> 00:06:28,480
Let me therefore
emphasize what are

102
00:06:28,480 --> 00:06:34,640
the normalized forms of P and Q.

103
00:06:34,640 --> 00:06:40,750
If I do the symmetric and
anti-symmetric combination

104
00:06:40,750 --> 00:06:48,400
of the annihilation
and creation operator,

105
00:06:48,400 --> 00:06:51,730
I over squared root 2.

106
00:06:51,730 --> 00:06:56,230
I call those a1 and a2.

107
00:07:00,720 --> 00:07:11,740
And they are nothing
else than Q and P

108
00:07:11,740 --> 00:07:20,560
normalized by the characteristic
momentum of spatial coordinates

109
00:07:20,560 --> 00:07:23,560
of the harmonic oscillator.

110
00:07:23,560 --> 00:07:29,460
So what is important
here is that a1 and a2--

111
00:07:29,460 --> 00:07:30,840
forget about P,Q now.

112
00:07:30,840 --> 00:07:32,280
They are equivalent.

113
00:07:32,280 --> 00:07:35,250
But for the
electromagnetic field,

114
00:07:35,250 --> 00:07:41,670
a1 and a2 have a very
direct interpretation.

115
00:07:41,670 --> 00:07:47,180
They are called the two
quadrature operators.

116
00:07:47,180 --> 00:07:51,160
And what I mean by
that becomes clear

117
00:07:51,160 --> 00:07:55,430
when I use the
Heisenberg representation

118
00:07:55,430 --> 00:07:56,610
for the electric field.

119
00:08:00,390 --> 00:08:05,530
And I'm here using
the formula which

120
00:08:05,530 --> 00:08:12,780
is given in the book of
Weissbluth, in page 175.

121
00:08:12,780 --> 00:08:14,516
Some pages copied
from this book have

122
00:08:14,516 --> 00:08:17,160
been posted on the website.

123
00:08:17,160 --> 00:08:22,470
So we have our
normalization factor,

124
00:08:22,470 --> 00:08:30,950
which is related to the electric
field of a single photon.

125
00:08:30,950 --> 00:08:32,402
We have the polarization factor.

126
00:08:36,820 --> 00:08:42,309
But now we have an
expression which

127
00:08:42,309 --> 00:08:50,710
involves the quadrature
operators, a1 and a2.

128
00:08:56,810 --> 00:08:59,990
Just to be specific, we
are not in a cavity here.

129
00:08:59,990 --> 00:09:04,090
Therefore, we have propagating
waves cosine (kr) sine (kr).

130
00:09:04,090 --> 00:09:08,600
But you can also immediately
use a similar expression

131
00:09:08,600 --> 00:09:11,970
for the case of a cavity.

132
00:09:15,190 --> 00:09:19,040
Let us specify that r equals 0.

133
00:09:19,040 --> 00:09:24,670
And then we realize what the
two quadrature operators are.

134
00:09:24,670 --> 00:09:33,290
a2 is the operator which
creates and annihilates

135
00:09:33,290 --> 00:09:36,560
an electromagnetic
field, so to speak,

136
00:09:36,560 --> 00:09:42,670
photons, which have
an electric field.

137
00:09:42,670 --> 00:09:46,970
Which oscillates
as cosine omega t.

138
00:09:46,970 --> 00:09:52,756
And a1 is the quadrature
operator for the sine omega t

139
00:09:52,756 --> 00:09:53,255
component.

140
00:09:56,250 --> 00:10:00,590
So therefore, if you simply
analyze the electric field,

141
00:10:00,590 --> 00:10:05,830
what is cosine omega 2 is
related to the a2 quadrature

142
00:10:05,830 --> 00:10:06,740
operator.

143
00:10:06,740 --> 00:10:11,270
The sine omega 2 oscillation
is related to the a1 quadrature

144
00:10:11,270 --> 00:10:11,770
operator.

145
00:10:15,240 --> 00:10:18,850
So life would be
easier, but more boring,

146
00:10:18,850 --> 00:10:23,330
if you could create a
pure cosine, or pure sine

147
00:10:23,330 --> 00:10:25,640
oscillation of the
electromagnetic field.

148
00:10:25,640 --> 00:10:31,210
But you can't because a1
and a2 do not commute.

149
00:10:31,210 --> 00:10:33,730
And there is an
uncertainty relation

150
00:10:33,730 --> 00:10:39,750
that delta a1 times delta a2
is larger or equal to 1/2.

151
00:10:39,750 --> 00:10:45,189
And we have the equal sign
for coherent states alpha.

152
00:10:51,240 --> 00:10:57,250
So therefore, if we look
at the electric field,

153
00:10:57,250 --> 00:10:58,810
you know, everything
moves around

154
00:10:58,810 --> 00:11:02,390
periodically in r because
it's a traveling wave and t

155
00:11:02,390 --> 00:11:04,270
because it's an
oscillating wave.

156
00:11:04,270 --> 00:11:08,490
So let's not confuse things
with simply peak r equals 0.

157
00:11:08,490 --> 00:11:09,730
We've already done that.

158
00:11:09,730 --> 00:11:12,500
But now let's peak t equals 0.

159
00:11:12,500 --> 00:11:14,980
That t equals 0.

160
00:11:14,980 --> 00:11:17,370
The sine omega t is 0.

161
00:11:17,370 --> 00:11:24,360
And therefore, the distribution
for a2, the expectation value,

162
00:11:24,360 --> 00:11:29,330
and the variance for the
quadrature operator, a2,

163
00:11:29,330 --> 00:11:32,360
can be simply read off by
looking at the electric field.

164
00:11:36,340 --> 00:11:48,530
So, in other words,
at t equals 0,

165
00:11:48,530 --> 00:11:51,010
the electric field,
which is obtained

166
00:11:51,010 --> 00:11:56,800
by projecting our quasi
probabilities on the y-axis,

167
00:11:56,800 --> 00:12:02,110
gives the expectation
value for a1

168
00:12:02,110 --> 00:12:11,205
and the variant,
delta a1 squared.

169
00:12:16,580 --> 00:12:18,190
Somebody says 2.

170
00:12:18,190 --> 00:12:20,811
That t equals 0.

171
00:12:20,811 --> 00:12:21,310
Yes.

172
00:12:26,530 --> 00:12:27,030
Yes.

173
00:12:30,140 --> 00:12:33,740
And now, if you want to see what
the other quadrature component

174
00:12:33,740 --> 00:12:34,780
is, well.

175
00:12:34,780 --> 00:12:38,790
We just wait a quarter period
until the sine, which was 0,

176
00:12:38,790 --> 00:12:39,740
is maximum.

177
00:12:39,740 --> 00:12:42,650
And the cosine omega t is 0.

178
00:12:42,650 --> 00:12:53,360
So therefore, it's
pi over 2 omega,

179
00:12:53,360 --> 00:12:58,465
using the projection on
the y-axis, gives us a1.

180
00:13:01,060 --> 00:13:03,725
And delta a1 squared.

181
00:13:12,000 --> 00:13:16,780
Or, alternatively, we
don't need to wait.

182
00:13:16,780 --> 00:13:19,910
We can do t equals 0.

183
00:13:19,910 --> 00:13:21,685
And we can project
onto the x-axis.

184
00:13:43,570 --> 00:13:46,100
Let me just throw few
things into this diagram.

185
00:13:46,100 --> 00:13:48,750
If you had a classical
motion, which

186
00:13:48,750 --> 00:13:59,220
would simply be cosine omega t.

187
00:13:59,220 --> 00:14:11,280
Then that would mean, if
you had a motion which

188
00:14:11,280 --> 00:14:21,100
where only cosine
omega t-- yes, it

189
00:14:21,100 --> 00:14:24,590
would be a point on the y-axis.

190
00:14:24,590 --> 00:14:27,310
However, classically
we can never

191
00:14:27,310 --> 00:14:30,180
have something which
is just cosine omega t.

192
00:14:30,180 --> 00:14:33,800
The point has to
blurred into a circle.

193
00:14:33,800 --> 00:14:37,400
This is the coherent state.

194
00:14:37,400 --> 00:14:42,940
So this coherent state has
now-- let me just call it 1,

195
00:14:42,940 --> 00:14:44,870
for the sake of the
argument-- the point

196
00:14:44,870 --> 00:14:46,980
would be the
classical oscillator.

197
00:14:46,980 --> 00:14:48,810
It is just cosine omega t.

198
00:14:48,810 --> 00:14:50,390
Everything is deterministic.

199
00:14:50,390 --> 00:14:51,310
No uncertainty.

200
00:14:51,310 --> 00:14:54,110
No nothing.

201
00:14:54,110 --> 00:14:57,450
Of course, it means that
the time of evolution,

202
00:14:57,450 --> 00:14:58,920
it goes in a circle.

203
00:14:58,920 --> 00:15:02,170
But this is what everything
does in an harmonic oscillator

204
00:15:02,170 --> 00:15:07,540
when time evolves into the
e to the i omega t factor.

205
00:15:07,540 --> 00:15:09,260
So let's not get
confused with it.

206
00:15:09,260 --> 00:15:11,405
Let's just look at t equals 0.

207
00:15:14,080 --> 00:15:17,400
And we input [INAUDIBLE]
if r equals 0,

208
00:15:17,400 --> 00:15:20,790
the classical
oscillator is one point.

209
00:15:20,790 --> 00:15:23,520
But now, we have a spread here.

210
00:15:23,520 --> 00:15:26,790
This says that trying
to mechanically

211
00:15:26,790 --> 00:15:29,050
the amplitude of the
cosine omega t term

212
00:15:29,050 --> 00:15:31,236
is not entirely false.

213
00:15:31,236 --> 00:15:32,360
It's not [INAUDIBLE] sharp.

214
00:15:32,360 --> 00:15:34,010
There are fluctuations.

215
00:15:34,010 --> 00:15:38,760
And in addition, we have
ellipse in this direction,

216
00:15:38,760 --> 00:15:40,900
which we project
onto the x-axis.

217
00:15:40,900 --> 00:15:44,490
And this tells us what the
distribution in our ensemble,

218
00:15:44,490 --> 00:15:46,110
in our kind of
mechanical ensemble,

219
00:15:46,110 --> 00:15:49,500
is for the amplitudes of
the sine omega t motion.

220
00:15:52,580 --> 00:15:55,910
So the best we can do
is try to mechanically--

221
00:15:55,910 --> 00:16:00,350
if you want to have something,
which is really just cosine

222
00:16:00,350 --> 00:16:02,280
omega t.

223
00:16:02,280 --> 00:16:08,500
We have to squeeze it, that the
cosine omega t amplitude is now

224
00:16:08,500 --> 00:16:10,430
extremely sharp.

225
00:16:10,430 --> 00:16:14,410
But the sine omega t
amplitude in the ensemble

226
00:16:14,410 --> 00:16:17,360
is completely smeared out.

227
00:16:17,360 --> 00:16:18,960
So this is what
we're talking about.

228
00:16:24,360 --> 00:16:27,740
Now, what I think has
confused some of you

229
00:16:27,740 --> 00:16:30,990
is what I thought was
a wonderful example.

230
00:16:30,990 --> 00:16:33,310
The classical
squeezing experiment.

231
00:16:33,310 --> 00:16:36,340
I mean, these are visuals which
will be in your head forever,

232
00:16:36,340 --> 00:16:39,630
when you saw Professor
Pritchard with a circle pendulum

233
00:16:39,630 --> 00:16:40,790
he's just pulling.

234
00:16:40,790 --> 00:16:44,340
And then the circle
squeezes into an ellipse.

235
00:16:44,340 --> 00:16:48,820
And it seems that something
here is squeezed in real space.

236
00:16:48,820 --> 00:16:50,830
But this is actually wrong.

237
00:16:50,830 --> 00:16:53,040
But how you should have
looked at the experiment,

238
00:16:53,040 --> 00:16:55,590
and I made a comment
about it, but maybe not

239
00:16:55,590 --> 00:16:57,090
emphatically enough.

240
00:16:57,090 --> 00:17:00,830
You should have really thought
about a single pendulum.

241
00:17:00,830 --> 00:17:04,579
And this single pendulum, if
it has an [INAUDIBLE] phase,

242
00:17:04,579 --> 00:17:09,400
is in a superposition of sine
omega t and cosine omega t.

243
00:17:09,400 --> 00:17:13,599
And if you pull on the string,
if you shorten and lengthen

244
00:17:13,599 --> 00:17:17,180
the pendulum, it's
sine 2 omega t.

245
00:17:17,180 --> 00:17:22,079
You will amplify the prefactor
in front of sine omega t.

246
00:17:22,079 --> 00:17:25,020
And you will exponentially
de-amplify the factor

247
00:17:25,020 --> 00:17:27,440
in front of cosine omega t.

248
00:17:27,440 --> 00:17:31,420
So therefore, what will happen
if this pendulum oscillates--

249
00:17:31,420 --> 00:17:37,080
and let me say with a phase,
well, sine omega t plus delta,

250
00:17:37,080 --> 00:17:38,650
if delta is 90 degrees.

