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WOLFGANG KETTERLE:
So, good afternoon.

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00:00:28,860 --> 00:00:30,940
Let me start by
presenting something

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00:00:30,940 --> 00:00:33,960
I learned today during lunch.

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00:00:33,960 --> 00:00:35,720
I had lunch with
three colleagues

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00:00:35,720 --> 00:00:38,330
and we discussed entanglement.

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00:00:38,330 --> 00:00:42,560
And, well, I'm telling you the
story a little bit differently,

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00:00:42,560 --> 00:00:46,404
but the question
which came up is

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00:00:46,404 --> 00:00:47,820
you have two
harmonic oscillators.

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00:00:51,370 --> 00:00:54,010
One harmonic oscillator
is in the ground state,

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00:00:54,010 --> 00:00:57,290
one is in the excited state.

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00:00:57,290 --> 00:01:02,970
And then you have the
harmonic oscillators

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00:01:02,970 --> 00:01:09,640
where the excitation
is here and there.

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00:01:09,640 --> 00:01:12,660
Is that an entangled
state or not?

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00:01:16,460 --> 00:01:17,335
It's a real question.

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00:01:30,550 --> 00:01:32,250
I want to think about it.

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00:01:32,250 --> 00:01:34,831
AUDIENCE: What's the physical
nature of these oscillators?

24
00:01:34,831 --> 00:01:36,604
AUDIENCE: Can you
distinguish these?

25
00:01:36,604 --> 00:01:37,520
WOLFGANG KETTERLE: OK.

26
00:01:37,520 --> 00:01:40,130
Who's working with ion types?

27
00:01:40,130 --> 00:01:43,360
There's a few people I know,
a few ion types here at MIT.

28
00:01:43,360 --> 00:01:49,444
If this is ion-- what is
a good ion? [INAUDIBLE]?

29
00:01:49,444 --> 00:01:50,360
AUDIENCE: [INAUDIBLE].

30
00:01:50,360 --> 00:01:51,943
WOLFGANG KETTERLE:
Strontium plus, OK.

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00:01:51,943 --> 00:01:56,560
So if it's strontium plus, which
can be in the quantum state v

32
00:01:56,560 --> 00:02:03,090
equals 1 or v equals 0, and you
have another strontium plus ion

33
00:02:03,090 --> 00:02:06,940
in a second ion
[INAUDIBLE]-- and it can be

34
00:02:06,940 --> 00:02:09,590
in the [INAUDIBLE]
first excited state--

35
00:02:09,590 --> 00:02:11,110
isn't it wonderfully entangled?

36
00:02:11,110 --> 00:02:13,430
Two systems far away.

37
00:02:13,430 --> 00:02:16,360
You can manipulate them, you
have two particles, and then

38
00:02:16,360 --> 00:02:17,490
two quantum states.

39
00:02:17,490 --> 00:02:21,460
It fulfills with flying
colors all the qualities

40
00:02:21,460 --> 00:02:25,710
you want to see of
an entangled state.

41
00:02:25,710 --> 00:02:30,120
But now I can say, if I call
this harmonic oscillator,

42
00:02:30,120 --> 00:02:34,010
I have two optical fibers.

43
00:02:34,010 --> 00:02:37,890
And you know, a single-mode
fiber defines a harmonic

44
00:02:37,890 --> 00:02:40,270
oscillator, namely the
electromagnetic field

45
00:02:40,270 --> 00:02:42,000
in the single mode.

46
00:02:42,000 --> 00:02:45,880
And if I put one
excitation into this fiber,

47
00:02:45,880 --> 00:02:49,550
I have a photon here and
here I have zero photon.

48
00:02:49,550 --> 00:02:53,660
Or here, I have one photon
and I have zero photons.

49
00:02:53,660 --> 00:02:56,350
So if you look at
the fiber, you would

50
00:02:56,350 --> 00:02:58,550
say the fiber has two states.

51
00:02:58,550 --> 00:03:01,290
It can have an
excitation or not.

52
00:03:01,290 --> 00:03:05,470
And if you use now that
approach, you would say,

53
00:03:05,470 --> 00:03:10,760
well, I have an entangled
state between two fibers.

54
00:03:10,760 --> 00:03:14,920
However, I can realize
exactly that state

55
00:03:14,920 --> 00:03:20,070
by having a beam splitter,
by having one photon coming

56
00:03:20,070 --> 00:03:22,750
on the beam splitter,
and then the photon

57
00:03:22,750 --> 00:03:27,210
is coupled into
one of the fibers.

58
00:03:27,210 --> 00:03:29,700
So in other words,
there is an ambiguity

59
00:03:29,700 --> 00:03:33,990
which I want to point out
between whether you have

60
00:03:33,990 --> 00:03:37,880
a state of one particle--
one photon into two fibers--

61
00:03:37,880 --> 00:03:40,000
or whether you have
two fibers which

62
00:03:40,000 --> 00:03:42,220
can be in two different states.

63
00:03:42,220 --> 00:03:46,430
So there is often an ambiguity,
what do you call the state

64
00:03:46,430 --> 00:03:49,470
and what do you
call the particle?

65
00:03:49,470 --> 00:03:51,840
So at our lunch
conversation it came up

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00:03:51,840 --> 00:03:57,070
in a somewhat different context,
but people would actually

67
00:03:57,070 --> 00:04:01,070
say that splitting
a photon-- sending

68
00:04:01,070 --> 00:04:03,480
one photon across
a beam splitter

69
00:04:03,480 --> 00:04:06,240
fulfills, if you
interpret it in that way,

70
00:04:06,240 --> 00:04:09,350
fulfills the definition
of entanglement.

71
00:04:09,350 --> 00:04:12,680
I know our wiki page and
Professor Schwan would probably

72
00:04:12,680 --> 00:04:16,529
not agree, but welcome to
the frontier of research

73
00:04:16,529 --> 00:04:18,990
where different people
have different opinions

74
00:04:18,990 --> 00:04:23,950
and certain definitions
are still being worked out.

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00:04:23,950 --> 00:04:25,580
But there is one
thing which is real.

76
00:04:29,160 --> 00:04:31,420
We have here two different
harmonic oscillators,

77
00:04:31,420 --> 00:04:35,340
ions oscillating in a harmonic
oscillator potential or photons

78
00:04:35,340 --> 00:04:36,940
in a single mode.

79
00:04:36,940 --> 00:04:40,470
I remember from 15
years ago, there

80
00:04:40,470 --> 00:04:45,550
were papers that show if you
put the ion in a cavity or such,

81
00:04:45,550 --> 00:04:47,640
you can really
transfer the quantum

82
00:04:47,640 --> 00:04:52,290
state from the single
photon from this harmonic

83
00:04:52,290 --> 00:04:54,520
oscillator-- the
harmonic oscillator

84
00:04:54,520 --> 00:04:56,710
of the electromagnetic
field-- to the harmonic

85
00:04:56,710 --> 00:04:59,140
oscillator of the ion.

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00:04:59,140 --> 00:05:02,960
So if we want to hold
onto the definition,

87
00:05:02,960 --> 00:05:07,090
which I'm not sure if you should
that this is a single photon,

88
00:05:07,090 --> 00:05:09,300
a single particle
cannot be entangled,

89
00:05:09,300 --> 00:05:12,655
well I know and people reminded
me at lunch the there is

90
00:05:12,655 --> 00:05:18,220
a protocol to [INAUDIBLE] map
the single photon in the two

91
00:05:18,220 --> 00:05:21,910
modes after the beam splitter
onto an entangled state

92
00:05:21,910 --> 00:05:23,860
of ions.

93
00:05:23,860 --> 00:05:26,790
So if you focus here on
the photon and say a single

94
00:05:26,790 --> 00:05:30,100
particle cannot be entangled,
you're facing the problem that

95
00:05:30,100 --> 00:05:32,975
there is a protocol which would
transfer something which you

96
00:05:32,975 --> 00:05:37,214
call not entangled to
something which is entangled.

97
00:05:37,214 --> 00:05:38,130
I hope you enjoy that.

98
00:05:38,130 --> 00:05:41,400
It's not clear what
is the particle

99
00:05:41,400 --> 00:05:42,700
and what is the excitation.

100
00:05:42,700 --> 00:05:43,741
There's really ambiguity.

101
00:05:46,257 --> 00:05:46,840
Any questions?

102
00:05:50,570 --> 00:05:56,100
OK, so that's what part
of my lunch was about.

103
00:05:56,100 --> 00:06:02,140
I thought since I wanted to
remind you of the schedule.

104
00:06:02,140 --> 00:06:05,670
In a few hours I will
actually be heading to Europe

105
00:06:05,670 --> 00:06:09,750
and say there for next week, so
therefore no classes next week.

106
00:06:09,750 --> 00:06:15,100
We've already made up for one
class with the Friday class,

107
00:06:15,100 --> 00:06:17,660
and the week later we
also have three classes

108
00:06:17,660 --> 00:06:19,660
and we've made up for the week.

109
00:06:19,660 --> 00:06:22,640
I've just put down here
for you the PSET schedule.

110
00:06:26,290 --> 00:06:30,300
I kept the Monday due date so
there is one PSET every week.

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00:06:30,300 --> 00:06:32,385
We have wonderful
material for those PSETs.

112
00:06:34,960 --> 00:06:38,510
The due date is now for this
week not the day of the class,

113
00:06:38,510 --> 00:06:43,110
but you will find a way to
get your homework delivered

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00:06:43,110 --> 00:06:46,800
or email it to the TA.

115
00:06:46,800 --> 00:06:48,820
Any questions about schedule?

116
00:06:48,820 --> 00:06:51,430
I think that has been
clearly announced.

117
00:06:54,920 --> 00:07:01,030
There was a question
in the last class

118
00:07:01,030 --> 00:07:04,650
about if you have
a density matrix,

119
00:07:04,650 --> 00:07:08,670
if any arbitrary density
matrix can be regarded

120
00:07:08,670 --> 00:07:12,800
as a partial trace
of a pure state.

121
00:07:12,800 --> 00:07:15,190
I was not immediately
clear-- I thought it was,

122
00:07:15,190 --> 00:07:16,340
but I wasn't sure.

123
00:07:16,340 --> 00:07:19,610
But the proof is so simple
that it fits on three lines.

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00:07:19,610 --> 00:07:23,450
You can simply take
a density matrix,

125
00:07:23,450 --> 00:07:25,670
you can double up
the Hilbert space,

126
00:07:25,670 --> 00:07:29,760
and define our pure state
in a Hilbert space of twice

127
00:07:29,760 --> 00:07:31,260
the dimension, and
you immediately

128
00:07:31,260 --> 00:07:34,190
see that this density
matrix is the partial trace

129
00:07:34,190 --> 00:07:35,450
of a pure state.

130
00:07:35,450 --> 00:07:37,990
So therefore, if you
want, you can always

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00:07:37,990 --> 00:07:43,760
say the density matrix is sort
of entangled-- you are entitled

132
00:07:43,760 --> 00:07:47,210
to the opinion that the density
matrix always originated

133
00:07:47,210 --> 00:07:51,040
into a pure state, but the
entanglement [INAUDIBLE]

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00:07:51,040 --> 00:07:53,680
to the other subsystem
have been broken.

135
00:07:53,680 --> 00:07:57,053
And I showed you last
class that if you

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00:07:57,053 --> 00:08:00,750
are one of the Bell states,
the most entangled states

137
00:08:00,750 --> 00:08:03,810
in a fully-entangled state
between two particles,

138
00:08:03,810 --> 00:08:05,880
but you just look
at one particle,

139
00:08:05,880 --> 00:08:09,170
one particle isn't just a random
states with a density matrix

140
00:08:09,170 --> 00:08:11,195
which is a unity matrix.

141
00:08:18,220 --> 00:08:25,680
Now let's get back to
where we left it on Monday.

142
00:08:25,680 --> 00:08:27,390
I introduced to
you-- and I just want

143
00:08:27,390 --> 00:08:32,490
to remind you of that, because
[INAUDIBLE] I introduced

144
00:08:32,490 --> 00:08:38,630
to you this famous phenomenon
of Hong-Ou-Mandel interference

145
00:08:38,630 --> 00:08:40,380
which is the
following situation.

146
00:08:40,380 --> 00:08:43,670
If you have 50/50 beam split and
you have two identical photons,

147
00:08:43,670 --> 00:08:46,080
you will never get
single photons out.

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00:08:46,080 --> 00:08:48,270
The photons are bosons.

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00:08:48,270 --> 00:08:49,650
They want to be together.

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00:08:49,650 --> 00:08:52,090
And so you will always
get two photons,

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00:08:52,090 --> 00:08:55,620
but you don't throw on which
side of the beam splitter.

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00:08:55,620 --> 00:09:00,070
Now, this is important
because we can now

153
00:09:00,070 --> 00:09:02,860
use the Hong-Ou-Mandel
interferences

154
00:09:02,860 --> 00:09:06,170
effect to entangle atoms.

155
00:09:06,170 --> 00:09:09,440
So the element we will take
from this beam splitter

156
00:09:09,440 --> 00:09:11,150
is the following.

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00:09:11,150 --> 00:09:14,990
We will have atoms emit photons.

158
00:09:14,990 --> 00:09:20,090
But then we create
entanglement probablistically.

159
00:09:20,090 --> 00:09:23,020
When these photons have passed
through beam splitters--

160
00:09:23,020 --> 00:09:24,740
and I tell you
everything about it--

161
00:09:24,740 --> 00:09:29,550
and both detectors make a click,
if both detectors make a click

162
00:09:29,550 --> 00:09:32,360
you know for sure
that the two photons

163
00:09:32,360 --> 00:09:34,600
at the input of the beam
splitter were not identical.

