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00:00:21,012 --> 00:00:22,095
PROFESSOR: Good afternoon.

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00:00:27,390 --> 00:00:32,180
We continue our discussion
of quantum states of light.

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00:00:32,180 --> 00:00:36,210
We talked at length
about coherent state,

11
00:00:36,210 --> 00:00:39,720
and when you talk about
quantum states of light,

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00:00:39,720 --> 00:00:41,400
each mode of the
electromagnetic field

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00:00:41,400 --> 00:00:43,040
is an harmonic oscillator.

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00:00:43,040 --> 00:00:46,690
We also encountered,
naturally, the number states.

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00:00:46,690 --> 00:00:51,243
And we realized--
yesterday, actually,

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00:00:51,243 --> 00:00:54,140
in the last class-- that
those number states have

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00:00:54,140 --> 00:00:56,190
non-classical properties.

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00:00:56,190 --> 00:00:59,490
For instance, they have a g2
function, the second order

19
00:00:59,490 --> 00:01:01,790
correlation function,
which is smaller

20
00:01:01,790 --> 00:01:05,230
than 1, which is impossible
for classic light,

21
00:01:05,230 --> 00:01:08,670
as you're proving in one of
your homework assignment.

22
00:01:08,670 --> 00:01:12,580
So at that point, we have
encountered coherent states,

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00:01:12,580 --> 00:01:15,830
which are as close as
possible to classical states.

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00:01:15,830 --> 00:01:19,360
And we have found the number
states as non-classical states.

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00:01:19,360 --> 00:01:22,950
Well, are there other
interesting states?

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00:01:22,950 --> 00:01:26,450
I wouldn't ask you this question
if the answer would not be yes,

27
00:01:26,450 --> 00:01:28,690
and this is what we
want to discuss today.

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00:01:28,690 --> 00:01:32,420
We want to talk about
non-classical states of light,

29
00:01:32,420 --> 00:01:36,710
which we can engineer,
actually, in the laboratory,

30
00:01:36,710 --> 00:01:38,970
by sending laser light
through nonlinear crystals.

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00:01:45,130 --> 00:01:51,030
Those go by the name,
squeezed states.

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00:01:51,030 --> 00:01:57,540
Just to give you
the cartoon picture,

33
00:01:57,540 --> 00:02:02,350
in our two-dimensional diagram,
with the quasi-probabilities,

34
00:02:02,350 --> 00:02:10,990
we have coherent
states, where the area

35
00:02:10,990 --> 00:02:14,640
of this disk, delta x
delta p, is h-bar over two.

36
00:02:14,640 --> 00:02:17,970
It's uncertainty limited.

37
00:02:17,970 --> 00:02:22,140
What we can do we now, is--
we cannot go beyond this.

38
00:02:22,140 --> 00:02:25,750
This is the fundamental
limit of quantum physics.

39
00:02:25,750 --> 00:02:31,390
However, we can take this
circle and we can squeeze it.

40
00:02:31,390 --> 00:02:36,090
We can squeeze it
horizontally, we can squeeze it

41
00:02:36,090 --> 00:02:38,170
into an elongated
vertical shape,

42
00:02:38,170 --> 00:02:41,140
or we can squeeze
it at any angle.

43
00:02:41,140 --> 00:02:43,280
That's what we call,
squeezed states.

44
00:02:51,090 --> 00:02:56,570
And those states have
non-classical properties.

45
00:02:56,570 --> 00:02:59,125
They are important
for metrology they

46
00:02:59,125 --> 00:03:01,110
are important for teleportation.

47
00:03:01,110 --> 00:03:02,580
There are lots and
lots of reasons

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00:03:02,580 --> 00:03:05,620
why you want to know about them.

49
00:03:05,620 --> 00:03:10,930
But again, as so often, I
feel I cannot convey to you

50
00:03:10,930 --> 00:03:14,670
the excitement of doing
squeezing in the quantum

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00:03:14,670 --> 00:03:15,460
domain.

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00:03:15,460 --> 00:03:17,343
And many, many
physicists now, they

53
00:03:17,343 --> 00:03:20,510
hear about squeezing just
in the quantum domain.

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00:03:20,510 --> 00:03:22,810
But I want to start with
classical squeezing.

55
00:03:28,760 --> 00:03:31,140
I will actually show you
video of an experiment

56
00:03:31,140 --> 00:03:32,430
on classical squeezing.

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00:03:32,430 --> 00:03:36,020
You can see squeezing
with your own eyes.

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00:03:36,020 --> 00:03:38,760
But this is just sort
of to set the stage,

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00:03:38,760 --> 00:03:41,860
to also get a feel
of what squeezing is.

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00:03:41,860 --> 00:03:44,655
And then we'll do quantum
mechanical squeezing.

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00:03:47,670 --> 00:03:49,280
But maybe-- tongue
in cheek-- I would

62
00:03:49,280 --> 00:03:52,400
say, since classical harmonic
oscillators and quantum

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00:03:52,400 --> 00:03:54,730
harmonic oscillators
have a lot in common,

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00:03:54,730 --> 00:03:57,310
the step from classical
squeezing to quantum

65
00:03:57,310 --> 00:03:59,520
of mechanical squeezing
is actually rather small.

66
00:04:02,040 --> 00:04:04,420
It's nice to squeeze light.

67
00:04:04,420 --> 00:04:06,890
It's nice to have those
non-classical states.

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00:04:06,890 --> 00:04:09,560
But the question is,
how can you detect it?

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00:04:09,560 --> 00:04:13,890
If you can't detect it, you
can't take advantage of it.

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00:04:13,890 --> 00:04:16,730
And the detection has
to be face-coherent.

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00:04:16,730 --> 00:04:18,380
I will tell you what that is.

72
00:04:18,380 --> 00:04:22,770
And it goes by the name,
homodyne detection.

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00:04:22,770 --> 00:04:25,970
And finally, we
can take everything

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00:04:25,970 --> 00:04:31,440
we have learned together, and
discuss how, in the laboratory,

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00:04:31,440 --> 00:04:34,681
teleportation of a
quantum state is done.

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00:04:34,681 --> 00:04:37,140
There is a nice
teleportation scheme,

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00:04:37,140 --> 00:04:39,190
and I want to use
that as an example

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00:04:39,190 --> 00:04:41,540
that the language and the
concepts I've introduced

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00:04:41,540 --> 00:04:43,290
are useful.

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00:04:43,290 --> 00:04:45,250
Concepts like,
squeezing operator,

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00:04:45,250 --> 00:04:49,330
displacement operator--
those methods

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00:04:49,330 --> 00:04:54,657
allowing us to, in
a very clear way,

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00:04:54,657 --> 00:04:56,490
discuss schemes which
lead to teleportation.

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00:04:59,720 --> 00:05:01,340
That's the menu for today.

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00:05:05,540 --> 00:05:07,645
Let's start with
classical squeezing.

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00:05:23,900 --> 00:05:30,590
For squeezing, we need
an harmonic oscillator,

87
00:05:30,590 --> 00:05:34,010
means for parabolic potential,
we have potential v of x.

88
00:05:37,760 --> 00:05:44,681
And then we study
the motion of-- that

89
00:05:44,681 --> 00:05:47,910
should be x squared-- the
motion of a particle in there.

90
00:05:47,910 --> 00:05:49,800
Before I even get
started any equation,

91
00:05:49,800 --> 00:05:53,660
let me explain what the effect
of squeezing will be about.

92
00:05:53,660 --> 00:05:55,560
If you have an
harmonic oscillator,

93
00:05:55,560 --> 00:05:59,910
you have, actually, the
motion of a pendulum.

94
00:05:59,910 --> 00:06:03,240
It has two quadrature
components, the cosine motion

95
00:06:03,240 --> 00:06:04,430
and the sine motion.

96
00:06:04,430 --> 00:06:07,470
And they are 90
degrees out of phase.

97
00:06:07,470 --> 00:06:10,780
What happens now is,
if you parametrically

98
00:06:10,780 --> 00:06:13,590
drive the harmonic
oscillator-- you modulate

99
00:06:13,590 --> 00:06:15,930
the harmonic oscillator
potential-- it's to omega.

100
00:06:18,440 --> 00:06:22,390
I will show you mathematically,
it's very, very easy to show,

101
00:06:22,390 --> 00:06:25,460
that depending on the
phase of the drive,

102
00:06:25,460 --> 00:06:29,720
you will actually exponentially
amplify the sine motion,

103
00:06:29,720 --> 00:06:31,900
and exponentially damp
the cosine motion.

104
00:06:31,900 --> 00:06:34,030
Or if you change, vice versa.

105
00:06:34,030 --> 00:06:39,290
So by driving the system, you
can amplify one quadrature

106
00:06:39,290 --> 00:06:45,490
component, and exponentially
die out the other quadrature

107
00:06:45,490 --> 00:06:46,709
component.

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00:06:46,709 --> 00:06:48,375
And that is called,
classical squeezing.

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00:06:54,010 --> 00:06:55,640
Let's do the math.

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00:06:55,640 --> 00:06:56,680
It's very simple.

111
00:06:59,870 --> 00:07:11,830
Our equation of motion
has the two solutions

112
00:07:11,830 --> 00:07:13,940
I've just mentioned.

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00:07:13,940 --> 00:07:19,780
It has a solution
with cosine omega 0,

114
00:07:19,780 --> 00:07:21,950
and one with sine omega 0 t.

115
00:07:25,670 --> 00:07:28,300
And we have two coefficients.

116
00:07:28,300 --> 00:07:30,030
The cosine is called, c.

117
00:07:30,030 --> 00:07:31,830
The sine coefficient
is called, s.

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00:07:41,550 --> 00:07:45,740
I have to call it c 0, because
I want to call that c and s.

119
00:07:49,070 --> 00:08:02,610
So what we have here is, we have
the two quadrature components

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00:08:02,610 --> 00:08:06,770
of the motion in an
harmonic oscillator.

121
00:08:06,770 --> 00:08:10,450
And graphically, we need that
for the electromagnetic field,

122
00:08:10,450 --> 00:08:11,560
as well.

123
00:08:11,560 --> 00:08:14,210
When we have our two
axes, like, you know,

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00:08:14,210 --> 00:08:17,820
the complex plane for the
cosine of probabilities,

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00:08:17,820 --> 00:08:21,730
I call one the s-axis.

126
00:08:21,730 --> 00:08:22,630
One is the c-axis.

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00:08:32,830 --> 00:08:36,710
That's just something
which confuses me.

128
00:08:40,350 --> 00:08:51,720
If you have only one--
just give me one second.

129
00:08:51,720 --> 00:08:54,220
Cosine-- Yeah.

130
00:08:57,070 --> 00:09:03,820
If you have only cosine
motion, the s component is 0,

131
00:09:03,820 --> 00:09:07,560
and the harmonic oscillator
would just oscillate here.

132
00:09:07,560 --> 00:09:10,700
If you have only
a sine component,

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00:09:10,700 --> 00:09:12,085
you stay on the x-axis.

134
00:09:14,590 --> 00:09:21,550
And now, if you have an equal
amount of cosine and sine,

135
00:09:21,550 --> 00:09:27,120
then you can describe the
trajectory to go in a circle.

136
00:09:29,720 --> 00:09:30,220
OK.

137
00:09:30,220 --> 00:09:33,630
This is just the undriven
harmonic oscillator.

138
00:09:33,630 --> 00:09:36,510
I don't want to dwell
on it any longer.

139
00:09:36,510 --> 00:09:39,370
But what we are
doing now is, we are

140
00:09:39,370 --> 00:09:41,990
adding a small parametric drive.

141
00:09:54,280 --> 00:10:01,330
Parametric drive means we
modulate the spring constant,

142
00:10:01,330 --> 00:10:08,090
or we replace the original
harmonic potential,

143
00:10:08,090 --> 00:10:14,710
which was this, by an
extra modulation term.

144
00:10:14,710 --> 00:10:17,930
So we have a small
parameter, epsilon.

145
00:10:17,930 --> 00:10:20,610
And as I pointed
out, the modulation

146
00:10:20,610 --> 00:10:22,450
is at twice the
resonance frequency.

147
00:10:25,060 --> 00:10:28,300
Now we want to solve
the equation of motion

148
00:10:28,300 --> 00:10:31,320
for the harmonic oscillator,
using this added potential.