251
00:17:38,650 --> 00:17:39,940
Cosine if delta is 0.

252
00:17:39,940 --> 00:17:40,930
It's sine.

253
00:17:40,930 --> 00:17:45,580
And let's say this pendulum
oscillates at 45 degrees.

254
00:17:45,580 --> 00:17:49,100
Sine omega t plus 45 degrees.

255
00:17:49,100 --> 00:17:54,170
If you now parametrically
derive it with squeezing action,

256
00:17:54,170 --> 00:17:58,504
it would now mean
that you, let's

257
00:17:58,504 --> 00:17:59,795
just make it a sine convention.

258
00:17:59,795 --> 00:18:01,470
You de-amplify the cosine.

259
00:18:01,470 --> 00:18:02,850
You amplify the sine.

260
00:18:02,850 --> 00:18:06,320
And after a while, instead of
oscillating with sine omega

261
00:18:06,320 --> 00:18:09,130
t plus 45 degrees,
it will oscillate

262
00:18:09,130 --> 00:18:12,676
with an amplified
amplitude at sine omega t.

263
00:18:12,676 --> 00:18:13,800
This is what you have done.

264
00:18:16,790 --> 00:18:20,430
And this is a
mechanical analogy.

265
00:18:20,430 --> 00:18:26,070
There is, of course,
no squeezing in any way

266
00:18:26,070 --> 00:18:29,010
because in a
classical pendulum, we

267
00:18:29,010 --> 00:18:32,590
start with one definite value.

268
00:18:32,590 --> 00:18:34,250
If you prepare the system well.

269
00:18:34,250 --> 00:18:37,840
And then, we just
change the motion.

270
00:18:37,840 --> 00:18:38,910
We amplify.

271
00:18:38,910 --> 00:18:42,430
We pick out a phase and
that's what we are doing.

272
00:18:42,430 --> 00:18:46,580
Now the true ways how classical
squeezing can come in.

273
00:18:46,580 --> 00:18:53,320
One it is if the motion of
the pendulum is-- maybe there

274
00:18:53,320 --> 00:18:54,640
is an uncertainty.

275
00:18:54,640 --> 00:18:58,230
Maybe Professor Pritchard did
experiments with an ion trap.

276
00:18:58,230 --> 00:19:00,770
And actually, 20 years
ago, he published

277
00:19:00,770 --> 00:19:02,770
a [INAUDIBLE] [? letter ?]
on classic squeezing.

278
00:19:02,770 --> 00:19:04,330
And you think, how
can you publish

279
00:19:04,330 --> 00:19:06,330
[INAUDIBLE] [? letter ?]
on classical squeezing?

280
00:19:06,330 --> 00:19:10,000
Well, he had developed the
world's most accurate ion trap,

281
00:19:10,000 --> 00:19:15,540
measuring atomic masses with
10 and 11 digit positions.

282
00:19:15,540 --> 00:19:19,340
And what was actually
one limiting factor

283
00:19:19,340 --> 00:19:25,080
was for Kelvin, the
thermal distribution

284
00:19:25,080 --> 00:19:27,420
of harmonic oscillator modes.

285
00:19:27,420 --> 00:19:33,870
And so, he didn't have
just one clean amplitude.

286
00:19:33,870 --> 00:19:36,540
The sine omega t
amplitudes had a spread

287
00:19:36,540 --> 00:19:40,670
because of the thermal
distribution he started from.

288
00:19:40,670 --> 00:19:44,490
And so what he then did is,
by simply classical squeezing

289
00:19:44,490 --> 00:19:46,640
by doing classically
with the ion trap

290
00:19:46,640 --> 00:19:50,330
exactly what he did with the
pendulum, derive sine 2 omega

291
00:19:50,330 --> 00:19:59,410
t, he could now take this
classic distribution.

292
00:19:59,410 --> 00:20:03,080
This is a classic distribution.

293
00:20:03,080 --> 00:20:05,820
In one axis, it's a
distribution of amplitudes

294
00:20:05,820 --> 00:20:07,230
of the cosine motion.

295
00:20:07,230 --> 00:20:10,500
And here it's the
classic distribution

296
00:20:10,500 --> 00:20:12,610
of the amplitudes
of the sine motion.

297
00:20:12,610 --> 00:20:16,710
And he was squeezing
it into this direction.

298
00:20:16,710 --> 00:20:19,950
So he had a narrower
definition of the coefficient

299
00:20:19,950 --> 00:20:22,150
for the cosine omega t motion.

300
00:20:22,150 --> 00:20:24,330
And as I will tell
you today, you

301
00:20:24,330 --> 00:20:26,420
can now do a
homodyne measurement,

302
00:20:26,420 --> 00:20:28,940
which is reducing the noise.

303
00:20:28,940 --> 00:20:32,130
So essentially, he prepared,
quote unquote "Effectively

304
00:20:32,130 --> 00:20:37,460
a [? code ?] ensemble by
squeezing the uncertainty

305
00:20:37,460 --> 00:20:38,870
in the cosine prefactor."

306
00:20:38,870 --> 00:20:41,640
Or at the expense of
increasing the prefactor,

307
00:20:41,640 --> 00:20:44,420
the uncertainty, the variance,
in prefactor of the sine omega

308
00:20:44,420 --> 00:20:44,950
t motion.

309
00:20:48,490 --> 00:20:53,100
Finally, you all saw
something visually.

310
00:20:53,100 --> 00:20:58,420
You saw how a circular motion
became a linear motion.

311
00:20:58,420 --> 00:21:01,060
So what was going on here?

312
00:21:01,060 --> 00:21:05,230
Well, I mentioned to you that
the circular pendulum actually

313
00:21:05,230 --> 00:21:06,080
has two modes.

314
00:21:06,080 --> 00:21:08,690
These are two modes of
the harmonic oscillator.

315
00:21:08,690 --> 00:21:10,180
And I'm not talking
about two modes

316
00:21:10,180 --> 00:21:11,305
of the harmonic oscillator.

317
00:21:11,305 --> 00:21:13,670
Everything we're discussing
here is about one mode

318
00:21:13,670 --> 00:21:15,750
of the harmonic oscillator.

319
00:21:15,750 --> 00:21:18,320
The circular motion
of the pendulum

320
00:21:18,320 --> 00:21:21,710
was just a nice
visualization trick

321
00:21:21,710 --> 00:21:25,030
that, if the pendulum
moves in a circle,

322
00:21:25,030 --> 00:21:27,480
you have a degenerate
harmonic oscillator.

323
00:21:27,480 --> 00:21:30,230
One is excited
with sine omega t.

324
00:21:30,230 --> 00:21:33,540
The other one is excited
with cosine omega t.

325
00:21:33,540 --> 00:21:36,720
And instead of doing
two experiments,

326
00:21:36,720 --> 00:21:39,110
if you would start
with sine omega t,

327
00:21:39,110 --> 00:21:42,560
and you parametrically
drive it, you amplified it.

328
00:21:42,560 --> 00:21:44,710
If you start with
cosine omega t,

329
00:21:44,710 --> 00:21:47,560
you could bring the
pendulum to a stop.

330
00:21:47,560 --> 00:21:50,200
But instead of doing
two experiments,

331
00:21:50,200 --> 00:21:52,090
Professor Pritchard
just did one.

332
00:21:52,090 --> 00:21:55,990
And he showed that the sine
omega t motions became larger.

333
00:21:55,990 --> 00:22:00,580
And the orthogonal cosine
omega t motion shrank.

334
00:22:00,580 --> 00:22:04,180
And therefore, you saw that
the circular motion, which

335
00:22:04,180 --> 00:22:06,440
was a superposition
of sine and cosine,

336
00:22:06,440 --> 00:22:11,709
became [? pure ?] sine motion.

337
00:22:11,709 --> 00:22:13,250
But the fact that
there was something

338
00:22:13,250 --> 00:22:15,410
we could see in
the spatial domain

339
00:22:15,410 --> 00:22:20,300
was simply due to the fact that
we had two experiments in one.

340
00:22:20,300 --> 00:22:22,850
Two versions of the same
harmonic oscillator,

341
00:22:22,850 --> 00:22:24,670
one in x and one in y.

342
00:22:24,670 --> 00:22:26,710
And then, when we
did the experiment,

343
00:22:26,710 --> 00:22:31,300
we saw something visually
in the spatial domain.

344
00:22:31,300 --> 00:22:34,820
So that's why we saw squeezing
in the spatial domain.

345
00:22:34,820 --> 00:22:36,830
But you should really
think about it.

346
00:22:36,830 --> 00:22:39,490
What the whole
action is, is it's

347
00:22:39,490 --> 00:22:46,420
an interplay of de-amplifying
prefactors of cosine amplifying

348
00:22:46,420 --> 00:22:48,110
prefactors of sine.

349
00:22:48,110 --> 00:22:51,110
And if the prefactor
has a distribution,

350
00:22:51,110 --> 00:22:53,900
by de-amplifying
it, you also shrink

351
00:22:53,900 --> 00:22:56,652
the reach of the distribution.

352
00:22:56,652 --> 00:22:58,110
And this is what
we call squeezing.

353
00:23:00,910 --> 00:23:01,644
Yes, [INAUDIBLE]?

354
00:23:01,644 --> 00:23:02,727
AUDIENCE: Stupid question.

355
00:23:02,727 --> 00:23:05,840
So the operators, a1
and a2 here, right?

356
00:23:05,840 --> 00:23:12,989
You use those instead of a and
a dagger because you use cosine

357
00:23:12,989 --> 00:23:14,905
and sine rather than
[? e  to the i omega t ?]

358
00:23:14,905 --> 00:23:16,330
[INAUDIBLE]?

359
00:23:16,330 --> 00:23:20,610
Because they should
contain [INAUDIBLE], right?

360
00:23:20,610 --> 00:23:22,470
PROFESSOR: Let me go
back to the definition.

361
00:23:22,470 --> 00:23:30,110
They are actually exactly
they correspond exactly

362
00:23:30,110 --> 00:23:34,874
to position and momentum of the
mechanical harmonic oscillator.

363
00:23:34,874 --> 00:23:35,415
AUDIENCE: Oh.

364
00:23:35,415 --> 00:23:36,305
That makes sense.

365
00:23:36,305 --> 00:23:38,085
Another thing is,
technically speaking,

366
00:23:38,085 --> 00:23:40,440
we could call the
[? Basco ?] electric field

367
00:23:40,440 --> 00:23:47,000
e cos kr minus omega
t as maximum squeezed

368
00:23:47,000 --> 00:23:49,000
if we only had the
cos components?

369
00:23:56,240 --> 00:24:05,205
PROFESSOR: Well,
not if I use it--

370
00:24:05,205 --> 00:24:07,010
it depends how I
define squeezing.

371
00:24:10,450 --> 00:24:14,320
So you would now
give a definition

372
00:24:14,320 --> 00:24:24,050
of squeezing which says
that the variance in a1

373
00:24:24,050 --> 00:24:30,510
is now unequal to
the variance in a2.

374
00:24:30,510 --> 00:24:33,080
So the classical
oscillator is the point.

375
00:24:33,080 --> 00:24:35,400
It has 0 variance in a1.

376
00:24:35,400 --> 00:24:37,630
0 variance in a2.

377
00:24:37,630 --> 00:24:40,220
But as I said, you can
actually apply all the way

378
00:24:40,220 --> 00:24:42,670
to a classical
oscillator if you add

379
00:24:42,670 --> 00:24:44,960
technical noise
or thermal noise.

380
00:24:44,960 --> 00:24:48,740
Then your system is prepared,
not with a sharp value,

381
00:24:48,740 --> 00:24:51,510
but with a distribution
which is simply

382
00:24:51,510 --> 00:24:54,920
may be [? false ?] distribution,
due to the preparation.

383
00:24:54,920 --> 00:25:00,600
So recall, it's
squeezing when the noise

384
00:25:00,600 --> 00:25:02,770
in the amplitude
of the sine motion

385
00:25:02,770 --> 00:25:07,990
is not equal to the noise in the
amplitude of the cosine motion.

386
00:25:07,990 --> 00:25:10,180
Some people say, if it's
a little bit narrower,

387
00:25:10,180 --> 00:25:18,220
they apply squeezing
to the situation

388
00:25:18,220 --> 00:25:27,700
that we are uncertainty limited.

389
00:25:27,700 --> 00:25:29,830
And then we squeeze.

390
00:25:29,830 --> 00:25:38,180
But of course, you can always
reduce the noise in your system

391
00:25:38,180 --> 00:25:40,060
by just preparing the system.

392
00:25:40,060 --> 00:25:42,210
By cooling the system.

393
00:25:42,210 --> 00:25:49,170
By selecting the system for
measurements, until you reach

394
00:25:49,170 --> 00:25:50,710
the quantum limit.

395
00:25:50,710 --> 00:25:54,750
So you can get a smaller
delta a1, a smaller delta a2,

396
00:25:54,750 --> 00:25:57,330
without squeezing, just
better preparation.

397
00:25:57,330 --> 00:25:59,800
Or by selecting your ensemble.