164
00:09:34,600 --> 00:09:38,140
Because if they were
identical, only one detector

165
00:09:38,140 --> 00:09:39,730
receives light.

166
00:09:39,730 --> 00:09:44,200
So we will actually used
this home interference

167
00:09:44,200 --> 00:09:50,540
to project out photon states
where the two photons are not

168
00:09:50,540 --> 00:09:55,060
identical because they have
difference polarization.

169
00:09:55,060 --> 00:09:57,770
Any questions?

170
00:09:57,770 --> 00:09:59,473
Yes?

171
00:09:59,473 --> 00:10:01,397
AUDIENCE: Does [INAUDIBLE]
include the phase.

172
00:10:01,397 --> 00:10:03,802
Like if you were in phase
lab with those photons,

173
00:10:03,802 --> 00:10:04,764
would it be still?

174
00:10:08,747 --> 00:10:10,580
WOLFGANG KETTERLE:
Frequency very important.

175
00:10:10,580 --> 00:10:13,700
Polarization important,
timing is important.

176
00:10:13,700 --> 00:10:15,833
A single photon has no phase.

177
00:10:19,530 --> 00:10:20,860
It's a global phase.

178
00:10:20,860 --> 00:10:23,030
And if you have
many single photons,

179
00:10:23,030 --> 00:10:25,330
your many photon state
is a product state

180
00:10:25,330 --> 00:10:26,970
of those single photon states.

181
00:10:26,970 --> 00:10:29,540
And all of the
individual phases--

182
00:10:29,540 --> 00:10:31,980
if you want to hold onto
this concept for a second--

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00:10:31,980 --> 00:10:34,070
just become one big
multiplicity phase.

184
00:10:37,950 --> 00:10:38,800
Yes, Nancy?

185
00:10:38,800 --> 00:10:39,716
AUDIENCE: [INAUDIBLE]?

186
00:10:44,030 --> 00:10:46,640
WOLFGANG KETTERLE: Oh,
everything [INAUDIBLE].

187
00:10:46,640 --> 00:10:48,465
easy experiment.

188
00:10:48,465 --> 00:10:52,370
But I think the beam splitter
is probably not-- I mean,

189
00:10:52,370 --> 00:10:54,860
beam splitter is a piece of
wonderfully polished optics.

190
00:10:54,860 --> 00:10:58,000
And in Germany and
elsewhere, people

191
00:10:58,000 --> 00:11:00,250
have really learned how
to do exquisite optics.

192
00:11:00,250 --> 00:11:02,370
I don't think you're
limited by optics right now.

193
00:11:02,370 --> 00:11:04,090
What I would think
is difficult, if you

194
00:11:04,090 --> 00:11:05,660
have two photons
on a beam splitter,

195
00:11:05,660 --> 00:11:07,950
they have to be in
the same spatial mode.

196
00:11:07,950 --> 00:11:11,260
So if you have to fibers and the
fibers are not fully aligned,

197
00:11:11,260 --> 00:11:13,640
or you have another mode
which is [INAUDIBLE],

198
00:11:13,640 --> 00:11:17,420
or if you have a lens which is
distorting your Gaussian mode,

199
00:11:17,420 --> 00:11:19,700
you mix other modes
in so all this

200
00:11:19,700 --> 00:11:22,350
creates non-distinguishability.

201
00:11:22,350 --> 00:11:24,376
But I think equality
of optics is probably

202
00:11:24,376 --> 00:11:25,500
the least of your concerns.

203
00:11:35,490 --> 00:11:40,880
So actually for today, because
I'm out of town next week,

204
00:11:40,880 --> 00:11:43,990
I want you to have
something to think about it,

205
00:11:43,990 --> 00:11:46,675
I already pre-wrote
some of the slides.

206
00:11:49,990 --> 00:11:52,520
So I can go a little bit faster.

207
00:11:52,520 --> 00:11:53,290
Give me feedback.

208
00:11:53,290 --> 00:11:55,840
If you think I'm going too
fast, I will not do it again.

209
00:11:55,840 --> 00:11:59,100
But I felt I can probably
have a reasonable speed

210
00:11:59,100 --> 00:12:03,510
by making annotations to
what I've already prepared.

211
00:12:03,510 --> 00:12:07,910
So the situation
is the following,

212
00:12:07,910 --> 00:12:12,940
we want to entangle two atoms.

213
00:12:12,940 --> 00:12:15,140
We have two atoms which
are both in the excited

214
00:12:15,140 --> 00:12:16,520
state in the scatter light.

215
00:12:19,210 --> 00:12:24,100
And they can go to two
different ground states.

216
00:12:24,100 --> 00:12:26,150
Think about two different
type of ion states.

217
00:12:26,150 --> 00:12:29,550
I call them U and
D, Up and Down.

218
00:12:29,550 --> 00:12:32,790
And when because
of selection rules,

219
00:12:32,790 --> 00:12:35,112
one state can be reached
with one polarization.

220
00:12:35,112 --> 00:12:37,570
The other state can be reached
with the other polarization.

221
00:12:37,570 --> 00:12:42,520
And I call this polarization H
and V, horizontal and vertical.

222
00:12:42,520 --> 00:12:44,890
In reality, this will be
circular polarization,

223
00:12:44,890 --> 00:12:48,210
so you may want to transform
from linear to circular

224
00:12:48,210 --> 00:12:48,870
after the fact.

225
00:12:48,870 --> 00:12:51,250
But for just the
discussion right now,

226
00:12:51,250 --> 00:12:54,270
I assume that horizontal
polarization leads

227
00:12:54,270 --> 00:12:56,460
to one state,
vertical polarization

228
00:12:56,460 --> 00:12:58,190
leads to the other state.

229
00:12:58,190 --> 00:13:03,620
So therefore, if you have
two atoms-- and Chris Monroe

230
00:13:03,620 --> 00:13:06,120
in Michigan did the
experiment where he really

231
00:13:06,120 --> 00:13:08,240
had two different
vacuum chambers with two

232
00:13:08,240 --> 00:13:11,020
different ions, they
both emitted photons,

233
00:13:11,020 --> 00:13:13,627
and then the photons came
together on a beam splitter.

234
00:13:13,627 --> 00:13:15,960
So that's what you should
sort of have in mind, two ions

235
00:13:15,960 --> 00:13:16,980
here and there.

236
00:13:16,980 --> 00:13:18,760
They both emit a photon.

237
00:13:18,760 --> 00:13:26,820
And after the have
both emitted a photon,

238
00:13:26,820 --> 00:13:30,450
each ion is in a
superposition before you

239
00:13:30,450 --> 00:13:31,750
make any measurements.

240
00:13:31,750 --> 00:13:34,360
It can be in up with
a horizontal photon

241
00:13:34,360 --> 00:13:36,400
or it can be in down
with a vertical photon.

242
00:13:36,400 --> 00:13:39,520
So that is the state of the ion.

243
00:13:39,520 --> 00:13:43,735
And so we have two of those.

244
00:13:46,340 --> 00:13:48,560
And then you want
to detect photons.

245
00:13:48,560 --> 00:13:51,310
And the goal now is by
using beam splitters and all

246
00:13:51,310 --> 00:13:54,460
those tricks, that the
detection of photons

247
00:13:54,460 --> 00:13:58,180
and the outcome of a measurement
will project this product

248
00:13:58,180 --> 00:14:01,440
state of two atoms
with their photon

249
00:14:01,440 --> 00:14:04,300
into a bell state for the atoms.

250
00:14:04,300 --> 00:14:06,280
So measurement on
photons can take

251
00:14:06,280 --> 00:14:08,380
two atoms which were
completely remote,

252
00:14:08,380 --> 00:14:10,070
had nothing to do
with each other,

253
00:14:10,070 --> 00:14:12,390
and suddenly they're
in a Bell state.

254
00:14:12,390 --> 00:14:16,000
And this is done by the
probabalistic measurement.

255
00:14:16,000 --> 00:14:37,130
And so let's-- so
lets develop it.

256
00:14:41,920 --> 00:14:55,660
Our initial state is UH plus DV.

257
00:14:59,220 --> 00:15:01,390
This is system one.

258
00:15:01,390 --> 00:15:07,360
And we have the direct product
with our second ion trap

259
00:15:07,360 --> 00:15:10,600
apparatus which
has done the same.

260
00:15:10,600 --> 00:15:13,610
The ion has emitted a photon and
we don't know anything about it

261
00:15:13,610 --> 00:15:15,530
at this point.

262
00:15:15,530 --> 00:15:18,086
And now it is important, if you
want to have entanglement you

263
00:15:18,086 --> 00:15:19,710
cannot go and measure
the polarization,

264
00:15:19,710 --> 00:15:23,045
because this would project an
individual ion into a state

265
00:15:23,045 --> 00:15:24,750
and would not have entangled it.

266
00:15:24,750 --> 00:15:30,280
So what we are doing is we
are allowing the two photons,

267
00:15:30,280 --> 00:15:34,250
one photon here,
another photon here,

268
00:15:34,250 --> 00:15:36,360
they come together
at a beam splitter.

269
00:15:41,170 --> 00:15:45,700
Actually, before I do it let me
just perform the product here.

270
00:15:45,700 --> 00:15:51,380
So what we get is-- well,
it's just multiplying it out.

271
00:15:51,380 --> 00:15:56,900
UU for the atoms,
HH for the photons.

272
00:15:56,900 --> 00:16:07,986
UDHV plus DUVH, plus DDVV.

273
00:16:11,420 --> 00:16:13,790
So what you want
to do is we want

274
00:16:13,790 --> 00:16:18,110
to send this beam--
the photons of course,

275
00:16:18,110 --> 00:16:22,400
the atoms stay put--
onto a beam splitter.

276
00:16:22,400 --> 00:16:27,050
Now, I can do it more rigorously
by keeping all the terms

277
00:16:27,050 --> 00:16:29,080
until the end of
the calculation.

278
00:16:29,080 --> 00:16:33,390
But I hope it's rather obvious
that we want to have it--

279
00:16:33,390 --> 00:16:35,200
if we want to send
it on a beam splitter

280
00:16:35,200 --> 00:16:39,080
and then we want to detect
that both detectors make click.

281
00:16:39,080 --> 00:16:42,750
That requires the photons
to be distinguishable.

282
00:16:42,750 --> 00:16:45,760
So therefore, those
two terms will not

283
00:16:45,760 --> 00:16:48,670
contribute because the photons
have the same polarization.

284
00:16:58,890 --> 00:17:04,599
So we are now using a
protective measurement.

285
00:17:04,599 --> 00:17:08,985
We want to find out what is
the wave function of the atoms.

286
00:17:13,079 --> 00:17:26,640
Measurement of one photon
in the modes A and B,

287
00:17:26,640 --> 00:17:30,040
which are the output modes
of our beam splitter.

288
00:17:36,040 --> 00:17:45,310
So it needs one more line.

289
00:17:45,310 --> 00:17:50,610
The output of the
beam splitter is--

290
00:17:50,610 --> 00:17:59,920
so I'm only focusing now on the
two terms which can give rise

291
00:17:59,920 --> 00:18:04,660
to two clicks at the
two different detectors.

292
00:18:04,660 --> 00:18:12,000
So I will factor out UD.

293
00:18:12,000 --> 00:18:17,520
And now we have the two photons
hitting the beam splitter

294
00:18:17,520 --> 00:18:22,170
and each photon goes into
a 50/50 superposition.

295
00:18:22,170 --> 00:18:27,360
So I take know this
horizontal photon.

296
00:18:30,837 --> 00:18:33,045
And since we don't mess
around with the polarization,

297
00:18:33,045 --> 00:18:37,010
it's still a horizontal
photon, but the photon

298
00:18:37,010 --> 00:18:46,620
can be now in mode B or in
mode A in one of the output

299
00:18:46,620 --> 00:18:50,080
modes of the beam splitter.

300
00:18:50,080 --> 00:19:02,610
And I label that
as 0,1 plus 1,0.

301
00:19:02,610 --> 00:19:11,640
Now for the second photon, it
had a vertical polarization,

302
00:19:11,640 --> 00:19:16,190
so it's a vertical photon.

303
00:19:16,190 --> 00:19:20,630
The second photon is coming
onto the beam splitter

304
00:19:20,630 --> 00:19:23,156
in the other input port,
because we have taken the two

305
00:19:23,156 --> 00:19:24,780
photons from two
different experiments,

306
00:19:24,780 --> 00:19:26,404
from two different
experimental setups,

307
00:19:26,404 --> 00:19:30,250
and now we combine
them so that we'll also

308
00:19:30,250 --> 00:19:33,830
be now in a superposition
of 0,1 and 1,0.

309
00:19:33,830 --> 00:19:37,775
But you know that one mode
transforms with a plus sign

310
00:19:37,775 --> 00:19:40,400
and the other mode transforms at
the beam splitter with a minus

311
00:19:40,400 --> 00:19:40,899
sign.

312
00:19:44,010 --> 00:19:46,450
So that's how you should
look at those terms.

313
00:19:46,450 --> 00:19:51,500
And of course, from of
this here, from this term

314
00:19:51,500 --> 00:20:05,090
we get something very
analogous, 0,1 plus 1,0.

315
00:20:05,090 --> 00:20:08,850
And for the other
spatial mode, the minus

316
00:20:08,850 --> 00:20:14,540
sign from the beam
splitter, 1,0.

317
00:20:14,540 --> 00:20:16,110
OK, now we are done.

318
00:20:16,110 --> 00:20:19,690
Now we are detecting photons.

