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PROFESSOR: Most likely this
will be a two-hour lecture

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00:00:27,480 --> 00:00:29,930
with a short break, so I
am teaching until 3 o'clock

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00:00:29,930 --> 00:00:33,250
and we have of the
classroom until 3 o'clock.

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00:00:33,250 --> 00:00:35,580
Let's jump right
into cold fermions.

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00:00:38,640 --> 00:00:40,600
Almost everything
I say in this talk

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00:00:40,600 --> 00:00:43,380
is some advice in a review paper
which Professor [? Swiller ?]

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00:00:43,380 --> 00:00:46,200
and myself wrote
a few years ago.

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00:00:46,200 --> 00:00:48,250
The paper will be posted
to the group website

16
00:00:48,250 --> 00:00:51,690
later this afternoon.

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00:00:51,690 --> 00:00:57,100
OK, when we cool down fermions,
Lithium-6 and Boson sodiums

18
00:00:57,100 --> 00:00:59,440
we immediately
notice a difference.

19
00:00:59,440 --> 00:01:01,910
The sodium cloud shrink,
shrink, shrinks and forms

20
00:01:01,910 --> 00:01:04,190
a small Bose-Einstein
condensate,

21
00:01:04,190 --> 00:01:07,140
whereas the lithium
cloud stops to shrink

22
00:01:07,140 --> 00:01:09,850
when we reach the
degeneracy temperature.

23
00:01:09,850 --> 00:01:12,650
And, of course, you know
it's at low temperature.

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00:01:12,650 --> 00:01:15,170
At high temperature the gases
have very similar behavior,

25
00:01:15,170 --> 00:01:19,190
but at low temperature they
look very, very different.

26
00:01:19,190 --> 00:01:22,200
And that can be
directly observed.

27
00:01:22,200 --> 00:01:26,220
We know that bosons do something
very special at low temperature

28
00:01:26,220 --> 00:01:28,460
and we talked a lot
about it in this course.

29
00:01:28,460 --> 00:01:30,616
Namely, they become
superfluid and form

30
00:01:30,616 --> 00:01:33,190
a Bose-Einstein
condensate, whereas,

31
00:01:33,190 --> 00:01:35,610
for single-component
fermions, there

32
00:01:35,610 --> 00:01:38,520
is some interesting physics
in form of Fermi sea,

33
00:01:38,520 --> 00:01:41,130
but nothing really special.

34
00:01:41,130 --> 00:01:44,830
So, in order to do something
more interesting with fermions

35
00:01:44,830 --> 00:01:47,010
you need two kinds of fermions.

36
00:01:47,010 --> 00:01:50,160
And two kinds of
fermions can form pairs.

37
00:01:50,160 --> 00:01:53,330
To think, for a moment, two
atoms can form a molecule

38
00:01:53,330 --> 00:01:56,750
and those Bosonic
pairs can condense

39
00:01:56,750 --> 00:02:01,320
into Bose-Einstein condensate
and becomes superfluid again.

40
00:02:01,320 --> 00:02:04,010
If that would be all to
Fermionic superfluidity

41
00:02:04,010 --> 00:02:05,980
I would be almost done
right now and would say,

42
00:02:05,980 --> 00:02:10,789
"OK a Bosonic atom is
a composite particle

43
00:02:10,789 --> 00:02:13,870
made of nuclei and
nucleus and electrons.

44
00:02:13,870 --> 00:02:16,540
Well, a bosonic molecule
is a composite particle,

45
00:02:16,540 --> 00:02:19,270
but in the end they
do both the same."

46
00:02:19,270 --> 00:02:21,310
But the special
things about fermions

47
00:02:21,310 --> 00:02:24,720
is there is another form of
pairing, which is much, much,

48
00:02:24,720 --> 00:02:26,310
weaker.

49
00:02:26,310 --> 00:02:29,690
And this is the form which
leads to superconductivity

50
00:02:29,690 --> 00:02:31,030
of electrons.

51
00:02:31,030 --> 00:02:32,890
It's Cooper pairing.

52
00:02:32,890 --> 00:02:35,370
And this is really
much more subtle

53
00:02:35,370 --> 00:02:38,170
because the pairs
are much larger

54
00:02:38,170 --> 00:02:41,640
than the inter-particle
separation.

55
00:02:41,640 --> 00:02:45,020
Now, in atomic physics,
we have a wonderful tool

56
00:02:45,020 --> 00:02:49,940
to study pairing because
atoms can form molecules,

57
00:02:49,940 --> 00:02:53,580
weakly bound molecules or
tightly bound molecules.

58
00:02:53,580 --> 00:02:55,850
And we can control
that in experiments

59
00:02:55,850 --> 00:02:57,790
with the Feshbach resonance.

60
00:02:57,790 --> 00:03:01,320
Assume you have two
atoms, which collide,

61
00:03:01,320 --> 00:03:03,390
and they can form a molecule.

62
00:03:03,390 --> 00:03:05,800
And if you apply
a magnetic field

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00:03:05,800 --> 00:03:07,570
the two free atoms
and the molecule

64
00:03:07,570 --> 00:03:09,560
may have a different
Zeeman shift,

65
00:03:09,560 --> 00:03:12,730
and therefore, there
may be a level crossing.

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00:03:12,730 --> 00:03:15,610
And this is called, as a
function of magnetic field,

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00:03:15,610 --> 00:03:17,160
the Feshbach resonance.

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00:03:17,160 --> 00:03:18,750
If the crossing isn't
crossing nothing

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00:03:18,750 --> 00:03:20,750
happens because the two
configurations would not

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00:03:20,750 --> 00:03:21,600
interact.

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00:03:21,600 --> 00:03:25,670
So, physics happens when
there are interactions

72
00:03:25,670 --> 00:03:29,200
for some hyperfine
coupling, for instance,

73
00:03:29,200 --> 00:03:33,210
which takes the crossing
into an anti-crossing.

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00:03:33,210 --> 00:03:35,120
And now you see the
following physics

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00:03:35,120 --> 00:03:38,830
around this Feshbach resonance,
that we have two atoms here.

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00:03:38,830 --> 00:03:41,950
Here, we have a stable molecule.

77
00:03:41,950 --> 00:03:44,430
And there are many, many more
molecular states down there,

78
00:03:44,430 --> 00:03:47,827
but for this restricted Hilbert
space this is the lowest state.

79
00:03:47,827 --> 00:03:48,785
It's a stable molecule.

80
00:03:51,980 --> 00:03:55,770
The atoms, if you compare the
solid line with a dashed line,

81
00:03:55,770 --> 00:03:59,384
attract each other because the
solid line has lower energy,

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00:03:59,384 --> 00:04:01,550
whereas, on the other side
of the Feshbach resonance

83
00:04:01,550 --> 00:04:04,130
they repel each other.

84
00:04:04,130 --> 00:04:07,530
So therefore, if you
look at the scattering

85
00:04:07,530 --> 00:04:09,950
lengths of the normalized
force between atoms

86
00:04:09,950 --> 00:04:13,190
we find this disparity feature
around the Feshbach resonance,

87
00:04:13,190 --> 00:04:15,640
and atomic physics
wouldn't be the same

88
00:04:15,640 --> 00:04:17,320
without Feshbach resonances.

89
00:04:17,320 --> 00:04:20,720
We can now tune attractive
repulsive interactions.

90
00:04:20,720 --> 00:04:22,810
In a richer situation
there may also

91
00:04:22,810 --> 00:04:27,160
be Zero-crossings of the
scattering lengths, which

92
00:04:27,160 --> 00:04:29,940
allows us to realize
non-interacting systems

93
00:04:29,940 --> 00:04:31,620
and such.

94
00:04:31,620 --> 00:04:33,930
And Feshbach
resonances were first

95
00:04:33,930 --> 00:04:38,460
observed at MIT
some 15 years ago.

96
00:04:38,460 --> 00:04:41,130
So coming back to fermions.

97
00:04:41,130 --> 00:04:45,040
This Feshbach resonance gives
us something very interesting

98
00:04:45,040 --> 00:04:47,270
regarding to pairing.

99
00:04:47,270 --> 00:04:50,680
On that side where we
have stable molecules,

100
00:04:50,680 --> 00:04:53,930
those molecules are
bosons and those bosons

101
00:04:53,930 --> 00:04:58,000
form a Bose-Einstein
condensate of bound pairs.

102
00:04:58,000 --> 00:05:01,200
Over here, we have attraction.

103
00:05:01,200 --> 00:05:05,220
But the attraction is not strong
enough to form a bound state,

104
00:05:05,220 --> 00:05:07,160
at least two atoms
in vacuum would not

105
00:05:07,160 --> 00:05:08,720
find form a molecular state.

106
00:05:08,720 --> 00:05:10,860
The attraction is too weak.

107
00:05:10,860 --> 00:05:14,400
But we know from condensed
matter physics textbook

108
00:05:14,400 --> 00:05:18,100
if you have many fermions,
if you have a Fermi sea,

109
00:05:18,100 --> 00:05:20,780
then the Fermi sea
plus attraction

110
00:05:20,780 --> 00:05:24,060
equals BCS Cooper pairs.

111
00:05:24,060 --> 00:05:27,190
So now we have an opportunity,
by simply changing

112
00:05:27,190 --> 00:05:31,800
the magnetic field, to go from
real, strongly-bound pairs

113
00:05:31,800 --> 00:05:34,890
to weakly-bound pairs, and
eventually to Cooper pairs,

114
00:05:34,890 --> 00:05:37,640
which would not form
pairs in vacuum.

115
00:05:37,640 --> 00:05:39,630
And this is a kind
of physics, which

116
00:05:39,630 --> 00:05:42,840
was for the first time
explored with ultra cold atoms,

117
00:05:42,840 --> 00:05:44,731
in the BEC-BCS crossover.

118
00:05:44,731 --> 00:05:45,230
Jenny.

119
00:05:45,230 --> 00:05:46,271
AUDIENCE:Just to clarify.

120
00:05:46,271 --> 00:05:49,514
So in the repulsive regime, you
said the atoms form molecules,

121
00:05:49,514 --> 00:05:50,942
but the atoms repel each other.

122
00:05:50,942 --> 00:05:53,560
So you're saying atoms that
don't form molecules repel each

123
00:05:53,560 --> 00:05:54,750
other or [INAUDIBLE]--

124
00:05:54,750 --> 00:05:55,560
PROFESSOR: Yeah.

125
00:05:55,560 --> 00:05:59,600
Actually if I go back to that,
here they form molecules,

126
00:05:59,600 --> 00:06:01,350
here they repel each other.

127
00:06:01,350 --> 00:06:04,840
And actually a lot
of people think,

128
00:06:04,840 --> 00:06:07,350
the repulsive side of
the Feshbach resonance

129
00:06:07,350 --> 00:06:08,610
means there is repulsion.

130
00:06:08,610 --> 00:06:10,790
But look, the atoms
attract each other

131
00:06:10,790 --> 00:06:12,670
so much that they
form a molecule.

132
00:06:12,670 --> 00:06:15,590
You can actually see the
repulsion of the atoms

133
00:06:15,590 --> 00:06:19,580
comes because there is a very
attractive molecular state

134
00:06:19,580 --> 00:06:22,950
below, and those two
states repel each other.

135
00:06:22,950 --> 00:06:25,020
So the repulsion
up here is actually

136
00:06:25,020 --> 00:06:28,300
caused by the molecular
state down there.

137
00:06:28,300 --> 00:06:30,270
So in other words,
if you go from

138
00:06:30,270 --> 00:06:32,790
the attractive to
the repulsive side--

139
00:06:32,790 --> 00:06:37,070
many people use it, in terms
of pairs-- you go actually

140
00:06:37,070 --> 00:06:39,190
from very weak
attraction, which is not

141
00:06:39,190 --> 00:06:40,940
sufficient to form
the bound state,

142
00:06:40,940 --> 00:06:43,090
to stronger and
stronger attraction,

143
00:06:43,090 --> 00:06:47,871
which now forms a stronger, and
more strongly bound molecule.