398
00:25:59,800 --> 00:26:05,180
So squeezing in a narrower
sense only makes sense

399
00:26:05,180 --> 00:26:08,150
when you hit the limit of what
quantum mechanics allows you.

400
00:26:08,150 --> 00:26:13,010
And now you want to distribute
the variance unequally

401
00:26:13,010 --> 00:26:19,120
between a1 and a2
because then you

402
00:26:19,120 --> 00:26:21,960
can get something in
delta a1 or delta a2.

403
00:26:21,960 --> 00:26:25,830
Which is better then
1 over square root 2.

404
00:26:25,830 --> 00:26:30,002
And this is now really
quantum mechanically squeezed.

405
00:26:30,002 --> 00:26:31,210
But they're both definitions.

406
00:26:31,210 --> 00:26:32,730
Classical squeezing exists.

407
00:26:32,730 --> 00:26:37,534
It's just not as common as
quantum mechanical squeezing.

408
00:26:37,534 --> 00:26:38,200
Other questions?

409
00:26:41,020 --> 00:26:41,710
Yes?

410
00:26:41,710 --> 00:26:43,670
AUDIENCE: I remember
that you said

411
00:26:43,670 --> 00:26:45,875
that in the classical
squeezing, you

412
00:26:45,875 --> 00:26:48,170
are attenuating one amplitude.

413
00:26:48,170 --> 00:26:51,230
And you were amplifying
the other amplitude.

414
00:26:51,230 --> 00:26:53,050
So, in this picture
then, shouldn't we

415
00:26:53,050 --> 00:26:56,370
have the ellipse come down
from the circle on the y-axis?

416
00:26:59,500 --> 00:27:00,500
PROFESSOR: OK.

417
00:27:00,500 --> 00:27:02,750
AUDIENCE: [INAUDIBLE] not
just changing delta a,

418
00:27:02,750 --> 00:27:05,742
but also changing b [INAUDIBLE].

419
00:27:05,742 --> 00:27:07,575
PROFESSOR: Let me just
get a sketch up here.

420
00:27:10,080 --> 00:27:11,960
So this is a2.

421
00:27:11,960 --> 00:27:13,680
This is a1.

422
00:27:13,680 --> 00:27:16,340
a2 is for the cosine omega t.

423
00:27:16,340 --> 00:27:18,360
And a1 for the sine omega t.

424
00:27:18,360 --> 00:27:19,740
So just to be specific.

425
00:27:19,740 --> 00:27:23,320
So you want to prepare an
harmonic oscillator, which

426
00:27:23,320 --> 00:27:26,500
is just sine omega t.

427
00:27:26,500 --> 00:27:28,990
This is a point here.

428
00:27:28,990 --> 00:27:35,990
If we are now parametrically--
so this has a value at t

429
00:27:35,990 --> 00:27:37,610
equals 0.

430
00:27:37,610 --> 00:27:43,050
Our a1 is not.

431
00:27:43,050 --> 00:27:47,920
So if you are now squeezing your
classical harmonic oscillator,

432
00:27:47,920 --> 00:27:54,900
you would have a situation
where a1 of t is s

433
00:27:54,900 --> 00:28:01,310
naught times e to the plus
or minus t, depending on

434
00:28:01,310 --> 00:28:04,610
whether you do the parametric
[? drive ?] at sine 2 omega

435
00:28:04,610 --> 00:28:07,110
t or cosine 2 omega t.

436
00:28:07,110 --> 00:28:10,590
So therefore, what
would happen is

437
00:28:10,590 --> 00:28:16,580
this point will be amplified.

438
00:28:16,580 --> 00:28:21,800
That would mean it would
just move out on the x-axis.

439
00:28:21,800 --> 00:28:24,550
So this would be
for the plus sign.

440
00:28:24,550 --> 00:28:29,320
Or, for the other case, you
would damp the motion to 0.

441
00:28:29,320 --> 00:28:32,579
And this is a minus sign here.

442
00:28:32,579 --> 00:28:33,120
AUDIENCE: OK.

443
00:28:33,120 --> 00:28:37,130
And you were also
saying the variance--

444
00:28:37,130 --> 00:28:39,080
PROFESSOR: A point
does not have variance.

445
00:28:39,080 --> 00:28:40,750
AUDIENCE: [INAUDIBLE].

446
00:28:40,750 --> 00:28:43,300
PROFESSOR: So if you
want to build a variance,

447
00:28:43,300 --> 00:28:45,500
you need, let's
say, three points.

448
00:28:45,500 --> 00:28:46,900
One is the average value.

449
00:28:46,900 --> 00:28:48,180
One is the left outlier.

450
00:28:48,180 --> 00:28:49,870
One is the right outlier.

451
00:28:49,870 --> 00:28:54,130
And what happens is now,
as you amplify the motion,

452
00:28:54,130 --> 00:28:57,040
you would also amplify,
magnify the distance

453
00:28:57,040 --> 00:28:58,830
between the points.

454
00:28:58,830 --> 00:29:03,040
And if you de-amplify
it with a minus sign,

455
00:29:03,040 --> 00:29:04,660
the distance between
the points would

456
00:29:04,660 --> 00:29:08,700
shrink because all the
three points converged to 0.

457
00:29:08,700 --> 00:29:10,280
So pretty much what
I just told you

458
00:29:10,280 --> 00:29:12,190
for the three point ensemble.

459
00:29:12,190 --> 00:29:16,480
You can now use it and construct
any initial condition you want.

460
00:29:16,480 --> 00:29:18,200
And see what happens
due to squeezing.

461
00:29:24,080 --> 00:29:26,941
Other questions?

462
00:29:26,941 --> 00:29:27,440
Good.

463
00:29:39,770 --> 00:29:46,387
So the question now is,
how to measure squeezing.

464
00:29:46,387 --> 00:29:47,845
How to take advantage
of squeezing.

465
00:30:02,090 --> 00:30:06,490
So the situation we are
facing is the following.

466
00:30:06,490 --> 00:30:16,360
Let's assume we have done
a nice squeezing job.

467
00:30:16,360 --> 00:30:21,190
And that means that
we have a sharp value.

468
00:30:21,190 --> 00:30:24,270
We have created a
narrow distribution

469
00:30:24,270 --> 00:30:26,910
of the cosign omega
2 coefficients.

470
00:30:26,910 --> 00:30:30,790
So the cosine omega 2
motion is rather sharp.

471
00:30:30,790 --> 00:30:36,240
But we also know that
the electric field itself

472
00:30:36,240 --> 00:30:38,340
is sharp at this moment.

473
00:30:38,340 --> 00:30:52,970
But since this ellipse
rotates, the electric field

474
00:30:52,970 --> 00:30:57,290
will have enormous uncertainty
a [INAUDIBLE] period later.

475
00:30:57,290 --> 00:31:02,430
So if we want to take
advantage that we have squeezed

476
00:31:02,430 --> 00:31:05,520
the electromagnetic
field, they are now

477
00:31:05,520 --> 00:31:08,140
a couple of ideas
which we can use.

478
00:31:08,140 --> 00:31:11,250
One is we could just
measure the electric fields

479
00:31:11,250 --> 00:31:12,430
stroboscopically.

480
00:31:12,430 --> 00:31:14,310
We would just make
a setup, where

481
00:31:14,310 --> 00:31:16,000
we look at this system
in a [INAUDIBLE]

482
00:31:16,000 --> 00:31:17,070
measurement process.

483
00:31:17,070 --> 00:31:19,760
Only, we only measure
the electric field

484
00:31:19,760 --> 00:31:24,052
when the ellipse is like
this or is like this.

485
00:31:24,052 --> 00:31:26,510
And therefore, we have a sharp
value of the electric field.

486
00:31:29,650 --> 00:31:33,150
But instead of doing a
stroboscopic measurement,

487
00:31:33,150 --> 00:31:34,940
we can do something else.

488
00:31:34,940 --> 00:31:52,520
Remember, we have a
distribution of cosine omega t

489
00:31:52,520 --> 00:31:55,390
and sine omega t.

490
00:31:55,390 --> 00:32:02,090
And we have a distribution
of co efficiency and s.

491
00:32:02,090 --> 00:32:06,600
And we know we are interested
in the cosine omega t.

492
00:32:06,600 --> 00:32:09,250
So how to pick that
out has actually

493
00:32:09,250 --> 00:32:13,930
been solved in
early [? radius ?].

494
00:32:13,930 --> 00:32:15,560
You do a homodyne detection.

495
00:32:15,560 --> 00:32:21,290
In other words, you take
a reference oscillator,

496
00:32:21,290 --> 00:32:23,460
which is strong.

497
00:32:23,460 --> 00:32:31,020
B of t is b naught times
cosine omega naught plus delta.

498
00:32:31,020 --> 00:32:40,700
And if you if you
multiply the two signals,

499
00:32:40,700 --> 00:32:42,705
your signal you are
interested in, or at least

500
00:32:42,705 --> 00:32:44,560
you're interested
in one comportment,

501
00:32:44,560 --> 00:32:49,650
you multiply it with
your local oscillator.

502
00:32:49,650 --> 00:32:51,630
And then you
[? indicate ?] over time.

503
00:32:57,130 --> 00:33:06,460
Then, of course, when you
peak the phase delta to be 0,

504
00:33:06,460 --> 00:33:09,840
cosine omega t times
cosine omega [? dt ?]

505
00:33:09,840 --> 00:33:11,620
gives cosine squared omega t.

506
00:33:11,620 --> 00:33:12,740
It averages to 1/2.

507
00:33:12,740 --> 00:33:15,790
So you would times average
where cosine omega t

508
00:33:15,790 --> 00:33:18,260
times sine omega
t averages to 0.

509
00:33:18,260 --> 00:33:23,050
So for delta equals
0, you project out

510
00:33:23,050 --> 00:33:24,990
the cosine component.

511
00:33:24,990 --> 00:33:27,540
And for delta equal
to 90 degrees,

512
00:33:27,540 --> 00:33:29,530
you peak out the sine component.

513
00:33:33,150 --> 00:33:37,240
So therefore, you can
have a measurement,

514
00:33:37,240 --> 00:33:40,110
it's a phase
sensitive measurement,

515
00:33:40,110 --> 00:33:43,725
by multiplying your signal with
the local oscillator, where

516
00:33:43,725 --> 00:33:47,860
you're only sensitive to the
component you have squeezed.

517
00:33:47,860 --> 00:33:50,320
And therefore, your
measurement uncertainty

518
00:33:50,320 --> 00:33:52,397
has now been reduced by
the squeezing factor.

519
00:34:01,163 --> 00:34:03,598
AUDIENCE: [INAUDIBLE] only
that should be the [INAUDIBLE].

520
00:34:07,500 --> 00:34:09,310
PROFESSOR: Yes,
actually, homodyne

521
00:34:09,310 --> 00:34:12,520
means we use the same frequency.

522
00:34:12,520 --> 00:34:18,030
Heterodyning would mean we
use two different frequencies.

523
00:34:18,030 --> 00:34:19,400
But I'm not talking about that.

524
00:34:22,250 --> 00:34:24,820
So we have to use exactly
the same frequency here.

525
00:34:24,820 --> 00:34:27,285
AUDIENCE: So instead, this
[INAUDIBLE] oscillation

526
00:34:27,285 --> 00:34:30,243
also needs a laser
to [INAUDIBLE]?

527
00:34:30,243 --> 00:34:32,219
PROFESSOR: Yes.

528
00:34:32,219 --> 00:34:35,989
So, to address your question,
Angie, what usually happens

529
00:34:35,989 --> 00:34:40,330
is you start with one
laser in those experiments.

530
00:34:40,330 --> 00:34:42,514
You frequence the
top of the laser.

531
00:34:47,080 --> 00:34:48,850
If you wanted to
do some squeezing,

532
00:34:48,850 --> 00:34:53,130
you remember that we need a
parametric oscillator, where

533
00:34:53,130 --> 00:34:57,240
one energetic photon
gives us two photons.

534
00:34:57,240 --> 00:35:01,920
So what you do is you start
with a laser, [INAUDIBLE].

535
00:35:01,920 --> 00:35:03,970
A 1064 nanometer.

536
00:35:03,970 --> 00:35:06,260
You frequency double
it to green laser.

537
00:35:06,260 --> 00:35:09,580
The green laser pumps your
parametric oscillator.

538
00:35:09,580 --> 00:35:12,300
And then you get,
for down conversion,

539
00:35:12,300 --> 00:35:15,470
you can squeeze light at 1064.

540
00:35:15,470 --> 00:35:18,150
But this is because you
first stopper the laser.

541
00:35:18,150 --> 00:35:20,350
And then you break the
photon into new pieces.

542
00:35:20,350 --> 00:35:24,460
It has exactly the same
frequency as your laser

543
00:35:24,460 --> 00:35:25,490
you started with.

544
00:35:25,490 --> 00:35:28,980
And this laser is in the local
oscillator, or the reference

545
00:35:28,980 --> 00:35:31,360
clock, for your
whole experiment.