319
00:20:19,690 --> 00:20:22,620
So we want to look at this
wave function and ask,

320
00:20:22,620 --> 00:20:26,830
what happens if in port
A-- the output port--

321
00:20:26,830 --> 00:20:31,522
we find one horizontal photon,
and in B it must be vertical

322
00:20:31,522 --> 00:20:33,480
because the photons have
to be distinguishable.

323
00:20:36,980 --> 00:20:42,650
So where do we have
one photon in mode A?

324
00:20:42,650 --> 00:20:46,370
In horizontal it is here.

325
00:20:46,370 --> 00:20:50,540
And here it is this
product with the minus sign

326
00:20:50,540 --> 00:20:55,180
where we have-- no,
sorry, the plus sign.

327
00:20:55,180 --> 00:20:57,160
Here we have one
photon in the vertical,

328
00:20:57,160 --> 00:20:59,270
so this is with a plus sign.

329
00:20:59,270 --> 00:21:01,400
But we have a
second possibility,

330
00:21:01,400 --> 00:21:11,470
one photon in mode A horizontal
happens here with a minus sign.

331
00:21:11,470 --> 00:21:17,090
And one photon in mode
B vertical happens here.

332
00:21:17,090 --> 00:21:21,710
So in other words, form
the term which gives rise

333
00:21:21,710 --> 00:21:25,680
to this measurement process,
we have a plus UD here,

334
00:21:25,680 --> 00:21:30,680
and because of the
minus sign, a minus DU.

335
00:21:30,680 --> 00:21:34,600
So therefore, if
you would detect

336
00:21:34,600 --> 00:21:37,020
the photon with the polarization
at this point, which

337
00:21:37,020 --> 00:21:45,030
we don't do, this polarization
gives rise to UD minus DU.

338
00:21:45,030 --> 00:21:50,870
We have a second possibility to
have a click at each detector,

339
00:21:50,870 --> 00:21:52,955
and this is when we reverse
the two polarizations.

340
00:21:55,910 --> 00:22:00,390
But in this case, you can
just add the wave function.

341
00:22:00,390 --> 00:22:09,126
You get the atomic state,
the sign doesn't matter.

342
00:22:09,126 --> 00:22:11,750
So you have to measurements, two
possibilities in polarization,

343
00:22:11,750 --> 00:22:13,705
but they both have
the same outcome.

344
00:22:16,690 --> 00:22:23,730
They generate this state, so
the result of all that is I've

345
00:22:23,730 --> 00:22:31,510
shown you that the two
atoms are left now in--

346
00:22:31,510 --> 00:22:35,520
and now I put back in the
correct normalization--

347
00:22:35,520 --> 00:22:40,850
in one of the Bell states.

348
00:22:40,850 --> 00:22:48,140
So whenever you detect the two
single photons in coincidence

349
00:22:48,140 --> 00:22:50,410
on both counters,
at that moment you

350
00:22:50,410 --> 00:22:54,320
know what you have is
an entangled state,

351
00:22:54,320 --> 00:22:55,480
a Bell state of atoms.

352
00:22:58,200 --> 00:23:02,660
Now, the big issue
here is efficiency.

353
00:23:02,660 --> 00:23:05,360
Your photon detectors
are not highly efficient.

354
00:23:05,360 --> 00:23:08,825
And I was sort of only
looking at combinations

355
00:23:08,825 --> 00:23:11,610
where we have this
double detection.

356
00:23:11,610 --> 00:23:14,760
Most components of
the wave function

357
00:23:14,760 --> 00:23:17,430
will give the
Hong-Ou-Mandel effect.

358
00:23:17,430 --> 00:23:19,210
The two photons go in one way.

359
00:23:19,210 --> 00:23:21,890
So therefore, you have
to prepare your system,

360
00:23:21,890 --> 00:23:24,760
have photons emitted,
and in most of the cases

361
00:23:24,760 --> 00:23:26,280
you will not get
the state you want,

362
00:23:26,280 --> 00:23:28,220
so you have to keep on trying.

363
00:23:28,220 --> 00:23:30,590
And usually those
experiments are

364
00:23:30,590 --> 00:23:35,430
heavily limited by the very,
very dismal success probability

365
00:23:35,430 --> 00:23:37,670
of creating the state you want.

366
00:23:37,670 --> 00:23:40,550
And when you want to extend
it, not just to two atoms

367
00:23:40,550 --> 00:23:44,520
entangled, to three atoms
and four atoms entangled,

368
00:23:44,520 --> 00:23:48,400
then you get a smaller
number to the power n

369
00:23:48,400 --> 00:23:49,910
for your efficiency.

370
00:23:49,910 --> 00:23:53,990
So this probabalistic
preparation doesn't scale up,

371
00:23:53,990 --> 00:23:58,410
but it is a very simple scheme,
it's a very powerful scheme,

372
00:23:58,410 --> 00:24:07,700
and it has been used to create
teleportation of atomic states

373
00:24:07,700 --> 00:24:13,555
and do some tests of Bells
inequality and the EPR Paradox.

374
00:24:16,592 --> 00:24:21,990
And here is the
abstract of a paper.

375
00:24:21,990 --> 00:24:27,280
So-- well, this was just
2008, so just a few years ago.

376
00:24:27,280 --> 00:24:37,940
And what you see here is
that here are one meter apart

377
00:24:37,940 --> 00:24:40,480
two [INAUDIBLE] ions.

378
00:24:40,480 --> 00:24:41,820
They emit light.

379
00:24:41,820 --> 00:24:43,920
The light goes through
an optical fiber.

380
00:24:43,920 --> 00:24:46,930
And now, after
the beam splitter,

381
00:24:46,930 --> 00:24:48,940
when you detect one
photon you don't

382
00:24:48,940 --> 00:24:50,730
know which ion has
emitted the photon.

383
00:24:56,330 --> 00:25:01,760
So this is exactly the
setup which I described.

384
00:25:04,570 --> 00:25:09,990
Of course, in an
optical fiber you

385
00:25:09,990 --> 00:25:15,540
have the two good polarizations
are horizontal and vertical

386
00:25:15,540 --> 00:25:18,120
in the polarization
maintaining fiber.

387
00:25:18,120 --> 00:25:20,710
So therefore, the
light which originally

388
00:25:20,710 --> 00:25:23,840
was emitted in a circular base
is-- angular momentum selection

389
00:25:23,840 --> 00:25:26,324
rules-- transformed into
linearly polarized light

390
00:25:26,324 --> 00:25:27,240
using the [INAUDIBLE].

391
00:25:32,697 --> 00:25:33,280
Any questions?

392
00:25:35,940 --> 00:25:36,840
Yes, Nikki?

393
00:25:36,840 --> 00:25:39,520
AUDIENCE: [INAUDIBLE] create
the initial state [INAUDIBLE]

394
00:25:39,520 --> 00:25:44,011
atom [INAUDIBLE] one
state [INAUDIBLE]

395
00:25:44,011 --> 00:25:48,252
make sure that the ions
state is a pure state, not

396
00:25:48,252 --> 00:25:49,999
a mixed state, [INAUDIBLE]?

397
00:25:53,991 --> 00:25:55,630
WOLFGANG KETTERLE:
Good question.

398
00:25:55,630 --> 00:25:58,440
So first of all, if
you think about it

399
00:25:58,440 --> 00:26:02,470
you will discover more and
more experimental challenges.

400
00:26:02,470 --> 00:26:07,390
What people must have used there
is sort of a storm laser pulse

401
00:26:07,390 --> 00:26:10,530
that you have more than
overkill to make sure

402
00:26:10,530 --> 00:26:12,890
that with a very,
very short time window

403
00:26:12,890 --> 00:26:15,310
both ions are excited.

404
00:26:15,310 --> 00:26:19,690
The ions are in a pure state,
and then they emit a photon.

405
00:26:19,690 --> 00:26:22,310
And if you have
one system prepared

406
00:26:22,310 --> 00:26:29,150
which can emit a photon, but it
has a branching ratio of 50/50,

407
00:26:29,150 --> 00:26:31,470
if it's an isolated
system, it will

408
00:26:31,470 --> 00:26:34,740
have a superposition state of
photon in one polarization--

409
00:26:34,740 --> 00:26:39,600
let's say a sigma plus photon
going to a magnetic quantum

410
00:26:39,600 --> 00:26:41,890
number state m equals
plus 1 and a sigma

411
00:26:41,890 --> 00:26:44,360
minus photon going
to m equals minus 1.

412
00:26:44,360 --> 00:26:48,060
And this is a pure state.

413
00:26:48,060 --> 00:26:52,620
The mixture only comes
if you're not careful.

414
00:26:52,620 --> 00:26:54,120
If you have a
magnetic field and you

415
00:26:54,120 --> 00:26:55,930
don't shield your
magnetic field well,

416
00:26:55,930 --> 00:26:58,580
or you have some
magnetized materials

417
00:26:58,580 --> 00:27:00,520
and you were not
aware of it, then that

418
00:27:00,520 --> 00:27:04,360
means that you get
different phase shifts which

419
00:27:04,360 --> 00:27:08,090
you can't calculate for, so
now you have a random phase.

420
00:27:08,090 --> 00:27:11,120
You don't know it, and
if you have ignorance

421
00:27:11,120 --> 00:27:13,960
you have to trace out or you
have to average over the phase.

422
00:27:13,960 --> 00:27:17,210
And then your pure state
becomes a density matrix.

423
00:27:17,210 --> 00:27:19,365
But the quantum
mechanical process itself

424
00:27:19,365 --> 00:27:23,720
of a particle having
different branching ratios

425
00:27:23,720 --> 00:27:25,710
is a pure system.

426
00:27:25,710 --> 00:27:28,280
It undergoes unitary
time evolution,

427
00:27:28,280 --> 00:27:30,930
and it stays in a pure state.

428
00:27:30,930 --> 00:27:37,700
And the state which is
populated by two ions, both

429
00:27:37,700 --> 00:27:41,330
emitting a photon
simultaneously,

430
00:27:41,330 --> 00:27:46,630
is exactly the state I've
written down for you.

431
00:27:46,630 --> 00:27:50,030
But you're absolutely right
that decoherence is an issue.

432
00:27:50,030 --> 00:27:52,620
You have to be very careful
with magnetic fields.

433
00:27:52,620 --> 00:27:57,820
You probably want to work with
atoms which are non-magnetic.

434
00:27:57,820 --> 00:28:01,450
So you want to where the spin
is only a nuclear spin which

435
00:28:01,450 --> 00:28:05,650
is much less sensitive with
a magnetic field, and so on.

436
00:28:05,650 --> 00:28:06,895
Other questions?

437
00:28:06,895 --> 00:28:11,780
AUDIENCE: So as soon as
both detectors click, then

438
00:28:11,780 --> 00:28:16,340
[INAUDIBLE] so that
you can no longer

439
00:28:16,340 --> 00:28:18,820
make use of that entanglement?

440
00:28:18,820 --> 00:28:22,540
WOLFGANG KETTERLE: No, no wait.

441
00:28:22,540 --> 00:28:23,970
Look here.

442
00:28:23,970 --> 00:28:26,430
We have a state which has atoms
in a state and two photons.

443
00:28:29,490 --> 00:28:33,070
So this state has
four particles.

444
00:28:33,070 --> 00:28:36,980
We detect two particles
which are the photons.

445
00:28:36,980 --> 00:28:40,060
And the atoms are untouched.

446
00:28:40,060 --> 00:28:43,170
The atoms are then afterwards
in an entangled state.

447
00:28:47,080 --> 00:28:49,980
So we have a bigger
system, we do a measurement

448
00:28:49,980 --> 00:28:51,880
of part of the
system, and we have

449
00:28:51,880 --> 00:28:54,320
arranged things in
the skillful way

450
00:28:54,320 --> 00:28:56,910
that the moment we know the
outcome of the measurement

451
00:28:56,910 --> 00:28:59,980
is such and such, we
know in which quantum

452
00:28:59,980 --> 00:29:02,280
state the rest of the system is.

453
00:29:02,280 --> 00:29:04,560
And this protocol
means that the atoms

454
00:29:04,560 --> 00:29:07,900
are left after the
outcome of the measurement

455
00:29:07,900 --> 00:29:10,570
is such and such, the atoms
are left in a pure Bell state.

456
00:29:23,060 --> 00:29:27,200
OK I've used the word
Bell, Bell states

457
00:29:27,200 --> 00:29:31,710
so often, I think it's time
to talk about what Mr. Bell is

458
00:29:31,710 --> 00:29:34,225
famous for, namely
the Bell's Inequality.

459
00:29:39,150 --> 00:29:43,730
So I said already
in the introduction

460
00:29:43,730 --> 00:29:48,460
for the quantumness of
light and entanglement

461
00:29:48,460 --> 00:29:53,910
that it is the EPR Paradox
in the Bell's inequalities

462
00:29:53,910 --> 00:29:56,970
which a lot of people,
including myself, think

463
00:29:56,970 --> 00:29:59,250
is the deeper essence
of quantum physics.

464
00:29:59,250 --> 00:30:05,000
It really shows that
quantum physics is not just

465
00:30:05,000 --> 00:30:07,100
classical physics
with wave character,

466
00:30:07,100 --> 00:30:10,080
it goes way beyond it.

467
00:30:10,080 --> 00:30:15,140
So I want to demonstrate
that with Bell's inequality.

468
00:30:15,140 --> 00:30:18,370
And the formulation
which is very simple,

469
00:30:18,370 --> 00:30:20,800
and I want to present it
here, is an inequality,

470
00:30:20,800 --> 00:30:22,560
which is-- I mean the name.

471
00:30:22,560 --> 00:30:26,420
Many Bell's inequalities now,
you have different states,

472
00:30:26,420 --> 00:30:29,860
different detectors, and you
can derive Bell's inequalities

473
00:30:29,860 --> 00:30:33,000
which are then violated
by quantum physics.