144
00:06:47,871 --> 00:06:48,370
Mark.

145
00:06:48,370 --> 00:06:52,693
AUDIENCE: Are there ways you can
tell whether the molecules are

146
00:06:52,693 --> 00:06:55,639
repulsive, or attractive just
by physics, or [INAUDIBLE].

147
00:06:58,627 --> 00:07:01,085
PROFESSOR: You mean what the
force is between the molecules

148
00:07:01,085 --> 00:07:02,590
now?

149
00:07:02,590 --> 00:07:05,370
It turns out that the
molecules are repulsive.

150
00:07:05,370 --> 00:07:09,080
In some picture here,
when the molecule forms,

151
00:07:09,080 --> 00:07:11,540
you can see it's a
loosely bound molecule.

152
00:07:11,540 --> 00:07:15,530
And the repulsion
between atoms turns

153
00:07:15,530 --> 00:07:17,700
into repulsion of molecules.

154
00:07:17,700 --> 00:07:21,510
If you use an [INAUDIBLE]
which has approximations,

155
00:07:21,510 --> 00:07:23,710
you would actually find
that the scattering lengths

156
00:07:23,710 --> 00:07:26,040
for molecules is
exactly two times

157
00:07:26,040 --> 00:07:28,040
the scattering length for atoms.

158
00:07:28,040 --> 00:07:29,790
But if you work
it out correctly,

159
00:07:29,790 --> 00:07:33,250
the factor is different than 2.

160
00:07:33,250 --> 00:07:34,350
Colin.

161
00:07:34,350 --> 00:07:38,760
AUDIENCE: You said that
the molecules are boson.

162
00:07:38,760 --> 00:07:42,540
I guess they're only bosons
if the size of the molecule--

163
00:07:42,540 --> 00:07:44,160
PROFESSOR: I come to that later.

164
00:07:44,160 --> 00:07:46,420
We come to that later, yes.

165
00:07:46,420 --> 00:07:50,870
AUDIENCE: My question was, I
guess the size of the molecule

166
00:07:50,870 --> 00:07:55,570
scales pretty strongly with how
far away you from resonance,

167
00:07:55,570 --> 00:07:59,650
C6 R6 balance for---

168
00:07:59,650 --> 00:08:01,670
PROFESSOR: In the
universal regime, which

169
00:08:01,670 --> 00:08:04,680
is open channel dominated--
sorry for talking shop--

170
00:08:04,680 --> 00:08:07,060
in that regime, the
size of the molecule

171
00:08:07,060 --> 00:08:09,430
is equal to one-half of
the scattering lengths.

172
00:08:09,430 --> 00:08:11,360
So as the scattering
lengths diverges,

173
00:08:11,360 --> 00:08:13,090
the molecule is very, very big.

174
00:08:13,090 --> 00:08:14,600
And as we move
away, the molecule

175
00:08:14,600 --> 00:08:16,330
gets smaller and smaller.

176
00:08:16,330 --> 00:08:18,990
But once we get out of
this quadratic regime,

177
00:08:18,990 --> 00:08:22,360
out of the universe of the
halo regime, then, of course,

178
00:08:22,360 --> 00:08:24,630
the molecular size is
completely determined

179
00:08:24,630 --> 00:08:25,750
by the close channel.

180
00:08:25,750 --> 00:08:27,720
And it is whatever it
is in the close channel.

181
00:08:27,720 --> 00:08:29,830
And there's no relation
to the scattering lengths.

182
00:08:33,450 --> 00:08:34,559
OK.

183
00:08:34,559 --> 00:08:37,679
So in other words,
it is now possible

184
00:08:37,679 --> 00:08:43,799
using Feshbach resonances-- Yes.

185
00:08:43,799 --> 00:08:46,080
I'm just missing colors,
those dots should be red.

186
00:08:46,080 --> 00:08:48,710
Do you have a color problem.

187
00:08:48,710 --> 00:08:51,131
The red has turned into black.

188
00:08:51,131 --> 00:08:52,130
At least on the monitor.

189
00:08:59,840 --> 00:09:00,870
That is much nicer.

190
00:09:04,810 --> 00:09:06,440
On the internal
monitor it's red,

191
00:09:06,440 --> 00:09:11,746
it's just that the
computer does not show red.

192
00:09:11,746 --> 00:09:12,800
Let me just try that.

193
00:09:22,070 --> 00:09:25,150
No, it's not cooperative.

194
00:09:25,150 --> 00:09:26,075
Just one more second.

195
00:09:36,112 --> 00:09:37,570
I have many more
slides with color,

196
00:09:37,570 --> 00:09:41,075
so I just want to make sure
that you recognize the features.

197
00:09:56,060 --> 00:09:57,880
I don't see any easy fix.

198
00:09:57,880 --> 00:10:00,700
So maybe we should
live with just

199
00:10:00,700 --> 00:10:02,400
imagining some of
the black is red.

200
00:10:11,940 --> 00:10:14,580
So in other words, by
using a magnetic field,

201
00:10:14,580 --> 00:10:19,340
we can go from
strongly-bound pairs, which

202
00:10:19,340 --> 00:10:22,830
form a Bose-Einsten condensate,
to weakly-bound pairs, which

203
00:10:22,830 --> 00:10:26,940
resemble Cooper pairs of
electrons in superconductors.

204
00:10:26,940 --> 00:10:29,620
So this is the physics
of the BEC-BCS crossover,

205
00:10:29,620 --> 00:10:33,410
and in the middle of it we find
a new form of superfluidity,

206
00:10:33,410 --> 00:10:36,050
which has not really
been realized before.

207
00:10:36,050 --> 00:10:38,380
Namely, you can see
these are molecules

208
00:10:38,380 --> 00:10:40,720
which are too big be
to be called molecules.

209
00:10:40,720 --> 00:10:42,340
Or these Cooper
pairs, which are too

210
00:10:42,340 --> 00:10:43,930
small to be called Cooper pairs.

211
00:10:43,930 --> 00:10:46,800
It's really pairs which
sort of elbow each other.

212
00:10:46,800 --> 00:10:48,730
They just fill all the space.

213
00:10:48,730 --> 00:10:55,250
The pairs are exactly as large
as the spacing between atoms.

214
00:10:55,250 --> 00:10:56,340
OK.

215
00:10:56,340 --> 00:11:00,300
In class I always try to teach
you something conceptional.

216
00:11:00,300 --> 00:11:03,210
And the conceptional part
I want to address now

217
00:11:03,210 --> 00:11:07,570
is, let's assume we are
on the right-hand side,

218
00:11:07,570 --> 00:11:11,160
on the attractive side of
the Feshbach resonance.

219
00:11:11,160 --> 00:11:13,606
Two atoms weakly
attract each other

220
00:11:13,606 --> 00:11:15,230
with their negative
scattering lengths,

221
00:11:15,230 --> 00:11:18,530
but there is no bound
molecular state.

222
00:11:18,530 --> 00:11:23,050
So how should you now understand
that if you have many fermions,

223
00:11:23,050 --> 00:11:28,440
that those fermions form
Cooper pairs in pairs?

224
00:11:28,440 --> 00:11:30,700
And the question I
would have for you,

225
00:11:30,700 --> 00:11:33,690
is-- definitely probably
heard about it--

226
00:11:33,690 --> 00:11:37,000
is this paring mechanism,
which requires the Fermi

227
00:11:37,000 --> 00:11:41,180
sea Is that many-body pairing?

228
00:11:41,180 --> 00:11:45,100
That somehow correlation between
many particles are necessary,

229
00:11:45,100 --> 00:11:47,250
and it's a genuine
many body effect?

230
00:11:47,250 --> 00:11:50,400
Or can you still
understand Cooper

231
00:11:50,400 --> 00:11:54,180
pairing as mainly
a two-body effect?

232
00:11:58,050 --> 00:12:00,190
So in other words,
how much many-body

233
00:12:00,190 --> 00:12:04,319
is really involved
in creating pairs

234
00:12:04,319 --> 00:12:05,610
in the presence of a Fermi sea?

235
00:12:08,820 --> 00:12:10,150
Does anybody know the answer?

236
00:12:13,810 --> 00:12:15,760
What you hear from
most people is

237
00:12:15,760 --> 00:12:18,265
that it's a real
many-body effect.

238
00:12:18,265 --> 00:12:21,740
Two particles cannot found
form a pair in vacuum.

239
00:12:21,740 --> 00:12:25,670
So it really many-body physics,
and you say "Wow, I really

240
00:12:25,670 --> 00:12:28,020
have to understand
complicated physics."

241
00:12:28,020 --> 00:12:29,238
AUDIENCE: [INAUDIBLE]

242
00:12:29,238 --> 00:12:29,988
PROFESSOR: Pardon?

243
00:12:29,988 --> 00:12:31,956
AUDIENCE: [INAUDIBLE]

244
00:12:36,205 --> 00:12:38,330
PROFESSOR: It has something
to do with a Fermi sea.

245
00:12:38,330 --> 00:12:41,434
What-- [? Timor ?] first.

246
00:12:41,434 --> 00:12:43,844
AUDIENCE: I may know the answer.

247
00:12:43,844 --> 00:12:48,182
Maybe you would like to--

248
00:12:48,182 --> 00:12:49,330
PROFESSOR: Take it away.

249
00:12:49,330 --> 00:12:50,913
AUDIENCE: There's
always a bound state

250
00:12:50,913 --> 00:12:55,292
in two dimensions, I think,
between both [INAUDIBLE]

251
00:12:55,292 --> 00:12:58,897
the Fermi sea helps you
get the two dimensions.

252
00:12:58,897 --> 00:12:59,980
PROFESSOR: That's correct.

253
00:12:59,980 --> 00:13:02,710
So what I want to show you is
exactly what [? Timor ?] said.

254
00:13:02,710 --> 00:13:04,760
You can really
understand Cooper pairs.

255
00:13:04,760 --> 00:13:07,140
You can understand the
exponential dependence

256
00:13:07,140 --> 00:13:09,250
on the scattering
lengths by simply

257
00:13:09,250 --> 00:13:12,290
looking at one particle,
at two particles,

258
00:13:12,290 --> 00:13:15,010
with a very weak potential.

259
00:13:15,010 --> 00:13:17,800
But you have to look
at it in two dimension.

260
00:13:17,800 --> 00:13:20,170
And it is eventually
the Fermi sea

261
00:13:20,170 --> 00:13:22,590
which provides a two-dimensional
density of states.

262
00:13:22,590 --> 00:13:24,570
So in other words,
the binding of Cooper

263
00:13:24,570 --> 00:13:26,740
pairs, at least
qualitatively, can

264
00:13:26,740 --> 00:13:29,840
be understood with a
single particle Schrodinger

265
00:13:29,840 --> 00:13:32,800
equation with undergraduate
physics knowledge.

266
00:13:32,800 --> 00:13:33,300
Colin

267
00:13:33,300 --> 00:13:36,228
AUDIENCE: I guess
the picture that

268
00:13:36,228 --> 00:13:39,156
was sort of when
you think of PCS

269
00:13:39,156 --> 00:13:41,912
and a real solid, like a
three dimensional solid,

270
00:13:41,912 --> 00:13:46,470
is that one electron
traveling through the lattice

271
00:13:46,470 --> 00:13:50,340
repels all the electrons
of similar spins

272
00:13:50,340 --> 00:13:51,768
[INAUDIBLE] exclusion.

273
00:13:51,768 --> 00:13:53,690
And then the electron
of another spin

274
00:13:53,690 --> 00:13:55,490
moving with opposite
momentum, sort of,

275
00:13:55,490 --> 00:13:58,700
sees that as a
deficit of that spin

276
00:13:58,700 --> 00:14:01,200
and sort of attracted
to that region.

277
00:14:01,200 --> 00:14:05,176
So in that sense isn't, sort
of this effective clearing out

278
00:14:05,176 --> 00:14:09,993
of other particles to make room
for [INAUDIBLE] many bodies

279
00:14:09,993 --> 00:14:11,592
associated with
Fermi sea [INAUDIBLE]

280
00:14:11,592 --> 00:14:14,310
PROFESSOR: Very good.