546
00:35:31,360 --> 00:35:35,380
So everything in your
experiment, the double laser,

547
00:35:35,380 --> 00:35:37,910
the parametrically
down converted beams.

548
00:35:37,910 --> 00:35:43,230
Everything is related to the
single laser you started with.

549
00:35:43,230 --> 00:35:44,950
And everything is
phase coherent.

550
00:35:44,950 --> 00:35:47,044
So that's how, usually,
the experiment is done.

551
00:35:51,330 --> 00:35:53,770
Before I tell you what we're
doing quantum mechanically,

552
00:35:53,770 --> 00:35:55,660
let me just also
get another question

553
00:35:55,660 --> 00:35:59,040
out of the system, which I've
been asked several times.

554
00:35:59,040 --> 00:36:02,010
People ask me,
well, the problem is

555
00:36:02,010 --> 00:36:05,890
that the ellipse
rotates like this.

556
00:36:05,890 --> 00:36:08,680
Isn't there a way--
now I need my hand--

557
00:36:08,680 --> 00:36:12,450
that we can have an
ellipse rotating like this?

558
00:36:12,450 --> 00:36:13,950
And that would be great.

559
00:36:13,950 --> 00:36:17,980
But this is sort of unnatural
because the harmonic oscillator

560
00:36:17,980 --> 00:36:18,970
does that.

561
00:36:18,970 --> 00:36:23,240
So if you wanted to do that,
you need an operator which

562
00:36:23,240 --> 00:36:26,990
is really, at every cycle of
the electromagnetic field,

563
00:36:26,990 --> 00:36:29,300
is when the light want
to sort of do this.

564
00:36:29,300 --> 00:36:30,080
No.

565
00:36:30,080 --> 00:36:31,380
Always push it back.

566
00:36:31,380 --> 00:36:32,810
And this is impractical.

567
00:36:32,810 --> 00:36:35,200
You need, really, an oscillator
which would completely

568
00:36:35,200 --> 00:36:39,080
change the quadrature components
of your harmonic oscillator

569
00:36:39,080 --> 00:36:42,730
in every single cycle of
the electromagnetic field.

570
00:36:42,730 --> 00:36:46,570
But sort of what homodyne
detection is, instead

571
00:36:46,570 --> 00:36:50,860
of now forcing the light to
stay aligned to sort of do

572
00:36:50,860 --> 00:36:53,050
this, which is
very unnatural, we

573
00:36:53,050 --> 00:36:55,700
allow the light
to freely evolve.

574
00:36:55,700 --> 00:36:59,560
But we have now an observer,
our local oscillator,

575
00:36:59,560 --> 00:37:04,180
which is rotating
synchronously with the ellipse.

576
00:37:04,180 --> 00:37:07,610
So we have a local oscillator
which is cosine omega t.

577
00:37:07,610 --> 00:37:10,920
It does, so to speak, exactly
what the ellipse is doing.

578
00:37:10,920 --> 00:37:13,070
So in that sense,
the local oscillator

579
00:37:13,070 --> 00:37:16,960
allows us, now, to
observe the ellipse always

580
00:37:16,960 --> 00:37:17,957
from its narrow side.

581
00:37:17,957 --> 00:37:20,040
Because the local oscillator
is [? cooperating ?].

582
00:37:22,504 --> 00:37:24,670
But the mathematics is
pretty much the [? Fourier ?]

583
00:37:24,670 --> 00:37:25,170
transform.

584
00:37:25,170 --> 00:37:30,262
The mathematics is a
[? Fourier ?] transform.

585
00:37:30,262 --> 00:37:32,470
The physics is the physics
of a [INAUDIBLE] detector.

586
00:37:38,780 --> 00:37:48,750
OK, now the only question
remaining is, how do we mix?

587
00:37:48,750 --> 00:37:53,850
How do we get a
product of our signal

588
00:37:53,850 --> 00:37:57,290
and the strong local oscillator?

589
00:37:57,290 --> 00:38:00,430
In an old radio, it's
done by an element.

590
00:38:00,430 --> 00:38:03,100
Maybe a diode, which
is a nonlinear circuit.

591
00:38:03,100 --> 00:38:06,360
If you drive a nonlinear
element with two input sources,

592
00:38:06,360 --> 00:38:10,290
you get something which
involves the product of the two.

593
00:38:15,630 --> 00:38:20,980
OK, so the principal
of homodyne detection

594
00:38:20,980 --> 00:38:28,120
is now that we want to mix
light at the beam splitter.

595
00:38:28,120 --> 00:38:32,080
So the device which
does all that for us

596
00:38:32,080 --> 00:38:38,720
is, after using so many
words, I would say it's

597
00:38:38,720 --> 00:38:40,320
disgustingly simple.

598
00:38:40,320 --> 00:38:44,070
It's really a half [INAUDIBLE].

599
00:38:44,070 --> 00:38:45,960
We talked about
beam splitters a lot

600
00:38:45,960 --> 00:38:48,780
here because beam
splitters perform

601
00:38:48,780 --> 00:38:51,340
wonderful unitary
transformations.

602
00:38:51,340 --> 00:38:55,480
And we'll exploit them
for many purposes.

603
00:38:55,480 --> 00:39:03,640
For the purpose of
this lecture, I simply

604
00:39:03,640 --> 00:39:08,540
assume that we have a
50-50 beam splitter.

605
00:39:08,540 --> 00:39:13,420
So, what I'm telling you
now about the beam splitter

606
00:39:13,420 --> 00:39:15,260
will be generalized.

607
00:39:15,260 --> 00:39:19,970
Either later today, or in
the lecture on Wednesday.

608
00:39:19,970 --> 00:39:26,100
So a beam splitter has two
input ports and output ports.

609
00:39:26,100 --> 00:39:29,790
We have light impinging
on the beam splitter.

610
00:39:29,790 --> 00:39:34,920
We call those modes a and b.

611
00:39:34,920 --> 00:39:46,070
And after the beam splitter,
we have two output modes.

612
00:39:46,070 --> 00:39:50,820
And let me call them a0 and b0.

613
00:39:50,820 --> 00:39:57,030
And what we measure is the
output of the beam splitter.

614
00:39:57,030 --> 00:39:58,840
So we have two photo detectors.

615
00:39:58,840 --> 00:40:00,080
And we measure the output.

616
00:40:12,700 --> 00:40:14,790
OK.

617
00:40:14,790 --> 00:40:22,070
The output modes, a
naught and b naught,

618
00:40:22,070 --> 00:40:33,070
are simply obtained by
taking the input modes

619
00:40:33,070 --> 00:40:35,020
and propagating them.

620
00:40:35,020 --> 00:40:37,480
We have two modes, a, b.

621
00:40:37,480 --> 00:40:41,160
You can say, what
goes vertically is a.

622
00:40:41,160 --> 00:40:42,055
Vertical is a.

623
00:40:42,055 --> 00:40:42,930
We call it a naught.

624
00:40:42,930 --> 00:40:43,710
Here, it's b.

625
00:40:43,710 --> 00:40:45,250
It becomes b naught.

626
00:40:45,250 --> 00:40:52,480
And what we have
to do is we have

627
00:40:52,480 --> 00:41:04,915
two transform the
operator a naught.

628
00:41:04,915 --> 00:41:06,040
Let me just make a comment.

629
00:41:08,332 --> 00:41:09,790
Sometimes in the
beam splitter, you

630
00:41:09,790 --> 00:41:14,210
want to think a quantum state
comes and is transformed.

631
00:41:14,210 --> 00:41:17,180
But instead of transforming
the state of a photon,

632
00:41:17,180 --> 00:41:19,450
I can also transform
the operator,

633
00:41:19,450 --> 00:41:20,705
which creates a photon.

634
00:41:20,705 --> 00:41:22,030
One is the Heisenberg picture.

635
00:41:22,030 --> 00:41:23,530
One is a Schrodinger picture.

636
00:41:23,530 --> 00:41:26,080
So right now, I'll use
the Heisenberg picture.

637
00:41:26,080 --> 00:41:28,160
I have two operators, a, b.

638
00:41:30,920 --> 00:41:34,500
Before the beam splitter is set,
I have two operators, a naught,

639
00:41:34,500 --> 00:41:35,200
b naught.

640
00:41:35,200 --> 00:41:38,870
Afterwards, I have
two other operators.

641
00:41:38,870 --> 00:41:40,040
I have operators a, b.

642
00:41:40,040 --> 00:41:42,520
Afterwards, i have operators
a naught, b naught.

643
00:41:42,520 --> 00:41:45,390
And, if you do the
time evolution,

644
00:41:45,390 --> 00:41:47,560
it's a unitary transformation.

645
00:41:47,560 --> 00:41:53,670
And for operators, we have
to multiply the operators

646
00:41:53,670 --> 00:41:55,560
from the left and
right hand side,

647
00:41:55,560 --> 00:42:00,350
with unitary operator,
u and u dagger.

648
00:42:00,350 --> 00:42:03,660
And we really talk
about the beam splitter

649
00:42:03,660 --> 00:42:06,720
in its full beauty
in a short while.

650
00:42:09,610 --> 00:42:11,930
It will be a special case
of what we discuss later.

651
00:42:11,930 --> 00:42:15,590
But I think it's pretty obvious
that a 50-50 beam splitters is

652
00:42:15,590 --> 00:42:18,420
simply creating two modes.

653
00:42:18,420 --> 00:42:21,040
One of the symmetric
combination.

654
00:42:21,040 --> 00:42:23,265
And one of the anti
symmetric combination.

655
00:42:29,620 --> 00:42:30,120
OK.

656
00:42:30,120 --> 00:42:36,900
So in our homodyne
detector, we are now

657
00:42:36,900 --> 00:42:42,965
measuring the number of
photons in the mode a naught.

658
00:42:45,550 --> 00:42:47,850
We measure with
the upper detector,

659
00:42:47,850 --> 00:42:50,560
the number of photons
in the mode b naught.

660
00:42:50,560 --> 00:42:53,980
But what we are then
doing is we run it

661
00:42:53,980 --> 00:42:56,810
through a different [INAUDIBLE].

662
00:42:56,810 --> 00:42:58,350
We want to cancel
certain noises,

663
00:42:58,350 --> 00:43:00,060
as you will see in a moment.

664
00:43:00,060 --> 00:43:07,000
This is why we
obtain the difference

665
00:43:07,000 --> 00:43:09,800
signal, which we call I minus.

666
00:43:09,800 --> 00:43:13,420
So let's now calculate
what I minus is.

667
00:43:13,420 --> 00:43:20,110
Well, the number of photons
in the mode a naught

668
00:43:20,110 --> 00:43:21,825
is a naught dagger a naught.

669
00:43:21,825 --> 00:43:27,470
That's the operator for
the number of photons.

670
00:43:27,470 --> 00:43:32,180
We subtract b naught
dagger b naught.

671
00:43:32,180 --> 00:43:36,770
And, as you can
immediately see, is

672
00:43:36,770 --> 00:43:41,030
that it because of the
beam splitter, this

673
00:43:41,030 --> 00:43:46,280
is now involving a
product of a and b.

674
00:43:46,280 --> 00:43:49,800
ab dagger plus a dagger b.

675
00:43:49,800 --> 00:43:52,780
So in other words,
when you ask yourself,

676
00:43:52,780 --> 00:43:57,390
how can we-- remember, I said
we peak out the cosine omega t

677
00:43:57,390 --> 00:44:00,180
component by
multiplying our signal

678
00:44:00,180 --> 00:44:03,090
with the local oscillator.

679
00:44:03,090 --> 00:44:05,550
And you would say,
how do you multiply

680
00:44:05,550 --> 00:44:09,540
two modes of the
electromagnetic field?

681
00:44:09,540 --> 00:44:12,040
Well, just send it
to a beam splitter.

682
00:44:12,040 --> 00:44:15,190
In a beam splitter, you
create the sum of them.

683
00:44:15,190 --> 00:44:18,760
But then your photo detector
is square [INAUDIBLE] detector.

684
00:44:18,760 --> 00:44:21,290
You measure the electric
fields squared or you

685
00:44:21,290 --> 00:44:24,350
have to measure the
operator, a dagger a.

686
00:44:24,350 --> 00:44:28,380
And now, you get the [INAUDIBLE]
between ab dagger and a dagger

687
00:44:28,380 --> 00:44:31,250
b.

688
00:44:31,250 --> 00:44:34,320
So this is how we
multiply two operators.

689
00:44:34,320 --> 00:44:40,340
How we get a signal, which is
a proportional to a or a dagger

690
00:44:40,340 --> 00:44:41,857
times b or b dagger.

691
00:44:49,820 --> 00:44:52,790
By the way, if it wouldn't
take the difference,

692
00:44:52,790 --> 00:44:57,210
we get terms of aa
dagger bb dagger.

693
00:44:57,210 --> 00:44:59,730
Just the mode a or the
mode b by themselves.

694
00:44:59,730 --> 00:45:01,150
And we want to get rid of them.

695
00:45:01,150 --> 00:45:03,615
And by taking the difference
between the two photo

696
00:45:03,615 --> 00:45:07,290
detectors, those parts of
the signal are becoming mode.