474
00:30:33,000 --> 00:30:37,120
And what I want to present
here is the CHSH inequality.

475
00:30:37,120 --> 00:30:44,860
That's Clauser,
Horn, Shimony, and--

476
00:30:44,860 --> 00:30:46,150
AUDIENCE: Holt.

477
00:30:46,150 --> 00:30:48,340
WOLFGANG KETTERLE:
Yes, thank you.

478
00:30:48,340 --> 00:30:52,160
So the situation
is the following.

479
00:30:52,160 --> 00:30:56,800
We have something
which decays, something

480
00:30:56,800 --> 00:30:58,100
which emits two photons.

481
00:30:58,100 --> 00:31:00,820
Maybe an atom in
an excited state,

482
00:31:00,820 --> 00:31:04,130
and it does a click clack,
a two photon cascade.

483
00:31:04,130 --> 00:31:07,230
One photon goes to
Bob, one goes to Alice.

484
00:31:07,230 --> 00:31:10,440
Or I always try to
stress similarities

485
00:31:10,440 --> 00:31:12,710
between light and atoms.

486
00:31:12,710 --> 00:31:14,190
We discussed the
experiment where

487
00:31:14,190 --> 00:31:17,650
you take a mercury
molecule, you dissociate it,

488
00:31:17,650 --> 00:31:20,330
and then Bob and Alice
each get an atom.

489
00:31:20,330 --> 00:31:24,120
And you can say these
photons has a polarization

490
00:31:24,120 --> 00:31:26,010
or the atom has a spin.

491
00:31:26,010 --> 00:31:30,290
But Bob and Alice can now use
different Stern Gerlach figures

492
00:31:30,290 --> 00:31:33,290
in x and y, at obscure
angles, circular bases,

493
00:31:33,290 --> 00:31:34,500
I mean you name it.

494
00:31:34,500 --> 00:31:37,980
But what happens is because
it's a spin up, spin

495
00:31:37,980 --> 00:31:41,130
down, horizontal,
vertical polarization,

496
00:31:41,130 --> 00:31:43,656
it's a two-level system
which is immediate.

497
00:31:43,656 --> 00:31:45,030
And after a Stern
Gerlach filter,

498
00:31:45,030 --> 00:31:46,900
you have only two combinations.

499
00:31:46,900 --> 00:31:49,500
We call it plus and minus.

500
00:31:49,500 --> 00:31:52,540
So in its most general
form, what we assume

501
00:31:52,540 --> 00:31:56,604
is that Bob does measurements
in a basis, if you think.

502
00:31:56,604 --> 00:31:58,770
Think about a spatial
orientation of a Stern Gerlach

503
00:31:58,770 --> 00:32:02,160
filter called S, where the
outcome is plus minus 1,

504
00:32:02,160 --> 00:32:04,990
t where the outcome
is plus minus 1,

505
00:32:04,990 --> 00:32:06,785
and Alice has her own choices.

506
00:32:11,600 --> 00:32:14,110
And we want to
assume now that this

507
00:32:14,110 --> 00:32:18,020
is everything is
classical probability.

508
00:32:18,020 --> 00:32:21,316
That if you just write
down this expression--

509
00:32:21,316 --> 00:32:22,690
don't ask me where
it comes from.

510
00:32:22,690 --> 00:32:24,350
This is probably something
which people wrote up

511
00:32:24,350 --> 00:32:25,933
after they found
something interesting

512
00:32:25,933 --> 00:32:27,970
and try to prove
it or simplify it.

513
00:32:27,970 --> 00:32:31,700
You just write down
QS, RS, RT minus QT.

514
00:32:31,700 --> 00:32:38,260
You can then rewrite it
by factoring out S and T.

515
00:32:38,260 --> 00:32:42,290
And now the next step
is because Q and R

516
00:32:42,290 --> 00:32:45,460
are-- they're are in each
measurement 1 or minus 1,

517
00:32:45,460 --> 00:32:47,610
either this is 0 or that is 0.

518
00:32:50,628 --> 00:32:56,210
No sorry, one is 0, one is 2.

519
00:32:56,210 --> 00:33:00,090
And therefore, this kind
of funny combination

520
00:33:00,090 --> 00:33:04,140
of letters for every measurement
is either plus or minus 2.

521
00:33:06,710 --> 00:33:12,142
Now we do many measurements,
and the probability

522
00:33:12,142 --> 00:33:19,450
for a certain outcome
that the variable Q--

523
00:33:19,450 --> 00:33:22,215
there's a certain probability
that the outcome is Q,

524
00:33:22,215 --> 00:33:25,840
so you sort of use this pretty
much probabilistic thing

525
00:33:25,840 --> 00:33:28,670
that the system has a
certain probability to B.

526
00:33:28,670 --> 00:33:30,340
And this is of course
the assumption,

527
00:33:30,340 --> 00:33:32,820
the probability is that
the particular comes

528
00:33:32,820 --> 00:33:37,420
with a certain probability
in the state q, r, s, t.

529
00:33:37,420 --> 00:33:40,060
Of course you should
scream, q, r, s, t commute,

530
00:33:40,060 --> 00:33:41,780
but this is classical now.

531
00:33:41,780 --> 00:33:44,220
They don't commute with each
other quantum mechanically,

532
00:33:44,220 --> 00:33:46,680
but [INAUDIBLE] in a moment.

533
00:33:46,680 --> 00:33:51,120
So then you simply put--
by multiplying each event

534
00:33:51,120 --> 00:33:54,750
with its portability, you
put now brackets around it.

535
00:33:54,750 --> 00:33:56,880
These are expectation
values, and what

536
00:33:56,880 --> 00:34:01,030
you have is an inequality
that this expression, which

537
00:34:01,030 --> 00:34:03,790
is the correlation between
certain measurements

538
00:34:03,790 --> 00:34:07,030
between the quantity
QS, RS, and so on

539
00:34:07,030 --> 00:34:09,659
is smaller or equal than 2.

540
00:34:09,659 --> 00:34:11,158
I mean it's very, very basic.

541
00:34:11,158 --> 00:34:12,449
And that's why I wrote it down.

542
00:34:12,449 --> 00:34:14,199
I don't want to spend
a lot of time on it.

543
00:34:14,199 --> 00:34:17,000
It's just classical
probabilistic reasoning.

544
00:34:24,010 --> 00:34:28,170
So now what are we doing
quantum mechanically?

545
00:34:33,639 --> 00:34:37,960
We want to prove that quantum
mechanics cannot be reduced

546
00:34:37,960 --> 00:34:39,840
to this classical reasoning.

547
00:34:39,840 --> 00:34:42,360
So we want to show that
this inequality is violated.

548
00:34:48,190 --> 00:34:52,750
We have a source or
entangled photons.

549
00:34:52,750 --> 00:34:56,500
We've talked about how
we can entangle photons.

550
00:34:56,500 --> 00:35:08,550
And now we measure the
quantities Q, R, S, and T.

551
00:35:08,550 --> 00:35:11,930
And just to give you an
example about the many possible

552
00:35:11,930 --> 00:35:17,684
choices, Q can be a
linear polarizer and R

553
00:35:17,684 --> 00:35:18,850
can be a circular polarizer.

554
00:35:23,360 --> 00:35:28,380
S can be a linear
polarizer at 45 degree,

555
00:35:28,380 --> 00:35:36,740
and T can be a
[INAUDIBLE] plate followed

556
00:35:36,740 --> 00:35:42,860
by a linear polarizer
at 45 degrees.

557
00:35:42,860 --> 00:35:46,530
Well, that sounds like
many trig functions,

558
00:35:46,530 --> 00:35:48,079
but we're not going into it.

559
00:35:48,079 --> 00:35:49,620
You have to choose
your polarization.

560
00:35:49,620 --> 00:35:53,100
There's a certain scheme that
linear polarization is not

561
00:35:53,100 --> 00:35:54,600
orthogonal to
circular polarization,

562
00:35:54,600 --> 00:35:57,860
so there's a certain
theme behind it.

563
00:35:57,860 --> 00:36:03,300
And if you would work
it out by just looking

564
00:36:03,300 --> 00:36:05,680
at entangled photons,
the entangled photons

565
00:36:05,680 --> 00:36:10,600
are in a state,
let's say HV plus VH.

566
00:36:10,600 --> 00:36:13,150
They are correlated
in polarization

567
00:36:13,150 --> 00:36:17,630
or spin up, spin down with the
down spin [INAUDIBLE] state.

568
00:36:17,630 --> 00:36:19,130
And you can just
work out what is

569
00:36:19,130 --> 00:36:21,990
the polarization
when you detect it.

570
00:36:21,990 --> 00:36:43,120
And what happens is that a
simple but tedious calculation

571
00:36:43,120 --> 00:36:52,850
says that all those quantities
QS, RS, and and RT are equal.

572
00:36:56,030 --> 00:37:03,430
Here we have a minus sign and
they are all 1 over square root

573
00:37:03,430 --> 00:37:03,930
2.

574
00:37:07,830 --> 00:37:12,410
And that means that instead
of the classical inequality,

575
00:37:12,410 --> 00:37:16,620
that this quantity
is smaller than 2.

576
00:37:16,620 --> 00:37:21,270
We find that this is
2 times square root 2.

577
00:37:24,760 --> 00:37:35,860
And the fact that
this is larger than 2

578
00:37:35,860 --> 00:37:39,300
has been experimentally
confirmed

579
00:37:39,300 --> 00:37:40,955
with larger and
larger precision.

580
00:37:44,394 --> 00:37:46,560
Actually, the person who
gave this [? CUA ?] seminar

581
00:37:46,560 --> 00:37:49,040
a week ago,
[INAUDIBLE], was part

582
00:37:49,040 --> 00:37:52,440
of the team with [INAUDIBLE]
who did one of the very, very

583
00:37:52,440 --> 00:37:55,910
first measurements violations
of Bell's inequality

584
00:37:55,910 --> 00:37:58,660
in the '80s some 30 years ago.

585
00:37:58,660 --> 00:38:02,275
So I mean, this happened
rather recently.

586
00:38:05,240 --> 00:38:09,700
So the math is trivial,
the result seems trivial.

587
00:38:09,700 --> 00:38:14,490
It just shows that the world
is quantum mechanically and not

588
00:38:14,490 --> 00:38:16,270
classical.

589
00:38:16,270 --> 00:38:19,700
And a lot of papers
have been written

590
00:38:19,700 --> 00:38:23,430
and discussions have been had
about what does it really mean?

591
00:38:23,430 --> 00:38:25,400
What does it mean
about the world?

592
00:38:25,400 --> 00:38:27,660
What does it tell
us about the world?

593
00:38:27,660 --> 00:38:39,410
So so the implication from
the established violation

594
00:38:39,410 --> 00:38:48,250
of Bell's inequality
and the CHSH inequality

595
00:38:48,250 --> 00:39:06,330
is that when we assumed that the
state has definite values of Q,

596
00:39:06,330 --> 00:39:19,010
R, S, T before observation,
we have to give up that.

597
00:39:19,010 --> 00:39:31,340
Or we have to give up that a
measurement performed by Alice

598
00:39:31,340 --> 00:39:44,480
does not influence Bob's
experiment, Bob's measurement.

599
00:39:44,480 --> 00:39:56,350
So in other words, what I
formulated here is the locality

600
00:39:56,350 --> 00:40:04,300
principle that what happens in
one location cannot influence

601
00:40:04,300 --> 00:40:06,160
what happens in
the other location.

602
00:40:06,160 --> 00:40:09,530
Well you can say maybe
there's some secret channel

603
00:40:09,530 --> 00:40:11,260
sending [INAUDIBLE]
the speed of light.

604
00:40:11,260 --> 00:40:13,060
But people [INAUDIBLE]
to create links

605
00:40:13,060 --> 00:40:16,590
to put the detector so far
away that even a [INAUDIBLE]

606
00:40:16,590 --> 00:40:19,620
at the speed of
light could not have

607
00:40:19,620 --> 00:40:22,680
influenced the
other measurement.

608
00:40:26,030 --> 00:40:27,470
So there has been
after the first

609
00:40:27,470 --> 00:40:28,928
Bell's inequality
experiment, there

610
00:40:28,928 --> 00:40:32,740
have been a series
of experiments

611
00:40:32,740 --> 00:40:35,180
to avoid all of those
loopholes that there

612
00:40:35,180 --> 00:40:38,180
is some unknown secret
communication between the two

613
00:40:38,180 --> 00:40:39,880
detectors.

614
00:40:39,880 --> 00:40:46,130
The first one is the
principal of-- sometimes

615
00:40:46,130 --> 00:40:47,660
I think a more
philosophical word

616
00:40:47,660 --> 00:40:49,480
than physical word, "reality".

617
00:40:49,480 --> 00:40:52,425
That those quantities
are real and they

618
00:40:52,425 --> 00:40:56,310
should exist before
we measure them.

619
00:40:56,310 --> 00:40:59,290
So the outcome of
Bell's inequality

620
00:40:59,290 --> 00:41:03,210
is-- the violation of Bell's
inequality is, at least one

621
00:41:03,210 --> 00:41:09,470
of them, at least one
of these principles--

622
00:41:09,470 --> 00:41:14,600
reality-- at least one
of these principles

623
00:41:14,600 --> 00:41:17,040
does not hold in nature.

624
00:41:17,040 --> 00:41:21,095
So either reality or
locality are violated.

625
00:41:30,290 --> 00:41:30,950
Any questions?

626
00:41:37,000 --> 00:41:42,800
And maybe just to connect
it to what I've said before,

627
00:41:42,800 --> 00:41:49,840
one reason you saw that here
in the title, Bell's inequality

628
00:41:49,840 --> 00:41:52,520
with two remote
atomic [INAUDIBLE].