281
00:14:14,310 --> 00:14:16,720
So we are talking about
two different things here.

282
00:14:16,720 --> 00:14:19,220
One is we need an attraction.

283
00:14:19,220 --> 00:14:21,700
In the atomic system
we get an attraction

284
00:14:21,700 --> 00:14:25,750
because of van der Waal's
molecular forces, that's it.

285
00:14:25,750 --> 00:14:28,180
For electrons which
have Coulomb charge,

286
00:14:28,180 --> 00:14:29,780
where is the attraction?

287
00:14:29,780 --> 00:14:32,320
And you wonderfully
described that there

288
00:14:32,320 --> 00:14:35,060
is attraction between electrons.

289
00:14:35,060 --> 00:14:36,630
It's like the couch effect.

290
00:14:36,630 --> 00:14:39,500
If you sit on the couch,
you indent the couch,

291
00:14:39,500 --> 00:14:42,120
and the person next to
you will move towards you.

292
00:14:42,120 --> 00:14:44,980
Similarly, when an electron
goes through a lattice

293
00:14:44,980 --> 00:14:48,390
it leaves a polarization
of the lattice behind

294
00:14:48,390 --> 00:14:51,910
and the next electron is
attracted to this trajectory.

295
00:14:51,910 --> 00:14:57,600
So what you are explaining is
the phonon-mediated process

296
00:14:57,600 --> 00:14:59,450
which leads to attraction.

297
00:14:59,450 --> 00:15:02,039
But if I have two fermions
which attract each other

298
00:15:02,039 --> 00:15:03,330
I don't need this complication.

299
00:15:03,330 --> 00:15:05,121
AUDIENCE: But couldn't
you even get it

300
00:15:05,121 --> 00:15:12,070
without a lattice
[INAUDIBLE] the Fermi gas.

301
00:15:12,070 --> 00:15:14,872
If I have one fermion,
and I move it,

302
00:15:14,872 --> 00:15:19,320
the location that it was has
been-- All the other fermions

303
00:15:19,320 --> 00:15:23,144
have been pushed away
from that so there's

304
00:15:23,144 --> 00:15:25,534
a sort of a density
depletion there.

305
00:15:25,534 --> 00:15:27,541
So couldn't the
other spin component

306
00:15:27,541 --> 00:15:29,358
see that as
effective attraction?

307
00:15:29,358 --> 00:15:33,730
PROFESSOR: Density
depletion are density waves,

308
00:15:33,730 --> 00:15:35,665
and these are
sound waves-- maybe

309
00:15:35,665 --> 00:15:39,020
there is a possibility to map
it on phonons in the lattice.

310
00:15:39,020 --> 00:15:40,580
I haven't heard about it.

311
00:15:40,580 --> 00:15:42,960
So I've never heard about
such an attraction which

312
00:15:42,960 --> 00:15:46,280
would happen in a Fermi gas,
which would give the Coulomb

313
00:15:46,280 --> 00:15:49,522
repulsion for some correlation
attractive component.

314
00:15:49,522 --> 00:15:51,490
What is well-described,
of course,

315
00:15:51,490 --> 00:15:54,910
is the couch model or the
phonon model you mentioned.

316
00:15:54,910 --> 00:15:57,920
But I'm saying, this is just
giving you the attraction.

317
00:15:57,920 --> 00:16:00,160
But the question
is now, and that's

318
00:16:00,160 --> 00:16:02,490
the next expression
I raised, once we

319
00:16:02,490 --> 00:16:04,380
have an attraction
between two particles,

320
00:16:04,380 --> 00:16:08,140
do we need many-body physics
or single-particle physics

321
00:16:08,140 --> 00:16:09,660
to have pairing?

322
00:16:09,660 --> 00:16:11,480
So let me address that.

323
00:16:11,480 --> 00:16:13,380
Now this is out of
the review paper,

324
00:16:13,380 --> 00:16:15,130
you could be much more about it.

325
00:16:15,130 --> 00:16:19,480
But it simply reminds you--
undergraduate problem--

326
00:16:19,480 --> 00:16:22,910
what happens when you have
two particles with a very

327
00:16:22,910 --> 00:16:25,316
weak attractive potential?

328
00:16:25,316 --> 00:16:26,940
And, of course, since
you can eliminate

329
00:16:26,940 --> 00:16:29,470
the center of mass motion,
when I say two particles

330
00:16:29,470 --> 00:16:31,830
it's the same as one
particle attracted

331
00:16:31,830 --> 00:16:34,660
to a central potential.

332
00:16:34,660 --> 00:16:37,100
What you learn, or what could
may have already learned

333
00:16:37,100 --> 00:16:39,940
in your undergraduate
days, is that in 1D,

334
00:16:39,940 --> 00:16:45,750
if you have an infinitesimal
depths of the square of L,

335
00:16:45,750 --> 00:16:47,130
you found a bound state.

336
00:16:47,130 --> 00:16:51,420
The bound state will decay very
slowly, but it's a bound state.

337
00:16:51,420 --> 00:16:55,570
In two-dimension you
also find a bound state.

338
00:16:55,570 --> 00:16:58,720
In two dimensions
logarithm appears.

339
00:16:58,720 --> 00:17:01,440
Often when logarithm appears
it's sort of the limit.

340
00:17:01,440 --> 00:17:04,300
The logarithm is the
weakest form of binding.

341
00:17:04,300 --> 00:17:07,599
And if you go one step
further, for a very small

342
00:17:07,599 --> 00:17:11,000
attractive potential, you
do not have a bound state.

343
00:17:11,000 --> 00:17:13,730
So in three -dimension
you need a minimum.

344
00:17:13,730 --> 00:17:15,902
You need a critical
energy of the square

345
00:17:15,902 --> 00:17:19,740
of L-- critical depths
before your bound state--

346
00:17:19,740 --> 00:17:24,099
whereas, infinitesimal
attraction

347
00:17:24,099 --> 00:17:27,900
is enough to bind particles
in one or two-dimension.

348
00:17:27,900 --> 00:17:30,580
So therefore, when we
talk about fermions

349
00:17:30,580 --> 00:17:34,670
with very weak attraction--
in 1D and 2D, no problem.

350
00:17:34,670 --> 00:17:37,550
They can form pairs by that,
pairs even in vacuum. --

351
00:17:37,550 --> 00:17:41,520
but, in 3D we need something
else because an infinitesimal

352
00:17:41,520 --> 00:17:45,586
attraction is not
enough to form a pair.

353
00:17:45,586 --> 00:17:47,335
Now let me ask you the
following question.

354
00:17:51,260 --> 00:17:53,790
Let me just cover the slide
because it gives the solution-.

355
00:17:53,790 --> 00:17:56,480
You all know that in
three dimension, when

356
00:17:56,480 --> 00:18:00,080
you have radio symmetry,
you can transform

357
00:18:00,080 --> 00:18:01,710
the three-dimensional
Schrodinger

358
00:18:01,710 --> 00:18:04,020
equation into an
angular equation,

359
00:18:04,020 --> 00:18:07,580
but then also into a
one-dimensional equation

360
00:18:07,580 --> 00:18:10,950
for the radial wave function.

361
00:18:10,950 --> 00:18:14,170
And of course, this
one-dimensional radial equation

362
00:18:14,170 --> 00:18:17,410
has the same potential as the
original three-dimensional

363
00:18:17,410 --> 00:18:19,170
equations.

364
00:18:19,170 --> 00:18:21,320
So I don't know, I want
to hear from one of you.

365
00:18:21,320 --> 00:18:24,790
Why does now an
infinitesimal square

366
00:18:24,790 --> 00:18:27,940
of L potential with
attraction give bound states

367
00:18:27,940 --> 00:18:31,260
in 1D, but not in 3D?

368
00:18:31,260 --> 00:18:33,440
Because in 3D, the
radial wave equation

369
00:18:33,440 --> 00:18:36,472
uses the same infinitesimal
potential as in 1D.

370
00:18:36,472 --> 00:18:36,972
Jenny.

371
00:18:36,972 --> 00:18:39,860
AUDIENCE:Because you have
to break down an effective

372
00:18:39,860 --> 00:18:42,870
potential that takes into
account the centrifugal

373
00:18:42,870 --> 00:18:44,870
PROFESSOR: Perfect, but
let's use L equals zero.

374
00:18:44,870 --> 00:18:48,360
Then the centrifugal
correction is zero.

375
00:18:48,360 --> 00:18:49,490
Otherwise you are right.

376
00:18:49,490 --> 00:18:50,900
It's an effective potential.

377
00:18:50,900 --> 00:18:53,720
But let's talk about L
equals zero, S states.

378
00:18:53,720 --> 00:18:58,660
Then there is no correction
due to the centrifugal forces.

379
00:19:01,396 --> 00:19:02,770
Colin.

380
00:19:02,770 --> 00:19:04,656
AUDIENCE: Radial
derivative is no longer

381
00:19:04,656 --> 00:19:09,357
just a simple second
derivative [INAUDIBLE].

382
00:19:09,357 --> 00:19:10,190
PROFESSOR: No it is.

383
00:19:10,190 --> 00:19:12,990
You have a more complicated
radial derivative,

384
00:19:12,990 --> 00:19:16,000
but then you go to
an equation which

385
00:19:16,000 --> 00:19:19,290
just says D2DR squared
over function U

386
00:19:19,290 --> 00:19:21,530
and it's exactly the
one-dimensional Schrodinger

387
00:19:21,530 --> 00:19:23,510
equation, which you write down.

388
00:19:23,510 --> 00:19:25,820
You can transform
the radial equation

389
00:19:25,820 --> 00:19:27,910
with a more
complicated derivative

390
00:19:27,910 --> 00:19:31,910
into something which is exactly
a 1D Schrodinger equation.

391
00:19:31,910 --> 00:19:32,410
Yes.

392
00:19:32,410 --> 00:19:34,366
AUDIENCE: You can
[INAUDIBLE] relationship

393
00:19:34,366 --> 00:19:35,344
between [INAUDIBLE]

394
00:19:38,544 --> 00:19:39,210
PROFESSOR: Sure.

395
00:19:39,210 --> 00:19:41,157
Uncertainty of relation
always can immediately

396
00:19:41,157 --> 00:19:42,990
help you if you have
an uncertainty relation

397
00:19:42,990 --> 00:19:45,260
for for kinetic
energy and you realize

398
00:19:45,260 --> 00:19:48,770
that you don't have enough
potential, but maybe

399
00:19:48,770 --> 00:19:51,020
mathematically, what
is different between

400
00:19:51,020 --> 00:19:54,790
the identically-looking
1D Schrodering

401
00:19:54,790 --> 00:19:59,130
equation and the
radial Schrodinger

402
00:19:59,130 --> 00:20:01,084
equation for the
wave function U?

403
00:20:01,084 --> 00:20:03,375
AUDIENCE: I don't know how
this is going to lead to it,

404
00:20:03,375 --> 00:20:06,579
but I noticed one difference
is that in in the three

405
00:20:06,579 --> 00:20:08,610
dimensional case you
can't go less than zero.

406
00:20:08,610 --> 00:20:09,580
PROFESSOR: Exactly.

407
00:20:09,580 --> 00:20:13,630
In the three dimensional case
you have a wave function psi,

408
00:20:13,630 --> 00:20:17,860
but then you do a substitution
that the 1D Schrodinger

409
00:20:17,860 --> 00:20:20,710
equation is not for
psi, it is for U.

410
00:20:20,710 --> 00:20:25,800
And U is the radial
part divided by R.

411
00:20:25,800 --> 00:20:27,760
So you transform to a
different function, which

412
00:20:27,760 --> 00:20:31,570
is one more power of R. And
the requirement because of this

413
00:20:31,570 --> 00:20:34,090
is that your radial
wave function

414
00:20:34,090 --> 00:20:36,450
of this one-dimensional
radial equation

415
00:20:36,450 --> 00:20:39,220
has to have a node
in the center.