149
00:10:45,180 --> 00:10:51,960
The way how we want to solve
it is, we assume epsilon

150
00:10:51,960 --> 00:10:53,420
is very small.

151
00:10:53,420 --> 00:10:56,420
So if the pendulum is
swinging with cosine omega t,

152
00:10:56,420 --> 00:10:58,770
it will take a while
for the epsilon

153
00:10:58,770 --> 00:11:02,520
term-- for the small term--
to change the motion.

154
00:11:02,520 --> 00:11:05,710
So therefore, we assume
that we can actually

155
00:11:05,710 --> 00:11:12,550
go back and use our
original solution.

156
00:11:12,550 --> 00:11:16,386
And assume that
over a short term,

157
00:11:16,386 --> 00:11:18,010
the epsilon term is
not doing anything.

158
00:11:18,010 --> 00:11:20,710
So for a short time, it looks
like an harmonic oscillator

159
00:11:20,710 --> 00:11:24,660
with a sine omega 0, and
cosine omega 0 t oscillation.

160
00:11:24,660 --> 00:11:27,830
But over any longer
period of time,

161
00:11:27,830 --> 00:11:30,530
the small term will
have an effect.

162
00:11:30,530 --> 00:11:34,560
And therefore, the
coefficients c of t, c, and s

163
00:11:34,560 --> 00:11:37,680
are no longer constant, but
change as a function of time.

164
00:11:54,600 --> 00:11:57,560
We want to solve, now,
the equation of motion.

165
00:11:57,560 --> 00:12:07,440
That means we use this,
here, as our ansatz.

166
00:12:07,440 --> 00:12:15,300
And we calculate the
second derivative.

167
00:12:15,300 --> 00:12:23,100
We assume that the coefficients
c and s are changing slowly.

168
00:12:23,100 --> 00:12:28,130
Therefore, the second derivative
of c and s can be neglected.

169
00:12:31,900 --> 00:12:35,370
By taking the derivative
of the second derivative

170
00:12:35,370 --> 00:12:38,010
of the cosine term
and the sine term,

171
00:12:38,010 --> 00:12:48,950
of course we simply get,
minus omega 0 squared, x of t.

172
00:12:55,160 --> 00:12:57,510
And now we have the
second-order derivatives.

173
00:13:00,100 --> 00:13:03,070
Since we neglect the
second-order derivative of c

174
00:13:03,070 --> 00:13:05,580
and s, the other
terms we get when

175
00:13:05,580 --> 00:13:08,310
we take the second derivative
is, first derivative

176
00:13:08,310 --> 00:13:11,230
of c times first
derivative of cosine.

177
00:13:11,230 --> 00:13:15,190
First derivative of s times
first derivative of sine.

178
00:13:15,190 --> 00:13:21,460
So we get two more terms,
which are, minus omega 0 c dot,

179
00:13:21,460 --> 00:13:25,880
times sine omega 0 t.

180
00:13:25,880 --> 00:13:36,690
Plus omega 0 s dot,
times cosine omega 0 t.

181
00:13:36,690 --> 00:13:42,305
This is the second derivative
of our ansatz for x.

182
00:13:48,240 --> 00:13:53,780
This has to be equal
to the force provided

183
00:13:53,780 --> 00:13:55,760
by the potential.

184
00:13:55,760 --> 00:13:58,440
So taking the potential--

185
00:14:28,424 --> 00:14:30,340
We need, now, the
derivative of the potential,

186
00:14:30,340 --> 00:14:33,260
for the potential of
use across this line.

187
00:14:33,260 --> 00:14:35,020
The first part is the
unperturbed harmonic

188
00:14:35,020 --> 00:14:38,460
oscillator, which
gives us simply,

189
00:14:38,460 --> 00:14:42,430
omega 0 squared times x.

190
00:14:42,430 --> 00:14:49,670
And the second term, due to
the parametric drive, is 2

191
00:14:49,670 --> 00:14:52,430
sine omega 0 t.

192
00:14:52,430 --> 00:14:56,835
And now, for x,
we use our ansatz

193
00:14:56,835 --> 00:15:01,640
for x, which is the
slowly-changing amplitude c

194
00:15:01,640 --> 00:15:08,080
times cosine omega 0 t,
plus s times sine omega 0 t.

195
00:15:15,010 --> 00:15:17,130
Those two terms cancel out.

196
00:15:27,040 --> 00:15:30,980
So now we have products
of trig function.

197
00:15:30,980 --> 00:15:34,140
Sine 2 omega times cosine omega.

198
00:15:34,140 --> 00:15:38,270
Well, you know if you take the
product of two trig functions,

199
00:15:38,270 --> 00:15:43,680
it becomes a trig function
of the sum or the difference

200
00:15:43,680 --> 00:15:45,110
of the argument.

201
00:15:45,110 --> 00:15:49,620
So if you take sine 2 omega
0 times cosine omega 0,

202
00:15:49,620 --> 00:15:52,520
and we use trigonometric
identities,

203
00:15:52,520 --> 00:15:58,290
we get an oscillation at 3
omega 0, which is 2 plus 1.

204
00:15:58,290 --> 00:16:00,380
And one at the difference,
which is omega 0.

205
00:16:03,100 --> 00:16:06,500
Let me write down the terms
which are of interest to us.

206
00:16:06,500 --> 00:16:09,220
Namely, the ones at omega 0.

207
00:16:09,220 --> 00:16:15,580
So let me factor out epsilon
omega 0 squared over 2.

208
00:16:15,580 --> 00:16:23,395
Then we have the term c times
sine omega 0 t, plus s times

209
00:16:23,395 --> 00:16:23,895
cosine.

210
00:16:27,880 --> 00:16:33,640
And then we have
terms at 3 omega

211
00:16:33,640 --> 00:16:37,680
0, which we are
going to neglect.

212
00:16:44,720 --> 00:16:51,530
Now we compare the two
sides of the equations.

213
00:16:51,530 --> 00:16:53,380
We have sine omega 0 term.

214
00:16:53,380 --> 00:16:56,710
We have cosine omega 0 term.

215
00:16:56,710 --> 00:16:59,040
And the two sides
of the equations

216
00:16:59,040 --> 00:17:03,770
are only consistent if the two
coefficients of the sine term,

217
00:17:03,770 --> 00:17:05,800
and the sine term, are the same.

218
00:17:05,800 --> 00:17:08,634
So therefore, we
obtain two equations.

219
00:17:13,680 --> 00:17:15,920
One for c dot, one for s dot.

220
00:17:28,030 --> 00:17:31,480
And these are first-order
differential equations.

221
00:17:31,480 --> 00:17:34,050
The solution is
clearly an exponential.

222
00:17:34,050 --> 00:17:37,380
But one has a plus sign,
one has a minus sign.

223
00:17:37,380 --> 00:17:40,710
So the c component, the
c quadrature component,

224
00:17:40,710 --> 00:17:43,840
is exponentially amplified
with this time constant.

225
00:17:43,840 --> 00:17:46,990
Whereas the sine component is
exponentially de-amplified.

226
00:17:55,290 --> 00:17:57,800
This finishes the
mathematical discussion

227
00:17:57,800 --> 00:17:59,210
of classical squeezing.

228
00:17:59,210 --> 00:18:08,220
We find that s of t, and c of
t, are exponential functions.

229
00:18:08,220 --> 00:18:18,290
In one case, it's
exponentially increasing.

230
00:18:18,290 --> 00:18:25,810
In the other case, it is
exponentially decreasing.

231
00:18:25,810 --> 00:18:32,940
And that means that, well,
if we go to our diagram,

232
00:18:32,940 --> 00:18:37,625
here-- and let's assume
we had an arbitrary

233
00:18:37,625 --> 00:18:40,570
superposition of cosine
and sine amplitude.

234
00:18:46,880 --> 00:18:47,870
This is cosine.

235
00:18:47,870 --> 00:18:49,210
This is sine.

236
00:18:49,210 --> 00:18:55,880
We had sort of a
cosine oscillation,

237
00:18:55,880 --> 00:18:59,500
and a sine oscillation.

238
00:18:59,500 --> 00:19:08,920
Which means that, as
a phasor, the system

239
00:19:08,920 --> 00:19:11,590
was moving on an ellipse.

240
00:19:11,590 --> 00:19:15,770
If the sine component is
exponentially de-amplified,

241
00:19:15,770 --> 00:19:20,190
and the cosine component
is exponentially amplified,

242
00:19:20,190 --> 00:19:25,110
that means whatever we start
with is squashed horizontally,

243
00:19:25,110 --> 00:19:26,810
is squashed vertically.

244
00:19:26,810 --> 00:19:31,480
And is amplified horizontally.

245
00:19:31,480 --> 00:19:34,809
In the end, it will
become a narrow strip.

246
00:19:34,809 --> 00:19:36,100
So this is classical squeezing.

247
00:19:40,520 --> 00:19:44,740
You may want to ask, why did
I neglect the 3 omega 0 term.

248
00:19:44,740 --> 00:19:50,110
Well, I have to, otherwise
I don't have a solution.

249
00:19:50,110 --> 00:19:53,730
Because I have to be consistent
with my approximations.

250
00:19:53,730 --> 00:19:57,200
So what I did here
is, I had an equation

251
00:19:57,200 --> 00:20:03,540
where I have the clear
vision that the solution has

252
00:20:03,540 --> 00:20:06,300
a slowly varying c
and s coefficient.

253
00:20:06,300 --> 00:20:08,730
And then I simply use that.

254
00:20:08,730 --> 00:20:13,150
I take the second-order
derivative,

255
00:20:13,150 --> 00:20:15,410
and I have only
Fourier components

256
00:20:15,410 --> 00:20:17,930
with omega 0, the
sine, and cosine.

257
00:20:17,930 --> 00:20:22,740
Now I've made an
approximation, here.

258
00:20:22,740 --> 00:20:25,530
For the derivative
of the potential,

259
00:20:25,530 --> 00:20:27,690
the first line is exact.

260
00:20:27,690 --> 00:20:30,040
But in order to match
the approximation

261
00:20:30,040 --> 00:20:32,450
I've done on the
other side, I can only

262
00:20:32,450 --> 00:20:37,910
focus on two Fourier components
resonant with omega 0,

263
00:20:37,910 --> 00:20:39,450
which I have here.

264
00:20:39,450 --> 00:20:42,140
So in other words,
the 3 omega 0 term

265
00:20:42,140 --> 00:20:44,900
would lead to additional
accelerations.

266
00:20:44,900 --> 00:20:49,910
Which I have not included
in the treatment.

267
00:20:49,910 --> 00:20:51,512
So it's consistent
with the ansatz.

268
00:20:51,512 --> 00:20:52,970
It's consistent
with the assumption

269
00:20:52,970 --> 00:20:56,490
that we have
resonant oscillations

270
00:20:56,490 --> 00:20:58,515
with a slowly
changing amplitude.

271
00:20:58,515 --> 00:21:00,640
There will be a small
[INAUDIBLE] for your omega 0,

272
00:21:00,640 --> 00:21:03,190
but it will be small.

273
00:21:03,190 --> 00:21:04,340
Any questions about that?

274
00:21:08,730 --> 00:21:20,580
Let me then show you
an animation of that.

275
00:21:27,394 --> 00:21:28,060
Classroom files.

276
00:21:31,184 --> 00:21:31,850
[VIDEO PLAYBACK]

277
00:21:31,850 --> 00:21:34,775
-We have Dave Pritchard,
professor of physics at MIT,

278
00:21:34,775 --> 00:21:37,860
demonstrating what squeezing is.

279
00:21:37,860 --> 00:21:41,450
Right now, we see a wave that's
going around in a circle.

280
00:21:41,450 --> 00:21:42,250
What's next?

281
00:21:42,250 --> 00:21:44,660
What's going to happen
now, Professor Pritchard?

282
00:21:44,660 --> 00:21:49,890
-Well, if we drive it in
twice the basic period,

283
00:21:49,890 --> 00:21:53,440
then we will parametrically
amplify one quadrature

284
00:21:53,440 --> 00:21:57,239
component, and we will
un-amplify the other one.

285
00:21:57,239 --> 00:21:58,780
So now I'm going to
start doing that.