629
00:41:52,520 --> 00:41:54,770
Those experiments got
a lot of attention

630
00:41:54,770 --> 00:41:56,670
because Bell's
inequality is something

631
00:41:56,670 --> 00:41:58,440
all physicists talk about.

632
00:41:58,440 --> 00:42:01,010
And when you do it
with photons, there

633
00:42:01,010 --> 00:42:04,260
is always the
detection loophole.

634
00:42:04,260 --> 00:42:07,050
You don't have perfect
photon to detectors.

635
00:42:07,050 --> 00:42:09,670
And some people said,
well, maybe the violation

636
00:42:09,670 --> 00:42:13,300
of Bell's inequality comes only
from the photons we detect.

637
00:42:13,300 --> 00:42:15,480
The undetected
photons would make up

638
00:42:15,480 --> 00:42:17,480
for the violation of
Bell's inequality,

639
00:42:17,480 --> 00:42:23,040
and Bell's inequality
is not violated.

640
00:42:23,040 --> 00:42:26,950
I mean, that almost sounds like
that nature wants to fool us.

641
00:42:26,950 --> 00:42:29,280
The photons team up
and say the photons

642
00:42:29,280 --> 00:42:31,640
which are detected
behave very differently

643
00:42:31,640 --> 00:42:33,400
from the photons which
are not detected.

644
00:42:33,400 --> 00:42:36,270
But these are at least
logical loopholes.

645
00:42:36,270 --> 00:42:39,950
And graduate students spend
half of their Ph.D., or maybe

646
00:42:39,950 --> 00:42:42,490
several graduate students
spend their whole Ph.D.

647
00:42:42,490 --> 00:42:45,950
In building such an
experiment which is now--

648
00:42:45,950 --> 00:42:47,440
you get a famous
paper out of it.

649
00:42:47,440 --> 00:42:51,910
And your research has
attention, so you know.

650
00:42:51,910 --> 00:42:56,735
I didn't sleep so much, so
maybe I'm not too serious now.

651
00:42:56,735 --> 00:42:58,164
But this is great research.

652
00:42:58,164 --> 00:42:59,830
I mean, this is really
pushing the limit

653
00:42:59,830 --> 00:43:03,500
of our understanding
of quantum physics.

654
00:43:03,500 --> 00:43:05,760
And Chris Monroe is a
wonderful physicist.

655
00:43:05,760 --> 00:43:10,540
And we are in the
same [INAUDIBLE].

656
00:43:10,540 --> 00:43:13,800
So if you have a
Bell state of atoms,

657
00:43:13,800 --> 00:43:16,430
and I told you how you get it
also with a lousy efficiency

658
00:43:16,430 --> 00:43:17,750
but you get a Bell state.

659
00:43:17,750 --> 00:43:20,320
If you now do a measurement
of those Bell state

660
00:43:20,320 --> 00:43:23,920
and you measure violation
of Bell's inequality, atoms,

661
00:43:23,920 --> 00:43:24,770
they don't run away.

662
00:43:24,770 --> 00:43:27,610
You can detect the atoms
with 100% probability.

663
00:43:27,610 --> 00:43:30,990
You can shine a laser light
on them, hundreds, thousands,

664
00:43:30,990 --> 00:43:33,780
millions of photons until they
have scattered enough light

665
00:43:33,780 --> 00:43:35,810
that you know
100%, I've detected

666
00:43:35,810 --> 00:43:37,410
the atom in it's quantum state.

667
00:43:37,410 --> 00:43:43,250
So Bell states with atoms, one
real world application of them

668
00:43:43,250 --> 00:43:46,200
is to test violations
of Bell's inequality

669
00:43:46,200 --> 00:43:50,460
with atoms which do not fall
into the detection loophole.

670
00:44:13,310 --> 00:44:20,480
Our next unit is
partially motivated

671
00:44:20,480 --> 00:44:25,050
to show you what all this
entanglement can do for us.

672
00:44:25,050 --> 00:44:28,530
So I mentioned that
entanglement is a resource.

673
00:44:28,530 --> 00:44:32,060
It's something useful
like energy is a resource.

674
00:44:32,060 --> 00:44:35,780
And so maybe ultimately there
should be a stock market,

675
00:44:35,780 --> 00:44:38,920
and you can buy so and so
many bits of entanglement.

676
00:44:38,920 --> 00:44:40,670
And you have to pay
for it because there's

677
00:44:40,670 --> 00:44:43,400
something you can do.

678
00:44:43,400 --> 00:44:48,330
And let me introduce
what can be done with it.

679
00:44:48,330 --> 00:44:51,830
If you have one photon
at frequency omega,

680
00:44:51,830 --> 00:44:55,600
and you have an
observation time-- you

681
00:44:55,600 --> 00:44:59,040
do a measurement
over time T, then

682
00:44:59,040 --> 00:45:02,230
the precision at which
you measure the frequency

683
00:45:02,230 --> 00:45:07,230
is fundamentally limited by, you
can say time energy uncertainty

684
00:45:07,230 --> 00:45:10,890
by the Fourier theorem, that the
uncertainty and the frequency

685
00:45:10,890 --> 00:45:14,260
of a single measurement
is 1 over the time T

686
00:45:14,260 --> 00:45:19,150
you had to detect or to measure
the frequency of the photon.

687
00:45:19,150 --> 00:45:22,220
But now we have n photons.

688
00:45:22,220 --> 00:45:25,690
And that means that the
uncertainty in the measurement,

689
00:45:25,690 --> 00:45:29,490
which is now small
delta omega, is

690
00:45:29,490 --> 00:45:32,180
delta omega for a single photon.

691
00:45:37,880 --> 00:45:42,010
But you know, when you do n
measurements, and average n

692
00:45:42,010 --> 00:45:48,420
measurements, you gain
by the square root of n.

693
00:45:48,420 --> 00:45:57,674
And this is regarded as
the fundamental shot noise

694
00:45:57,674 --> 00:45:58,590
limit of measurements.

695
00:46:01,880 --> 00:46:06,310
But now assume we can
do something fancy.

696
00:46:06,310 --> 00:46:09,290
We can make a super photon.

697
00:46:09,290 --> 00:46:13,730
We take our n photons
of frequency omega

698
00:46:13,730 --> 00:46:19,130
and make one photon
of frequency n omega.

699
00:46:19,130 --> 00:46:22,340
Now we have only one photon.

700
00:46:22,340 --> 00:46:34,720
The frequency uncertainty of
the measurement is delta omega,

701
00:46:34,720 --> 00:46:37,870
but this is now the uncertainty
off the n times more

702
00:46:37,870 --> 00:46:39,510
energetic photon.

703
00:46:39,510 --> 00:46:42,470
So therefore, if you're
interested in the quantity

704
00:46:42,470 --> 00:46:46,080
omega, which is the frequency
of the single photon,

705
00:46:46,080 --> 00:46:50,790
we have now made an improvement
over the standard shot noise

706
00:46:50,790 --> 00:46:51,870
limit by square root n.

707
00:46:58,350 --> 00:47:01,470
So everybody follows?

708
00:47:01,470 --> 00:47:03,310
So that's just a
[INAUDIBLE] experiment.

709
00:47:03,310 --> 00:47:06,710
If you can take n
photons-- I've told you

710
00:47:06,710 --> 00:47:12,470
how we've talked about the
[INAUDIBLE] oscillator, how

711
00:47:12,470 --> 00:47:14,040
we can pump a crystal.

712
00:47:14,040 --> 00:47:16,282
And in your homework
assignment you

713
00:47:16,282 --> 00:47:17,990
do a nice calculation
with a Hamiltonian.

714
00:47:17,990 --> 00:47:21,550
One big photon in, it
breaks into two photons.

715
00:47:21,550 --> 00:47:24,370
The reverse process is
frequency dot doubling.

716
00:47:24,370 --> 00:47:29,360
So if you would not measure n
photons individually, but first

717
00:47:29,360 --> 00:47:32,880
ate up their frequency by
making a photon of n times

718
00:47:32,880 --> 00:47:35,950
the frequency and then
look a single photon,

719
00:47:35,950 --> 00:47:39,850
you have now, you measure
all the photons together.

720
00:47:39,850 --> 00:47:42,480
And therefore, your
accuracy improves

721
00:47:42,480 --> 00:47:46,034
by a factor of n and not just
by a factor of square root n.

722
00:47:48,880 --> 00:47:50,950
So that tells us something.

723
00:47:50,950 --> 00:47:56,430
If we do something with the
photons, if we entangle them,

724
00:47:56,430 --> 00:48:02,290
there is a possibility
to vastly improve

725
00:48:02,290 --> 00:48:05,215
the standard limit
of measurements.

726
00:48:05,215 --> 00:48:07,090
And so people who are
really interested in it

727
00:48:07,090 --> 00:48:09,970
are people who push the
limits of precision, people

728
00:48:09,970 --> 00:48:12,800
who build atomic
clocks and want to get

729
00:48:12,800 --> 00:48:15,210
the last little bit of
accuracy which is possible.

730
00:48:15,210 --> 00:48:19,070
So they have already exhausted
all technical possibilities,

731
00:48:19,070 --> 00:48:24,570
and the next thing is now, well
maybe entanglement and sort

732
00:48:24,570 --> 00:48:30,450
of subtleties of quantum
physics can come to their help.

733
00:48:35,630 --> 00:48:39,130
So this shows you
what is possible.

734
00:48:39,130 --> 00:48:42,750
But now I want to tell
you how it can be done.

735
00:48:49,110 --> 00:48:53,380
So what I first want to do
is, before we can improve

736
00:48:53,380 --> 00:48:55,960
over the fundamental
shot noise limit,

737
00:48:55,960 --> 00:48:58,850
I have to show you how
the fundamental shot noise

738
00:48:58,850 --> 00:49:01,260
limit naturally emerges.

739
00:49:01,260 --> 00:49:06,850
So I want to introduce to you
beam splitter interferometers.

740
00:49:06,850 --> 00:49:09,900
Interferometers are there
to measure phase shifts.

741
00:49:09,900 --> 00:49:11,950
You split a beam,
you combine it,

742
00:49:11,950 --> 00:49:13,860
and if there's a phase
shift you notice if.

743
00:49:13,860 --> 00:49:15,740
You get interference fringes.

744
00:49:15,740 --> 00:49:20,860
And I want to show you that very
naturally to standard quantum

745
00:49:20,860 --> 00:49:23,020
limit, the shot noise emerges.

746
00:49:23,020 --> 00:49:26,360
But then we are ready to
look at our description

747
00:49:26,360 --> 00:49:29,710
of the interferometer and
say, where can we now change

748
00:49:29,710 --> 00:49:35,505
the rules to put in more
quantum-ness or entanglement

749
00:49:35,505 --> 00:49:37,005
and eventually get
higher precision.

750
00:49:40,020 --> 00:49:43,420
Since the standard quantum
limit is well know,

751
00:49:43,420 --> 00:49:48,230
I want to rather
quickly go over that.

752
00:49:48,230 --> 00:49:56,280
So the goal is that we want
to measure a phase shift.

753
00:49:56,280 --> 00:50:00,990
And my first simple
deviation of the phase shift

754
00:50:00,990 --> 00:50:07,820
is that in this picture of
the quasi probabilities,

755
00:50:07,820 --> 00:50:10,340
the coherent state is
a circle like that.

756
00:50:17,635 --> 00:50:20,100
The width of the
circle in natural units

757
00:50:20,100 --> 00:50:28,100
is 1, but the radius of that
is alpha of the coherent state,

758
00:50:28,100 --> 00:50:30,730
which is square root n.

759
00:50:30,730 --> 00:50:36,210
So now if you ask how well
with this uncertainty can

760
00:50:36,210 --> 00:50:39,910
I observe the phase of the
photon which circulates

761
00:50:39,910 --> 00:50:42,920
in this quasi
probability plane, you

762
00:50:42,920 --> 00:50:47,860
find that the phase is
1 over square root n.

763
00:50:47,860 --> 00:50:51,540
Or based on your homework
assignment number one,

764
00:50:51,540 --> 00:50:54,630
you can say we have some
Heisenberg uncertainty

765
00:50:54,630 --> 00:50:58,300
between photon
number and the phase.

766
00:50:58,300 --> 00:51:03,160
And in a coherent state,
the standard deviation

767
00:51:03,160 --> 00:51:05,180
in the photon number
is square root n,

768
00:51:05,180 --> 00:51:07,720
and you again get the
standard quantum limit.

769
00:51:18,570 --> 00:51:22,280
Let's now obtain the
standard quantum limit

770
00:51:22,280 --> 00:51:24,690
from a real measurement
device, because this

771
00:51:24,690 --> 00:51:27,570
is what we want to
generalize for entangles,

772
00:51:27,570 --> 00:51:30,810
[INAUDIBLE] state, and
all the special things.

773
00:51:30,810 --> 00:51:33,630
So an interferometer
is the following.

774
00:51:33,630 --> 00:51:35,350
It consists of two
beam splitters.

775
00:51:38,050 --> 00:51:41,430
After the second beam splitter
we have two detectors.

776
00:51:41,430 --> 00:51:44,407
And the quality we
will measure will

777
00:51:44,407 --> 00:51:46,240
be the difference of
the two photo currents.

778
00:51:49,010 --> 00:51:57,330
And if you don't
do any phase shift,

779
00:51:57,330 --> 00:51:59,040
we know the two beams
splitters are just

780
00:51:59,040 --> 00:52:00,690
the identity transformation.