416
00:20:39,220 --> 00:20:40,450
And now you see what happens.

417
00:20:40,450 --> 00:20:43,520
If you start at zero, your
wave function shoots up.

418
00:20:43,520 --> 00:20:45,620
If you don't have
enough attraction

419
00:20:45,620 --> 00:20:49,150
it will never curve down and
become a normalizable state.

420
00:20:49,150 --> 00:20:52,500
Whereas, in 1D you do
not have the requirement

421
00:20:52,500 --> 00:20:55,210
that the wave function
if your 1D Schrodinger

422
00:20:55,210 --> 00:20:56,560
equation has a node.

423
00:20:56,560 --> 00:20:59,280
So you have the same equation,
the same infinitesimal

424
00:20:59,280 --> 00:21:02,980
potential, but the
radial equation

425
00:21:02,980 --> 00:21:06,410
has a different
boundary condition.

426
00:21:06,410 --> 00:21:08,540
So that's the difference.

427
00:21:08,540 --> 00:21:09,500
And that's important.

428
00:21:09,500 --> 00:21:12,380
It changes the
physics completely.

429
00:21:12,380 --> 00:21:13,450
OK.

430
00:21:13,450 --> 00:21:17,388
Now it looks very different and
it seems you have to-- Niki.

431
00:21:17,388 --> 00:21:19,660
AUDIENCE: What about 2D?

432
00:21:19,660 --> 00:21:22,025
PROFESSOR: Limiting case,
we come to that in a second.

433
00:21:22,025 --> 00:21:25,795
In 2D an infinitesimal
potential is still enough.

434
00:21:28,420 --> 00:21:31,530
Now I want to show you how you
can take care of all dimensions

435
00:21:31,530 --> 00:21:35,040
with one equation and not
solve three different equations

436
00:21:35,040 --> 00:21:37,730
with three different
kinetic energy terms.

437
00:21:37,730 --> 00:21:40,285
That goes as follows.

438
00:21:40,285 --> 00:21:42,410
Recognize that this is just
Schrodinger's equation,

439
00:21:42,410 --> 00:21:45,960
kinetic energy minus
energy equals potential.

440
00:21:45,960 --> 00:21:48,060
But now we can
Fourier transform it.

441
00:21:48,060 --> 00:21:50,280
And the nice thing about
Fourier transformation

442
00:21:50,280 --> 00:21:54,590
is that the second derivative,
spatial derivative,

443
00:21:54,590 --> 00:21:57,160
is simply q squared.

444
00:21:57,160 --> 00:22:05,000
So therefore, if you describe
now everything in Fourier

445
00:22:05,000 --> 00:22:10,540
space, you have q squared
plus k squared times psi,

446
00:22:10,540 --> 00:22:14,450
but you divide by q
squared plus k squared.

447
00:22:14,450 --> 00:22:15,520
Here you have a product.

448
00:22:15,520 --> 00:22:17,700
The product turns
into convolution.

449
00:22:17,700 --> 00:22:20,580
Anyway, it may be not the most
familiar way you've seen it,

450
00:22:20,580 --> 00:22:24,510
but this is just Schrodinger's
equation in Fourier components.

451
00:22:24,510 --> 00:22:26,750
And now we want to
simplify that we

452
00:22:26,750 --> 00:22:28,440
have a short-range potential.

453
00:22:28,440 --> 00:22:31,970
So we assume we have a
short range potential.

454
00:22:31,970 --> 00:22:36,680
If you had a delta function,
the momentum components

455
00:22:36,680 --> 00:22:40,290
of the potential would
be constant for all R.

456
00:22:40,290 --> 00:22:42,440
But, we get into
something unphysical

457
00:22:42,440 --> 00:22:45,100
if you don't put in a cutoff,
so we put in a cutoff.

458
00:22:45,100 --> 00:22:47,050
But with that
cutoff, you can now

459
00:22:47,050 --> 00:22:49,630
re-write the
equation as follows.

460
00:22:49,630 --> 00:22:51,200
And now we want to do one thing.

461
00:22:51,200 --> 00:22:53,530
We want to integrate over q.

462
00:22:53,530 --> 00:22:57,210
Then on the right-hand side we
also have an integral over q.

463
00:22:57,210 --> 00:22:59,760
And then we divide
by the common factor.

464
00:22:59,760 --> 00:23:01,750
And what you get is this.

465
00:23:01,750 --> 00:23:03,910
I know I'm going fast,
but this is nothing else

466
00:23:03,910 --> 00:23:06,540
than re-writing Schrodinger's
equation for you.

467
00:23:06,540 --> 00:23:07,550
But now look at it.

468
00:23:10,260 --> 00:23:15,120
It has an integral
over all energies

469
00:23:15,120 --> 00:23:20,020
with the density of
states, or of E, of energy.

470
00:23:20,020 --> 00:23:22,130
E is k square.

471
00:23:22,130 --> 00:23:24,000
This is the energy
of the bound state,

472
00:23:24,000 --> 00:23:26,180
and we want the bound state.

473
00:23:26,180 --> 00:23:27,900
And on the left
hand side we have

474
00:23:27,900 --> 00:23:30,370
our infinitesimal potential.

475
00:23:30,370 --> 00:23:32,210
And one thing is
clear, if you have

476
00:23:32,210 --> 00:23:34,600
an infinitesimal
potential, we can only

477
00:23:34,600 --> 00:23:37,490
get an infinitesimally
bound state.

478
00:23:37,490 --> 00:23:40,290
So now we do the following.

479
00:23:40,290 --> 00:23:53,450
If the left hand side
goes to 0, If you

480
00:23:53,450 --> 00:23:55,270
go to an infinitesimal
attraction,

481
00:23:55,270 --> 00:23:57,680
the left hand side diverges.

482
00:23:57,680 --> 00:24:01,020
And the question is now,
if the energy goes to 0,

483
00:24:01,020 --> 00:24:04,370
does the right hand
side diverge or not?

484
00:24:04,370 --> 00:24:07,500
In other words,
if this integral,

485
00:24:07,500 --> 00:24:11,320
where you said E equals
0, does not diverge,

486
00:24:11,320 --> 00:24:14,840
you cannot fulfill this
equation with an infinitesimal

487
00:24:14,840 --> 00:24:17,090
attraction in the
square of l potential.

488
00:24:17,090 --> 00:24:19,830
So therefore, the
condition for bound state

489
00:24:19,830 --> 00:24:24,490
is now that this expression
here with E equals 0, diverges.

490
00:24:28,430 --> 00:24:31,010
And this combines
now all the cases.

491
00:24:31,010 --> 00:24:33,340
And you have now an
integral equation,

492
00:24:33,340 --> 00:24:37,050
and you can now solve
it for infinitesimal V0.

493
00:24:37,050 --> 00:24:41,870
You can solve it, what is the
energy as a functional of V0?

494
00:24:41,870 --> 00:24:48,460
And in two dimensions, when the
density of states versus energy

495
00:24:48,460 --> 00:24:51,542
is constant, you have
the weakest divergence

496
00:24:51,542 --> 00:24:52,500
on the right hand side.

497
00:24:52,500 --> 00:24:54,340
It's logarithmic divergence.

498
00:24:54,340 --> 00:24:56,310
In three dimensions,
this doesn't diverge.

499
00:24:56,310 --> 00:24:59,030
In one dimension it
diverges big time.

500
00:24:59,030 --> 00:25:00,830
Two dimensions is the limit.

501
00:25:00,830 --> 00:25:02,710
There's a logarithmic
divergence.

502
00:25:02,710 --> 00:25:04,890
And if you do the math,
which is elementary,

503
00:25:04,890 --> 00:25:07,730
but I'm not doing it,
you find that the energy

504
00:25:07,730 --> 00:25:11,930
of the bound state depends
exponentially on minus 1

505
00:25:11,930 --> 00:25:13,450
over the attraction.

506
00:25:13,450 --> 00:25:16,430
So for infinitesimal
attraction, this

507
00:25:16,430 --> 00:25:18,870
turns to E to the
minus infinity.

508
00:25:18,870 --> 00:25:21,380
It's the weakest bound
state you can imagine,

509
00:25:21,380 --> 00:25:22,500
and this happens in 2D.

510
00:25:25,010 --> 00:25:29,160
OK, let's go from one particle
physics back to fermions.

511
00:25:29,160 --> 00:25:33,690
And let me now present to
you how Leon Cooper really

512
00:25:33,690 --> 00:25:38,185
found the key ingredients to
the long standing problem, "How

513
00:25:38,185 --> 00:25:40,680
do electrons pair in
a superconductor?"

514
00:25:40,680 --> 00:25:43,820
And he did the following.

515
00:25:43,820 --> 00:25:46,520
He made an artificial
simplification,

516
00:25:46,520 --> 00:25:49,240
which you can see is ingenious.

517
00:25:49,240 --> 00:25:50,690
He assumed the following.

518
00:25:50,690 --> 00:25:55,290
He just wanted to understand,
how do two electrons pair?

519
00:25:55,290 --> 00:25:58,060
But he simply assumed
that those two electrons

520
00:25:58,060 --> 00:26:00,480
are on top of a Fermi sea.

521
00:26:00,480 --> 00:26:03,790
So he pretty much said, "I
want to solve the problem where

522
00:26:03,790 --> 00:26:09,860
two electrons just scatter
through all available states

523
00:26:09,860 --> 00:26:14,910
and then figure out if an
infinitesimal attraction

524
00:26:14,910 --> 00:26:18,190
between those electrons
leads to binding."

525
00:26:18,190 --> 00:26:21,740
Of course, in the real world,
the electrons in the Fermi sea

526
00:26:21,740 --> 00:26:23,860
are identical to
those electrons,

527
00:26:23,860 --> 00:26:26,720
to those two special electrons,
and they're constantly

528
00:26:26,720 --> 00:26:27,700
exchanged.

529
00:26:27,700 --> 00:26:30,430
But he simply made
this artificial model.

530
00:26:30,430 --> 00:26:32,680
And then, of course,
we can immediately

531
00:26:32,680 --> 00:26:35,090
write down the
equivalent equation.

532
00:26:35,090 --> 00:26:38,850
It's exactly the same
formalism, except for when

533
00:26:38,850 --> 00:26:41,770
we integrate over
all energies here.

534
00:26:41,770 --> 00:26:44,090
We integrate, yes.

535
00:26:44,090 --> 00:26:48,020
We integrate over the
three-dimensional energy.

536
00:26:48,020 --> 00:26:53,120
But now, we integrate
only over a small region

537
00:26:53,120 --> 00:26:56,230
on top of the Fermi sea.

538
00:26:56,230 --> 00:26:58,310
And everything looks the same.

539
00:26:58,310 --> 00:27:01,830
We just have sometimes
subtract the Fermi energy.

540
00:27:01,830 --> 00:27:05,340
All energies are now
measured not relative to 0,

541
00:27:05,340 --> 00:27:08,400
but relative to
the Fermi energy.

542
00:27:08,400 --> 00:27:11,870
If you compare it to the
equation I just showed you,

543
00:27:11,870 --> 00:27:14,000
forcing the particle
bound state,

544
00:27:14,000 --> 00:27:17,020
you pretty much see that
everything is the same,

545
00:27:17,020 --> 00:27:20,560
except for energies are measured
relative to the Fermi energy.

546
00:27:20,560 --> 00:27:24,340
And secondly, we're not
integrating from zero,

547
00:27:24,340 --> 00:27:25,420
as before.

548
00:27:25,420 --> 00:27:28,830
We start integrating
from the Fermi energy.

549
00:27:28,830 --> 00:27:33,760
And that means now, that in
this area of integration,

550
00:27:33,760 --> 00:27:37,860
the three-dimensional density
of states is constant,

551
00:27:37,860 --> 00:27:41,960
namely it's the density for
where the energy is the Fermi

552
00:27:41,960 --> 00:27:45,220
energy and can be pulled
out of the integral.