286
00:22:02,510 --> 00:22:05,550
And then you notice that its
motion turns into an ellipse.

287
00:22:05,550 --> 00:22:09,440
We've amplified this
quadrature component,

288
00:22:09,440 --> 00:22:11,605
but we've un-amplifed that one.

289
00:22:11,605 --> 00:22:14,830
And that's squeezing.

290
00:22:14,830 --> 00:22:16,605
[END VIDEO PLAYBACK]

291
00:22:16,605 --> 00:22:18,230
PROFESSOR: Feel free
to try it at home.

292
00:22:18,230 --> 00:22:20,284
[LAUGHTER]

293
00:22:20,284 --> 00:22:21,700
PROFESSOR: Actually,
you may start

294
00:22:21,700 --> 00:22:24,060
to think about
this demonstration.

295
00:22:24,060 --> 00:22:29,210
What he has shown was, when you
have a circular pendulum which

296
00:22:29,210 --> 00:22:31,560
goes in a circle or
an ellipse, and you

297
00:22:31,560 --> 00:22:36,290
start pulling on the rope
with a certain phase,

298
00:22:36,290 --> 00:22:39,540
that one quadrature component
will be de-amplified.

299
00:22:39,540 --> 00:22:41,460
The other one will be amplified.

300
00:22:41,460 --> 00:22:44,620
And as a result, no
matter what the circular

301
00:22:44,620 --> 00:22:48,140
or the elliptical motion was,
after driving it for a while,

302
00:22:48,140 --> 00:22:52,204
it will only swing
in one direction.

303
00:22:52,204 --> 00:22:54,120
And this is the collection
you have amplified.

304
00:22:57,890 --> 00:23:00,350
There is one thing which
should give you pause.

305
00:23:00,350 --> 00:23:03,670
I have discussed, actually,
a single harmonic oscillator.

306
00:23:03,670 --> 00:23:06,510
What Dave Pritchard
demonstrated was actually

307
00:23:06,510 --> 00:23:08,300
two harmonic oscillators.

308
00:23:08,300 --> 00:23:12,250
The harmonic oscillator has
an x motion and a y motion.

309
00:23:12,250 --> 00:23:14,810
However, you can
say, this was just

310
00:23:14,810 --> 00:23:17,120
sort of a trick for
the demonstration,

311
00:23:17,120 --> 00:23:19,370
because when you have
a circular motion,

312
00:23:19,370 --> 00:23:21,970
initially, you
have the sine omega

313
00:23:21,970 --> 00:23:26,744
and the cosine omega 0 component
present simultaneously.

314
00:23:26,744 --> 00:23:28,410
And you can see what
happens to the sine

315
00:23:28,410 --> 00:23:30,970
and the cosine component
in one experiment.

316
00:23:30,970 --> 00:23:33,181
So in that sense, he did
two experiments at once.

317
00:23:33,181 --> 00:23:35,180
He showed what happens
when you have, initially,

318
00:23:35,180 --> 00:23:37,500
a sine component, and what
happens when you initially

319
00:23:37,500 --> 00:23:40,660
have a cosine component.

320
00:23:40,660 --> 00:23:41,200
OK.

321
00:23:41,200 --> 00:23:44,630
So we know what
classical squeezing is.

322
00:23:44,630 --> 00:23:49,300
And what we have learned, also--
and this helps me now a lot

323
00:23:49,300 --> 00:23:52,610
to motivate how we squeeze
in quantum mechanics--

324
00:23:52,610 --> 00:23:56,100
you have realized that what
is really essential here is,

325
00:23:56,100 --> 00:24:00,610
to drive it to omega 0.

326
00:24:00,610 --> 00:24:03,020
What we need now to do
squeezing in the quantum domain,

327
00:24:03,020 --> 00:24:08,210
if we want to squeeze light,
we need something at 2 omega 0.

328
00:24:08,210 --> 00:24:18,910
So let's now squeeze
quantum mechanically.

329
00:24:21,780 --> 00:24:22,700
Go back here.

330
00:24:30,190 --> 00:24:34,315
The second sub-section is
now, squeezed quantum states.

331
00:24:51,580 --> 00:24:53,330
What we want to
discuss is, we want

332
00:24:53,330 --> 00:24:58,520
to discuss a quantum
harmonic oscillator.

333
00:24:58,520 --> 00:25:07,940
We want to have some form of
parametric drive at 2 omega 0.

334
00:25:07,940 --> 00:25:13,240
And this will result
in squeezed states.

335
00:25:19,180 --> 00:25:28,510
Now, what does it require, if
you want to bring in 2 omega 0?

336
00:25:28,510 --> 00:25:31,480
Well, let's not forget
our harmonic oscillators

337
00:25:31,480 --> 00:25:34,470
are modes of the
electromagnetic field.

338
00:25:34,470 --> 00:25:38,440
If you now want to couple a mode
of the electromagnetic field,

339
00:25:38,440 --> 00:25:44,390
at 2 omega 0, with our
harmonic oscillator at omega 0,

340
00:25:44,390 --> 00:25:48,070
we need a coupling between
two electromagnetic fields.

341
00:25:48,070 --> 00:25:52,000
So therefore, we need nonlinear
interactions between photons.

342
00:25:55,150 --> 00:25:56,270
So this was a tautology.

343
00:25:56,270 --> 00:25:58,347
We need nonlinear
physics, which leads

344
00:25:58,347 --> 00:25:59,680
to interactions between photons.

345
00:25:59,680 --> 00:26:02,520
Linear physics means,
each harmonic oscillator

346
00:26:02,520 --> 00:26:03,950
is independent.

347
00:26:03,950 --> 00:26:07,130
So we need some
nonlinear process

348
00:26:07,130 --> 00:26:14,214
six which will be equivalent
to have interactions

349
00:26:14,214 --> 00:26:14,880
between photons.

350
00:26:24,500 --> 00:26:27,900
The device which we
will provide that

351
00:26:27,900 --> 00:26:30,550
is an optical
parametric oscillator.

352
00:26:43,690 --> 00:26:47,250
I could spend a long
time explaining to you

353
00:26:47,250 --> 00:26:49,810
how those nonlinear
crystals work.

354
00:26:49,810 --> 00:26:53,360
What is the polarization,
what is the polarizability,

355
00:26:53,360 --> 00:26:56,110
how do you drive it,
what is the nonlinearity.

356
00:26:56,110 --> 00:27:01,120
But for the discussion
in this class,

357
00:27:01,120 --> 00:27:03,360
which focuses on
fundamental concepts,

358
00:27:03,360 --> 00:27:06,070
I can actually bypass
it by just saying,

359
00:27:06,070 --> 00:27:08,910
assume you have a system--
and this is actually

360
00:27:08,910 --> 00:27:11,010
what the optical
parametric oscillator does,

361
00:27:11,010 --> 00:27:17,060
is you pump it with
photons at 2 omega 0.

362
00:27:17,060 --> 00:27:26,240
And then the crystal generates
two photons at omega 0.

363
00:27:26,240 --> 00:27:30,490
Which of course, is consistent
with energy conservation.

364
00:27:30,490 --> 00:27:34,250
And if you fulfill some
phase-matching condition,

365
00:27:34,250 --> 00:27:37,590
it's also consistent with
momentum conservation.

366
00:27:37,590 --> 00:27:40,100
But I don't want to go into
phase-matching at this point.

367
00:27:43,290 --> 00:27:48,800
Technically, this is
done as simple as that.

368
00:27:48,800 --> 00:27:52,120
You have to pick
the right crystal.

369
00:27:52,120 --> 00:27:55,940
Actually, a crystal which does
mixing between three photon

370
00:27:55,940 --> 00:27:58,850
fields cannot have
inversion symmetry,

371
00:27:58,850 --> 00:28:03,160
otherwise this
nonlinear term is 0.

372
00:28:03,160 --> 00:28:05,110
What you need is
a special crystal.

373
00:28:05,110 --> 00:28:07,780
KDP is a common choice.

374
00:28:07,780 --> 00:28:10,415
And this crystal will now
do for us the following.

375
00:28:13,320 --> 00:28:14,900
You shine in laser light.

376
00:28:14,900 --> 00:28:19,000
Let's say, at 532
nanometer, a green light.

377
00:28:19,000 --> 00:28:31,080
And then this photon breaks up
into two photons of omega 0.

378
00:28:31,080 --> 00:28:33,490
This is how it's done
in the laboratory.

379
00:28:33,490 --> 00:28:36,170
The piece of art is, you have
to pick the right crystal.

380
00:28:36,170 --> 00:28:38,030
It has to be cut
at the right angle.

381
00:28:38,030 --> 00:28:40,840
You may have to heat
it, and make sure

382
00:28:40,840 --> 00:28:44,440
that you select the
temperature for which

383
00:28:44,440 --> 00:28:47,270
some form of resonant condition
is fulfilled, to do that.

384
00:28:47,270 --> 00:28:48,890
But in essence,
that's what you do.

385
00:28:48,890 --> 00:28:52,230
One laser beam, put in a
crystal, and then the photon

386
00:28:52,230 --> 00:28:54,310
is broken into two equal parts.

387
00:28:54,310 --> 00:28:56,300
And these are our two
photons at omega 0.

388
00:29:03,300 --> 00:29:03,800
OK.

389
00:29:07,150 --> 00:29:09,890
I hope you enjoy the
elegance-- we can completely

390
00:29:09,890 --> 00:29:15,200
bypass all the material physics
by putting operators on it.

391
00:29:15,200 --> 00:29:17,300
We call this mode, b.

392
00:29:17,300 --> 00:29:19,730
And we call this mode, a.

393
00:29:19,730 --> 00:29:24,840
So the whole parametric process,
the down conversion process

394
00:29:24,840 --> 00:29:28,230
of one photon into
two, is now described

395
00:29:28,230 --> 00:29:31,310
by the following Hamiltonian.

396
00:29:31,310 --> 00:29:36,820
We destroy a photon in
mode b, a 2 omega 0.

397
00:29:36,820 --> 00:29:41,090
And now we create
two photons at omega.

398
00:29:41,090 --> 00:29:42,760
We destroy a photon
at 2 omega 0,

399
00:29:42,760 --> 00:29:44,980
create two photons at omega 0.

400
00:29:44,980 --> 00:29:50,330
And since the Hamiltonian
has to be Hamiltonian,

401
00:29:50,330 --> 00:29:52,780
the opposite, the
time-reverse process,

402
00:29:52,780 --> 00:29:54,250
has to be possible, too.

403
00:29:54,250 --> 00:29:58,350
And that means we destroy
two photons at omega 0,

404
00:29:58,350 --> 00:30:00,460
and create one
photon at 2 omega 0.

405
00:30:04,200 --> 00:30:07,100
So now we forget about
nonlinear crystals,

406
00:30:07,100 --> 00:30:10,220
about non-inversion
symmetry in materials.

407
00:30:10,220 --> 00:30:12,417
We just take this
Hamiltonian and play with it.

408
00:30:20,060 --> 00:30:21,720
By simply looking
at the Hamiltonian,

409
00:30:21,720 --> 00:30:23,400
what is the time
evolution of a photon

410
00:30:23,400 --> 00:30:25,090
field under this Hamiltonian.

411
00:30:25,090 --> 00:30:27,740
We figure out what happens
when you send light

412
00:30:27,740 --> 00:30:30,170
through a crystal, and
what is the output.

413
00:30:30,170 --> 00:30:33,060
And I want to show you now
that the output of that

414
00:30:33,060 --> 00:30:37,380
is squeezed light, which is
exactly what I promised you

415
00:30:37,380 --> 00:30:39,340
with these quasi-probabilities.

416
00:30:39,340 --> 00:30:44,270
We have a coherent state,
which is a nice circle.

417
00:30:44,270 --> 00:30:49,990
We time-evolve the coherent
state, our nice round circle,

418
00:30:49,990 --> 00:30:51,620
with this Hamiltonian.

419
00:30:51,620 --> 00:30:52,870
And what we get is an ellipse.

420
00:30:55,910 --> 00:30:59,190
And if you want intuition,
look at the classical example

421
00:30:59,190 --> 00:31:01,490
we did before, which
really tells you

422
00:31:01,490 --> 00:31:05,910
in a more intuitive
way what is happening.