697
00:45:07,290 --> 00:45:09,940
And are subtracted out.

698
00:45:09,940 --> 00:45:12,780
So therefore, this is
called balanced homodyne

699
00:45:12,780 --> 00:45:15,100
because we have the
balanced beam splitter.

700
00:45:15,100 --> 00:45:16,390
We measure to two signals.

701
00:45:16,390 --> 00:45:18,650
And then we take the
difference of the two signals.

702
00:45:24,380 --> 00:45:24,880
OK.

703
00:45:29,570 --> 00:45:31,590
There's one more
thing we have to add.

704
00:45:31,590 --> 00:45:37,050
And then we find we understand
the balanced homodyne

705
00:45:37,050 --> 00:45:37,640
detection.

706
00:45:37,640 --> 00:45:41,740
We want to explore it.

707
00:45:41,740 --> 00:45:44,590
That our mode b,
remember we want

708
00:45:44,590 --> 00:45:46,990
to measure mode a. a squeezed.

709
00:45:46,990 --> 00:45:50,460
a has quantum properties.
a has only single photons.

710
00:45:50,460 --> 00:45:56,360
And b is just a trick to
project out the cosine omega t.

711
00:45:56,360 --> 00:46:10,670
And we do that by choosing
for b a strong coherent state,

712
00:46:10,670 --> 00:46:13,430
described by a coherent
state parameter beta.

713
00:46:24,850 --> 00:46:27,790
Of course, whenever we have
a strong coherent state,

714
00:46:27,790 --> 00:46:32,850
that means that we
can replace b by beta,

715
00:46:32,850 --> 00:46:37,370
and b dagger by beta star.

716
00:46:37,370 --> 00:46:39,440
This is sort of
the classical field

717
00:46:39,440 --> 00:46:41,600
limit of a strong
coherent state.

718
00:46:45,010 --> 00:46:53,740
And now we can ask, what this
our different signal I minus?

719
00:46:53,740 --> 00:46:59,180
Well, the coherent state depends
on the phase angle theta.

720
00:46:59,180 --> 00:47:00,670
So it will be phase sensitive.

721
00:47:00,670 --> 00:47:04,170
It will depend on
the angle theta.

722
00:47:04,170 --> 00:47:05,730
But now there is one more thing.

723
00:47:05,730 --> 00:47:09,180
And that is, if you use a
stronger coherent state,

724
00:47:09,180 --> 00:47:14,680
of course both of our outputs
go up in proportion to beta.

725
00:47:14,680 --> 00:47:19,180
So therefore, we want
to go to a normalization

726
00:47:19,180 --> 00:47:22,200
by dividing by beta.

727
00:47:22,200 --> 00:47:24,270
So this is now our
normalized output.

728
00:47:39,740 --> 00:47:41,550
I'm looking at this.

729
00:47:41,550 --> 00:47:46,120
So this is our operator
for the signal I minus.

730
00:47:46,120 --> 00:47:48,270
And that is a.

731
00:47:48,270 --> 00:47:56,060
b has been replaced by, b dagger
has been replaced by b star.

732
00:47:56,060 --> 00:48:01,540
That gives us a times
e to the-- because

733
00:48:01,540 --> 00:48:04,050
of the complex
conjugation-- minus i

734
00:48:04,050 --> 00:48:11,870
theta plus a dagger times e
to the i theta divided by 2.

735
00:48:16,990 --> 00:48:25,540
So therefore, if we
choose the phase to be 0,

736
00:48:25,540 --> 00:48:28,200
we measure this
symmetric combination,

737
00:48:28,200 --> 00:48:30,020
a plus a dagger over 2.

738
00:48:40,140 --> 00:48:45,570
And this is-- just
comparing notes--

739
00:48:45,570 --> 00:48:49,440
simply the a1 quadrature
operator divided by square root

740
00:48:49,440 --> 00:48:51,190
2.

741
00:48:51,190 --> 00:48:57,250
And if you put a phase shifter,
just a dispersive element,

742
00:48:57,250 --> 00:48:59,270
into the strong local
oscillator and shift

743
00:48:59,270 --> 00:49:05,040
the phase by pi over 2, by
delaying the light pulse

744
00:49:05,040 --> 00:49:09,200
by a quarter wavelength, what
we are now projecting out

745
00:49:09,200 --> 00:49:15,810
is the other
quadrature component.

746
00:49:18,800 --> 00:49:22,250
So therefore, using
the beam splitter

747
00:49:22,250 --> 00:49:24,460
and the local
oscillator, we can now

748
00:49:24,460 --> 00:49:28,960
measure the expectation
value for a1 and for a2.

749
00:49:31,800 --> 00:49:34,990
I've posted a few papers
on the website which

750
00:49:34,990 --> 00:49:37,140
show some of the pioneering
experiments where

751
00:49:37,140 --> 00:49:39,440
people did exactly that.

752
00:49:39,440 --> 00:49:46,070
So then, when they observed the
quadrature comportment, which

753
00:49:46,070 --> 00:49:50,230
was squeezed, they usually
did it for a squeezed vacuum,

754
00:49:50,230 --> 00:49:53,540
so a1 and a2 had 0
expectation value.

755
00:49:53,540 --> 00:49:56,840
But the interesting part is,
how much noise was there?

756
00:49:56,840 --> 00:50:02,200
If you don't squeeze, you find
that your normalized noise

757
00:50:02,200 --> 00:50:05,860
simply corresponds to the
classical photon shot noise.

758
00:50:05,860 --> 00:50:08,820
But if you are
squeezed, you find

759
00:50:08,820 --> 00:50:10,895
that one quadrature
comportment has

760
00:50:10,895 --> 00:50:13,320
a smaller noise than shot noise.

761
00:50:13,320 --> 00:50:15,600
So [INAUDIBLE] photons
in your different signal.

762
00:50:15,600 --> 00:50:17,396
And your noise is less
than square root n.

763
00:50:17,396 --> 00:50:18,770
Where as in the
other quadrature,

764
00:50:18,770 --> 00:50:21,080
the component is larger.

765
00:50:21,080 --> 00:50:24,890
So you can look up
on the website papers

766
00:50:24,890 --> 00:50:29,220
where people slowly varied the
phase of the local oscillator.

767
00:50:29,220 --> 00:50:31,600
And you see, this
is shot noise, how

768
00:50:31,600 --> 00:50:33,155
the noise is below shot noise.

769
00:50:33,155 --> 00:50:34,410
Exceeds shot noise.

770
00:50:34,410 --> 00:50:35,540
It's below shot noise.

771
00:50:35,540 --> 00:50:37,200
And exceeds shot noise.

772
00:50:37,200 --> 00:50:41,110
And this was the first
evidence for the generation

773
00:50:41,110 --> 00:50:41,880
of squeezed light.

774
00:50:49,280 --> 00:50:49,780
Questions?

775
00:51:00,540 --> 00:51:01,040
OK.

776
00:51:01,040 --> 00:51:05,750
So we have discussed
the detection

777
00:51:05,750 --> 00:51:09,960
of squeezed light using
balanced homodyne detector.

778
00:51:09,960 --> 00:51:15,750
And balanced means that the
beam splitter was 50-50.

779
00:51:15,750 --> 00:51:19,720
So now we are ready to
discuss the unbalanced case.

780
00:51:30,250 --> 00:51:38,300
So let's get our
unbalanced beam splitter

781
00:51:38,300 --> 00:51:41,680
and, just for the
sake of the argument,

782
00:51:41,680 --> 00:51:46,840
let's say it has a really
good transmission of tt.

783
00:51:46,840 --> 00:51:49,930
So transmission
coefficient t squared

784
00:51:49,930 --> 00:51:53,090
tells you what fraction of
the power is transmitted.

785
00:51:53,090 --> 00:51:56,435
And let's just assume for
the sake of the argument, 99%

786
00:51:56,435 --> 00:51:58,370
are transmitted.

787
00:51:58,370 --> 00:52:01,480
So therefore, if we
start with our signal

788
00:52:01,480 --> 00:52:04,540
a, which may be
[INAUDIBLE] squeezed

789
00:52:04,540 --> 00:52:06,130
quantum state, number state.

790
00:52:06,130 --> 00:52:09,020
You name it.

791
00:52:09,020 --> 00:52:12,245
a naught is pretty
much the same as a.

792
00:52:12,245 --> 00:52:15,180
We haven't lost so much.

793
00:52:15,180 --> 00:52:20,170
But now we use our
local oscillator,

794
00:52:20,170 --> 00:52:21,105
which is very strong.

795
00:52:24,740 --> 00:52:29,680
Only a very small fraction
of it will be reflected.

796
00:52:29,680 --> 00:52:32,540
But we can always compensate
for the small reflection

797
00:52:32,540 --> 00:52:36,370
by making b even stronger.

798
00:52:36,370 --> 00:52:38,490
Let's see what we get.

799
00:52:38,490 --> 00:52:42,630
So the mode, a naught,
the output mode,

800
00:52:42,630 --> 00:52:46,550
is now the transmission
coefficient times a.

801
00:52:49,480 --> 00:52:50,530
a and a.

802
00:52:50,530 --> 00:52:53,070
The operators are amplitudes.

803
00:52:53,070 --> 00:52:56,730
So we have to use, if you
have a 99% beam splitter,

804
00:52:56,730 --> 00:52:59,990
we have to take the
square root of 0.99.

805
00:52:59,990 --> 00:53:02,540
This is the transmission
coefficient.

806
00:53:02,540 --> 00:53:05,480
And now we have the
transmission coefficient

807
00:53:05,480 --> 00:53:08,920
for the strong local
oscillator, which

808
00:53:08,920 --> 00:53:12,800
we will again approximate
by its eigenvalue.

809
00:53:12,800 --> 00:53:13,950
Coherent state eigenvalue.

810
00:53:18,320 --> 00:53:21,150
And the reflection coefficient
is 1 minus t squared.

811
00:53:25,820 --> 00:53:33,156
So if we make the approximation
that t is approximately 1,

812
00:53:33,156 --> 00:53:34,780
we can take it out
off the parenthesis.

813
00:53:44,780 --> 00:53:45,980
And we obtain that.

814
00:53:48,750 --> 00:53:53,690
If t is close to 1,
we can neglect it.

815
00:53:53,690 --> 00:53:58,300
And what we see now is
what we have obtained

816
00:53:58,300 --> 00:54:03,210
is actually the original
quantum state a.

817
00:54:03,210 --> 00:54:03,910
Or I should say.

818
00:54:03,910 --> 00:54:04,720
The operator.

819
00:54:04,720 --> 00:54:06,030
I'm in the Heisenberg picture.

820
00:54:06,030 --> 00:54:11,320
a is the mode
operator for the input

821
00:54:11,320 --> 00:54:14,740
of the unbalanced
homodyne detector.

822
00:54:14,740 --> 00:54:19,460
And what we have simply done is
we have displaced the operator.

823
00:54:19,460 --> 00:54:22,620
We have displaced
the mode operator a.

824
00:54:22,620 --> 00:54:26,920
And the displacement operator
has an [? argument ?]

825
00:54:26,920 --> 00:54:33,670
of 1 minus t squared beta.

826
00:54:33,670 --> 00:54:36,360
And this is a
Hermitian conjugate.

827
00:54:36,360 --> 00:54:41,880
1 minus t squared beta.

828
00:54:41,880 --> 00:54:45,820
So what I've shown you here
is that the local oscillator

829
00:54:45,820 --> 00:54:52,740
and the beam splitter is the
realization, or implementation,

830
00:54:52,740 --> 00:54:54,140
of the displacement operator.

831
00:55:02,440 --> 00:55:08,090
So in the limit that
t goes to unity,

832
00:55:08,090 --> 00:55:12,000
we are not losing anything
of out quantum state.

833
00:55:12,000 --> 00:55:15,680
But by reflecting
in the amplitude

834
00:55:15,680 --> 00:55:20,520
of a strong coherent state, we
simply take our quantum state,

835
00:55:20,520 --> 00:55:24,810
and we displace it in this
two dimensional plane.

836
00:55:24,810 --> 00:55:28,500
So that's what this beam
splitter does for us.

837
00:55:35,557 --> 00:55:36,140
Any questions?

838
00:55:44,440 --> 00:55:46,420
In the next few
classes, we really

839
00:55:46,420 --> 00:55:48,360
take advantage of
those elements.

840
00:55:48,360 --> 00:55:51,020
We know now that the
displacement operator is just

841
00:55:51,020 --> 00:55:52,180
a beam splitter.

842
00:55:52,180 --> 00:55:55,270
We know when we have
squeezed some light

843
00:55:55,270 --> 00:55:57,490
by using balanced
homodyne detection,

844
00:55:57,490 --> 00:55:59,870
we can just do measurement
of one quadrature component

845
00:55:59,870 --> 00:56:01,100
or the other.

846
00:56:01,100 --> 00:56:04,390
So what I hope for those of
you who haven't heard about it,

847
00:56:04,390 --> 00:56:06,990
what you take away
from that is that these

848
00:56:06,990 --> 00:56:10,150
are extremely simple elements.