781
00:52:00,690 --> 00:52:05,840
But now we put in a phase
shift, and the question

782
00:52:05,840 --> 00:52:08,770
is, what is the smallest
phase shift we can measure?

783
00:52:08,770 --> 00:52:10,665
What is the accuracy
measuring this phase?

784
00:52:13,230 --> 00:52:16,410
We have discussed at
length all the elements.

785
00:52:16,410 --> 00:52:19,920
So we have a beam
splitter, a phase shifter,

786
00:52:19,920 --> 00:52:23,070
and a second beam splitter.

787
00:52:23,070 --> 00:52:27,510
You know that the phase
shifter is a rotation in z,

788
00:52:27,510 --> 00:52:32,860
the beam splitter is a
rotation in-- was it X or Y?

789
00:52:32,860 --> 00:52:39,000
Looks like I think Y
because of the i's.

790
00:52:39,000 --> 00:52:42,610
Just a warning here,
in this section

791
00:52:42,610 --> 00:52:45,810
I use a beam splitter which uses
a different phase convention.

792
00:52:45,810 --> 00:52:51,540
But they are all equally
apart from some phases.

793
00:52:51,540 --> 00:52:55,490
So by multiplying
the three matrices,

794
00:52:55,490 --> 00:52:59,780
we get the transformatrix for
the [INAUDIBLE] interferometer.

795
00:52:59,780 --> 00:53:05,540
So the output CD
is this matrix here

796
00:53:05,540 --> 00:53:11,700
which is a simple rotation
times the input state.

797
00:53:11,700 --> 00:53:14,960
And our measurement
is the difference

798
00:53:14,960 --> 00:53:18,080
in photon numbers in
the two output modes.

799
00:53:18,080 --> 00:53:28,510
So it's D dega D minus C dega C.

800
00:53:28,510 --> 00:53:35,870
So what we can do now is we can
obtain C and D from the input

801
00:53:35,870 --> 00:53:39,340
modes A and B. So
therefore we can now

802
00:53:39,340 --> 00:53:46,380
express our signal in
terms of the input modes.

803
00:53:46,380 --> 00:53:49,700
When we know what we put
into the interferometer,

804
00:53:49,700 --> 00:53:53,510
because we know all its elements
we know what we can get out.

805
00:53:53,510 --> 00:53:54,510
And this is done here.

806
00:53:58,440 --> 00:54:02,780
And the phase which appeared in
the rotation matrix, the phase

807
00:54:02,780 --> 00:54:05,890
shift in the
interferometer appears now

808
00:54:05,890 --> 00:54:10,610
as a cosine phi and
sine phi contribution.

809
00:54:10,610 --> 00:54:13,710
And we have sort off-
the cosine and sine phi

810
00:54:13,710 --> 00:54:17,580
have two operators
as a pre-factor.

811
00:54:17,580 --> 00:54:21,640
In one case, it's A dega A minus
B dega B. In the other case

812
00:54:21,640 --> 00:54:25,950
it's a cross-term, A
dega B plus B dega A,

813
00:54:25,950 --> 00:54:29,830
Now we will be more specific
what is signal and noise.

814
00:54:29,830 --> 00:54:33,200
I just want to tell you in
the standard way of operating

815
00:54:33,200 --> 00:54:38,070
the interferometer you have
only one beam in the mode A.

816
00:54:38,070 --> 00:54:40,870
You take a beam, you split
it, you recombine it.

817
00:54:40,870 --> 00:54:43,830
B is nothing or the vacuum.

818
00:54:43,830 --> 00:54:45,670
It may introduce some noise.

819
00:54:45,670 --> 00:54:49,980
But if you have a lot of
intensity in the beam A,

820
00:54:49,980 --> 00:54:53,900
it is this term which
dominates, and you find fringes

821
00:54:53,900 --> 00:54:57,050
as a function of the
phase shift cosine phi.

822
00:54:57,050 --> 00:55:01,350
Whereas what comes here is
sort of more the vacuum mode.

823
00:55:01,350 --> 00:55:02,960
It gives rise to noise terms.

824
00:55:08,250 --> 00:55:16,900
So if you ask, what is
the expectation value

825
00:55:16,900 --> 00:55:29,090
for x, this x operator,
it varies co-sinusoidally.

826
00:55:29,090 --> 00:55:32,400
And there is a special
point for a phase shift

827
00:55:32,400 --> 00:55:35,695
of 90 degrees when we have the
steepest slope and the highest

828
00:55:35,695 --> 00:55:36,195
sensitivity.

829
00:55:42,180 --> 00:55:46,070
So these are sort of all
just setting the stage.

830
00:55:46,070 --> 00:55:50,140
Which we are interested in
is, if we now have an input

831
00:55:50,140 --> 00:55:52,370
state of light--
and as you know,

832
00:55:52,370 --> 00:55:53,634
there's all of this noise.

833
00:55:53,634 --> 00:55:56,259
There's the fundamental noise of
the vacuum, or coherent state,

834
00:55:56,259 --> 00:55:59,140
of Heisenberg's Uncertainty.

835
00:55:59,140 --> 00:56:02,180
And this means when we
do repeated measurements

836
00:56:02,180 --> 00:56:04,390
we will have a
variance in the phase.

837
00:56:04,390 --> 00:56:07,490
And we want to know, what
is the fundamental limit

838
00:56:07,490 --> 00:56:10,120
on the standard deviation in
the measurement of the phase.

839
00:56:12,950 --> 00:56:16,760
Well, the standard
deviation in the phase

840
00:56:16,760 --> 00:56:19,450
is nothing else than
the standard deviation

841
00:56:19,450 --> 00:56:24,710
of what we measure in
divided by how sensitive it

842
00:56:24,710 --> 00:56:25,380
is to the phase.

843
00:56:29,530 --> 00:56:35,580
So by taking this
expression for M,

844
00:56:35,580 --> 00:56:40,400
M is an operator X plus cosine
phi plus something times

845
00:56:40,400 --> 00:56:41,060
sine phi.

846
00:56:45,680 --> 00:56:50,370
We can now evaluate
this expression.

847
00:56:50,370 --> 00:56:53,900
This will actually appear
several times today,

848
00:56:53,900 --> 00:56:59,070
and it's also a key question to
one of your homework problems.

849
00:56:59,070 --> 00:57:05,300
So this is sort of what
defines the accuracy

850
00:57:05,300 --> 00:57:06,390
of the interferometer.

851
00:57:06,390 --> 00:57:09,900
And what we have here
are expectation values

852
00:57:09,900 --> 00:57:12,080
of operators.

853
00:57:12,080 --> 00:57:15,480
And now we can--
and that's what we

854
00:57:15,480 --> 00:57:17,840
will do for the rest
of this lecture.

855
00:57:17,840 --> 00:57:21,820
We will look at this expression
for different inputs states.

856
00:57:21,820 --> 00:57:25,350
Coherent state, single photons,
[INAUDIBLE] light, entangled

857
00:57:25,350 --> 00:57:25,950
state.

858
00:57:25,950 --> 00:57:27,366
So that's what
you're going to do.

859
00:57:29,640 --> 00:57:33,300
So if you take the
derivative of dn d phi,

860
00:57:33,300 --> 00:57:35,670
the cosine becomes a sine.

861
00:57:35,670 --> 00:57:39,460
A minus sign, the
sine becomes a cosine.

862
00:57:39,460 --> 00:57:43,420
And if you specialize
to the situation, which

863
00:57:43,420 --> 00:57:47,390
is where the slope
is very steep,

864
00:57:47,390 --> 00:57:51,560
a 90 degree phase shift
around this point, our phase

865
00:57:51,560 --> 00:58:03,650
sensitivity is given by
the expectation value

866
00:58:03,650 --> 00:58:08,910
of the variance of the operator
y divided by the operator x.

867
00:58:13,600 --> 00:58:18,500
So now we are ready
to plug things in.

868
00:58:18,500 --> 00:58:23,500
The first thing is of
course the coherent state.

869
00:58:23,500 --> 00:58:28,160
For the coherent state, our
input to the beam splitter,

870
00:58:28,160 --> 00:58:30,070
one is a coherent
state, the other one

871
00:58:30,070 --> 00:58:37,680
is a vacuum in mode A and B.

872
00:58:37,680 --> 00:58:47,780
We had expressions-- let me
just scroll back-- for x and y,

873
00:58:47,780 --> 00:58:51,400
expressed by the operators
a, b, a dega b dega.

874
00:58:58,860 --> 00:59:04,690
So therefore we can
calculate that now.

875
00:59:04,690 --> 00:59:10,990
And x is nothing
else than the number

876
00:59:10,990 --> 00:59:14,030
of photons in the
coherent state.

877
00:59:14,030 --> 00:59:16,360
And this is our signal.

878
00:59:16,360 --> 00:59:22,210
And y is 0.

879
00:59:22,210 --> 00:59:24,010
This was the noise term.

880
00:59:24,010 --> 00:59:27,300
And the variance
in y-- we're just

881
00:59:27,300 --> 00:59:30,640
doing commutators--
is given by n.

882
00:59:30,640 --> 00:59:45,830
So therefore, if we
calculate the square root

883
00:59:45,830 --> 00:59:52,650
of y square over x for
the coherent state input,

884
00:59:52,650 --> 00:59:55,972
we take the square root of y
squared, which is square root

885
00:59:55,972 --> 01:00:02,050
n, we divide by n, we obtain
the standard shot noise limit.

886
01:00:02,050 --> 01:00:03,425
Sure, what else
should we expect?

887
01:00:07,920 --> 01:00:11,370
So now we know how
to use the formalism.

888
01:00:16,900 --> 01:00:19,020
We can now apply
it too-- we can go

889
01:00:19,020 --> 01:00:24,150
from the most classical state
of light, the coherent state,

890
01:00:24,150 --> 01:00:30,150
to what I regard at least for
a single mode the most quantum

891
01:00:30,150 --> 01:00:32,810
state, namely a single photon.

892
01:00:32,810 --> 01:00:35,490
Remember when we talked
about the G2 function,

893
01:00:35,490 --> 01:00:39,540
G2 function can only
be smaller than 1

894
01:00:39,540 --> 01:00:41,520
for non-classical states.

895
01:00:41,520 --> 01:00:43,600
And for the single
photon it's 0.

896
01:00:43,600 --> 01:00:46,140
That's the biggest violation
of the classical equation

897
01:00:46,140 --> 01:00:48,580
that G2 has to be
larger or equal than 1.

898
01:00:48,580 --> 01:00:51,200
So now we're really dealing
with a quantum system.

899
01:00:51,200 --> 01:00:55,050
And the question is, what will
we get for the single photon?

900
01:00:58,435 --> 01:01:07,480
So for the single
photon we want to use

901
01:01:07,480 --> 01:01:10,680
the dual-rail representation.

902
01:01:10,680 --> 01:01:15,630
We want to use the powerful
formalism we have used.

903
01:01:15,630 --> 01:01:17,550
After the beam
splitter, the photon

904
01:01:17,550 --> 01:01:20,860
can be in one mode
or the other mode.

905
01:01:20,860 --> 01:01:24,306
We call this in the
dual-rail representation

906
01:01:24,306 --> 01:01:27,600
the logical 0 or the logical 1.

907
01:01:32,160 --> 01:01:40,200
So in this
representation, when we

908
01:01:40,200 --> 01:01:50,810
start with a photon in one input
mode, this is the logical 0,

909
01:01:50,810 --> 01:01:57,270
the beam splitter is
directing the photons

910
01:01:57,270 --> 01:02:01,191
into one mode or the other
mode, but this is a single qubit

911
01:02:01,191 --> 01:02:01,690
rotation.

912
01:02:05,720 --> 01:02:08,880
The phase shifter is another
single qubit operation.

913
01:02:08,880 --> 01:02:12,390
The second beam splitter
is another operation.

914
01:02:12,390 --> 01:02:19,720
So by just using
the rotation, we'll

915
01:02:19,720 --> 01:02:22,810
find out what is
the output state.

916
01:02:22,810 --> 01:02:26,870
So we start with the logical
0 with the photon in one mode.

917
01:02:26,870 --> 01:02:30,750
The beam splitter creates
a superposition state.

918
01:02:30,750 --> 01:02:32,560
Remember, we have
the interferometer,

919
01:02:32,560 --> 01:02:35,080
and in the lower arm
we put in a phase.

920
01:02:35,080 --> 01:02:40,180
So therefore, the lower arm gets
multiplied with a phase shift.

921
01:02:40,180 --> 01:02:44,080
And then it goes again through
a beam splitter, which is just

922
01:02:44,080 --> 01:02:49,530
another rotation, and with that
we obtain the output state.

923
01:02:49,530 --> 01:03:04,110
And so the output state
is now a superposition

924
01:03:04,110 --> 01:03:07,330
of the logical 0
and the logical 1,

925
01:03:07,330 --> 01:03:08,830
and we've picked
up a phase shift.

926
01:03:12,310 --> 01:03:17,370
So no we ask-- we have the
beam splitter, phase shift,

927
01:03:17,370 --> 01:03:19,860
recombiner, and
now we are asking,

928
01:03:19,860 --> 01:03:23,260
what is the probability to
detect the photon in one

929
01:03:23,260 --> 01:03:24,320
of the two output modes.

930
01:03:24,320 --> 01:03:27,400
We just put a counter there.

931
01:03:27,400 --> 01:03:35,550
And this probability is nothing
else than the probability

932
01:03:35,550 --> 01:03:39,520
to have a logical 1,
because a logical 1

933
01:03:39,520 --> 01:03:41,850
in the dual-rail
representation means

934
01:03:41,850 --> 01:03:43,270
the photon is in one mode.

935
01:03:43,270 --> 01:03:44,980
The logical 0 would
be the other mode.