423
00:31:05,910 --> 00:31:07,190
OK.

424
00:31:07,190 --> 00:31:10,600
We want to make one
simplifying assumption, here.

425
00:31:10,600 --> 00:31:13,910
And this is that
we pump the crystal

426
00:31:13,910 --> 00:31:17,320
at 2 omega 0 with a
strong laser beam.

427
00:31:17,320 --> 00:31:26,450
So we assume that the mode,
b, is a powerful laser beam.

428
00:31:26,450 --> 00:31:29,440
Or in other words, a
strong coherent state.

429
00:31:37,330 --> 00:31:42,030
We assume that the mode,
b, is in a coherent state.

430
00:31:44,600 --> 00:31:46,270
Coherent states
are always labeled

431
00:31:46,270 --> 00:31:48,940
with a complex parameter,
which I call beta, now.

432
00:31:48,940 --> 00:31:51,150
Well, it's mode b,
therefore I call it, beta.

433
00:31:51,150 --> 00:31:54,580
For mode a, I've
called it, alpha.

434
00:31:54,580 --> 00:31:58,610
The coherent state has an
amplitude, which I call,

435
00:31:58,610 --> 00:31:59,690
r over 2.

436
00:31:59,690 --> 00:32:00,670
And it has a phase.

437
00:32:03,550 --> 00:32:11,490
We know, of course, that the
operator, b, acting on beta,

438
00:32:11,490 --> 00:32:15,450
gives us beta times beta,
because a coherent state is

439
00:32:15,450 --> 00:32:17,315
an eigenstate of the
annihilation operator.

440
00:32:21,490 --> 00:32:25,590
But when we look at the
action of the operator b

441
00:32:25,590 --> 00:32:32,900
plus, the photon
creation operator,

442
00:32:32,900 --> 00:32:35,080
the coherent state
is not an eigenstate

443
00:32:35,080 --> 00:32:36,290
of the creation operator.

444
00:32:36,290 --> 00:32:38,520
It's only an eigenstate of
the annihilation operator.

445
00:32:38,520 --> 00:32:42,100
But what sort of happens
is, the coherent state

446
00:32:42,100 --> 00:32:45,370
is the sum over many,
many number states with n.

447
00:32:45,370 --> 00:32:49,740
And the creation operator
goes from n, to n plus 1,

448
00:32:49,740 --> 00:32:56,680
and has matrix elements which
are square root n plus 1.

449
00:32:56,680 --> 00:32:59,850
So in other words,
if n is large,

450
00:32:59,850 --> 00:33:03,740
and if we don't care about the
subtle difference between n,

451
00:33:03,740 --> 00:33:08,200
and n plus 1, in this
limit the coherent state

452
00:33:08,200 --> 00:33:12,880
is also an eigenstate of
the creation operator,

453
00:33:12,880 --> 00:33:18,220
with an eigenvalue,
which is beta star.

454
00:33:24,530 --> 00:33:28,940
This means that we have a
coherent state which is strong.

455
00:33:28,940 --> 00:33:31,020
Strong means, it has
a large amplitude

456
00:33:31,020 --> 00:33:33,370
of the electric field.

457
00:33:33,370 --> 00:33:36,820
The photon states which
are involved, n, are large.

458
00:33:36,820 --> 00:33:40,039
And we don't have whether
it be n, or n plus 1.

459
00:33:40,039 --> 00:33:42,080
This is actually, also,
I should mention it here,

460
00:33:42,080 --> 00:33:44,580
explicitly-- this is
sort of the step when

461
00:33:44,580 --> 00:33:47,030
we have a quantum
description of light.

462
00:33:47,030 --> 00:33:52,630
And we replace the operators,
p and p dega, by a c number,

463
00:33:52,630 --> 00:33:55,150
then we really go back
to classical physics.

464
00:33:55,150 --> 00:33:59,210
Then we pretend that we have a
classical electric field, which

465
00:33:59,210 --> 00:34:05,260
is described by the
imaginary part of beta.

466
00:34:05,260 --> 00:34:08,969
So when you have an Hamiltonian,
where you write down

467
00:34:08,969 --> 00:34:11,330
an electric field, and
the electric field is not

468
00:34:11,330 --> 00:34:14,100
changing-- you have an
external electric field.

469
00:34:14,100 --> 00:34:16,830
This is really the limit
of a quantum field,

470
00:34:16,830 --> 00:34:20,409
where you've eliminated
the operator by a c number.

471
00:34:20,409 --> 00:34:22,110
This is essentially
your electric field.

472
00:34:22,110 --> 00:34:25,320
And we do this
approximation, here.

473
00:34:25,320 --> 00:34:30,199
Because we are interested in
the quantum features of mode

474
00:34:30,199 --> 00:34:33,540
a-- a is our quantum
mode, with single photons,

475
00:34:33,540 --> 00:34:36,719
or with a vacuum state,
and we want to squeeze it.

476
00:34:36,719 --> 00:34:41,090
b is just, they have
parametric drive.

477
00:34:41,090 --> 00:34:46,900
With this approximation, we
have only the a operators.

478
00:34:58,175 --> 00:34:59,050
This is our operator.

479
00:35:07,950 --> 00:35:09,724
Any question?

480
00:35:09,724 --> 00:35:12,194
AUDIENCE: [INAUDIBLE] would
give us a [INAUDIBLE], right?

481
00:35:16,640 --> 00:35:17,780
PROFESSOR: Yes, thank you.

482
00:35:17,780 --> 00:35:20,450
That means, here should
be a minus sign, yes.

483
00:35:27,650 --> 00:35:28,816
OK.

484
00:35:28,816 --> 00:35:34,440
I've motivated our discussion
with this nonlinear crystal,

485
00:35:34,440 --> 00:35:37,900
which generates pair of photons.

486
00:35:37,900 --> 00:35:42,500
This is the Hamiltonian
which describes it.

487
00:35:42,500 --> 00:35:46,330
And if you want to have a time
evolution by this Hamiltonian,

488
00:35:46,330 --> 00:35:50,300
you put this Hamiltonian into
a time evolution operator.

489
00:35:50,300 --> 00:35:59,869
In other words, you--
e to the minus iHt

490
00:35:59,869 --> 00:36:00,785
is the time evolution.

491
00:36:05,730 --> 00:36:11,390
If you now evolve a quantum
state of light for a fixed

492
00:36:11,390 --> 00:36:16,890
time, t, we apply the
operator, e to the minus iHt,

493
00:36:16,890 --> 00:36:20,440
to the quantum state of light.

494
00:36:20,440 --> 00:36:26,020
What I've just said
is now the motivation

495
00:36:26,020 --> 00:36:31,700
for the definition of
the squeezing operator.

496
00:36:38,910 --> 00:36:42,620
The squeezing
operator, S of r, is

497
00:36:42,620 --> 00:36:52,794
defined to be the exponent
of minus r over 2,

498
00:36:52,794 --> 00:36:56,000
a squared minus a dega squared.

499
00:37:06,810 --> 00:37:10,560
This is related to
the discussion above.

500
00:37:10,560 --> 00:37:15,820
You would say, hey, you want
to do that time evolution,

501
00:37:15,820 --> 00:37:17,240
where is the i?

502
00:37:17,240 --> 00:37:20,630
Well, I've just made
a choice of phi.

503
00:37:20,630 --> 00:37:31,770
If phi is chosen
to be pi over 2,

504
00:37:31,770 --> 00:37:33,880
then the time evolution
with the Hamiltonian,

505
00:37:33,880 --> 00:37:36,340
above, gives me the
squeezing operator, below.

506
00:37:41,780 --> 00:37:44,810
So with that motivation
we are now studying,

507
00:37:44,810 --> 00:37:46,745
what is the squeezing
operator doing

508
00:37:46,745 --> 00:37:47,890
to quantum states of light?

509
00:37:51,980 --> 00:37:53,950
Any questions about that?

510
00:37:53,950 --> 00:37:55,720
I know I spent a
lot of time on it.

511
00:37:55,720 --> 00:37:58,930
I could have taught this
class by just saying, here

512
00:37:58,930 --> 00:38:01,370
is an operator, the
squeezing operator.

513
00:38:01,370 --> 00:38:03,120
Trust me, it does
wonderful things.

514
00:38:03,120 --> 00:38:05,110
And then we can
work out everything.

515
00:38:05,110 --> 00:38:08,530
But I find his
unsatisfying, so I

516
00:38:08,530 --> 00:38:11,940
wanted to show you what is
really behind this operator.

517
00:38:11,940 --> 00:38:14,885
And I want you to
have a feeling,

518
00:38:14,885 --> 00:38:17,260
where does this operator come
from, and what is it doing?

519
00:38:19,910 --> 00:38:24,300
In essence, what I've
introduced into our description

520
00:38:24,300 --> 00:38:28,270
is now an operator, which
is creating and destroying

521
00:38:28,270 --> 00:38:31,240
pairs of photons.

522
00:38:31,240 --> 00:38:33,480
And this will actually
do wonderful things

523
00:38:33,480 --> 00:38:35,500
to our quantum states.

524
00:38:46,037 --> 00:38:48,120
What are the properties
of the squeezing operator?

525
00:38:55,570 --> 00:38:57,516
What is important
is, it is unitary.

526
00:38:57,516 --> 00:38:58,890
It does a unitary
time evolution.

527
00:39:01,820 --> 00:39:04,610
You may not see
that immediately, so

528
00:39:04,610 --> 00:39:08,136
let me explain that.

529
00:39:08,136 --> 00:39:12,190
You know from your basic
quantum mechanics course,

530
00:39:12,190 --> 00:39:22,660
that e to the i operator A is
unitary, when A is Hermitian.

531
00:39:28,230 --> 00:39:33,210
So the squeezing operator--
with the definition

532
00:39:33,210 --> 00:39:38,220
above-- can be written
as, I factor out

533
00:39:38,220 --> 00:39:48,490
2 i's over 2 a squared
minus a dega squared.

534
00:39:48,490 --> 00:39:54,479
And you can immediately
verify that this part, here,

535
00:39:54,479 --> 00:39:55,020
is Hermitian.

536
00:39:59,360 --> 00:40:01,620
If you do the
Hermitian conjugate,

537
00:40:01,620 --> 00:40:03,490
a squared turns
into a dega squared.

538
00:40:03,490 --> 00:40:06,210
a dega squared turns
into a squared.

539
00:40:06,210 --> 00:40:08,370
So we have a problem
with a minus sign.

540
00:40:08,370 --> 00:40:10,990
But if you do the
complex conjugate of i,

541
00:40:10,990 --> 00:40:12,720
this takes care
of the minus sign.

542
00:40:12,720 --> 00:40:14,490
So this part is Hermitian.

543
00:40:14,490 --> 00:40:17,930
We multiply it with i,
therefore this whole operator.

544
00:40:17,930 --> 00:40:21,140
Thus a unitary transformation
in [INAUDIBLE].

545
00:40:27,222 --> 00:40:27,805
Any questions?

546
00:40:32,770 --> 00:40:33,270
OK.

547
00:40:37,280 --> 00:40:47,410
So after being familiar with
this operator, we want to know,

548
00:40:47,410 --> 00:40:50,370
what is this operator doing?

549
00:40:56,340 --> 00:41:00,190
I can describe, now, what this
operator does, in a Schrodinger

550
00:41:00,190 --> 00:41:03,400
picture, or in a
Heisenberg picture.

551
00:41:03,400 --> 00:41:05,410
I pick whatever is
more convenient.

552
00:41:05,410 --> 00:41:09,920
And for now, this is
the Heisenberg picture.

553
00:41:09,920 --> 00:41:15,740
In the Heisenberg picture, what
is changing are the operators.

554
00:41:15,740 --> 00:41:19,180
Therefore, in the
Heisenberg picture,

555
00:41:19,180 --> 00:41:25,330
this unitary transformation
transforms the operators.

556
00:41:25,330 --> 00:41:35,311
And we can study what happens
when we transform the operator,

557
00:41:35,311 --> 00:41:35,810
x.

558
00:41:41,710 --> 00:41:51,125
The unitary transformation
is done by-- the operator, x,

559
00:41:51,125 --> 00:41:54,900
is transformed by multiplying
from the left side

560
00:41:54,900 --> 00:41:58,610
with S, from the righthand
side with S dega.