849
00:56:10,150 --> 00:56:12,820
But by combining
them, we can really

850
00:56:12,820 --> 00:56:18,610
realize very sophisticated
schemes of quantum optics.

851
00:56:18,610 --> 00:56:20,990
To some extent, when I heard
about it for the first time,

852
00:56:20,990 --> 00:56:23,300
I think, the mathematics
look so fancy.

853
00:56:23,300 --> 00:56:27,470
I couldn't believe that such
simple elements can actually

854
00:56:27,470 --> 00:56:31,100
realize what those
operators describe.

855
00:56:31,100 --> 00:56:33,730
So I gave you the example for
the displacement operator.

856
00:56:45,780 --> 00:56:47,860
When I learned about
the beam splitter,

857
00:56:47,860 --> 00:56:51,250
and its underlying
physics, there

858
00:56:51,250 --> 00:56:54,540
was one thing which
really fascinated me.

859
00:56:54,540 --> 00:56:59,560
And this is the most simple
element you can think of.

860
00:56:59,560 --> 00:57:02,250
I mean, what is simpler
than a beam splitter?

861
00:57:02,250 --> 00:57:05,550
A beam splitter has two
inputs, two outputs.

862
00:57:05,550 --> 00:57:09,410
The simplest optical element
is just an attenuator.

863
00:57:09,410 --> 00:57:11,480
Put in a window
in your laser beam

864
00:57:11,480 --> 00:57:14,650
and you lose 4% of
your power preserves.

865
00:57:14,650 --> 00:57:16,920
Or put in just a little
bit of dirty optics

866
00:57:16,920 --> 00:57:18,770
and you lose a few percent.

867
00:57:18,770 --> 00:57:20,630
So what I want to
discuss with you now

868
00:57:20,630 --> 00:57:24,125
is, what really is an
attenuator quantum mechanically?

869
00:57:28,890 --> 00:57:35,710
Well if, classically,
the attenuator

870
00:57:35,710 --> 00:57:37,930
would do the following.

871
00:57:37,930 --> 00:57:43,180
An attenuator is a device
which has a transmission, which

872
00:57:43,180 --> 00:57:45,270
is a transmission
coefficient squared,

873
00:57:45,270 --> 00:57:46,990
which is smaller than unity.

874
00:57:49,810 --> 00:57:54,910
And, in a classical system,
if you have a coherent state,

875
00:57:54,910 --> 00:58:00,805
you would simply assume that the
coherent state gets multiplied

876
00:58:00,805 --> 00:58:02,320
by the transmission coefficient.

877
00:58:05,010 --> 00:58:17,300
In other words, that you
have your original state,

878
00:58:17,300 --> 00:58:20,880
described by this
phase, or alpha.

879
00:58:20,880 --> 00:58:25,430
And then, the action
of the attenuator

880
00:58:25,430 --> 00:58:35,420
would simply be to
scale everything down

881
00:58:35,420 --> 00:58:38,620
by a transmission
effect of alpha.

882
00:58:38,620 --> 00:58:40,750
So the picture you should
have is the following.

883
00:58:40,750 --> 00:58:42,320
You have a coherent state.

884
00:58:42,320 --> 00:58:45,400
It gets attenuated by the
transmission coefficient.

885
00:58:45,400 --> 00:58:48,940
But if there are fluctuations
about the coherent state,

886
00:58:48,940 --> 00:58:51,140
also the fluctuations
get attenuated.

887
00:58:51,140 --> 00:58:55,450
Because everything gets
attenuated by this attenuator.

888
00:58:55,450 --> 00:59:00,960
So if you look at that, you
should immediately say, no.

889
00:59:00,960 --> 00:59:03,180
This is quantum
mechanically forbidden.

890
00:59:03,180 --> 00:59:07,160
Because a coherent state with
a minimum uncertainty state,

891
00:59:07,160 --> 00:59:11,170
this shaded area cannot
be smaller than 1/2.

892
00:59:11,170 --> 00:59:13,600
But here it has become smaller.

893
00:59:13,600 --> 00:59:15,770
So what I've shown
to you here is

894
00:59:15,770 --> 00:59:18,480
it's a violation of
quantum mechanics.

895
00:59:21,620 --> 00:59:25,190
It would actually mean, let
me just give you the example.

896
00:59:25,190 --> 00:59:40,520
It would mean that if-- yeah.

897
00:59:43,860 --> 00:59:45,600
The coherent state
is quantum limited.

898
00:59:45,600 --> 00:59:48,080
And if you calculate what is
the fluctuations in the photo

899
00:59:48,080 --> 00:59:49,580
number, it's a shot noise.

900
00:59:49,580 --> 00:59:53,970
So just to give you sort of
simple, intuitive example,

901
00:59:53,970 --> 00:59:59,460
if you have 10,000 photons plus
minus 100, it's a shot noise.

902
00:59:59,460 --> 01:00:00,680
Square root n.

903
01:00:00,680 --> 01:00:03,740
If you could now attenuate
it by a factor of 100,

904
01:00:03,740 --> 01:00:08,050
and you would go from
10,000 plus minus 100,

905
01:00:08,050 --> 01:00:11,910
to 100 plus minus 1.

906
01:00:11,910 --> 01:00:13,790
That's much better
than the shot noise.

907
01:00:13,790 --> 01:00:15,560
I mean, this is what
I'm just telling you

908
01:00:15,560 --> 01:00:18,040
what a simpleminded
attenuator would do.

909
01:00:18,040 --> 01:00:20,590
And you would immediately say,
that's too good to be true.

910
01:00:20,590 --> 01:00:22,550
I cannot get sub
shot noise light.

911
01:00:26,640 --> 01:00:27,490
So what is wrong?

912
01:00:34,060 --> 01:00:34,740
Impossible.

913
01:00:34,740 --> 01:00:37,060
Not allowed.

914
01:00:37,060 --> 01:00:40,820
Well, we've just tried
to formulate something

915
01:00:40,820 --> 01:00:41,730
intuitively.

916
01:00:41,730 --> 01:00:42,820
And we have to be careful.

917
01:00:47,320 --> 01:00:55,740
Well, we know
already one way how

918
01:00:55,740 --> 01:01:00,362
we can attenuate an input beam.

919
01:01:00,362 --> 01:01:04,300
And maybe we should go back to
the situation and analyze it.

920
01:01:04,300 --> 01:01:08,570
We know what if we can attenuate
it with a beam splitter.

921
01:01:08,570 --> 01:01:12,850
And this beam splitter has a
transmission coefficient of t.

922
01:01:12,850 --> 01:01:17,260
And then we get our
transmitted coherent light.

923
01:01:17,260 --> 01:01:21,720
There is something
getting reflected.

924
01:01:21,720 --> 01:01:26,820
But now we realize that
this beam splitter is not

925
01:01:26,820 --> 01:01:29,480
just a device which
has one input.

926
01:01:29,480 --> 01:01:31,060
It has another input.

927
01:01:31,060 --> 01:01:34,600
And you may say, well, I don't
care about the other input.

928
01:01:34,600 --> 01:01:36,340
I don't want to use it.

929
01:01:36,340 --> 01:01:39,890
Well, if you don't
want to use the input,

930
01:01:39,890 --> 01:01:41,620
it has the vacuum state.

931
01:01:45,000 --> 01:01:51,110
So therefore, if you would
realize the attenuator

932
01:01:51,110 --> 01:02:00,110
with the beam splitter, it
would mean that, in addition,

933
01:02:00,110 --> 01:02:03,190
and this is what the math
really shows, in addition

934
01:02:03,190 --> 01:02:07,140
to the attenuated coherent
state, which mathematically

935
01:02:07,140 --> 01:02:12,350
is also attenuating
the fluctuations,

936
01:02:12,350 --> 01:02:16,240
you have to add something.

937
01:02:16,240 --> 01:02:19,750
Which is the
reflection coefficient

938
01:02:19,750 --> 01:02:22,945
times the vacuum state.

939
01:02:27,880 --> 01:02:30,120
Which is this.

940
01:02:30,120 --> 01:02:35,370
And if you now
correctly do the math,

941
01:02:35,370 --> 01:02:40,960
if you add the two
together, you find

942
01:02:40,960 --> 01:02:50,730
you get a coherent state which
has an amplitude of t alpha.

943
01:02:50,730 --> 01:02:53,510
But has the correct
[INAUDIBLE] fluctuation

944
01:02:53,510 --> 01:02:55,960
is again a minimum
uncertainty state.

945
01:02:55,960 --> 01:02:58,900
So the disk of your
attenuated state

946
01:02:58,900 --> 01:03:01,780
hasn't exactly the same area
as the unattenuated state.

947
01:03:05,830 --> 01:03:08,700
So I've shown you
the physical part.

948
01:03:08,700 --> 01:03:11,220
I have shown you the
graphical solution.

949
01:03:11,220 --> 01:03:12,700
The math is very simple.

950
01:03:12,700 --> 01:03:15,300
But I really want you
to do the math yourself.

951
01:03:15,300 --> 01:03:16,730
This is a new
homework problem we

952
01:03:16,730 --> 01:03:19,065
designed to illustrate
the physics.

953
01:03:19,065 --> 01:03:21,630
But what it tells
you is the following.

954
01:03:21,630 --> 01:03:25,370
If you take a neutral
density filter out

955
01:03:25,370 --> 01:03:28,340
of the lab [INAUDIBLE] and say,
this is not a beam splitter.

956
01:03:28,340 --> 01:03:30,230
This is an attenuator.

957
01:03:30,230 --> 01:03:31,290
Sorry.

958
01:03:31,290 --> 01:03:35,500
You cannot simply attenuate
a quantum mechanical mode.

959
01:03:35,500 --> 01:03:38,500
This is not a unitary
time evolution.

960
01:03:38,500 --> 01:03:42,130
What you attenuate [INAUDIBLE]
is, without you knowing it,

961
01:03:42,130 --> 01:03:45,590
it couples the electromagnetic
wave to whatever.

962
01:03:45,590 --> 01:03:47,790
To the heat path
of the [INAUDIBLE],

963
01:03:47,790 --> 01:03:49,180
which is in your attenuator.

964
01:03:49,180 --> 01:03:50,950
I don't even want
to describe it.

965
01:03:50,950 --> 01:03:53,820
But you're not
circumventing the limitation

966
01:03:53,820 --> 01:03:55,160
of the beam splitter.

967
01:03:55,160 --> 01:03:58,630
Whenever you attenuate,
whenever you have a laser beam,

968
01:03:58,630 --> 01:04:01,480
and it undergoes losses,
when you send a laser beam

969
01:04:01,480 --> 01:04:04,370
through the atmosphere, and
it undergoes some losses by,

970
01:04:04,370 --> 01:04:06,180
who knows,
[INAUDIBLE] scattering

971
01:04:06,180 --> 01:04:10,085
through the air or
something like this.

972
01:04:10,085 --> 01:04:13,670
That means you get an
attenuated coherent state.

973
01:04:13,670 --> 01:04:16,960
But you couple in the
fluctuations off the vacuum.

974
01:04:16,960 --> 01:04:19,810
And this establishes
shot noise now,

975
01:04:19,810 --> 01:04:22,750
at the lower level of intensity.

976
01:04:22,750 --> 01:04:27,780
So your attenuator is not
a single state device.

977
01:04:27,780 --> 01:04:29,010
Dissipation.

978
01:04:29,010 --> 01:04:32,110
Attenuation really
means you connect

979
01:04:32,110 --> 01:04:33,850
with other parts of
[INAUDIBLE] space.

980
01:04:33,850 --> 01:04:39,520
You cannot attenuate in a
small part [INAUDIBLE] space.

981
01:04:39,520 --> 01:04:40,650
This is impossible.

982
01:04:40,650 --> 01:04:42,300
This is not unitary
time evolution.

983
01:04:57,620 --> 01:05:05,320
So what I've just told you
has dramatic consequences

984
01:05:05,320 --> 01:05:11,560
for any form of non
classical, or squeezed, light.

985
01:05:11,560 --> 01:05:14,370
And that's the following.

986
01:05:14,370 --> 01:05:16,030
Most of you are
experimentalists.

987
01:05:16,030 --> 01:05:19,930
And you know that when you run
a laser at one or two watts,

988
01:05:19,930 --> 01:05:23,560
you send it through optics, and
shudders, and optical fibers.

989
01:05:23,560 --> 01:05:26,780
How much do you get at the
end of your experiment?

990
01:05:26,780 --> 01:05:28,560
Not even 50%.

991
01:05:28,560 --> 01:05:31,300
So, whenever you create
light, and then you

992
01:05:31,300 --> 01:05:34,570
want to do something, you
lose some of the light.

993
01:05:34,570 --> 01:05:39,060
And let's now assume that
you have done what was really

994
01:05:39,060 --> 01:05:43,550
a breakthrough in scientific
headlines within your lifetime.

995
01:05:43,550 --> 01:05:46,230
You have generated
squeezed light.