553
00:27:45,220 --> 00:27:47,640
And then we have the same
logarithmic divergence

554
00:27:47,640 --> 00:27:50,910
as we had in two dimensions.

555
00:27:50,910 --> 00:27:54,860
So therefore, the problem
of two electrons scattering

556
00:27:54,860 --> 00:27:58,040
on the surface of
the Fermi sea means

557
00:27:58,040 --> 00:28:00,890
that we have essentially
the same condition

558
00:28:00,890 --> 00:28:03,760
for the electrons as if they
would live in two dimensions.

559
00:28:07,580 --> 00:28:08,310
Question.

560
00:28:08,310 --> 00:28:11,030
If the world were
four-dimensional,

561
00:28:11,030 --> 00:28:13,960
and we would have
the same physics,

562
00:28:13,960 --> 00:28:17,870
we have two electrons on the
surface of the Fermi sea,

563
00:28:17,870 --> 00:28:21,320
would the physics of the two
electrons, in that regard,

564
00:28:21,320 --> 00:28:24,871
be 4 minus one dimension,
three dimension,

565
00:28:24,871 --> 00:28:26,120
or would it be two dimensions?

566
00:28:30,310 --> 00:28:33,190
AUDIENCE: [INAUDIBLE]

567
00:28:33,190 --> 00:28:36,550
PROFESSOR: Well, that's
exactly the question.

568
00:28:36,550 --> 00:28:40,450
Let's assume you had
four spatial dimensions.

569
00:28:40,450 --> 00:28:43,690
For this kind of analysis,
it's the effective density

570
00:28:43,690 --> 00:28:46,550
of states on the Fermi surface.

571
00:28:46,550 --> 00:28:48,766
Now, two dimensional?

572
00:28:48,766 --> 00:28:49,910
Or is it three-dimensional?

573
00:28:52,666 --> 00:28:55,600
AUDIENCE: It should
be three, right?

574
00:28:55,600 --> 00:28:57,556
You remove one
dimension by restricting

575
00:28:57,556 --> 00:29:00,980
yourself the area
in the Fermi sea?

576
00:29:00,980 --> 00:29:06,645
PROFESSOR: It is, if you say
the particles have momentum

577
00:29:06,645 --> 00:29:10,630
on the Fermi surface, if
the Fermi surface fixes

578
00:29:10,630 --> 00:29:14,350
one momentum, and now they
have three different momenta.

579
00:29:14,350 --> 00:29:17,800
So in that sense, the motion on
the surface in four dimension,

580
00:29:17,800 --> 00:29:19,380
is three-dimensional.

581
00:29:19,380 --> 00:29:22,350
But, it was sort of a trick
question, to some extent,

582
00:29:22,350 --> 00:29:24,635
because here we have
reformulated everything

583
00:29:24,635 --> 00:29:26,730
with energy.

584
00:29:26,730 --> 00:29:31,240
And the answer is now, the
density of state is constant.

585
00:29:31,240 --> 00:29:34,440
It is the four-dimensional
density of states,

586
00:29:34,440 --> 00:29:37,790
but evaluated at
the Fermi energy.

587
00:29:37,790 --> 00:29:39,910
So therefore, no matter
in what dimensions

588
00:29:39,910 --> 00:29:43,370
you are, because you are
restricted to the Fermi sea,

589
00:29:43,370 --> 00:29:50,460
you integrate over energy with
a constant density of states.

590
00:29:50,460 --> 00:29:52,742
And if you pull the
constant density state out

591
00:29:52,742 --> 00:29:55,500
of the integral you have
a logarithmic divergence.

592
00:29:55,500 --> 00:29:58,890
And you pretty much
get the same answer.

593
00:29:58,890 --> 00:30:00,840
So, therefore the
physics in four dimension

594
00:30:00,840 --> 00:30:01,990
would be the same.

595
00:30:01,990 --> 00:30:05,960
It would be the fermions
which live on this Fermi

596
00:30:05,960 --> 00:30:09,980
surface would just
marginally pair.

597
00:30:09,980 --> 00:30:14,880
AUDIENCE: Can you explain
the reduction of dimensions?

598
00:30:14,880 --> 00:30:16,350
Where does [INAUDIBLE]

599
00:30:23,630 --> 00:30:26,740
PROFESSOR: Well, the motion,
you can see on the Fermi surface

600
00:30:26,740 --> 00:30:28,560
is n minus one dimension.

601
00:30:28,560 --> 00:30:30,970
But the energetic
analysis always

602
00:30:30,970 --> 00:30:33,240
uses a constant
density of state.

603
00:30:33,240 --> 00:30:34,810
And the constant
density of state

604
00:30:34,810 --> 00:30:37,880
is characteristic
for two dimensions.

605
00:30:37,880 --> 00:30:38,880
So you have both.

606
00:30:38,880 --> 00:30:41,350
If you analyze the
motion, yes, you

607
00:30:41,350 --> 00:30:43,670
have n minus one dimensions.

608
00:30:43,670 --> 00:30:46,260
But if you look at the
question of bound states,

609
00:30:46,260 --> 00:30:50,110
you encounter the same
situation as in two dimension,

610
00:30:50,110 --> 00:30:53,280
because the effective density
of states is constant.

611
00:30:57,510 --> 00:31:00,750
OK so with that, the
message I have for you is,

612
00:31:00,750 --> 00:31:03,510
Cooper pairing is a
single particle effect.

613
00:31:03,510 --> 00:31:06,755
A single particle trapped to
an arbitrarily weak potentially

614
00:31:06,755 --> 00:31:08,360
in two dimensions.

615
00:31:08,360 --> 00:31:10,500
And the only many-body
physics which

616
00:31:10,500 --> 00:31:12,380
allows those Cooper
pairs to form

617
00:31:12,380 --> 00:31:14,920
is, that you have
a Pauli blocking,

618
00:31:14,920 --> 00:31:17,490
that you have Fermi sea,
which prevents particles

619
00:31:17,490 --> 00:31:20,390
from going into
the deep Fermi sea.

620
00:31:20,390 --> 00:31:22,800
They stay on the surface,
and therefore, they

621
00:31:22,800 --> 00:31:24,210
form weak pairs.

622
00:31:24,210 --> 00:31:26,020
So the only many
body physics, which

623
00:31:26,020 --> 00:31:29,668
we need for that kind of
pairing is Pauli blocking.

624
00:31:29,668 --> 00:31:30,496
Mark.

625
00:31:30,496 --> 00:31:31,162
AUDIENCE: Sorry.

626
00:31:31,162 --> 00:31:35,150
Can you explain what is
ER in the previous slide?

627
00:31:35,150 --> 00:31:36,495
PROFESSOR: ER was the cutoff.

628
00:31:36,495 --> 00:31:38,800
We were cutting
off the potential

629
00:31:38,800 --> 00:31:41,580
at a short-- we didn't
use a delta potential,

630
00:31:41,580 --> 00:31:45,660
we used a square of l potential,
which extends out to R. And ER

631
00:31:45,660 --> 00:31:52,260
is the recoil energy associated
with R. Its 1 over R squared,

632
00:31:52,260 --> 00:31:54,800
with a few h bars and n to
give it the units of energy.

633
00:31:57,710 --> 00:31:58,680
Colin.

634
00:31:58,680 --> 00:32:01,650
AUDIENCE: If you
go back to that,

635
00:32:01,650 --> 00:32:05,143
I can imagine Taylor
expanding my density of states

636
00:32:05,143 --> 00:32:09,090
in any dimension
around Fermi energy.

637
00:32:09,090 --> 00:32:12,010
I'll always have my constant
term, and the first term is 3D,

638
00:32:12,010 --> 00:32:16,540
it's E to 1/2, or
some linear term.

639
00:32:16,540 --> 00:32:21,923
Doesn't that imply that
even if give the system

640
00:32:21,923 --> 00:32:25,787
any infinitesimal
amount of energy,

641
00:32:25,787 --> 00:32:31,580
that integral no
longer diverges and--

642
00:32:31,580 --> 00:32:32,540
PROFESSOR: No.

643
00:32:32,540 --> 00:32:35,000
The question of
divergence is robust.

644
00:32:35,000 --> 00:32:38,010
Depends only on the leading
term and not on the expansion.

645
00:32:41,690 --> 00:32:43,190
If you have a
divergence you are not

646
00:32:43,190 --> 00:32:47,610
changing it by a
small correction term.

647
00:32:47,610 --> 00:32:50,990
Because a small correction
term is on top of unity.

648
00:32:50,990 --> 00:32:54,700
If you start with the
density of states at 0,

649
00:32:54,700 --> 00:32:57,020
which goes with
epsilon to some power,

650
00:32:57,020 --> 00:32:59,460
you have a different situation.

651
00:32:59,460 --> 00:33:01,530
You catch really the
physics by saying

652
00:33:01,530 --> 00:33:04,760
you take a constant
density of states here.

653
00:33:04,760 --> 00:33:09,840
Anyway, let me just go on and
say we should now figure out--

654
00:33:09,840 --> 00:33:11,370
and this is also
important when it

655
00:33:11,370 --> 00:33:16,090
comes to pairing-- which pairs
are the most important ones.

656
00:33:16,090 --> 00:33:18,040
And the message I
want to give you here,

657
00:33:18,040 --> 00:33:20,250
the pairs which are
most important one,

658
00:33:20,250 --> 00:33:22,990
on the ones which
have zero momentum.

659
00:33:22,990 --> 00:33:27,680
Because those pairs can scatter
to all states on the Fermi

660
00:33:27,680 --> 00:33:31,460
surface without violating
momentum conservation.

661
00:33:31,460 --> 00:33:36,080
However, if you have the total
momentum of the two particles

662
00:33:36,080 --> 00:33:39,080
which is non-0, they
can only scatter

663
00:33:39,080 --> 00:33:42,010
on the segment of the sphere.

664
00:33:42,010 --> 00:33:45,640
And you got already the message
that the higher the density

665
00:33:45,640 --> 00:33:49,410
of state, the better
it is for the binding.

666
00:33:49,410 --> 00:33:52,260
So therefore, the pairs
which have the largest

667
00:33:52,260 --> 00:33:55,740
binding energies will be the
one with total momentum zero.

668
00:33:59,800 --> 00:34:04,370
OK, now we go from
Cooper to BCS.

669
00:34:04,370 --> 00:34:07,250
Those three guys figured
out how they can now

670
00:34:07,250 --> 00:34:10,360
take the physics
described by Cooper,

671
00:34:10,360 --> 00:34:14,090
and really fold it into a
correct many-body formalism.

672
00:34:14,090 --> 00:34:17,280
In other words, not to
artificially have 2 fermions,

673
00:34:17,280 --> 00:34:19,560
which scatter, and there
is just the Fermi sea

674
00:34:19,560 --> 00:34:20,710
of the other ones.

675
00:34:20,710 --> 00:34:22,940
They are now treating
the two electrons,

676
00:34:22,940 --> 00:34:27,020
which scatter, on equal footing
with the whole Fermi sea.

677
00:34:27,020 --> 00:34:31,800
So, now we have democracy, all
electrons are created equal.

678
00:34:31,800 --> 00:34:35,449
So I want to derive for
you the BCS wavefunction

679
00:34:35,449 --> 00:34:38,520
in the way which was not the
way done by Bardeen, Cooper,

680
00:34:38,520 --> 00:34:41,060
Schrieffer, but it makes
an immediate analogy

681
00:34:41,060 --> 00:34:43,215
with Bose-Einstein condensation.

682
00:34:43,215 --> 00:34:46,770
, So what happens is you
could now have the idea--

683
00:34:46,770 --> 00:34:50,239
of course I'm very suggestive
in leading you to this idea--

684
00:34:50,239 --> 00:34:52,770
two fermions form a pair.

685
00:34:52,770 --> 00:34:57,060
And if the pairs both
condense, then each pair

686
00:34:57,060 --> 00:34:58,940
would be identical.

687
00:34:58,940 --> 00:35:01,260
So maybe we should
take our n particles,

688
00:35:01,260 --> 00:35:06,395
our n fermions-- n half spin
up, n spin down-- each of them

689
00:35:06,395 --> 00:35:07,890
should form a pair.