936
01:03:49,150 --> 01:03:53,584
I felt if I would spend three or
four times as much time on it,

937
01:03:53,584 --> 01:03:55,250
it wouldn't increase
your understanding.

938
01:03:55,250 --> 01:03:57,340
You may have to sit down
and look through it,

939
01:03:57,340 --> 01:04:00,720
but it's just really
putting matrices together,

940
01:04:00,720 --> 01:04:02,800
[? single ?]
photons, mapping it.

941
01:04:02,800 --> 01:04:05,570
Every step is
trivial and is what

942
01:04:05,570 --> 01:04:09,730
we have done in a
different context before.

943
01:04:09,730 --> 01:04:12,720
So the result is
that the probability

944
01:04:12,720 --> 01:04:16,300
of finding the [? single ?]
photon in our detector

945
01:04:16,300 --> 01:04:19,800
has a cosine phi factor.

946
01:04:19,800 --> 01:04:21,160
So that's how we measure phase.

947
01:04:21,160 --> 01:04:25,790
Our counter, the probability
that photons arrive.

948
01:04:25,790 --> 01:04:29,390
If phi is 0, the
probability is one.

949
01:04:29,390 --> 01:04:36,980
If phi is 180 degrees, all the
photons go to the other mode.

950
01:04:36,980 --> 01:04:39,450
I mean, that's what you'd
expect an interferometer to do.

951
01:04:43,460 --> 01:04:46,140
But now comes the
interesting question.

952
01:04:46,140 --> 01:04:47,820
How precise can be
measured the phase?

953
01:04:52,230 --> 01:04:57,100
Remember, in the coherent
state, the coherent state

954
01:04:57,100 --> 01:05:00,300
had fluctuations in the photon
number of square root n,

955
01:05:00,300 --> 01:05:05,310
and this gave rise to the
standard quantum limit.

956
01:05:05,310 --> 01:05:08,480
A single photon is
a single photon.

957
01:05:08,480 --> 01:05:09,390
There is no noise.

958
01:05:09,390 --> 01:05:11,010
We always measure
a single photon

959
01:05:11,010 --> 01:05:14,020
if you would simply detect the
number of photons in the input

960
01:05:14,020 --> 01:05:15,340
state.

961
01:05:15,340 --> 01:05:18,070
But we are not doing that.

962
01:05:18,070 --> 01:05:20,957
We have sent the photon through
an interferometer in order

963
01:05:20,957 --> 01:05:21,920
to measure the phase.

964
01:05:21,920 --> 01:05:26,100
And now we have-- and
this is a result of this,

965
01:05:26,100 --> 01:05:28,630
I would say, trivial
calculation that this is now

966
01:05:28,630 --> 01:05:31,160
the probability to
observe a photon.

967
01:05:31,160 --> 01:05:34,190
Well, there is not a whole lot
we can learn from one photon,

968
01:05:34,190 --> 01:05:37,040
so we run the
experiment n times.

969
01:05:37,040 --> 01:05:39,740
And what we get is a
binomial distribution,

970
01:05:39,740 --> 01:05:43,830
just a coin toss
with probability P.

971
01:05:43,830 --> 01:05:46,310
What is the variance in P?

972
01:05:46,310 --> 01:05:48,170
It's not a Poissonian
distribution.

973
01:05:48,170 --> 01:05:55,060
It is a binomial distribution.

974
01:05:55,060 --> 01:05:58,800
Of course, if the probability is
1 you detect all your photons.

975
01:05:58,800 --> 01:06:00,050
There is no uncertainty.

976
01:06:00,050 --> 01:06:04,390
If the probability is 0 you
detect 0 with 0 uncertainty.

977
01:06:04,390 --> 01:06:06,970
So therefore, the
expression for the variance

978
01:06:06,970 --> 01:06:12,360
of the binomial distribution
is p times 1 minus p.

979
01:06:12,360 --> 01:06:15,700
If not, I admit I had
to refresh my memory

980
01:06:15,700 --> 01:06:21,140
about the binomial
distribution yesterday evening.

981
01:06:21,140 --> 01:06:24,860
So therefore, when we repeat
the experiment many times

982
01:06:24,860 --> 01:06:29,210
and we have sent into
our interferometer

983
01:06:29,210 --> 01:06:32,970
m photons and we
measure n clicks.

984
01:06:32,970 --> 01:06:36,350
And n over m is our measurement
for the probability.

985
01:06:39,140 --> 01:06:43,590
This measurement of the
probability has noise.

986
01:06:43,590 --> 01:06:47,702
And the noise is a variance
of the standard deviation

987
01:06:47,702 --> 01:06:48,910
of the binomial distribution.

988
01:06:51,460 --> 01:06:56,490
This expression for P
put into the variance

989
01:06:56,490 --> 01:06:58,990
of the binomial distribution,
and a little bit

990
01:06:58,990 --> 01:07:01,854
of triganomic
manipulation gives us

991
01:07:01,854 --> 01:07:03,270
the result on the
right-hand side.

992
01:07:06,320 --> 01:07:12,960
And we want to know what is
the uncertainty in the phase.

993
01:07:12,960 --> 01:07:14,880
I mean, this is what
we want to measure.

994
01:07:14,880 --> 01:07:17,120
Well, the uncertainty
in the phase

995
01:07:17,120 --> 01:07:26,770
is the uncertainty in our
measurement, which is delta P.

996
01:07:26,770 --> 01:07:30,167
And then we have to divide by
how sensitive the probability

997
01:07:30,167 --> 01:07:32,870
is with respect to the phase.

998
01:07:32,870 --> 01:07:36,580
So we take this expression
of the probability

999
01:07:36,580 --> 01:07:40,000
as a function of the
phase, take a derivative,

1000
01:07:40,000 --> 01:07:45,810
and then by doing the ratio
of this over that we find 1

1001
01:07:45,810 --> 01:07:47,600
over square root n.

1002
01:07:47,600 --> 01:07:50,270
So it doesn't make
any difference

1003
01:07:50,270 --> 01:07:53,600
if you put the n photons
into a coherent beam

1004
01:07:53,600 --> 01:07:56,530
and run our
interferometer or if you

1005
01:07:56,530 --> 01:07:59,620
go through the great
pain of preparing

1006
01:07:59,620 --> 01:08:02,910
each photon in a
non-classical flux state

1007
01:08:02,910 --> 01:08:05,330
and just operating
the interferometer

1008
01:08:05,330 --> 01:08:06,710
with single photons.

1009
01:08:06,710 --> 01:08:10,590
In both cases do we obtain
the standard quantum limit

1010
01:08:10,590 --> 01:08:12,290
1 over square root n.

1011
01:08:44,399 --> 01:08:48,319
Now we are really
ready to see how can we

1012
01:08:48,319 --> 01:08:50,689
improve on the shot noise?

1013
01:08:50,689 --> 01:08:53,569
It seems that unless we
do something special,

1014
01:08:53,569 --> 01:08:56,500
we will always
get square root n.

1015
01:08:56,500 --> 01:09:00,490
I've already shown you
at the beginning that's

1016
01:09:00,490 --> 01:09:02,850
the shot noise limit
is not fundamental.

1017
01:09:02,850 --> 01:09:05,340
Just take this
thought experiment

1018
01:09:05,340 --> 01:09:08,600
that you take the n photons
through a highly nonlinear

1019
01:09:08,600 --> 01:09:13,100
process, you create one
photon of frequency n omega,

1020
01:09:13,100 --> 01:09:16,960
and then you have more
precision of this measurement.

1021
01:09:16,960 --> 01:09:20,990
So there must be a way to have
a precision which scales 1

1022
01:09:20,990 --> 01:09:24,250
over n and not 1
over square root n.

1023
01:09:24,250 --> 01:09:30,100
The mathematical augment
related to our treatment

1024
01:09:30,100 --> 01:09:39,660
of the interferometer
is the following,

1025
01:09:39,660 --> 01:09:47,140
our signal was A dega
A minus B dega B. Well,

1026
01:09:47,140 --> 01:09:49,340
if one of the input
mode dominates,

1027
01:09:49,340 --> 01:09:52,160
A dega A is the [? photon ?]
[? number. ?] So this sort

1028
01:09:52,160 --> 01:09:55,100
of looks like the
[? photon ?] [? number ?].

1029
01:09:55,100 --> 01:09:59,150
But if we now find a
scheme-- and I will show you

1030
01:09:59,150 --> 01:10:02,770
that this is possible-- that
this cos term A dega B plus B

1031
01:10:02,770 --> 01:10:07,680
dega A, which is some form of
quantum noise, as we will see,

1032
01:10:07,680 --> 01:10:19,170
is 0, well what do we get now?

1033
01:10:19,170 --> 01:10:22,020
So let's hope that maybe
by some squeezing--

1034
01:10:22,020 --> 01:10:25,740
I will show you several versions
which show you how quantumness

1035
01:10:25,740 --> 01:10:32,790
can give us more
than short noise.

1036
01:10:32,790 --> 01:10:37,670
And one example will be
that by squeezing light.

1037
01:10:37,670 --> 01:10:39,680
We've learned already
that squeezed light

1038
01:10:39,680 --> 01:10:41,520
can suppress the noise.

1039
01:10:41,520 --> 01:10:46,580
If you squeeze light, something
which comes from the modes

1040
01:10:46,580 --> 01:10:49,020
where we apply squeezed
vacuum has been reduced.

1041
01:10:49,020 --> 01:10:51,500
I mean, it's what we
discussed already.

1042
01:10:51,500 --> 01:10:54,550
So the best we can do is
that the noise term is 0.

1043
01:10:59,330 --> 01:11:00,480
So what do we have now?

1044
01:11:00,480 --> 01:11:04,909
If we have a signal which is
finite and the noise is 0,

1045
01:11:04,909 --> 01:11:06,450
what is our precision
of measurement?

1046
01:11:09,772 --> 01:11:13,850
Well, it first looks like
the signal to noise ratio

1047
01:11:13,850 --> 01:11:15,820
is infinite.

1048
01:11:15,820 --> 01:11:28,930
But it's not quite that,
because if our signal is x,

1049
01:11:28,930 --> 01:11:31,410
it's a photon number.

1050
01:11:31,410 --> 01:11:36,750
And the signal x was sensitive
to the phase by cosine phi.

1051
01:11:40,130 --> 01:11:46,050
The sensitivity of our
measurement of the quantity

1052
01:11:46,050 --> 01:11:56,160
m with phase is
n times sine phi.

1053
01:11:56,160 --> 01:12:01,040
And this is smaller or equal
than n, where the equal sign is

1054
01:12:01,040 --> 01:12:06,540
obtained for 90 degrees.

1055
01:12:12,660 --> 01:12:15,940
So therefore, we have a
[? photon ?] number n.

1056
01:12:15,940 --> 01:12:19,540
We may absolutely know what the
initial [? photon ?] number is.

1057
01:12:19,540 --> 01:12:24,910
But now we want to measure
whether the phase is

1058
01:12:24,910 --> 01:12:27,200
different from pi over 2.

1059
01:12:27,200 --> 01:12:34,070
And the smallest change in our
signal which we can resolve

1060
01:12:34,070 --> 01:12:38,380
is that we get one photon less.

1061
01:12:45,110 --> 01:12:47,130
So that's the smallest
resolvable change

1062
01:12:47,130 --> 01:12:50,320
due to the [? photon ?]
nature of our detection.

1063
01:12:50,320 --> 01:12:54,980
And that implies that
the smallest phase

1064
01:12:54,980 --> 01:12:57,330
shift we can
resolve is 1 over n.

1065
01:13:00,100 --> 01:13:02,905
So this is more a
thought experiment,

1066
01:13:02,905 --> 01:13:07,210
now looking at the math and
seeing what is the best signal?

1067
01:13:07,210 --> 01:13:09,970
A dega A can never be larger
than [? photon ?] numbers,

1068
01:13:09,970 --> 01:13:11,240
so that's a resource.

1069
01:13:11,240 --> 01:13:13,860
We put in energy in
the terms of photons,

1070
01:13:13,860 --> 01:13:17,060
and then the best we can do is
that we don't have any noise.

1071
01:13:17,060 --> 01:13:19,170
And then we are simply
limited by the fact

1072
01:13:19,170 --> 01:13:24,630
that when we deviate from
the maximum output signal

1073
01:13:24,630 --> 01:13:27,380
because there is a phase
shift in the interferometer,

1074
01:13:27,380 --> 01:13:35,070
our sensitivity comes in grains,
is grainy by the [? photon ?]

1075
01:13:35,070 --> 01:13:35,570
number.

1076
01:13:39,940 --> 01:13:44,980
So I'm not telling you how
we achieve to get y equals 0.

1077
01:13:44,980 --> 01:13:50,420
The question is only
that at least the math

1078
01:13:50,420 --> 01:13:53,100
seems to make it possible.

1079
01:13:53,100 --> 01:14:01,215
So now we ask how to achieve
sub-shot noise precision.

1080
01:14:12,930 --> 01:14:16,615
So here we know this is at
least mathematically possible.

1081
01:14:20,210 --> 01:14:24,810
So we want to improve
this interferometer,

1082
01:14:24,810 --> 01:14:28,260
and the question is we
have to change something.

1083
01:14:28,260 --> 01:14:31,650
This interferometer, [INAUDIBLE]
we've seen [INAUDIBLE]

1084
01:14:31,650 --> 01:14:34,070
state is not giving
us any better result,

1085
01:14:34,070 --> 01:14:35,920
so we have three options now.

1086
01:14:35,920 --> 01:14:40,410
We can do something fancy in
the input state, use entangled

1087
01:14:40,410 --> 01:14:42,380
state.