996
01:05:46,230 --> 01:05:48,790
And now we want to use
the squeezed light.

997
01:05:48,790 --> 01:05:51,642
Shine it on atoms and do
a precision measurement,

998
01:05:51,642 --> 01:05:53,600
which is better than the
standard quantum limit

999
01:05:53,600 --> 01:05:55,183
because you have
squeezed the ellipse.

1000
01:05:55,183 --> 01:05:58,420
And you want to now exploit
the sharpness of the ellipse.

1001
01:05:58,420 --> 01:06:02,560
What happens to your
aspect ratio of the ellipse

1002
01:06:02,560 --> 01:06:05,900
when your beam is attenuated?

1003
01:06:05,900 --> 01:06:09,180
So let me just discuss
it with you graphically.

1004
01:06:09,180 --> 01:06:11,920
So let's assume we
have a squeezed state.

1005
01:06:11,920 --> 01:06:13,720
And we send it through
an optical fiber.

1006
01:06:18,020 --> 01:06:20,210
The result is you will
never send a squeezed state

1007
01:06:20,210 --> 01:06:21,310
through an optical fiber.

1008
01:06:21,310 --> 01:06:23,170
But I want you to realize that.

1009
01:06:23,170 --> 01:06:27,001
So let's assume we have our
squeezed state, symbolized

1010
01:06:27,001 --> 01:06:27,500
by that.

1011
01:06:32,260 --> 01:06:34,620
Let's use some red
color for the state.

1012
01:06:34,620 --> 01:06:37,240
And now, in the
time evolution, we

1013
01:06:37,240 --> 01:06:40,180
have some [INAUDIBLE]
scattering.

1014
01:06:40,180 --> 01:06:41,840
Some fiber absorption.

1015
01:06:41,840 --> 01:06:43,930
But you know already,
the absorption

1016
01:06:43,930 --> 01:06:47,310
is in reality a beam splitter,
which happens in the vacuum.

1017
01:06:47,310 --> 01:06:47,810
So

1018
01:06:47,810 --> 01:06:51,620
We have now some finite
transmission coefficients.

1019
01:06:51,620 --> 01:06:56,300
And that would mean,
which is the bad news,

1020
01:06:56,300 --> 01:07:00,940
that your ellipse gets shrunk.

1021
01:07:00,940 --> 01:07:01,440
It shrinks.

1022
01:07:08,680 --> 01:07:12,040
It shrinks by the
transmission factor t.

1023
01:07:14,590 --> 01:07:17,890
But that's what I did bad.

1024
01:07:17,890 --> 01:07:19,240
You lose some of your power.

1025
01:07:19,240 --> 01:07:28,150
But the really bad thing
is that if I multiply it

1026
01:07:28,150 --> 01:07:38,045
with the reflection coefficient,
you couple in the vacuum.

1027
01:07:38,045 --> 01:07:38,545
Oops.

1028
01:07:38,545 --> 01:07:41,030
I'm going to change to red.

1029
01:07:44,900 --> 01:07:49,750
And, as a result, since
the noise in the vacuum

1030
01:07:49,750 --> 01:07:52,620
is equal in both
quadrature components,

1031
01:07:52,620 --> 01:07:56,980
you've worked so hard to squeeze
it, to make it asymmetric.

1032
01:07:56,980 --> 01:08:01,550
But what you get now
is from your ellipse,

1033
01:08:01,550 --> 01:08:06,060
you get something which is
much more egg shaped now.

1034
01:08:06,060 --> 01:08:10,520
So in other words, you can
write it down with operators.

1035
01:08:10,520 --> 01:08:12,970
But once you understand
what is going on,

1036
01:08:12,970 --> 01:08:21,100
you immediately realize that
losses reduce the squeezing.

1037
01:08:24,750 --> 01:08:28,090
And this is a challenge to
all the experiments using

1038
01:08:28,090 --> 01:08:29,760
squeezed or non classical light.

1039
01:08:34,580 --> 01:08:36,810
And you see how
the scaling works.

1040
01:08:36,810 --> 01:08:39,529
If you have squeezed your
ellipse by a factor of 100,

1041
01:08:39,529 --> 01:08:44,729
even 1% of the vacuum
will spoil your squeezing.

1042
01:08:44,729 --> 01:08:47,100
So the more you have
squeezed, the more

1043
01:08:47,100 --> 01:08:52,149
non-classical the light is--
the more valuable it is.

1044
01:08:52,149 --> 01:08:55,660
The more sensitive it is,
to even very small losses.

1045
01:09:02,999 --> 01:09:03,499
Questions?

1046
01:09:10,880 --> 01:09:12,050
OK, good.

1047
01:09:12,050 --> 01:09:17,499
Now we have 10 more minutes.

1048
01:09:21,420 --> 01:09:23,950
What I want to do now
is, I want to show you

1049
01:09:23,950 --> 01:09:31,120
that the language
which we have used,

1050
01:09:31,120 --> 01:09:34,350
the methods we have introduced,
can now be used for something

1051
01:09:34,350 --> 01:09:37,330
which is really
cool-- teleportation.

1052
01:09:37,330 --> 01:09:41,470
I want to show you how
balanced homodyne detection,

1053
01:09:41,470 --> 01:09:44,800
quadrature measurement, and
displacement operator can

1054
01:09:44,800 --> 01:09:48,550
be put together to
generalize is scheme which

1055
01:09:48,550 --> 01:09:51,100
is a scheme for
teleporting quantum states.

1056
01:09:57,980 --> 01:10:01,930
This is an application
of squeezing, homodyning,

1057
01:10:01,930 --> 01:10:03,330
and all that.

1058
01:10:03,330 --> 01:10:04,670
Teleportation of light.

1059
01:10:15,580 --> 01:10:19,340
Let me just illustrate
what the problem is.

1060
01:10:19,340 --> 01:10:21,130
I know we have only
10 minutes, and I've

1061
01:10:21,130 --> 01:10:25,170
decided not to write down
all the math for you.

1062
01:10:25,170 --> 01:10:27,110
I went, actually, to
the Wikipedia page,

1063
01:10:27,110 --> 01:10:31,290
and corrected a few
mistakes in the equations

1064
01:10:31,290 --> 01:10:33,220
and edited a few explanations.

1065
01:10:33,220 --> 01:10:35,610
I think you can just sit
down and read it by yourself.

1066
01:10:35,610 --> 01:10:37,260
What I want to
explain to you, here,

1067
01:10:37,260 --> 01:10:41,310
is the physical concepts
behind teleportation,

1068
01:10:41,310 --> 01:10:46,150
and you the crucial steps
to realize teleportation.

1069
01:10:46,150 --> 01:10:47,685
So first, what is teleportation?

1070
01:10:50,460 --> 01:10:55,600
Well, teleportation has
a sender and a received,

1071
01:10:55,600 --> 01:10:57,750
which in quantum
information science

1072
01:10:57,750 --> 01:10:59,183
are called Alice and Bob.

1073
01:11:01,780 --> 01:11:07,850
Teleportation means Alice
has a quantum state, psi.

1074
01:11:07,850 --> 01:11:12,630
And she wants to send this
quantum state, psi, to Bob.

1075
01:11:12,630 --> 01:11:16,250
In other words, you have,
maybe, some squeezed light.

1076
01:11:16,250 --> 01:11:18,170
You have something--
a quantum state,

1077
01:11:18,170 --> 01:11:21,760
psi-- which is interesting.

1078
01:11:21,760 --> 01:11:33,100
And the question is, how can
Bob create an identical copy

1079
01:11:33,100 --> 01:11:35,094
of this quantum
state, psi, that he

1080
01:11:35,094 --> 01:11:37,260
can sort of [INAUDIBLE]
with the same quantum state.

1081
01:11:40,810 --> 01:11:43,320
Before you appreciate
teleportation,

1082
01:11:43,320 --> 01:11:47,880
you have to realize
what the problem is.

1083
01:11:51,550 --> 01:11:55,890
The problem is, really,
in fundamental properties

1084
01:11:55,890 --> 01:11:58,610
of quantum systems,
fundamental limitations-- what

1085
01:11:58,610 --> 01:12:00,600
you can do with
a quantum system.

1086
01:12:00,600 --> 01:12:03,530
First, I have to tell you
what is allowed, and what not.

1087
01:12:03,530 --> 01:12:06,940
In this game, teleportation
means, of course,

1088
01:12:06,940 --> 01:12:10,660
you will not take your atom and
your photons in the state psi,

1089
01:12:10,660 --> 01:12:13,160
just propagate them to
Bob, and Bob has them.

1090
01:12:13,160 --> 01:12:14,340
That's trivial.

1091
01:12:14,340 --> 01:12:16,780
I can send any
quantum state to you

1092
01:12:16,780 --> 01:12:20,070
by transmitting an atom
or a laser beam to you,

1093
01:12:20,070 --> 01:12:22,900
and you have the same
quantum state I had earlier.

1094
01:12:22,900 --> 01:12:24,250
This is not teleportation.

1095
01:12:24,250 --> 01:12:27,010
This is trivial propagation.

1096
01:12:27,010 --> 01:12:30,483
What is meant is that we don't
have a quantum channel-- I'm

1097
01:12:30,483 --> 01:12:33,350
not allowed to send my
quantum state to you.

1098
01:12:33,350 --> 01:12:36,560
But in teleportation, I can do
a measurement on the quantum

1099
01:12:36,560 --> 01:12:41,030
state, call you up through
a classical communication

1100
01:12:41,030 --> 01:12:43,380
channel, and tell you, the
result of my measurement

1101
01:12:43,380 --> 01:12:45,310
is such and such.

1102
01:12:45,310 --> 01:12:47,610
And then you would say, well,
if I have a spin system,

1103
01:12:47,610 --> 01:12:50,810
and I measure spin up--
I call you and say,

1104
01:12:50,810 --> 01:12:54,510
my measurement was spin up, and
you create a spin up system.

1105
01:12:54,510 --> 01:12:57,110
Isn't that teleportation?

1106
01:12:57,110 --> 01:13:00,460
The answer is no, because
maybe the state I had

1107
01:13:00,460 --> 01:13:02,710
was a superposition
of spin up and down.

1108
01:13:02,710 --> 01:13:05,300
And what I measured
is only spin up.

1109
01:13:05,300 --> 01:13:07,220
And by telling you
it's spin up, you

1110
01:13:07,220 --> 01:13:08,950
would never create a
superposition state,

1111
01:13:08,950 --> 01:13:10,690
you would just create
a spin up state.

1112
01:13:10,690 --> 01:13:14,900
So the effect is, that if I
do a measurement on my quantum

1113
01:13:14,900 --> 01:13:17,325
state, and report
my results to you,

1114
01:13:17,325 --> 01:13:20,280
you have insufficient
information.

1115
01:13:20,280 --> 01:13:23,150
Because a projective
measurement on a quantum

1116
01:13:23,150 --> 01:13:27,650
state inevitably leads
to loss of information.

1117
01:13:27,650 --> 01:13:30,180
A measurement on a
single quantum state

1118
01:13:30,180 --> 01:13:32,240
does not create
enough information

1119
01:13:32,240 --> 01:13:33,630
to recreate the quantum state.

1120
01:13:36,540 --> 01:13:39,830
Of course, there is
an obvious solution.

1121
01:13:39,830 --> 01:13:42,580
When we want to obtain
information about a quantum

1122
01:13:42,580 --> 01:13:45,830
state, we often, in
quantum mechanics,

1123
01:13:45,830 --> 01:13:48,210
have to do many measurements.

1124
01:13:48,210 --> 01:13:50,410
And we can take a
spin state, we can

1125
01:13:50,410 --> 01:13:52,160
measure what is
its x, what its y,

1126
01:13:52,160 --> 01:13:54,440
what is the c
component of the spin.

1127
01:13:54,440 --> 01:13:56,750
We can completely
characterize the spin state,

1128
01:13:56,750 --> 01:13:58,720
and then we know
everything about it.

1129
01:13:58,720 --> 01:14:02,040
But the problem is, we have
only one quantum state,

1130
01:14:02,040 --> 01:14:04,884
and we can't do
repeated measurements.

1131
01:14:04,884 --> 01:14:07,300
Then you would say, well, the
next thing is, why don't you

1132
01:14:07,300 --> 01:14:10,030
just take your quantum
state and Xerox it,

1133
01:14:10,030 --> 01:14:11,347
make many, many copies.

1134
01:14:11,347 --> 01:14:12,930
And then you have
an ensemble, and you

1135
01:14:12,930 --> 01:14:15,210
can take as many
measurements you want.

1136
01:14:15,210 --> 01:14:17,840
You can do an x
measurement, p measurement.

1137
01:14:17,840 --> 01:14:20,650
I mean, you can reconstruct
the complete wave function.

1138
01:14:20,650 --> 01:14:22,640
You can measure with an x basis.

1139
01:14:22,640 --> 01:14:24,800
You can measure with
a momentum basis.

1140
01:14:24,800 --> 01:14:27,320
You can collect all
the information.