690
00:35:07,890 --> 00:35:10,910
So one and two forms a pair,
three and four forms a pair,

691
00:35:10,910 --> 00:35:11,860
and such.

692
00:35:11,860 --> 00:35:13,955
And then we see,
our wavefunction

693
00:35:13,955 --> 00:35:17,890
of identical fermion
pairs slash bosons is just

694
00:35:17,890 --> 00:35:19,800
the product of those pairs.

695
00:35:19,800 --> 00:35:22,666
But the pairing wavefunction,
the molecular wavefunction

696
00:35:22,666 --> 00:35:24,040
if you want, so
[INAUDIBLE], it's

697
00:35:24,040 --> 00:35:27,060
the same for all particles.

698
00:35:27,060 --> 00:35:30,030
So this is just like in the
Bose-Einstein condensate.

699
00:35:30,030 --> 00:35:32,720
We have a product state of n
times the same wavefunction.

700
00:35:32,720 --> 00:35:35,070
Here we have a product
state of n over two

701
00:35:35,070 --> 00:35:36,907
times the same
pairing wavefunction.

702
00:35:40,780 --> 00:35:44,650
We should anti-symmetrize
that's important.

703
00:35:44,650 --> 00:35:47,310
Now let us see what we got.

704
00:35:47,310 --> 00:35:50,710
Well, we describe everything
in second quantization.

705
00:35:50,710 --> 00:35:54,070
We use field operators
which create particles

706
00:35:54,070 --> 00:35:55,560
at a certain
position, and then we

707
00:35:55,560 --> 00:35:57,080
have the wavefunction
of the pairs

708
00:35:57,080 --> 00:36:00,160
to make sure they are created
in the right position.

709
00:36:09,980 --> 00:36:12,070
We transform everything
our operators

710
00:36:12,070 --> 00:36:16,040
and do Fourier transform.

711
00:36:16,040 --> 00:36:18,330
We go from operators
and positions space,

712
00:36:18,330 --> 00:36:20,310
to operators in momentum space.

713
00:36:20,310 --> 00:36:22,517
But now, it's just formalism.

714
00:36:22,517 --> 00:36:23,600
You can read up about it..

715
00:36:23,600 --> 00:36:25,891
But the important thing is
now I define a pair creation

716
00:36:25,891 --> 00:36:26,520
operator.

717
00:36:26,520 --> 00:36:29,910
This pair creation operator
creates one of those pairs.

718
00:36:29,910 --> 00:36:33,190
And what it does is it creates.

719
00:36:33,190 --> 00:36:37,000
We want to focus on the
pairs we zero momentum.

720
00:36:37,000 --> 00:36:40,720
It forms a plain wave fermion
in the momentum state with minus

721
00:36:40,720 --> 00:36:43,620
k, with opposite
spin with plus k.

722
00:36:43,620 --> 00:36:46,450
So these are sort of two
opposite, plain waves

723
00:36:46,450 --> 00:36:47,540
with zero momentum.

724
00:36:47,540 --> 00:36:52,560
But now the leading factor
is the Fourier transform

725
00:36:52,560 --> 00:36:55,670
of this pairing wavefunction.

726
00:36:55,670 --> 00:36:58,210
So if you want to assume it's
a molecule or wavefunction,

727
00:36:58,210 --> 00:37:01,240
Fourier transformers, and
this is the leading factor.

728
00:37:01,240 --> 00:37:04,795
And with that definition, this
is now the creation operator

729
00:37:04,795 --> 00:37:06,975
of a molecule consisting
of two fermions.

730
00:37:10,230 --> 00:37:10,980
OK.

731
00:37:10,980 --> 00:37:15,290
So now, we can say, well
maybe a good idea of what

732
00:37:15,290 --> 00:37:19,100
describes our system
is now, so to speak,

733
00:37:19,100 --> 00:37:21,210
a Bose-Einstein
condensate, where

734
00:37:21,210 --> 00:37:24,530
we create n over two
pairs out of the vacuum.

735
00:37:27,290 --> 00:37:32,160
Well, first I want to show
you that this is actually

736
00:37:32,160 --> 00:37:34,080
the BCS wavefunction.

737
00:37:34,080 --> 00:37:37,380
Most people do not to use this
form for the BCS wavefunction,

738
00:37:37,380 --> 00:37:39,790
but I want to show you
mathematically identical

739
00:37:39,790 --> 00:37:41,530
the BCS wavefunction.

740
00:37:41,530 --> 00:37:45,670
But first I have to clarify is
that really a Bose-condensate?

741
00:37:45,670 --> 00:37:48,920
If I would have shown you this
equation without telling you

742
00:37:48,920 --> 00:37:54,170
anything about it, ya, this
looks -- n over 2 identical, b,

743
00:37:54,170 --> 00:37:56,460
boson-- Bosonic particles.

744
00:37:56,460 --> 00:37:58,260
But now we have to
be more careful.

745
00:37:58,260 --> 00:38:01,220
We have our definition
for the pairing operator.

746
00:38:01,220 --> 00:38:03,580
And now we can calculate.

747
00:38:03,580 --> 00:38:04,440
Are those bosons?

748
00:38:04,440 --> 00:38:06,070
Let's just look at
the communicator

749
00:38:06,070 --> 00:38:08,130
between b dega, b dega, bb.

750
00:38:08,130 --> 00:38:09,520
Yes fine fine.

751
00:38:09,520 --> 00:38:15,470
But now, the commutator
between b and b dega is not 1.

752
00:38:15,470 --> 00:38:19,220
It would be 1 if
the probability,

753
00:38:19,220 --> 00:38:24,360
that the population of one
of those plain waves k,

754
00:38:24,360 --> 00:38:27,380
is much, much
smaller than 1, then

755
00:38:27,380 --> 00:38:29,830
the commutator is full filled.

756
00:38:29,830 --> 00:38:35,920
But the population
of momentum states

757
00:38:35,920 --> 00:38:38,530
depends on the size of the pair.

758
00:38:38,530 --> 00:38:43,830
If the size is huge, you have
low lying momentum states,

759
00:38:43,830 --> 00:38:45,270
which have a huge population.

760
00:38:45,270 --> 00:38:49,530
If the size is small of
the pair, you can say,

761
00:38:49,530 --> 00:38:52,130
in a tightly-bound system,
particles have so much

762
00:38:52,130 --> 00:38:54,670
0 point energy that
every given momentum

763
00:38:54,670 --> 00:38:58,100
state has pretty
much 0 occupation.

764
00:38:58,100 --> 00:39:00,920
So in other words,
this criterion here,

765
00:39:00,920 --> 00:39:06,815
tells us that our
pairs are bosons if nk

766
00:39:06,815 --> 00:39:08,930
is much smaller than 1.

767
00:39:08,930 --> 00:39:13,350
But if nk is not smaller
than 1, then those pairs

768
00:39:13,350 --> 00:39:17,020
act out in a Fermionic way.

769
00:39:17,020 --> 00:39:20,120
And the criterion
happens exactly when

770
00:39:20,120 --> 00:39:23,090
the size of the pairs is
comparable to the particle

771
00:39:23,090 --> 00:39:25,060
spacing.

772
00:39:25,060 --> 00:39:27,560
So in other words,
what we have described

773
00:39:27,560 --> 00:39:30,440
with this wavefunction
is, we have a wavefunction

774
00:39:30,440 --> 00:39:32,410
which is a two
Bose-Einstein condensate

775
00:39:32,410 --> 00:39:33,460
for tightly bound pairs.

776
00:39:33,460 --> 00:39:35,320
That's what we expect.

777
00:39:35,320 --> 00:39:39,610
It still has this form,
but those pairs now

778
00:39:39,610 --> 00:39:42,780
feel the Fermi pressure.

779
00:39:42,780 --> 00:39:45,570
And therefore, this
is automatically

780
00:39:45,570 --> 00:39:46,905
taken account in the formalism.

781
00:39:46,905 --> 00:39:47,405
Colin

782
00:39:47,405 --> 00:39:48,280
AUDIENCE: [INAUDIBLE]

783
00:39:50,632 --> 00:39:51,465
PROFESSOR: This one.

784
00:39:51,465 --> 00:39:53,890
AUDIENCE: [INAUDIBLE]

785
00:39:53,890 --> 00:39:57,320
PROFESSOR:It's a commutator
not the anti-commutator.

786
00:39:57,320 --> 00:40:00,230
It's bb dega minus b dega b.

787
00:40:00,230 --> 00:40:02,720
I just want to be careful here
because, for fermions, you

788
00:40:02,720 --> 00:40:04,440
often use the anti-commutator.

789
00:40:04,440 --> 00:40:06,580
But I want to check
with these bosons,

790
00:40:06,580 --> 00:40:08,151
so I use the Bosonic,
the commutator,

791
00:40:08,151 --> 00:40:09,150
not the anti-commutator.

792
00:40:12,610 --> 00:40:15,260
Let me just show you
one thing real quick.

793
00:40:15,260 --> 00:40:20,870
Namely, that this wavefunction
is identical to the famous BCS

794
00:40:20,870 --> 00:40:22,830
wavefunction.

795
00:40:22,830 --> 00:40:26,860
Well, almost, not quite.

796
00:40:26,860 --> 00:40:30,040
You know that this
is a state which

797
00:40:30,040 --> 00:40:34,380
has n particles,
n over two pairs.

798
00:40:34,380 --> 00:40:36,250
Remember when we
talked about photons,

799
00:40:36,250 --> 00:40:40,640
we had Fock states with n
particles, with n photons.

800
00:40:40,640 --> 00:40:45,220
But we also had coherent states,
where alpha square was n.

801
00:40:45,220 --> 00:40:49,190
But this was sort of, the
classic coherent state.

802
00:40:49,190 --> 00:40:51,480
And this coherent
state could be expanded

803
00:40:51,480 --> 00:40:53,860
as a sum over Fock states.

804
00:40:53,860 --> 00:40:56,300
And one can now
say, to some extent,

805
00:40:56,300 --> 00:41:00,680
when it comes to certain
properties of light, n photons

806
00:41:00,680 --> 00:41:02,810
sometimes it's
important to distinguish

807
00:41:02,810 --> 00:41:04,970
whether it be a Fock
state or coherent state.

808
00:41:04,970 --> 00:41:06,540
But, for many
things, if you have

809
00:41:06,540 --> 00:41:09,337
n photons in the same
state, you can have a laser,

810
00:41:09,337 --> 00:41:10,920
then you have a
coherent state, or you

811
00:41:10,920 --> 00:41:12,470
have n photons in a cavity.

812
00:41:12,470 --> 00:41:14,500
And that may be similar.

813
00:41:14,500 --> 00:41:20,390
So let me show you what happens
when I take this Fock state,

814
00:41:20,390 --> 00:41:24,390
but construct what is now
the coherent state version.

815
00:41:24,390 --> 00:41:30,200
In the coherent state
version-- this here

816
00:41:30,200 --> 00:41:33,950
is a Fock state
with j particles--

817
00:41:33,950 --> 00:41:37,310
but now I take a
coherent superposition,

818
00:41:37,310 --> 00:41:39,395
and I've chosen my
factors such that I

819
00:41:39,395 --> 00:41:41,150
will get a coherent state.

820
00:41:41,150 --> 00:41:43,490
And indeed, I get
the coherent state.

821
00:41:46,770 --> 00:41:51,860
This is now sort of, what
the laser is for Fock state,

822
00:41:51,860 --> 00:41:55,320
is this coherent state for a
state with a specific Fermion

823
00:41:55,320 --> 00:41:56,290
number.

824
00:41:56,290 --> 00:41:58,505
And this state now,
when I just-- you

825
00:41:58,505 --> 00:42:00,790
can also say in many-body
physics-- we say,

826
00:42:00,790 --> 00:42:04,510
I just went from the
canonical ensemble,

827
00:42:04,510 --> 00:42:06,290
which has a fixed
number of atoms

828
00:42:06,290 --> 00:42:09,485
to the grand canonical ensemb.e,
where the number of atoms

829
00:42:09,485 --> 00:42:11,230
fluctuates.