561
00:42:04,420 --> 00:42:11,450
You are familiar with
expressions like, this,

562
00:42:11,450 --> 00:42:13,600
and how to disentangle them.

563
00:42:18,300 --> 00:42:19,920
If you have an e to
the i alpha, e to

564
00:42:19,920 --> 00:42:25,430
the minus alpha, if you
could move the alpha past x.

565
00:42:25,430 --> 00:42:29,500
So if A and x commute,
i A, minus i A

566
00:42:29,500 --> 00:42:30,940
would just give unity.

567
00:42:30,940 --> 00:42:33,720
So therefore, this
expression is just

568
00:42:33,720 --> 00:42:38,830
x, unless you have non-Hermitian
commutators between A and x.

569
00:42:38,830 --> 00:42:43,480
I think you have solved, in
your basic mechanics course,

570
00:42:43,480 --> 00:42:50,680
many such problems which
involve identities of that form.

571
00:42:50,680 --> 00:42:56,430
Then there are higher
order commutator,

572
00:42:56,430 --> 00:43:02,330
the commutator of A with
the commutator of a x.

573
00:43:05,950 --> 00:43:08,220
Unless one of those
commutator vanishes,

574
00:43:08,220 --> 00:43:09,725
you can get an infinite series.

575
00:43:23,840 --> 00:43:30,140
Our operator, A, is nothing else
than the annihilation operator,

576
00:43:30,140 --> 00:43:33,900
a squared minus the creation
operator, a dega squared.

577
00:43:33,900 --> 00:43:38,432
So we can express everything
in terms of a, and a dega.

578
00:43:45,370 --> 00:43:51,990
The position operator in
our harmonic oscillator

579
00:43:51,990 --> 00:43:54,000
can also be expressed
by a, and a dega.

580
00:43:58,010 --> 00:44:02,860
By doing elementary
manipulations

581
00:44:02,860 --> 00:44:06,110
on the righthand side,
and recouping terms,

582
00:44:06,110 --> 00:44:14,080
you find immediately that
the unitary transformation

583
00:44:14,080 --> 00:44:22,620
of the Heisenberg operator, x,
gives you an x operator back.

584
00:44:22,620 --> 00:44:28,440
But multiplied with an
exponential, e to the r.

585
00:44:28,440 --> 00:44:38,830
And if we would do the same to
the momentum operator, which

586
00:44:38,830 --> 00:44:43,130
is a minus a dega
over square root 2,

587
00:44:43,130 --> 00:44:49,770
we will find that the unitary
transformation of the momentum

588
00:44:49,770 --> 00:44:56,520
operator is de-amplifying
the momentum

589
00:44:56,520 --> 00:44:58,660
operator by an
exponential factor.

590
00:45:12,260 --> 00:45:15,660
If we would assume
that we have a vacuum

591
00:45:15,660 --> 00:45:17,740
state in the
harmonic oscillator,

592
00:45:17,740 --> 00:45:22,140
and while classically, it
would be at x equals 0,

593
00:45:22,140 --> 00:45:24,730
p equals 0, quantum
mechanically,

594
00:45:24,730 --> 00:45:29,590
we have single-point noise in
x, and single-point noise in p.

595
00:45:29,590 --> 00:45:34,390
Then you would find that
the squeezing operator is

596
00:45:34,390 --> 00:45:40,900
amplifying the
quantum noise in x.

597
00:45:40,900 --> 00:45:46,450
But it squeezes, or
reduces, the noise in p.

598
00:45:54,670 --> 00:46:06,460
If we apply this squeezing
operator to the vacuum state,

599
00:46:06,460 --> 00:46:14,470
we obtain what is usually
called, squeezed vacuum.

600
00:46:25,230 --> 00:46:38,080
And it means that, in this
quasi-probability diagram,

601
00:46:38,080 --> 00:46:47,250
the action of the
squeezing operator

602
00:46:47,250 --> 00:46:52,895
is turning the vacuum
state into an ellipse.

603
00:47:17,980 --> 00:47:21,640
What happens to energy, here?

604
00:47:21,640 --> 00:47:26,810
The vacuum state is the
lowest-energy state.

605
00:47:26,810 --> 00:47:32,890
If you now act with a
squeezing operator to it,

606
00:47:32,890 --> 00:47:38,150
we obtain a state which
has-- the same energy?

607
00:47:38,150 --> 00:47:41,578
Is it energy-conserving,
or very high energy?

608
00:47:41,578 --> 00:47:43,800
AUDIENCE: Higher [INAUDIBLE].

609
00:47:43,800 --> 00:47:44,910
PROFESSOR: Yes.

610
00:47:44,910 --> 00:47:45,484
Why?

611
00:47:45,484 --> 00:47:47,192
AUDIENCE: It's no
longer the [INAUDIBLE].

612
00:47:52,460 --> 00:47:53,950
PROFESSOR: Sure, yeah.

613
00:47:53,950 --> 00:47:55,244
It's a vacuum state.

614
00:47:55,244 --> 00:47:56,660
We act on the
vacuum state, but we

615
00:47:56,660 --> 00:47:58,660
get a state which is no
longer the vacuum state.

616
00:48:06,170 --> 00:48:09,970
The reason why we
have extra energy--

617
00:48:09,970 --> 00:48:12,950
the squeezed vacuum is
very, very energetic.

618
00:48:12,950 --> 00:48:19,630
Because the squeezing operator
had a dega squared, a squared.

619
00:48:19,630 --> 00:48:23,300
Well a squared, the annihilation
operator acting on the vacuum,

620
00:48:23,300 --> 00:48:24,920
gives 0.

621
00:48:24,920 --> 00:48:28,380
But what we are creating now,
we are acting on the vacuum,

622
00:48:28,380 --> 00:48:31,790
and we are creating
pairs of photons.

623
00:48:31,790 --> 00:48:34,430
So we are adding, literally,
energy to the system.

624
00:48:34,430 --> 00:48:38,240
And the energy, of course,
comes from the drive laser,

625
00:48:38,240 --> 00:48:42,230
from the laser 2 omega 0, which
delivers the energy in forms

626
00:48:42,230 --> 00:48:44,632
of photons which
are split into half,

627
00:48:44,632 --> 00:48:46,090
and they go into
our quantum field.

628
00:48:49,030 --> 00:48:51,555
In the limit of
infinite squeezing--

629
00:48:51,555 --> 00:48:53,270
I will show it to
you, mathematically,

630
00:48:53,270 --> 00:48:55,220
but it's nice to
discuss it already here.

631
00:48:55,220 --> 00:48:58,761
In the limit of
infinite squeezing,

632
00:48:58,761 --> 00:49:00,135
what is the state
we are getting?

633
00:49:03,630 --> 00:49:05,444
AUDIENCE: Eigenstate
of momentum.

634
00:49:05,444 --> 00:49:06,735
PROFESSOR: Eigenstate momentum.

635
00:49:09,970 --> 00:49:12,980
We get the p equals
0 eigenstate.

636
00:49:16,730 --> 00:49:19,175
What is the energy of the
p equals 0 eigenstate?

637
00:49:27,681 --> 00:49:28,647
AUDIENCE: Infinite.

638
00:49:28,647 --> 00:49:33,102
It has to contain
all number states.

639
00:49:33,102 --> 00:49:34,810
PROFESSOR: It contains
all number states?

640
00:49:34,810 --> 00:49:40,140
OK, you think immediately into
number states, which is great.

641
00:49:40,140 --> 00:49:43,100
But in a more
pedestrian way, the p

642
00:49:43,100 --> 00:49:46,560
equals 0 state has
no kinetic energy.

643
00:49:46,560 --> 00:49:50,240
But if a state is localized
in momentum, p equals 0,

644
00:49:50,240 --> 00:49:53,990
it has to be infinitely
smeared out on the x-axis.

645
00:49:53,990 --> 00:49:56,690
And don't forget, we have an
harmonic oscillator potential.

646
00:49:56,690 --> 00:50:00,720
If you have a particle which
is completely delocalized in x,

647
00:50:00,720 --> 00:50:03,850
it has infinite potential
energy at the wings.

648
00:50:03,850 --> 00:50:06,430
So therefore in the limit
of extreme squeezing,

649
00:50:06,430 --> 00:50:10,249
we involve an extreme
number of number states.

650
00:50:10,249 --> 00:50:12,540
Actually, I want to be more
specific-- of photon pairs.

651
00:50:12,540 --> 00:50:17,170
We have states with 2n, and
n can be infinitely large.

652
00:50:17,170 --> 00:50:21,700
But we'll see in the classical
picture, what we get here

653
00:50:21,700 --> 00:50:24,370
when we squeeze it is,
we get the p equals

654
00:50:24,370 --> 00:50:28,205
0 eigenstate, which has infinite
energy, due to the harmonic

655
00:50:28,205 --> 00:50:29,080
oscillator potential.

656
00:50:34,050 --> 00:50:36,080
If we would allow
with the system

657
00:50:36,080 --> 00:50:40,390
now, after we have squeezed
it, to evolve for a quarter

658
00:50:40,390 --> 00:50:42,610
period in the
harmonic oscillator,

659
00:50:42,610 --> 00:50:50,370
then the ellipse would turn
into an vertical ellipse.

660
00:50:50,370 --> 00:50:57,340
So this is now an
eigenstate of x.

661
00:50:57,340 --> 00:51:00,190
It's the x equals 0 eigenstate.

662
00:51:00,190 --> 00:51:05,560
But the x equals 0 eigenstate
has also infinite energy,

663
00:51:05,560 --> 00:51:08,130
because due to Heisenberg's
uncertainty relation,

664
00:51:08,130 --> 00:51:10,950
it involves momentum states
of infinite momentum.

665
00:51:15,610 --> 00:51:16,110
Questions?

666
00:51:33,254 --> 00:51:35,719
AUDIENCE: [INAUDIBLE] a is
the photon field, right?

667
00:51:35,719 --> 00:51:39,915
So p is roughly the
electrical field, right?

668
00:51:39,915 --> 00:51:40,540
PROFESSOR: Yes.

669
00:51:40,540 --> 00:51:44,492
AUDIENCE: So it's kind of that
the electric field counts 0,

670
00:51:44,492 --> 00:51:49,760
and x is kind of the a,
the-- and it-- because of

671
00:51:49,760 --> 00:51:51,902
[INAUDIBLE].

672
00:51:51,902 --> 00:51:54,866
The electrical
field is squeezed?

673
00:51:54,866 --> 00:51:56,348
PROFESSOR: Yes.

674
00:51:56,348 --> 00:51:58,324
AUDIENCE: It means we
have no electrical field?

675
00:52:00,830 --> 00:52:03,190
PROFESSOR: We'll come
to that in a moment.

676
00:52:03,190 --> 00:52:07,130
I want to do a little bit
more math, to show you.

677
00:52:07,130 --> 00:52:10,400
I wanted to derive for you
an expression of the squeeze

678
00:52:10,400 --> 00:52:12,990
state, in number
basis, and such.

679
00:52:18,150 --> 00:52:19,610
Your question
mentioned something

680
00:52:19,610 --> 00:52:20,776
which is absolutely correct.

681
00:52:20,776 --> 00:52:26,150
By squeezing that, we
have now the p-axis

682
00:52:26,150 --> 00:52:27,990
is the electric field axis.

683
00:52:27,990 --> 00:52:31,810
So now we have, actually, in
the limit of infinite squeezing,

684
00:52:31,810 --> 00:52:37,010
we have an electric field which
has no uncertainty anymore.

685
00:52:39,550 --> 00:52:45,330
By squeezing the coherent state
into a momentum eigenstate,

686
00:52:45,330 --> 00:52:49,630
we have created a sharp
value for the electric field.

687
00:52:49,630 --> 00:52:51,560
We have created an
electric field eigenstate.

688
00:52:54,620 --> 00:52:56,270
Well you would say,
it's pretty boring,

689
00:52:56,270 --> 00:52:58,650
because the only electric
field state we have created

690
00:52:58,650 --> 00:53:01,840
is electric field e equals 0.

691
00:53:01,840 --> 00:53:04,000
But in the next
half-hour, we want

692
00:53:04,000 --> 00:53:06,640
to discuss the
displacement operator,

693
00:53:06,640 --> 00:53:08,000
and I will tell you what it is.