1088
01:14:42,380 --> 01:14:46,200
We can use some very
fancy beam splitters,

1089
01:14:46,200 --> 01:14:48,750
or we can do something
special to the way

1090
01:14:48,750 --> 01:14:51,010
how we read out
the interferometer.

1091
01:14:51,010 --> 01:14:54,050
And all those three
are possibilities.

1092
01:14:54,050 --> 01:15:00,040
We can put in some
extra quantumness

1093
01:15:00,040 --> 01:15:01,390
into each of the three steps.

1094
01:15:07,450 --> 01:15:08,241
OK

1095
01:15:08,241 --> 01:15:09,157
AUDIENCE: [INAUDIBLE]?

1096
01:15:15,842 --> 01:15:17,300
WOLFGANG KETTERLE:
OK, the question

1097
01:15:17,300 --> 01:15:21,668
is can we do something
with the phase shift?

1098
01:15:21,668 --> 01:15:23,075
AUDIENCE: [INAUDIBLE].

1099
01:15:23,075 --> 01:15:25,200
WOLFGANG KETTERLE: I think
if we would do something

1100
01:15:25,200 --> 01:15:27,030
with the phase
shift, we would do--

1101
01:15:27,030 --> 01:15:29,510
you know, we would change
apples with oranges,

1102
01:15:29,510 --> 01:15:32,920
because we want to compare
different interferometers

1103
01:15:32,920 --> 01:15:34,850
by measuring the same thing.

1104
01:15:34,850 --> 01:15:36,910
And maybe as an
experimentalist say,

1105
01:15:36,910 --> 01:15:38,810
what I want to be
able to measure

1106
01:15:38,810 --> 01:15:42,110
I take a very, very thing glass
plate which just makes a very

1107
01:15:42,110 --> 01:15:44,520
small phase shift,
and I'm now comparing

1108
01:15:44,520 --> 01:15:46,240
different interferometers.

1109
01:15:46,240 --> 01:15:49,480
And when I put the glass
plate in and pull it out,

1110
01:15:49,480 --> 01:15:51,840
I want the person who reads
out the interferometer

1111
01:15:51,840 --> 01:15:53,104
to see a change.

1112
01:15:53,104 --> 01:15:55,270
So that's how I want to
compare the interferometers.

1113
01:15:59,310 --> 01:16:02,020
Actually, the question
you are raising

1114
01:16:02,020 --> 01:16:05,480
was in another way
also discussed today

1115
01:16:05,480 --> 01:16:09,820
at lunch with three of
my wonderful colleagues.

1116
01:16:09,820 --> 01:16:15,610
Namely what we discussed was
some recent papers, one of them

1117
01:16:15,610 --> 01:16:17,890
published in
Nature, which claims

1118
01:16:17,890 --> 01:16:20,570
precision better
than Heisenberg.

1119
01:16:20,570 --> 01:16:23,020
I mean, what I'm sort of
indicating to you is Heisenberg

1120
01:16:23,020 --> 01:16:25,140
should be the best
which can be achieved.

1121
01:16:25,140 --> 01:16:27,710
How can we do even
better than Heisenberg?

1122
01:16:27,710 --> 01:16:30,280
Well, and this was
actually related, Nancy,

1123
01:16:30,280 --> 01:16:31,310
to your question.

1124
01:16:31,310 --> 01:16:32,810
If you want to
measure magnetization

1125
01:16:32,810 --> 01:16:35,880
and you use some
nonlinear physics

1126
01:16:35,880 --> 01:16:44,160
where the photons, where
the magnetization involved

1127
01:16:44,160 --> 01:16:50,370
affects the photon field by
some higher power of what

1128
01:16:50,370 --> 01:16:54,410
you measure, you really
change what you measure.

1129
01:16:54,410 --> 01:16:56,060
Maybe I'm not
expressing it clearly.

1130
01:16:56,060 --> 01:16:58,270
If you measure magnetization
and you have a certain quantum

1131
01:16:58,270 --> 01:16:59,880
limit in measuring
the magnetization,

1132
01:16:59,880 --> 01:17:01,040
that's one thing.

1133
01:17:01,040 --> 01:17:03,450
But if you now bring
another material close

1134
01:17:03,450 --> 01:17:06,480
to your magnetization and this
material goes through a phase

1135
01:17:06,480 --> 01:17:09,790
shift, you sort of amplify
by a physical process

1136
01:17:09,790 --> 01:17:12,530
by a nonlinear Hamiltonian
what you want to measure.

1137
01:17:12,530 --> 01:17:15,100
And then of course
you can-- depending

1138
01:17:15,100 --> 01:17:16,940
on what kind of
nonlinear process

1139
01:17:16,940 --> 01:17:19,640
you're using-- you
can get a signal which

1140
01:17:19,640 --> 01:17:23,580
scales tremendously with
the number of photons you

1141
01:17:23,580 --> 01:17:25,190
put into your system.

1142
01:17:25,190 --> 01:17:27,630
So in other words, there
are loopholes like this,

1143
01:17:27,630 --> 01:17:31,260
and some of them led
to very fleshy papers.

1144
01:17:31,260 --> 01:17:33,630
And our lunch discussion
was that some of it

1145
01:17:33,630 --> 01:17:35,430
is really completely trivial.

1146
01:17:35,430 --> 01:17:42,580
And some papers
who claim that they

1147
01:17:42,580 --> 01:17:44,830
have seen a scaling
of the precision

1148
01:17:44,830 --> 01:17:47,680
with photon number which
is better than Heisenberg,

1149
01:17:47,680 --> 01:17:50,620
not 1 over n, maybe
1 over n square,

1150
01:17:50,620 --> 01:17:53,870
that some of those papers
were purely classical

1151
01:17:53,870 --> 01:17:56,750
and the only quantum
character of this paper

1152
01:17:56,750 --> 01:17:59,640
was the name of
Heisenberg in the title.

1153
01:17:59,640 --> 01:18:04,570
So it's not related to
any uncertainty relation,

1154
01:18:04,570 --> 01:18:07,200
but that's not what
we want to discuss.

1155
01:18:07,200 --> 01:18:10,280
Let me just spend
quickly-- let me see, yup.

1156
01:18:21,850 --> 01:18:25,800
We should the week with
something interesting and not

1157
01:18:25,800 --> 01:18:27,090
just shot noise.

1158
01:18:27,090 --> 01:18:30,820
So what I want to
use now is I want

1159
01:18:30,820 --> 01:18:35,580
to replace the beam
splitter by something

1160
01:18:35,580 --> 01:18:37,540
which involves Bell states.

1161
01:18:37,540 --> 01:18:40,450
So as the beam splitter
we do a massive creation

1162
01:18:40,450 --> 01:18:41,350
of bell states.

1163
01:18:41,350 --> 01:18:43,310
It's our entangler.

1164
01:18:43,310 --> 01:18:47,205
And our second beam
splitter is a Bell analyzer.

1165
01:18:47,205 --> 01:18:50,920
It is a disentangler.

1166
01:18:50,920 --> 01:18:55,730
So what I mean is the
following, we will actually

1167
01:18:55,730 --> 01:19:00,150
just put one photon
into this input beam,

1168
01:19:00,150 --> 01:19:02,990
and we will only
read out one channel.

1169
01:19:02,990 --> 01:19:06,760
So all these here are
only auxiliary modes.

1170
01:19:06,760 --> 01:19:10,360
This is how I make a special
quantum beam splitter.

1171
01:19:10,360 --> 01:19:12,710
And what we need
for this description

1172
01:19:12,710 --> 01:19:14,835
is essentially two gates.

1173
01:19:14,835 --> 01:19:18,080
We need a single qubit.

1174
01:19:18,080 --> 01:19:21,200
The single qubit
is the [INAUDIBLE]

1175
01:19:21,200 --> 01:19:23,730
which we have already discussed.

1176
01:19:23,730 --> 01:19:28,170
And in the dual-rail
representation

1177
01:19:28,170 --> 01:19:33,450
where its photon can be
in one of the two modes,

1178
01:19:33,450 --> 01:19:35,860
the [INAUDIBLE] is
simply connecting

1179
01:19:35,860 --> 01:19:38,300
the two modes in this way.

1180
01:19:38,300 --> 01:19:42,520
And we discussed
that the [INAUDIBLE]

1181
01:19:42,520 --> 01:19:46,430
can be described by a beam
splitter with a phase shift.

1182
01:19:46,430 --> 01:19:48,710
So in other words,
we need one element

1183
01:19:48,710 --> 01:19:52,490
which is a beam splitter
with a phase shift.

1184
01:19:52,490 --> 01:19:55,960
But this is only
acting on one qubits.

1185
01:19:55,960 --> 01:19:59,090
And now we want
to connect qubits

1186
01:19:59,090 --> 01:20:02,690
with a controlled NOT gate.

1187
01:20:02,690 --> 01:20:04,460
To remind you,
controlled NOT gate

1188
01:20:04,460 --> 01:20:08,050
is something where the
photons stay where they are.

1189
01:20:08,050 --> 01:20:14,910
But if the control bit is
1, it flips the target bit.

1190
01:20:14,910 --> 01:20:18,610
If the control bit is 0, nothing
happens to the target bit.

1191
01:20:18,610 --> 01:20:21,350
And we discussed
already that we can

1192
01:20:21,350 --> 01:20:23,860
realize one qubit with
one interferometer.

1193
01:20:23,860 --> 01:20:25,100
These are the dual-rails.

1194
01:20:25,100 --> 01:20:29,070
We always have one photon in two
states, this mode or that mode.

1195
01:20:29,070 --> 01:20:33,930
And know the other
qubit, one of the rails

1196
01:20:33,930 --> 01:20:36,330
can go through the
nonlinear Kerr medium.

1197
01:20:36,330 --> 01:20:40,600
If the bit is in c
through the phase

1198
01:20:40,600 --> 01:20:43,295
shift in the
nonlinear Kerr medium,

1199
01:20:43,295 --> 01:20:46,110
it flips the
[INAUDIBLE] down there,

1200
01:20:46,110 --> 01:20:47,490
and this the controlled knot.

1201
01:20:56,710 --> 01:20:59,800
So now we want to
use two qubits.

1202
01:20:59,800 --> 01:21:04,590
So I'm talking about,
in our fancy entangler

1203
01:21:04,590 --> 01:21:08,920
I'm just talking about
the first two rails here.

1204
01:21:08,920 --> 01:21:13,440
So we have those two rails.

1205
01:21:16,020 --> 01:21:23,630
The [INAUDIBLE] puts
us into a superposition

1206
01:21:23,630 --> 01:21:27,380
of the logical CO and 1.

1207
01:21:27,380 --> 01:21:39,690
And now after the
c-not gate, so we

1208
01:21:39,690 --> 01:21:44,440
are in a superposition
of-- just give me

1209
01:21:44,440 --> 01:21:54,610
a second-- we start here with 1.

1210
01:21:54,610 --> 01:21:56,445
Sorry, we start here with 1.

1211
01:21:59,065 --> 01:21:59,944
AUDIENCE: Excuse me.

1212
01:21:59,944 --> 01:22:01,156
WOLFGANG KETTERLE: Yeah?

1213
01:22:01,156 --> 01:22:02,010
I'm wrapping up.

1214
01:22:02,010 --> 01:22:03,798
This is my last.

1215
01:22:03,798 --> 01:22:06,702
AUDIENCE: One [INAUDIBLE]
minus 1 [INAUDIBLE].

1216
01:22:13,010 --> 01:22:15,870
WOLFGANG KETTERLE: The 0
[INAUDIBLE] in 0 plus 1.

1217
01:22:15,870 --> 01:22:20,910
And let me just finish that.

1218
01:22:25,080 --> 01:22:27,870
This is the target, year.

1219
01:22:27,870 --> 01:22:30,020
Sorry, it's correct.

1220
01:22:30,020 --> 01:22:31,790
We start with 0,
we start with 0.

1221
01:22:36,050 --> 01:22:39,920
So what we have here is
the product state of 0

1222
01:22:39,920 --> 01:22:44,020
plus 1 upstairs
and 0 downstairs.

1223
01:22:44,020 --> 01:22:47,640
But if we have a 1,
we flip the 0 to 1.

1224
01:22:47,640 --> 01:22:53,030
So therefore, what we have now
is the state 00 plus 1 1 over

1225
01:22:53,030 --> 01:22:55,000
square root 2.

1226
01:22:55,000 --> 01:22:58,320
And then we apply
our phase shifter.

1227
01:22:58,320 --> 01:23:04,796
And what we get out of this
state is 00 plus e to the 2i

1228
01:23:04,796 --> 01:23:09,780
phi times 1 1 over
square root 2.

1229
01:23:09,780 --> 01:23:12,740
So now by using sort of--
and now I should stop.

1230
01:23:12,740 --> 01:23:14,010
I know people are waiting.

1231
01:23:14,010 --> 01:23:18,720
But by using two of
those, I suddenly

1232
01:23:18,720 --> 01:23:21,840
have multiplied the phase
by 2, so something is now

1233
01:23:21,840 --> 01:23:23,500
sensitive to 2 phi.

1234
01:23:23,500 --> 01:23:27,920
And if I use n such, more
and more entanglement,

1235
01:23:27,920 --> 01:23:31,000
I will show you--
no class next week,

1236
01:23:31,000 --> 01:23:33,040
but in the following
week-- that we suddenly

1237
01:23:33,040 --> 01:23:38,370
have a term in our quantum
state which is e to the n phi.

1238
01:23:38,370 --> 01:23:40,240
And this gives us effect
of an in precision.

1239
01:23:43,040 --> 01:23:48,150
OK, have a good rest of
the week and we meet again

1240
01:23:48,150 --> 01:23:51,310
in a week and a half.