830
00:42:11,230 --> 00:42:15,540
But now I can simply
use the definitions.

831
00:42:15,540 --> 00:42:18,310
If I Taylor expand
this component

832
00:42:18,310 --> 00:42:22,300
because all higher powers of the
Fermionic creation [INAUDIBLE]

833
00:42:22,300 --> 00:42:23,850
operators give 0.

834
00:42:23,850 --> 00:42:27,490
The Taylor expansion ends
after the first term.

835
00:42:27,490 --> 00:42:32,980
And with one more step, I have
the famous BCS wavefunction.

836
00:42:32,980 --> 00:42:35,100
So therefore the
BCS wavefunction,

837
00:42:35,100 --> 00:42:38,420
which has been widely used
to describe superconductors,

838
00:42:38,420 --> 00:42:46,620
is really identical to
creating n over 2 pass b,

839
00:42:46,620 --> 00:42:49,300
identical pass b, in the vacuum.

840
00:42:49,300 --> 00:42:52,000
And I'm just telling you
that, because that means now,

841
00:42:52,000 --> 00:42:56,430
mathematically, the BCS
wavefunction, which many people

842
00:42:56,430 --> 00:42:59,020
say describe a very,
very different physics,

843
00:42:59,020 --> 00:43:01,522
is formally equivalent
to the wavefunction

844
00:43:01,522 --> 00:43:02,730
for Bose-Einstein condensate.

845
00:43:09,230 --> 00:43:09,910
OK.

846
00:43:09,910 --> 00:43:13,840
So what have you done so far?

847
00:43:13,840 --> 00:43:17,960
We talked about one
particle physics,

848
00:43:17,960 --> 00:43:21,690
how very weak attraction
leads to bound states.

849
00:43:21,690 --> 00:43:27,900
We realized that if fermions
live on top of a Fermi sea,

850
00:43:27,900 --> 00:43:31,636
it gives at least the binding,
the energetic consideration,

851
00:43:31,636 --> 00:43:33,930
a two dimension character.

852
00:43:33,930 --> 00:43:35,610
And we have two states.

853
00:43:35,610 --> 00:43:39,200
And we have bound states for
infinitesimal attraction.

854
00:43:39,200 --> 00:43:42,840
And now we have
used our intuition.

855
00:43:42,840 --> 00:43:46,770
If those pairs exists, if they
form kind of a Bose-Einstein

856
00:43:46,770 --> 00:43:51,410
condensate to construct
a trial wavefunction.

857
00:43:51,410 --> 00:43:54,110
And now I want to take
this trial wavefunction

858
00:43:54,110 --> 00:43:58,700
and show you how we can use
it to solve the Many-Body

859
00:43:58,700 --> 00:44:00,960
Hamiltonian.

860
00:44:00,960 --> 00:44:05,820
So far, single particle
physics, two particle physics,

861
00:44:05,820 --> 00:44:08,170
intuition to get a
trial wavefunction,

862
00:44:08,170 --> 00:44:10,680
but now we go to
many-body problem.

863
00:44:10,680 --> 00:44:11,455
Colin.

864
00:44:11,455 --> 00:44:14,466
AUDIENCE: Can you go
to the last slide?

865
00:44:14,466 --> 00:44:18,434
So the BCS wavefunction has the
normal component and the parody

866
00:44:18,434 --> 00:44:19,922
component.

867
00:44:19,922 --> 00:44:24,390
Where does the normal
component come into the Bose--

868
00:44:24,390 --> 00:44:25,620
PROFESSOR: This one?

869
00:44:25,620 --> 00:44:28,100
Oh we are talking
here about t equals 0.

870
00:44:28,100 --> 00:44:29,980
We are talking about
a wavefunction.

871
00:44:29,980 --> 00:44:31,771
We are not talking
about finite temperature

872
00:44:31,771 --> 00:44:34,540
where you have distribution
over excited states.

873
00:44:34,540 --> 00:44:37,690
This is really
temperature physics.

874
00:44:37,690 --> 00:44:38,690
Was that your question?

875
00:44:41,410 --> 00:44:42,685
100% superfluid fraction.

876
00:44:47,890 --> 00:44:51,294
OK, we are back
now. [? Timor. ?]

877
00:44:51,294 --> 00:44:52,960
AUDIENCE: Just before
you get into this.

878
00:44:52,960 --> 00:44:55,610
This maybe a silly
question, but why is it

879
00:44:55,610 --> 00:44:58,135
that up has to pair with down.

880
00:44:58,135 --> 00:45:01,777
It seems ok for up to
pair with up, right?

881
00:45:01,777 --> 00:45:02,360
PROFESSOR: No.

882
00:45:02,360 --> 00:45:05,090
What happens is, we
have assumed here

883
00:45:05,090 --> 00:45:06,770
that we have
short-range interaction.

884
00:45:06,770 --> 00:45:09,313
And up and up can never
come close together

885
00:45:09,313 --> 00:45:10,812
because of poly
exclusion principal.

886
00:45:10,812 --> 00:45:14,402
AUDIENCE: But then in
principal, if you have key wave

887
00:45:14,402 --> 00:45:16,330
attraction, it could work.

888
00:45:16,330 --> 00:45:20,050
PROFESSOR: You're talking
about now, different pairing

889
00:45:20,050 --> 00:45:20,890
mechanisms.

890
00:45:20,890 --> 00:45:26,080
I'm talking here about the plain
vanilla BCS s-wave pairing.

891
00:45:26,080 --> 00:45:28,000
And that means spin
up and spin down pair.

892
00:45:31,640 --> 00:45:33,220
Also, this is where
I can work out

893
00:45:33,220 --> 00:45:36,310
the analogy with BEC-BCS
cross-over and such.

894
00:45:36,310 --> 00:45:38,450
So I'm just giving you
the simplest example.

895
00:45:38,450 --> 00:45:41,980
But yes, in helium-3
at high magnetic field,

896
00:45:41,980 --> 00:45:46,670
you have one of the phases where
up and up pair for key waves.

897
00:45:46,670 --> 00:45:49,500
It's a triplet, superfluid.

898
00:45:49,500 --> 00:45:51,930
OK so we the Many-Body
Hamiltonian, and what

899
00:45:51,930 --> 00:45:55,390
I want you to sort
of, enjoy now is,

900
00:45:55,390 --> 00:45:58,220
I took you last class
through the bosons.

901
00:45:58,220 --> 00:46:02,000
And you should maybe see how
the Fermions are different,

902
00:46:02,000 --> 00:46:04,450
but we use similar concepts.

903
00:46:04,450 --> 00:46:06,792
And so you may ask yourself
at the end of the day,

904
00:46:06,792 --> 00:46:09,000
"Are the fermions really so
different from the bosons

905
00:46:09,000 --> 00:46:10,375
when it comes to
super fluidity?"

906
00:46:10,375 --> 00:46:11,510
But we're getting there.

907
00:46:14,300 --> 00:46:15,940
Let's make the
same approximation,

908
00:46:15,940 --> 00:46:19,830
we use the delta function here.

909
00:46:19,830 --> 00:46:21,410
And we Fourier transform.

910
00:46:21,410 --> 00:46:25,430
We go from position
space to momentum space.

911
00:46:25,430 --> 00:46:30,220
But now comes one
important approximation.

912
00:46:30,220 --> 00:46:31,570
Fermions are fairly complicated.

913
00:46:31,570 --> 00:46:34,310
Look I mean, we have
products of four operators

914
00:46:34,310 --> 00:46:37,910
and we integrate over three
momenta, Initial momenta,

915
00:46:37,910 --> 00:46:41,017
and to the initial momentum,
the final momentum,

916
00:46:41,017 --> 00:46:42,100
and the momentum transfer.

917
00:46:42,100 --> 00:46:44,430
So we have an integral
over three momenta.

918
00:46:44,430 --> 00:46:47,290
We have to do something,
this is too complicated.

919
00:46:47,290 --> 00:46:52,156
And the BCS approximation
is, the only focus

920
00:46:52,156 --> 00:46:58,400
in these sum of pair operators
that we scatter, or we have.

921
00:46:58,400 --> 00:47:02,340
k prime minus k prime
scatters into k minus k.

922
00:47:02,340 --> 00:47:04,360
So we pretty much
save whenever we have,

923
00:47:04,360 --> 00:47:06,470
in the interaction
term, two fermions.

924
00:47:06,470 --> 00:47:09,860
We are only looking
at pairs, where

925
00:47:09,860 --> 00:47:13,220
the center of mass
momentum is 0.

926
00:47:13,220 --> 00:47:15,810
You can say it's an
uncontrolled approximation,

927
00:47:15,810 --> 00:47:18,260
but it's based on intuition.

928
00:47:18,260 --> 00:47:20,860
So the hope is that
we still capture

929
00:47:20,860 --> 00:47:25,014
the essence of super fluidity by
doing this major simplification

930
00:47:25,014 --> 00:47:25,805
of the Hamiltonian.

931
00:47:29,210 --> 00:47:31,580
So now I want to show you
that this Hamiltonian can

932
00:47:31,580 --> 00:47:33,030
be solved in two ways.

933
00:47:33,030 --> 00:47:34,730
And this really
connects me with things

934
00:47:34,730 --> 00:47:36,890
you have learned previously.

935
00:47:36,890 --> 00:47:40,380
One is, we can use as
a variational ansatz

936
00:47:40,380 --> 00:47:44,080
these pair of wavefunctions
which I just constructed.

937
00:47:44,080 --> 00:47:46,850
That's actually what Bardeen
Cooper Schrieffer did, but they

938
00:47:46,850 --> 00:47:50,020
used, as a tidal wave
function, the BCS wavefunction

939
00:47:50,020 --> 00:47:53,510
with the use and [? Weiss, ?]
which is identical.

940
00:47:53,510 --> 00:47:55,820
But then I want to
show you, partially

941
00:47:55,820 --> 00:47:58,209
because I'm feeling
nostalgic about the bosons,

942
00:47:58,209 --> 00:48:00,375
that you can also use the
Bogoliubov transformation.

943
00:48:00,375 --> 00:48:03,430
The same method we
learned for the bosons.

944
00:48:03,430 --> 00:48:05,530
So variation on
this just means use

945
00:48:05,530 --> 00:48:09,220
this BCS wavefunction,
which I just showed you,

946
00:48:09,220 --> 00:48:11,900
plug it into the
Hamiltonian, and all

947
00:48:11,900 --> 00:48:18,350
you do is, you vary the
coefficients u and v until you

948
00:48:18,350 --> 00:48:21,540
find the minimum of the energy.

949
00:48:21,540 --> 00:48:26,110
And this gives you the
famous BCS solution.

950
00:48:26,110 --> 00:48:31,941
And this is the plain
vanilla formalism

951
00:48:31,941 --> 00:48:33,940
for superconductors in
Fermionic super fluidity.

952
00:48:36,520 --> 00:48:39,200
OK I don't have time
to go into details,

953
00:48:39,200 --> 00:48:41,640
but I think you've
got the big picture.

954
00:48:41,640 --> 00:48:44,710
But let me know just
emphasize the similarities

955
00:48:44,710 --> 00:48:47,280
we had with the bosons.

956
00:48:47,280 --> 00:48:53,670
We have a Hamiltonian, even
with the BCS approximations

957
00:48:53,670 --> 00:48:58,240
that we got writ over 1,
sum over momenta by focusing

958
00:48:58,240 --> 00:49:01,000
on the pairs with c
[INAUDIBLE] mass momentum.

959
00:49:01,000 --> 00:49:03,460
You still have products
of four operators.

960
00:49:03,460 --> 00:49:05,310
We want to get rid of it.

961
00:49:05,310 --> 00:49:09,560
But now we use a
mean field idea.