694
00:53:08,000 --> 00:53:10,170
That we can now
move the ellipses,

695
00:53:10,170 --> 00:53:13,280
and move the circles,
anywhere where we want.

696
00:53:13,280 --> 00:53:14,820
So once we have
an electric field

697
00:53:14,820 --> 00:53:18,226
state which has a sharp value
of the electric field at e

698
00:53:18,226 --> 00:53:22,450
equals 0, we can
just translate it.

699
00:53:22,450 --> 00:53:25,710
But before you get too
excited about having

700
00:53:25,710 --> 00:53:28,440
an eigenstate of
the electric field,

701
00:53:28,440 --> 00:53:30,590
I want you to think
about what happened

702
00:53:30,590 --> 00:53:33,920
after one quarter-period it
of the harmonic oscillator

703
00:53:33,920 --> 00:53:36,590
frequency.

704
00:53:36,590 --> 00:53:40,160
It turns upside down,
and your electric field

705
00:53:40,160 --> 00:53:42,880
has an infinite variance.

706
00:53:45,850 --> 00:53:48,470
That's what quantum
mechanics tells us.

707
00:53:48,470 --> 00:53:53,120
We can create
electric fields which

708
00:53:53,120 --> 00:53:56,640
are very precise, but
only for a short moment.

709
00:53:56,640 --> 00:53:59,350
So in other words, this
electric field state

710
00:53:59,350 --> 00:54:02,300
which we have created
would have a sharp value.

711
00:54:02,300 --> 00:54:04,940
A moment later, it would
be very smeared out,

712
00:54:04,940 --> 00:54:06,820
then it has a sharp
value again, and then

713
00:54:06,820 --> 00:54:09,020
it's smeared out again.

714
00:54:09,020 --> 00:54:12,852
I mean, that's what
squeezed states are.

715
00:54:12,852 --> 00:54:14,030
Other questions?

716
00:54:14,030 --> 00:54:15,405
AUDIENCE: That's
why [INAUDIBLE].

717
00:54:18,080 --> 00:54:20,530
PROFESSOR: That's why we
need homodyne detection.

718
00:54:20,530 --> 00:54:23,710
Yes, exactly.

719
00:54:23,710 --> 00:54:27,520
If we have squeezed something,
which is sort of narrow,

720
00:54:27,520 --> 00:54:28,870
that's great for measurement.

721
00:54:28,870 --> 00:54:32,190
Now we can do a
measurement of, maybe,

722
00:54:32,190 --> 00:54:36,086
a LIGO measurement for
gravitational waves with higher

723
00:54:36,086 --> 00:54:37,460
precision, because
we have a more

724
00:54:37,460 --> 00:54:39,900
precise value in
our quantum state.

725
00:54:39,900 --> 00:54:41,885
But we have to look at
it at the right time.

726
00:54:45,110 --> 00:54:47,510
We have to look
at it synchronized

727
00:54:47,510 --> 00:54:49,280
with the harmonic motion.

728
00:54:49,280 --> 00:54:54,290
Homodyne detection means we
look only at the sine component,

729
00:54:54,290 --> 00:54:55,970
or at the cosine component.

730
00:54:55,970 --> 00:54:58,390
Or if I want to simplify
it, what you want to do

731
00:54:58,390 --> 00:55:00,140
is, if you have a
state like this,

732
00:55:00,140 --> 00:55:02,470
you want to measure the
electric field, so to speak,

733
00:55:02,470 --> 00:55:03,280
stroboscopically.

734
00:55:03,280 --> 00:55:05,071
You want to look at
your system always when

735
00:55:05,071 --> 00:55:06,860
the ellipse is like this.

736
00:55:06,860 --> 00:55:10,240
The stroboscopic measurement
is, as I will show you,

737
00:55:10,240 --> 00:55:12,450
in essence, a
lock-in measurement,

738
00:55:12,450 --> 00:55:13,740
which is phase-sensitive.

739
00:55:13,740 --> 00:55:15,550
And this will be
homodyne detection.

740
00:55:15,550 --> 00:55:18,250
So we can only take
advantage of the squeezing,

741
00:55:18,250 --> 00:55:21,410
of having less uncertainty
in one quadrature component,

742
00:55:21,410 --> 00:55:23,924
if you do phase-sensitive
detection, which

743
00:55:23,924 --> 00:55:24,840
is homodyne detection.

744
00:55:27,470 --> 00:55:32,071
But now I'm already an
hour ahead of the course.

745
00:55:32,071 --> 00:55:32,570
OK.

746
00:55:32,570 --> 00:55:33,245
Back to basics.

747
00:55:43,790 --> 00:55:48,610
We want to explicitly
calculate, now,

748
00:55:48,610 --> 00:55:50,820
how does a squeezed
vacuum look like.

749
00:56:00,080 --> 00:56:03,600
We actually want to do it
twice, because it's useful.

750
00:56:03,600 --> 00:56:06,730
We have to see it in
two different basis.

751
00:56:06,730 --> 00:56:10,420
One is, I want to write down
the squeezed vacuum for you

752
00:56:10,420 --> 00:56:12,420
in a number representation.

753
00:56:12,420 --> 00:56:14,665
And then in a coherent
state representation.

754
00:56:19,150 --> 00:56:26,200
The squeezing operator is an
exponential function involving

755
00:56:26,200 --> 00:56:29,750
a squared, and a dega squared.

756
00:56:29,750 --> 00:56:34,610
And of course, we're now using
the Taylor expansion of that.

757
00:56:40,250 --> 00:56:42,475
We are acting on
the vacuum state.

758
00:56:45,180 --> 00:56:46,710
I will not do the calculation.

759
00:56:46,710 --> 00:56:47,970
It's again, elementary.

760
00:56:47,970 --> 00:56:54,710
You have n factorial, you have
terms with a dega acts on c,

761
00:56:54,710 --> 00:56:56,550
well, you pay 2 photons.

762
00:56:56,550 --> 00:56:59,900
If it acts again, it
adds 2 more photons,

763
00:56:59,900 --> 00:57:03,120
and the matrix element
of a dega acting on n

764
00:57:03,120 --> 00:57:05,030
is square root n plus 1.

765
00:57:05,030 --> 00:57:07,180
You just sort of
rearrange the terms.

766
00:57:07,180 --> 00:57:09,280
And what you find is,
what I will write you down

767
00:57:09,280 --> 00:57:11,190
in the next line.

768
00:57:11,190 --> 00:57:13,545
The important thing you
should immediately realize

769
00:57:13,545 --> 00:57:17,820
is, the squeeze state is
something very special.

770
00:57:17,820 --> 00:57:22,140
It is the superposition
of number states,

771
00:57:22,140 --> 00:57:27,020
but all number states are even
because our squeezing operator

772
00:57:27,020 --> 00:57:28,810
creates pairs of photons.

773
00:57:28,810 --> 00:57:31,140
This is what the parametric
down-conversion does.

774
00:57:31,140 --> 00:57:33,690
We inject photons
into the vacuum,

775
00:57:33,690 --> 00:57:37,300
but always exactly in pairs.

776
00:57:37,300 --> 00:57:39,620
And therefore, it's
not a random state.

777
00:57:39,620 --> 00:57:42,380
It's a highly correlated state
with very special properties.

778
00:57:44,920 --> 00:57:45,570
OK.

779
00:57:45,570 --> 00:57:52,770
If you do the calculation
and recoup the terms,

780
00:57:52,770 --> 00:57:57,400
you get factorials,
you get 2 to the n,

781
00:57:57,400 --> 00:58:00,540
you get another factorial.

782
00:58:00,540 --> 00:58:06,840
You get hyperbolic tangent--
sorry, to the power n.

783
00:58:13,710 --> 00:58:22,725
And the normalization is done
by the square root of the cos

784
00:58:22,725 --> 00:58:23,225
function.

785
00:58:31,440 --> 00:58:34,640
And the more we
squeeze, the larger

786
00:58:34,640 --> 00:58:38,850
are the amplitudes at
higher and higher n.

787
00:58:38,850 --> 00:58:41,700
But this is also obvious from
the graphic representation

788
00:58:41,700 --> 00:58:42,610
I've shown you.

789
00:58:49,540 --> 00:58:53,370
Let me add the coherent
state representation.

790
00:59:11,480 --> 00:59:18,640
The coherence states are
related to the number

791
00:59:18,640 --> 00:59:25,650
states in that way.

792
00:59:29,300 --> 00:59:43,530
If we transform now from number
states to coherent states,

793
00:59:43,530 --> 00:59:47,740
the straightforward
calculation gives, now,

794
00:59:47,740 --> 00:59:51,520
superposition over
coherent states.

795
00:59:51,520 --> 00:59:57,920
Coherent states
require an integral.

796
00:59:57,920 --> 01:00:12,310
e to the minus e to the
r over 2, divided by--

797
01:00:19,800 --> 01:00:22,260
Anyway, all this
expressions, they

798
01:00:22,260 --> 01:00:26,800
may not be in its general
form, too illuminating.

799
01:00:26,800 --> 01:00:28,910
But those things can
be done analytically.

800
01:00:31,730 --> 01:00:34,720
I just want to mention the
interesting limiting case

801
01:00:34,720 --> 01:00:37,845
of infinite squeezing.

802
01:00:37,845 --> 01:00:39,761
AUDIENCE: When you do
the integral over alpha,

803
01:00:39,761 --> 01:00:41,921
is this over like, a
magnitude of alpha,

804
01:00:41,921 --> 01:00:43,212
or a real part, or [INAUDIBLE]?

805
01:01:21,204 --> 01:01:22,620
PROFESSOR: I
remember, but I'm not

806
01:01:22,620 --> 01:01:26,330
100% sure that
alpha is real, here.

807
01:01:26,330 --> 01:01:28,230
I mean, it sort of
makes sense, because we

808
01:01:28,230 --> 01:01:29,890
start with the vacuum state.

809
01:01:29,890 --> 01:01:33,070
And if we squeeze
it, we are not really

810
01:01:33,070 --> 01:01:35,260
going into the
imaginary direction.

811
01:01:35,260 --> 01:01:38,551
So I think what is involved
here are only real alpha.

812
01:01:38,551 --> 01:01:40,717
AUDIENCE: For negative r,
we should get [INAUDIBLE].

813
01:01:46,570 --> 01:01:50,977
PROFESSOR: For negative r,
we need imaginary state.

814
01:01:50,977 --> 01:01:52,435
AUDIENCE: So we
should [INAUDIBLE].

815
01:01:59,060 --> 01:02:01,220
PROFESSOR: Let me double-check.

816
01:02:01,220 --> 01:02:02,260
I don't remember that.

817
01:02:05,570 --> 01:02:07,400
You know that,
sometimes, I admit

818
01:02:07,400 --> 01:02:09,870
it, the issue-- if
you research material,

819
01:02:09,870 --> 01:02:13,420
prepare a course some years
ago, you forget certain things.

820
01:02:13,420 --> 01:02:15,660
If I prepared the lecture,
and everything worked out

821
01:02:15,660 --> 01:02:16,868
yesterday, I would know that.

822
01:02:16,868 --> 01:02:19,740
But certain things
you don't remember.

823
01:02:19,740 --> 01:02:21,380
As far as I know,
it's the real axis.

824
01:02:21,380 --> 01:02:22,505
But I have to double-check.

825
01:02:25,786 --> 01:02:30,320
The limiting case
is interesting.

826
01:02:30,320 --> 01:02:35,850
If r goes to infinity, you
can show that this is simply

827
01:02:35,850 --> 01:02:39,410
the integral, d alpha
over coherent states.

828
01:02:47,780 --> 01:02:57,670
We have discussed,
graphically, the situation

829
01:02:57,670 --> 01:03:04,100
where we had-- so these
are quasi-probabilities.

830
01:03:07,750 --> 01:03:13,130
In that case of
infinite squeezing,

831
01:03:13,130 --> 01:03:15,900
we have the momentum
eigenstate, p equals 0.

832
01:03:20,520 --> 01:03:27,150
This is the limit of the
infinitely squeezed vacuum,

833
01:03:27,150 --> 01:03:31,270
and in a coherent
state representation,

834
01:03:31,270 --> 01:03:37,860
it is the integral over
coherent state alpha.