1141
01:14:27,320 --> 01:14:30,900
But the problem is-- otherwise
the whole teleportation

1142
01:14:30,900 --> 01:14:32,475
would not be an
issue at all-- there

1143
01:14:32,475 --> 01:14:34,490
is the no-cloning theorem.

1144
01:14:34,490 --> 01:14:38,160
You cannot duplicate
a quantum state.

1145
01:14:38,160 --> 01:14:41,930
If you have an atom in a certain
quantum state and another atom,

1146
01:14:41,930 --> 01:14:44,250
it is quantum-mechanically
forbidden--

1147
01:14:44,250 --> 01:14:46,670
there is no unitary
transformation,

1148
01:14:46,670 --> 01:14:49,590
no way of creating a
situation that you have

1149
01:14:49,590 --> 01:14:53,500
one unknown quantum
state, plus another atom,

1150
01:14:53,500 --> 01:14:57,120
or another light beam,
and after some interaction

1151
01:14:57,120 --> 01:14:59,300
you have two times the
same quantum state.

1152
01:14:59,300 --> 01:15:00,400
You cannot clone.

1153
01:15:03,620 --> 01:15:08,335
Therefore, based on all that, we
cannot clone the quantum state.

1154
01:15:08,335 --> 01:15:10,540
We are only left
with the one copy

1155
01:15:10,540 --> 01:15:12,830
of the quantum state,
which Alice has.

1156
01:15:12,830 --> 01:15:14,877
Alice is not allowed
to send it to Bob.

1157
01:15:14,877 --> 01:15:16,710
Maybe Bob is on the
other side of the ocean,

1158
01:15:16,710 --> 01:15:19,150
or on another planet.

1159
01:15:19,150 --> 01:15:22,610
All what Alice can do is,
she can do one measurement

1160
01:15:22,610 --> 01:15:27,010
and tell Bob, this
is my measurement.

1161
01:15:27,010 --> 01:15:29,980
So this is the problem
of teleportation.

1162
01:15:29,980 --> 01:15:32,890
How is that possible?

1163
01:15:32,890 --> 01:15:37,790
Well, the way how I put
it, it seems impossible.

1164
01:15:37,790 --> 01:15:41,360
But there's a way out of it.

1165
01:15:46,210 --> 01:15:47,770
Let me just write
down what I said.

1166
01:15:47,770 --> 01:16:01,630
So the goal is now-- Alice
performs measurement,

1167
01:16:01,630 --> 01:16:06,780
reports result to Bob.

1168
01:16:16,810 --> 01:16:28,030
And now, Bob will recreate
the quantum state.

1169
01:16:28,030 --> 01:16:31,600
What I've just said is, to
perform a single measurement

1170
01:16:31,600 --> 01:16:32,760
is not enough information.

1171
01:16:32,760 --> 01:16:34,490
So this is not enough.

1172
01:16:34,490 --> 01:16:36,680
We need one more resource.

1173
01:16:36,680 --> 01:16:45,610
And the resource,
which is now used

1174
01:16:45,610 --> 01:16:55,850
to use quantum teleportation,
is that you take some entangled

1175
01:16:55,850 --> 01:17:02,630
system, or-- and this why I talk
about it today-- or squeezed

1176
01:17:02,630 --> 01:17:04,874
light.

1177
01:17:04,874 --> 01:17:05,790
That's the same thing.

1178
01:17:05,790 --> 01:17:09,630
It's a form of entanglement.

1179
01:17:09,630 --> 01:17:13,410
I told you that when we
generate squeezed light

1180
01:17:13,410 --> 01:17:15,620
with a parametric
down conversion,

1181
01:17:15,620 --> 01:17:20,890
the parametric down conversion
takes a green photon

1182
01:17:20,890 --> 01:17:24,570
and creates two identical
infrared photons.

1183
01:17:24,570 --> 01:17:26,560
Until now, we have
discussed that

1184
01:17:26,560 --> 01:17:29,170
those identical
infrared photons go

1185
01:17:29,170 --> 01:17:31,650
into the same mode,
which is squeezed.

1186
01:17:31,650 --> 01:17:34,440
But now, a slight
extension of this concept

1187
01:17:34,440 --> 01:17:37,410
would mean, in parametric
down conversion,

1188
01:17:37,410 --> 01:17:40,490
you squeeze something, but
one photon goes to Alice,

1189
01:17:40,490 --> 01:17:41,450
one photon goes to Bob.

1190
01:17:47,990 --> 01:17:53,930
So now, Alice and Bob have
an additional resource.

1191
01:17:53,930 --> 01:17:57,280
They sort of own--
each of them-- half

1192
01:17:57,280 --> 01:18:00,970
of the two twin brothers,
which are the photons created

1193
01:18:00,970 --> 01:18:03,050
in the parametric down
conversion process.

1194
01:18:05,900 --> 01:18:09,240
And that will work.

1195
01:18:14,740 --> 01:18:15,980
So the idea is the following.

1196
01:18:19,890 --> 01:18:24,190
To create those twin
beams of photons

1197
01:18:24,190 --> 01:18:29,020
is simply done with an
optical parametric oscillator.

1198
01:18:29,020 --> 01:18:31,755
There is one extension, which
is described on the Wiki.

1199
01:18:31,755 --> 01:18:33,380
I won't have time to
explain it to you.

1200
01:18:33,380 --> 01:18:36,080
But it's a two-mode OPO.

1201
01:18:36,080 --> 01:18:39,210
It puts the two identical
photons-- not in the same beam,

1202
01:18:39,210 --> 01:18:41,270
as we did before, we had
the a dagger squared,

1203
01:18:41,270 --> 01:18:47,470
a squared operator-- they
go into two different modes.

1204
01:18:47,470 --> 01:18:50,300
You know the magic of
beam splitters by now.

1205
01:18:50,300 --> 01:18:51,960
Now, two beams come out.

1206
01:18:51,960 --> 01:18:55,470
One goes to Alice,
one goes to Bob.

1207
01:18:55,470 --> 01:19:00,640
If Alice would take her input
state-- this unknown state

1208
01:19:00,640 --> 01:19:06,580
which is handed to her
by Victor, somebody

1209
01:19:06,580 --> 01:19:10,030
else who participates in
the game-- we know already,

1210
01:19:10,030 --> 01:19:14,160
if Alice would
perform a measurement,

1211
01:19:14,160 --> 01:19:16,200
the quantum state
would be destroyed.

1212
01:19:16,200 --> 01:19:19,830
The result of the measurement--
let's talk about spin-1/2--

1213
01:19:19,830 --> 01:19:21,420
would only be one
spin projection.

1214
01:19:24,220 --> 01:19:28,600
It's not enough to
reproduce a state.

1215
01:19:28,600 --> 01:19:31,270
But what she is doing
is, she uses, again,

1216
01:19:31,270 --> 01:19:33,330
the magic of the beam splitter.

1217
01:19:33,330 --> 01:19:36,170
So one of those
twin brother photons

1218
01:19:36,170 --> 01:19:38,650
is now mixed with
the unknown quantum

1219
01:19:38,650 --> 01:19:41,110
state at the beam splitter.

1220
01:19:41,110 --> 01:19:45,390
And the output of
the beam splitter

1221
01:19:45,390 --> 01:19:49,040
is now entering the
balanced homodyne detection,

1222
01:19:49,040 --> 01:19:51,270
which we just discussed.

1223
01:19:51,270 --> 01:19:53,320
The output of this beam
splitter-- and there

1224
01:19:53,320 --> 01:19:56,300
are two outputs--
both of them is now

1225
01:19:56,300 --> 01:19:59,040
becoming part of a balanced
homodyne measurement.

1226
01:19:59,040 --> 01:20:00,670
And you see the ingredients.

1227
01:20:00,670 --> 01:20:02,900
So this, and this,
will be measured.

1228
01:20:02,900 --> 01:20:04,800
The other input for
the balanced homodyne

1229
01:20:04,800 --> 01:20:08,120
is a strong local oscillator.

1230
01:20:08,120 --> 01:20:10,720
The phase of the
local oscillator

1231
01:20:10,720 --> 01:20:14,810
is chosen, in one case, that you
measure the x or a 1 quadrature

1232
01:20:14,810 --> 01:20:18,280
component, or the p
or the a 2 component.

1233
01:20:18,280 --> 01:20:21,490
So now what Alice
has done is, by using

1234
01:20:21,490 --> 01:20:24,060
this balanced
homodyning, she has--

1235
01:20:24,060 --> 01:20:26,730
with this local
oscillator-- has actually

1236
01:20:26,730 --> 01:20:29,430
now performed two measurements.

1237
01:20:29,430 --> 01:20:34,070
The quantum state is destroyed,
but here she gets an x value,

1238
01:20:34,070 --> 01:20:35,345
and here she gets a p value.

1239
01:20:38,560 --> 01:20:40,340
So how the magic works out.

1240
01:20:40,340 --> 01:20:44,270
It's really just a few
lines of mathematics, now.

1241
01:20:44,270 --> 01:20:48,170
These were sort of, you
know, twin brothers.

1242
01:20:48,170 --> 01:20:50,610
But it wasn't clear
in which quantum

1243
01:20:50,610 --> 01:20:52,712
state the twin brothers are.

1244
01:20:52,712 --> 01:20:54,212
If you write it
down, and I can show

1245
01:20:54,212 --> 01:20:56,850
you the formula in a moment,
these are twin brothers.

1246
01:21:00,570 --> 01:21:06,790
But those twin brothers-- this
is Brother 1, this is Brother

1247
01:21:06,790 --> 01:21:12,680
2-- are in sort of a
continuum of states.

1248
01:21:12,680 --> 01:21:16,480
And when Alice this would
measure that this twin brother

1249
01:21:16,480 --> 01:21:20,930
is in state big X, that would
be a projective measurement.

1250
01:21:20,930 --> 01:21:24,490
And Bob's twin brother
would now, with certainty,

1251
01:21:24,490 --> 01:21:28,570
also be in the state, X.

1252
01:21:28,570 --> 01:21:34,460
So therefore, what happens
is, the measurement of x and p

1253
01:21:34,460 --> 01:21:38,270
is now producing-- is now,
through the measurement

1254
01:21:38,270 --> 01:21:43,760
process-- putting the other
photon, or the other beam,

1255
01:21:43,760 --> 01:21:46,690
the other twin brother, into
a specific quantum state.

1256
01:21:49,780 --> 01:21:53,280
And if you look for
few lines of math,

1257
01:21:53,280 --> 01:21:57,210
the magic is that the quantum
state-- which is now here,

1258
01:21:57,210 --> 01:22:03,620
with Bob-- turns out
to be a displaced copy

1259
01:22:03,620 --> 01:22:06,070
of the original state.

1260
01:22:06,070 --> 01:22:09,930
And the displacement
depends on x and p.

1261
01:22:09,930 --> 01:22:14,750
So if Alice now tells Bob,
hey, I measured x and p,

1262
01:22:14,750 --> 01:22:20,170
and Bob is now [INAUDIBLE]
his displacement operator--

1263
01:22:20,170 --> 01:22:21,670
remember, a
displacement operator is

1264
01:22:21,670 --> 01:22:26,910
nothing else than an
unbalanced beam splitter,

1265
01:22:26,910 --> 01:22:30,300
with a huge local
oscillator as an input.

1266
01:22:30,300 --> 01:22:35,300
So if Bob is now setting up
his displacement operator,

1267
01:22:35,300 --> 01:22:38,210
it makes a displacement
which depends on x and p.

1268
01:22:38,210 --> 01:22:43,270
He can take this other twin
brother, shift it back,

1269
01:22:43,270 --> 01:22:46,530
and he will exactly
regenerate the quantum

1270
01:22:46,530 --> 01:22:50,050
state which Alice had.

1271
01:22:50,050 --> 01:22:52,830
So this is, now,
how a quantum state

1272
01:22:52,830 --> 01:22:57,024
can be transmitted
without having any quantum

1273
01:22:57,024 --> 01:22:58,065
channel for transmission.

1274
01:22:58,065 --> 01:23:00,380
You're not propagating
the quantum state.

1275
01:23:00,380 --> 01:23:04,380
You use classical communication,
but the resource you use

1276
01:23:04,380 --> 01:23:07,130
is some EPR pairs,
or squeezed light.

1277
01:23:15,260 --> 01:23:16,760
It's a few lines
of equations, but I

1278
01:23:16,760 --> 01:23:18,300
don't have time
to go through it.

1279
01:23:18,300 --> 01:23:19,730
They're really
annotated in a way

1280
01:23:19,730 --> 01:23:23,240
that I think it will be an
enjoyful reading for you.

1281
01:23:23,240 --> 01:23:24,730
Any questions?

1282
01:23:24,730 --> 01:23:27,541
Time is over.

1283
01:23:27,541 --> 01:23:28,040
OK.

1284
01:23:28,040 --> 01:23:31,540
A reminder for those who
came late-- this week

1285
01:23:31,540 --> 01:23:34,530
we have three classes,
Monday, Wednesday, Friday.

1286
01:23:34,530 --> 01:23:36,580
Have a good afternoon.