962
00:49:09,560 --> 00:49:12,530
We always get rid of
products of four operators

963
00:49:12,530 --> 00:49:15,070
by grabbing one or two
of them, and saying,

964
00:49:15,070 --> 00:49:16,520
"we don't look at
the operator, we

965
00:49:16,520 --> 00:49:19,340
look at the expectation value,
which is the mean field."

966
00:49:19,340 --> 00:49:22,430
So we want to make a
mean field approximation.

967
00:49:22,430 --> 00:49:25,720
But when you go for
bosons more simply,

968
00:49:25,720 --> 00:49:29,934
you just took the operators,
c zero, zero momentum,

969
00:49:29,934 --> 00:49:31,350
and replacing with
the square root

970
00:49:31,350 --> 00:49:34,380
of n 0, the number of
atoms in the condensate.

971
00:49:34,380 --> 00:49:36,900
But now we have to be a
little bit more subtle.

972
00:49:36,900 --> 00:49:41,690
So what we should now define,
is a pairing mean field.

973
00:49:41,690 --> 00:49:46,350
Not an expectation value of
c 0, but an expectation value

974
00:49:46,350 --> 00:49:49,800
of this creational annihilation
operators four pairs.

975
00:49:49,800 --> 00:49:54,410
A little bit more abstract,
but it follows the same logic.

976
00:49:54,410 --> 00:49:57,640
And then we do the
decoupling approximation,

977
00:49:57,640 --> 00:50:00,580
which I lectured about to you
with a [? mod ?] insulator,

978
00:50:00,580 --> 00:50:05,290
that whenever we have products
of operators, of products

979
00:50:05,290 --> 00:50:09,770
of pair operators now,
we take the mean value,

980
00:50:09,770 --> 00:50:14,200
but we neglect the product
of the fluctuations.

981
00:50:14,200 --> 00:50:17,370
Exactly what I've shown you
before when we described

982
00:50:17,370 --> 00:50:20,530
the [? mod ?] insulator
to superfluid transition.

983
00:50:20,530 --> 00:50:21,030
OK.

984
00:50:24,320 --> 00:50:28,850
This is the pairing
field with an index k

985
00:50:28,850 --> 00:50:32,010
and eventually it's
important to define a delta.

986
00:50:32,010 --> 00:50:37,020
The famous gap in the BCS,
with the BCS COE, which

987
00:50:37,020 --> 00:50:40,750
is a sum over all
pairing fields with k.

988
00:50:40,750 --> 00:50:44,680
And it is this delta which
plays the role of the condensate

989
00:50:44,680 --> 00:50:48,320
wavefunction exactly
the same way as psi,

990
00:50:48,320 --> 00:50:51,810
the macroscopic wavefunction
in the [? Kospiesky ?] equation

991
00:50:51,810 --> 00:50:53,700
[INAUDIBLE].

992
00:50:53,700 --> 00:50:57,600
So you do the same thing,
we just write it down,

993
00:50:57,600 --> 00:51:02,170
we drop quadratic terms,
which are fluctuation terms.

994
00:51:02,170 --> 00:51:04,960
And the moment we
have now, simply

995
00:51:04,960 --> 00:51:07,340
bi-linear product of operators.

996
00:51:07,340 --> 00:51:09,750
We do exactly the same
as we did with bosons,

997
00:51:09,750 --> 00:51:13,560
we transform to a new
set of operators, which

998
00:51:13,560 --> 00:51:15,910
is the Bogoliubov
Transformation.

999
00:51:15,910 --> 00:51:17,720
Of course, this time,
we want to make sure

1000
00:51:17,720 --> 00:51:22,990
that those transforming
Fermionic operators,

1001
00:51:22,990 --> 00:51:27,360
we want to make sure that
after transformation we

1002
00:51:27,360 --> 00:51:30,700
fulfill Fermionic commutator.

1003
00:51:30,700 --> 00:51:32,300
And then we determine
coefficients

1004
00:51:32,300 --> 00:51:43,630
in such a way that
cross-terms cancel out,

1005
00:51:43,630 --> 00:51:47,030
and what we obtain now is
a diagonalized Hamiltonian.

1006
00:51:47,030 --> 00:51:50,590
It's diagonalized, it has
the ground state energy.

1007
00:51:50,590 --> 00:51:52,750
And this, you now
realize, this is

1008
00:51:52,750 --> 00:51:54,690
nothing like an
harmonic oscillator.

1009
00:51:54,690 --> 00:51:58,460
It's a quasi particle gas
of independent excitations.

1010
00:51:58,460 --> 00:52:02,260
Gamma k is an excitation
with energy E K,

1011
00:52:02,260 --> 00:52:05,430
and this means we've
diagonalized the problem.

1012
00:52:05,430 --> 00:52:07,507
I can't do justice
to the formalism.

1013
00:52:07,507 --> 00:52:09,090
I think the best I
can do here for you

1014
00:52:09,090 --> 00:52:12,270
is to show how the
same ideas come back

1015
00:52:12,270 --> 00:52:14,460
and solve the same problem.

1016
00:52:14,460 --> 00:52:16,322
Also everything is
now for fermions.

1017
00:52:16,322 --> 00:52:17,286
Mark.

1018
00:52:17,286 --> 00:52:30,995
AUDIENCE: [INAUDIBLE]
Separate to [INAUDIBLE]

1019
00:52:30,995 --> 00:52:36,563
At the very top of the line,
you have [INAUDIBLE] then

1020
00:52:36,563 --> 00:52:38,920
the next slide,
you have the delta,

1021
00:52:38,920 --> 00:52:44,240
which is the sum over k
expectation [INAUDIBLE] sum.

1022
00:52:44,240 --> 00:52:46,580
PROFESSOR: I derived it
myself because it confused me.

1023
00:52:46,580 --> 00:52:49,150
But I can just reassure you.

1024
00:52:49,150 --> 00:52:53,170
If you take the
original Hamiltonian,

1025
00:52:53,170 --> 00:52:56,540
pluck this into
here, and neglect

1026
00:52:56,540 --> 00:53:00,540
the product of the
smaller fluctuations,

1027
00:53:00,540 --> 00:53:02,600
you get exactly this equation.

1028
00:53:02,600 --> 00:53:05,660
It's really just accounting
and regrouping terms.

1029
00:53:05,660 --> 00:53:07,890
There is no assumption,
no concept involved,

1030
00:53:07,890 --> 00:53:10,550
it's really just plugging it in.

1031
00:53:10,550 --> 00:53:14,070
The physics is that we
neglect the correlation

1032
00:53:14,070 --> 00:53:17,530
of the fluctuations.

1033
00:53:17,530 --> 00:53:19,316
OK.

1034
00:53:19,316 --> 00:53:21,190
Yeah, I think I want to
wrap up the fermions.

1035
00:53:21,190 --> 00:53:23,290
We still want to talk a
little bit about ions,

1036
00:53:23,290 --> 00:53:27,450
but we have now solved it.

1037
00:53:27,450 --> 00:53:29,200
Either with the a
variational [? onsets ?]

1038
00:53:29,200 --> 00:53:30,900
or with the Bogoliubov
Transformation.

1039
00:53:30,900 --> 00:53:35,850
Two very different ways
lead to the same solution.

1040
00:53:35,850 --> 00:53:38,670
We have quasi particles.

1041
00:53:38,670 --> 00:53:41,509
Here, we determined
what the energies are.

1042
00:53:41,509 --> 00:53:43,300
They are a little bit
complicated equations

1043
00:53:43,300 --> 00:53:45,480
to find what the
quasi-particle energies are.

1044
00:53:45,480 --> 00:53:49,180
But everything is known, we
can now throw in temperature.

1045
00:53:49,180 --> 00:53:51,080
Colin wanted final temperature.

1046
00:53:51,080 --> 00:53:53,860
Well what happens with this
Hamiltonian final temperature?

1047
00:53:53,860 --> 00:53:56,840
Well, we just have
occupation number

1048
00:53:56,840 --> 00:53:59,740
for those quasi-particles,
the thermal occupation,

1049
00:53:59,740 --> 00:54:02,020
and we have to self-consistency.

1050
00:54:02,020 --> 00:54:05,560
Now look what happens in the
presence of quasi particles.

1051
00:54:05,560 --> 00:54:08,350
This eventually
leads to an equation

1052
00:54:08,350 --> 00:54:10,230
for the critical temperature.

1053
00:54:10,230 --> 00:54:12,850
So what we see
here is a function

1054
00:54:12,850 --> 00:54:15,900
of the inverse
scattering lengths.

1055
00:54:15,900 --> 00:54:18,910
Here we have a BEC
temperature which is constant.

1056
00:54:18,910 --> 00:54:23,440
And when we go on the BCS
side, where we no longer form

1057
00:54:23,440 --> 00:54:26,750
bound molecules, we
have is exponentially

1058
00:54:26,750 --> 00:54:30,731
decaying temperature for
very small attraction.

1059
00:54:30,731 --> 00:54:33,230
The transition temperature to
the Superfund superfluid state

1060
00:54:33,230 --> 00:54:34,188
is exponentially small.

1061
00:54:38,260 --> 00:54:42,510
All this is theory, but it has
been experimentally observed.

1062
00:54:42,510 --> 00:54:46,470
If you want to pair fermions,
we just need two states.

1063
00:54:46,470 --> 00:54:49,780
At MIT we use mainly Lithium-6.

1064
00:54:49,780 --> 00:54:52,180
Lithium-6 has many
hyper find states,

1065
00:54:52,180 --> 00:54:54,190
but we just use
the two lowest one.

1066
00:54:54,190 --> 00:54:58,250
In a high magnetic field, where
we have a Feshbach resonance,

1067
00:54:58,250 --> 00:55:00,940
we pair the particles--
well we needed the lasers

1068
00:55:00,940 --> 00:55:05,540
and equipment-- and then we
observe pair condensates.

1069
00:55:05,540 --> 00:55:07,690
They look exactly like
atomic condensates,

1070
00:55:07,690 --> 00:55:09,920
but their pairs inside.

1071
00:55:09,920 --> 00:55:11,470
That they're really
pairs inside we

1072
00:55:11,470 --> 00:55:14,050
found out with a f spectroscopy.

1073
00:55:14,050 --> 00:55:19,790
And eventually when you locate
the gases we find vortices.

1074
00:55:19,790 --> 00:55:22,880
And you know already,
vortices are the smoking gun

1075
00:55:22,880 --> 00:55:23,690
of super fluidity.

1076
00:55:26,330 --> 00:55:30,790
So with that I've rushed you
through an important and very

1077
00:55:30,790 --> 00:55:37,230
elegant chapter of physics,
the BEC BCS cross-over.

1078
00:55:37,230 --> 00:55:39,390
Let me just sort of
end with an outlook

1079
00:55:39,390 --> 00:55:42,900
that I've talked about
basic phenomenon in bosons

1080
00:55:42,900 --> 00:55:46,270
and fermions, both centered
about super fluidity,

1081
00:55:46,270 --> 00:55:48,270
because I think it was
pedagogical to show

1082
00:55:48,270 --> 00:55:51,540
you the differences
and the similarities.

1083
00:55:51,540 --> 00:55:55,440
But we can now regard those
ultra-cold bosons and fermions

1084
00:55:55,440 --> 00:55:58,210
as building blocks of
quantum simulators,

1085
00:55:58,210 --> 00:56:02,010
building blocks to
understand, in its simplest

1086
00:56:02,010 --> 00:56:04,840
manifestation,
interesting physics.

1087
00:56:04,840 --> 00:56:07,780
So you can see, the Bosic
gases are quantum simulator

1088
00:56:07,780 --> 00:56:09,630
for superfluid helium.

1089
00:56:09,630 --> 00:56:15,780
The Fermi gases are a quantum
simulator for superconductors.

1090
00:56:15,780 --> 00:56:17,694
We briefly talked
about optical lattices,

1091
00:56:17,694 --> 00:56:19,360
which can be regarded
as a [? quantum ?]

1092
00:56:19,360 --> 00:56:21,980
simulator for
crystalline materials.

1093
00:56:21,980 --> 00:56:25,220
But there is much more you can
do with those building blocks.