835
01:03:37,860 --> 01:03:42,180
I'm pretty sure alpha is
real here, seeing that now.

836
01:03:42,180 --> 01:03:50,410
There is a second limit, which
happens simply-- you can say,

837
01:03:50,410 --> 01:03:54,252
by rotation, or by time
evolution-- which is the x

838
01:03:54,252 --> 01:03:56,700
equals 0 eigenstate.

839
01:03:56,700 --> 01:04:01,880
And this is proportional
to the integral over alpha

840
01:04:01,880 --> 01:04:05,120
when we take the
coherent state i alpha,

841
01:04:05,120 --> 01:04:07,370
and we integrate from
minus to plus infinity.

842
01:04:13,811 --> 01:04:14,310
OK.

843
01:04:14,310 --> 01:04:17,070
So we have connected
our squeezed states,

844
01:04:17,070 --> 01:04:22,285
the squeezed vacuum, with number
states, with coherent states.

845
01:04:26,860 --> 01:04:29,830
Now we need one more thing.

846
01:04:29,830 --> 01:04:33,660
So far we've only
squeezed the vacuum,

847
01:04:33,660 --> 01:04:37,580
and we have defined
the squeezing operator

848
01:04:37,580 --> 01:04:42,590
that it takes a vacuum
state and elongates it.

849
01:04:42,590 --> 01:04:46,210
In order to generate
more general states,

850
01:04:46,210 --> 01:04:48,834
we want to get away
from the origin.

851
01:04:48,834 --> 01:04:50,750
And this is done by the
displacement operator.

852
01:05:10,990 --> 01:05:16,060
The definition of the
displacement operator

853
01:05:16,060 --> 01:05:19,130
is given here.

854
01:05:19,130 --> 01:05:21,890
The displacement by
a complex number,

855
01:05:21,890 --> 01:05:28,620
alpha, is done by putting
alpha, and alpha star,

856
01:05:28,620 --> 01:05:32,110
into an exponential function.

857
01:05:32,110 --> 01:05:38,030
In many quantum
mechanic courses,

858
01:05:38,030 --> 01:05:43,380
you show very easily the
elementary properties.

859
01:05:43,380 --> 01:05:48,820
If the displacement
operator is used

860
01:05:48,820 --> 01:05:55,210
to transform the annihilation
operator, it just does that.

861
01:06:06,061 --> 01:06:13,067
If you take the complex
conjugate of it--

862
01:06:13,067 --> 01:06:14,650
so in other words,
what that means is,

863
01:06:14,650 --> 01:06:16,850
it's called the
displacement operator,

864
01:06:16,850 --> 01:06:18,790
I just take that
as the definition.

865
01:06:18,790 --> 01:06:21,230
But you immediately
see why it's called

866
01:06:21,230 --> 01:06:23,160
the displacement
operator when we

867
01:06:23,160 --> 01:06:26,560
do the unitary transformation
of the annihilation operator,

868
01:06:26,560 --> 01:06:32,140
we get the annihilation operator
displaced by a complex number.

869
01:06:32,140 --> 01:06:33,740
So the action,
the transformation

870
01:06:33,740 --> 01:06:37,750
of the annihilation operator
is the annihilator operator

871
01:06:37,750 --> 01:06:41,020
itself, minus a c number.

872
01:06:41,020 --> 01:06:44,060
So therefore, we say, the
annihilation operator data

873
01:06:44,060 --> 01:06:45,175
has been displaced.

874
01:06:47,720 --> 01:06:53,290
So this is the action of
the displacement operator

875
01:06:53,290 --> 01:06:57,030
on an operator-- on the
annihilation operator.

876
01:07:00,780 --> 01:07:03,820
The question is now, how does
the displacement operator

877
01:07:03,820 --> 01:07:07,160
act on quantum states?

878
01:07:07,160 --> 01:07:12,360
And the simplest quantum
state we want to test out

879
01:07:12,360 --> 01:07:14,720
is the vacuum state.

880
01:07:14,720 --> 01:07:19,660
And well, not surprisingly,
the displacement operator,

881
01:07:19,660 --> 01:07:22,320
displacing the vector
state by alpha,

882
01:07:22,320 --> 01:07:24,849
is creating the
coherent state, alpha.

883
01:07:31,840 --> 01:07:35,310
This can be proven in one line.

884
01:07:35,310 --> 01:07:41,950
We take our displaced
vacuum, and we act on it

885
01:07:41,950 --> 01:07:43,380
with the annihilation operator.

886
01:07:48,160 --> 01:07:53,570
If we act with the annihilation
operator on something,

887
01:07:53,570 --> 01:07:56,730
and we get the same thing
back times an eigenvalue,

888
01:07:56,730 --> 01:07:58,010
we know it's a coherent state.

889
01:07:58,010 --> 01:08:00,350
Because this was the
definition of coherent states.

890
01:08:00,350 --> 01:08:04,080
So therefore, in order to show
that this is a coherent state,

891
01:08:04,080 --> 01:08:08,170
we want to show that it's an
eigenstate of the annihilation

892
01:08:08,170 --> 01:08:09,390
operator.

893
01:08:09,390 --> 01:08:11,200
So this is what we want to do.

894
01:08:14,945 --> 01:08:18,890
The proof is very easy.

895
01:08:18,890 --> 01:08:24,450
By multiplying this expression
with unity, which is DD dega,

896
01:08:24,450 --> 01:08:26,240
we have this.

897
01:08:26,240 --> 01:08:29,290
And now we can use the
result for the transformation

898
01:08:29,290 --> 01:08:31,390
of operators.

899
01:08:31,390 --> 01:08:36,850
Namely, that this is simply
the annihilation operator,

900
01:08:36,850 --> 01:08:38,210
plus alpha.

901
01:08:41,529 --> 01:08:45,359
If the annihilation operator
acts on the vacuum state,

902
01:08:45,359 --> 01:08:46,359
we get 0.

903
01:08:46,359 --> 01:08:51,890
If alpha acts on the vacuum
state, we get alpha times 0.

904
01:08:51,890 --> 01:08:58,825
So therefore, what
we obtain is that.

905
01:09:03,290 --> 01:09:07,050
When the annihilation
operator acts on this state,

906
01:09:07,050 --> 01:09:10,590
we get alpha times the state,
and therefore the state

907
01:09:10,590 --> 01:09:13,130
is a coherent state
with eigenvalue alpha.

908
01:09:21,540 --> 01:09:26,740
In a graphical way, if
you have a vacuum state

909
01:09:26,740 --> 01:09:33,330
the displacement operator, D
alpha, takes a vacuum state

910
01:09:33,330 --> 01:09:36,451
and creates a
coherent state alpha.

911
01:09:50,830 --> 01:09:59,340
If you want to have squeezed
states with a finite value,

912
01:09:59,340 --> 01:10:03,512
well, we just discussed
the electric field.

913
01:10:03,512 --> 01:10:04,970
Related to the
harmonic oscillator,

914
01:10:04,970 --> 01:10:07,210
we want squeeze
states, which are not

915
01:10:07,210 --> 01:10:13,420
centered at the origin, which
have a finite value of x or p.

916
01:10:13,420 --> 01:10:25,510
We can now create them by
first squeezing the vacuum,

917
01:10:25,510 --> 01:10:26,760
and then displacing the state.

918
01:10:37,584 --> 01:10:39,306
AUDIENCE: What's the
physical realization

919
01:10:39,306 --> 01:10:42,510
of the displacement operator?

920
01:10:42,510 --> 01:10:44,832
PROFESSOR: What is the
physical realization

921
01:10:44,832 --> 01:10:46,040
of the displacement operator?

922
01:10:54,850 --> 01:10:57,530
Just one second.

923
01:10:57,530 --> 01:11:02,090
The physical representation
of the displacement operator--

924
01:11:02,090 --> 01:11:04,480
we'll do that on Monday--
is the following.

925
01:11:04,480 --> 01:11:10,250
If you pass an arbitrary state
through a beam splitter--

926
01:11:10,250 --> 01:11:12,780
but it's a beam splitter
which has very, very

927
01:11:12,780 --> 01:11:17,600
high transmission-- and then,
from the-- I'll just show that.

928
01:11:17,600 --> 01:11:20,900
If you have a state-- this
is a beam splitter-- which

929
01:11:20,900 --> 01:11:23,870
has a very high transmission,
T is approximately 1,

930
01:11:23,870 --> 01:11:26,130
then the state passes through.

931
01:11:26,130 --> 01:11:30,750
But then from the other
side of the beam splitter,

932
01:11:30,750 --> 01:11:33,780
you come with a very
strong coherent state.

933
01:11:33,780 --> 01:11:37,510
You have a coherent state
which is characterized

934
01:11:37,510 --> 01:11:40,210
by a large complex number, beta.

935
01:11:40,210 --> 01:11:44,870
And then there is a
reflection coefficient,

936
01:11:44,870 --> 01:11:47,620
r, which is very small.

937
01:11:47,620 --> 01:11:53,080
It sort of reflects the coherent
state with an amplitude r beta.

938
01:11:53,080 --> 01:11:56,215
If you mix together the
transmitted state and r

939
01:11:56,215 --> 01:11:59,030
beta-- I will show
that to you explicitly,

940
01:11:59,030 --> 01:12:01,750
by doing a quantum treatment
of the beam splitter--

941
01:12:01,750 --> 01:12:03,990
what you get is,
the initial state

942
01:12:03,990 --> 01:12:07,330
is pretty much transmitted
without attenuation.

943
01:12:07,330 --> 01:12:12,100
But the reflected part of
the strong coherent state--

944
01:12:12,100 --> 01:12:15,010
you compensate for the
small r by a large beta--

945
01:12:15,010 --> 01:12:17,540
does actually an exact
displacement of r beta.

946
01:12:20,270 --> 01:12:21,270
It's actually great.

947
01:12:21,270 --> 01:12:23,560
The beam splitter is
a wonderful device.

948
01:12:23,560 --> 01:12:25,460
You think you have a
displacement operator

949
01:12:25,460 --> 01:12:29,560
formulated with a's
and a dega's, it

950
01:12:29,560 --> 01:12:31,250
looks like something extract.

951
01:12:31,250 --> 01:12:34,610
But you can go to the lab,
simply get one beam splitter,

952
01:12:34,610 --> 01:12:36,610
take a strong laser
beam, and whatever

953
01:12:36,610 --> 01:12:38,090
you send through
the beam splitter

954
01:12:38,090 --> 01:12:41,390
gets displaced, gets acted
upon by the beam splitter.

955
01:12:44,630 --> 01:12:45,490
Yes.

956
01:12:45,490 --> 01:12:48,020
AUDIENCE: You showed the
displacement operator, when

957
01:12:48,020 --> 01:12:51,760
you acted on the vacuum state,
will displace the vacuum

958
01:12:51,760 --> 01:12:53,336
state to a state alpha.

959
01:12:53,336 --> 01:12:55,766
Does it still hold
if you acted on,

960
01:12:55,766 --> 01:12:57,710
like another coherent state.

961
01:12:57,710 --> 01:12:59,710
Or in this case, a
squeeze state like that?

962
01:12:59,710 --> 01:13:01,250
PROFESSOR: Yes.

963
01:13:01,250 --> 01:13:04,280
I haven't shown it,
but it's really--

964
01:13:04,280 --> 01:13:08,060
it displaces-- When we
use this representation

965
01:13:08,060 --> 01:13:09,930
with quasi-probabilities,
it simply

966
01:13:09,930 --> 01:13:11,310
does a displacement
in the plane.

967
01:13:26,330 --> 01:13:26,830
But no.

968
01:13:26,830 --> 01:13:29,371
To be honest, when I say it does
a displacement on the plane,

969
01:13:29,371 --> 01:13:32,480
it reminds me that we
have three different ways

970
01:13:32,480 --> 01:13:34,420
of defining quasi-probabilities.

971
01:13:34,420 --> 01:13:36,805
The w, the p, and
the q representation.

972
01:13:39,480 --> 01:13:42,060
I know we use it all the
time, that we displace things

973
01:13:42,060 --> 01:13:43,130
in the plane.

974
01:13:43,130 --> 01:13:46,000
But I'm wondering if the
displacement operator does

975
01:13:46,000 --> 01:13:49,045
an exact displacement of
all representations, or only

976
01:13:49,045 --> 01:13:51,000
of the q representation.

977
01:13:51,000 --> 01:13:52,625
That's something I
don't know for sure.

978
01:13:52,625 --> 01:13:54,875
AUDIENCE: I was thinking it
could also, like-- I mean,

979
01:13:54,875 --> 01:13:57,260
are you going to be able to
displace all types of light,

980
01:13:57,260 --> 01:14:01,033
like thermal light, or any
representation of light

981
01:14:01,033 --> 01:14:03,366
that you could put in, is the
same displacement operator

982
01:14:03,366 --> 01:14:03,949
going to work?

983
01:14:03,949 --> 01:14:10,850
Or is its domain just the
vacuum and coherent states?

984
01:14:10,850 --> 01:14:13,730
PROFESSOR: The fact is,
the coherent states--

985
01:14:13,730 --> 01:14:15,415
I've shown you that
it's a vacuum state.

986
01:14:18,310 --> 01:14:23,600
I know that's the next thing to
show, the displacement operator

987
01:14:23,600 --> 01:14:26,190
if you have a displacement
by alpha followed

988
01:14:26,190 --> 01:14:29,530
by displacement by beta,
it is equal to displacement

989
01:14:29,530 --> 01:14:31,090
by alpha plus beta.

990
01:14:31,090 --> 01:14:33,470
So displacement
operator forms a group,

991
01:14:33,470 --> 01:14:35,950
and if you do two
displacements, they

992
01:14:35,950 --> 01:14:38,310
equal into one area
of displacement,

993
01:14:38,310 --> 01:14:42,220
which is the sum of
two complex numbers.

994
01:14:42,220 --> 01:14:44,850
What I'm just saying, if you
do the first displacement

995
01:14:44,850 --> 01:14:47,000
you can get an arbitrarily
coherent state.

996
01:14:47,000 --> 01:14:49,530
So therefore, the
displacement operator

997
01:14:49,530 --> 01:14:55,430
is exactly displacing
a coherent state

998
01:14:55,430 --> 01:14:58,510
by the argument of the
displacement operator.

999
01:14:58,510 --> 01:15:01,060
And now if you take an
arbitrary quantum state

1000
01:15:01,060 --> 01:15:04,835
and expand it into coherent
states-- coherent states are

1001
01:15:04,835 --> 01:15:07,319
not only complete,
they are over-complete.

1002
01:15:07,319 --> 01:15:09,360
All you have done is,
you've done a displacement.

1003
01:15:12,910 --> 01:15:16,216
Now the over-completeness,
of course,

1004
01:15:16,216 --> 01:15:17,590
means you have to
think about it,

1005
01:15:17,590 --> 01:15:21,120
because you can represent
states in more than one way

1006
01:15:21,120 --> 01:15:22,270
by coherent states.

1007
01:15:22,270 --> 01:15:24,030
But if you have
your representation,

1008
01:15:24,030 --> 01:15:25,831
you just displace
it, and this is

1009
01:15:25,831 --> 01:15:27,497
the result of the
displacement operator.

1010
01:15:32,170 --> 01:15:36,910
So since the q
representation is simply,

1011
01:15:36,910 --> 01:15:38,520
you take the
statistical operator

1012
01:15:38,520 --> 01:15:41,480
and look for the
elements in alpha,

1013
01:15:41,480 --> 01:15:43,780
and if you displace alpha,
the q representation

1014
01:15:43,780 --> 01:15:44,730
has been moved around.

1015
01:15:44,730 --> 01:15:47,590
So I'm sure that for
the q representation,

1016
01:15:47,590 --> 01:15:52,720
for the q quasi-probabilities,
the displacement operator

1017
01:15:52,720 --> 01:15:54,840
shifted around in this place.

1018
01:15:54,840 --> 01:16:00,685
For the w and p
representation, I'm not sure.

1019
01:16:00,685 --> 01:16:02,310
Maybe there's an
expert in the audience

1020
01:16:02,310 --> 01:16:06,830
who knows more
about it than I do.

1021
01:16:06,830 --> 01:16:08,840
OK.

1022
01:16:08,840 --> 01:16:10,520
We have just five minutes left.

1023
01:16:15,610 --> 01:16:18,410
I want to discuss now
the electric field

1024
01:16:18,410 --> 01:16:20,720
of squeezed states.

1025
01:16:20,720 --> 01:16:25,840
And for that, let
me load a picture.

1026
01:16:33,220 --> 01:16:34,145
Insert picture.

1027
01:16:40,200 --> 01:16:40,870
Classroom files.

1028
01:16:55,420 --> 01:17:02,561
Let us discuss, now, the
electric field of squeezed

1029
01:17:02,561 --> 01:17:03,060
states.

1030
01:17:18,010 --> 01:17:23,380
Just as a reminder, we can
discuss the electric field

1031
01:17:23,380 --> 01:17:27,740
by using the quasi-probability
representation.

1032
01:17:27,740 --> 01:17:30,880
And the electric field
is the projection

1033
01:17:30,880 --> 01:17:33,225
of the quasi-probabilities
on the vertical axis.

1034
01:17:36,150 --> 01:17:40,000
And then the time evolution
is, that everything rotates

1035
01:17:40,000 --> 01:17:42,880
with omega in this
complex plane.

1036
01:17:50,520 --> 01:17:51,900
We discussed it already.

1037
01:17:51,900 --> 01:17:56,700
For coherent state, we have
a circle which rotates.

1038
01:17:56,700 --> 01:18:00,220
Therefore, the projected
fuzziness of the electric field

1039
01:18:00,220 --> 01:18:01,710
is always the same.

1040
01:18:01,710 --> 01:18:05,400
And as time goes by, we have
a sinusoidal-bearing electric

1041
01:18:05,400 --> 01:18:07,972
field.

1042
01:18:07,972 --> 01:18:09,180
Let me just make one comment.

1043
01:18:09,180 --> 01:18:13,450
If you look into the literature,
some people actually say,

1044
01:18:13,450 --> 01:18:15,760
the electric field
is the projection

1045
01:18:15,760 --> 01:18:17,420
on the horizontal axis.

1046
01:18:17,420 --> 01:18:21,240
So there are people who say,
the electric field is given

1047
01:18:21,240 --> 01:18:25,285
by the x-coordinate of
the harmonic oscillator,

1048
01:18:25,285 --> 01:18:28,220
whereas I'm telling you,
it's the p-coordinate.

1049
01:18:28,220 --> 01:18:30,450
Well, if you think
one person is wrong,

1050
01:18:30,450 --> 01:18:33,330
I would suggest you just
wait a quarter-period

1051
01:18:33,330 --> 01:18:36,320
of the harmonic oscillator, and
then the other person is right.

1052
01:18:36,320 --> 01:18:38,320
It's really just a
phase convention.

1053
01:18:38,320 --> 01:18:41,560
What do you assume
to be t equals

1054
01:18:41,560 --> 01:18:44,385
0-- it's really arbitrary.

1055
01:18:44,385 --> 01:18:46,510
But here in this course,
I will use the projections

1056
01:18:46,510 --> 01:18:48,451
on the vertical axis.

1057
01:18:48,451 --> 01:18:48,950
OK.

1058
01:18:48,950 --> 01:18:53,320
If you project the number
state, we get always,

1059
01:18:53,320 --> 01:18:56,302
0 electric field, with
a large uncertainty.

1060
01:18:56,302 --> 01:18:57,385
So that's just a reminder.

1061
01:19:01,920 --> 01:19:05,200
But now we have
a squeezed state.

1062
01:19:05,200 --> 01:19:09,430
It's a displaced squeezed state.

1063
01:19:09,430 --> 01:19:12,740
If you project it
onto the y-axis,

1064
01:19:12,740 --> 01:19:17,070
we have first some
large uncertainty.

1065
01:19:17,070 --> 01:19:20,830
I think this plot assumes that
we rotate with negative time,

1066
01:19:20,830 --> 01:19:24,020
so I apologize for that.

1067
01:19:24,020 --> 01:19:26,080
You can just invert
time, if you want.

1068
01:19:26,080 --> 01:19:30,670
So after a quarter-period,
the ellipse is now horizontal,

1069
01:19:30,670 --> 01:19:34,690
and that means the electric
field is very sharp.

1070
01:19:34,690 --> 01:19:37,680
As time goes by, you
see that the uncertainty

1071
01:19:37,680 --> 01:19:40,040
of the electric field
is large, small, large,

1072
01:19:40,040 --> 01:19:41,140
small-- it modulates.

1073
01:19:46,470 --> 01:19:49,930
It can become very extreme,
when you do extreme squeezing,

1074
01:19:49,930 --> 01:19:53,050
so you have an extremely precise
value of the electric field,

1075
01:19:53,050 --> 01:19:57,360
here, but you've a large
uncertainty, there.

1076
01:19:57,360 --> 01:20:00,540
Sometimes you want
to accurately measure

1077
01:20:00,540 --> 01:20:02,560
the 0 crossing of
the electric field.

1078
01:20:02,560 --> 01:20:05,240
This may be something
which interests you,

1079
01:20:05,240 --> 01:20:06,870
for an experiment.

1080
01:20:06,870 --> 01:20:10,750
In that case, you
actually want to have

1081
01:20:10,750 --> 01:20:13,480
an ellipse which is
horizontally squeezed.

1082
01:20:13,480 --> 01:20:16,130
Now, whenever the
electric field is 0,

1083
01:20:16,130 --> 01:20:17,740
there is very little noise.

1084
01:20:17,740 --> 01:20:20,590
But after a quarter-period,
when the electric field

1085
01:20:20,590 --> 01:20:23,220
reaches its maximum,
you have a lot of noise.

1086
01:20:23,220 --> 01:20:27,330
So it's sort of your choice
which way you squeeze.

1087
01:20:27,330 --> 01:20:32,310
Whether you want the
electric field to be precise,

1088
01:20:32,310 --> 01:20:35,835
have little fluctuations
when it goes through 0,

1089
01:20:35,835 --> 01:20:37,335
or when it goes
through the maximum.

1090
01:20:46,510 --> 01:20:49,440
So what we have done
here is, we have first

1091
01:20:49,440 --> 01:20:53,420
created the squeezed
vacuum, and then we

1092
01:20:53,420 --> 01:20:56,590
have acted on it with a
displacement operator.

1093
01:21:04,680 --> 01:21:06,120
OK.

1094
01:21:06,120 --> 01:21:08,950
I think that's a
good moment to stop.

1095
01:21:08,950 --> 01:21:16,650
Let me just say what I wanted
to take from this picture.

1096
01:21:16,650 --> 01:21:22,130
The fact that the electric
field is precise only

1097
01:21:22,130 --> 01:21:29,780
at certain moments means that
we can only take advantage of it

1098
01:21:29,780 --> 01:21:32,200
when we do a
phase-sensitive detection.

1099
01:21:32,200 --> 01:21:36,280
We only want to
sort of, measure,

1100
01:21:36,280 --> 01:21:39,600
the electric field
when it's sharp.

1101
01:21:39,600 --> 01:21:51,120
Or-- this is equivalent-- we
should regard light is always

1102
01:21:51,120 --> 01:21:54,155
composed of two
quadrature components.

1103
01:21:54,155 --> 01:21:57,670
You can say, the cosine,
the sine oscillation, the x,

1104
01:21:57,670 --> 01:21:59,000
and the p.

1105
01:21:59,000 --> 01:22:03,170
And the squeezing is
squeezed in one quadrature,

1106
01:22:03,170 --> 01:22:06,300
by it is elongated in
the other quadrature.

1107
01:22:06,300 --> 01:22:08,540
Therefore, we want to
be phase-sensitive.

1108
01:22:08,540 --> 01:22:11,690
We want to pick out
either the cosine omega

1109
01:22:11,690 --> 01:22:14,110
T, or the sine
omega T oscillation.

1110
01:22:14,110 --> 01:22:15,990
This is sort of,
homodyne detection.

1111
01:22:15,990 --> 01:22:18,204
We'll discuss it on Monday.

1112
01:22:18,204 --> 01:22:18,745
Any question?

1113
01:22:21,250 --> 01:22:21,850
OK.

1114
01:22:21,850 --> 01:22:23,400
Good.