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PROFESSOR: OK, good afternoon.

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00:00:30,140 --> 00:00:34,820
So this week we talk
about quantum gases,

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ultra-cold atomic gases.

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00:00:37,930 --> 00:00:43,080
And sure, they're ideal Bose
gases, ideal Fermi gases,

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00:00:43,080 --> 00:00:45,360
but I will spend one
or two minutes on each,

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because that's exactly solvable.

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00:00:47,740 --> 00:00:49,250
It's in all the textbooks.

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00:00:49,250 --> 00:00:51,560
That's simple.

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00:00:51,560 --> 00:00:55,700
Quantum gases become interesting
because of interactions.

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00:00:55,700 --> 00:00:58,610
And in my lecture today
and my last lecture

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00:00:58,610 --> 00:01:01,070
I want to introduce you
to Bose gases and Fermi

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00:01:01,070 --> 00:01:03,110
gases with inter actions.

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00:01:03,110 --> 00:01:07,180
And they both turn superfluid,
and the superfluid properties

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00:01:07,180 --> 00:01:10,800
are determined by the
interactions between the atoms.

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00:01:10,800 --> 00:01:15,110
So the purpose of those
lectures is number one,

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00:01:15,110 --> 00:01:17,740
to acquaint to you with
important phenomena

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00:01:17,740 --> 00:01:21,690
in cold gases-- superfluidity,
superfluidity in lattices,

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00:01:21,690 --> 00:01:28,090
superfluidity just in
free gas, in a normal gas

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00:01:28,090 --> 00:01:31,270
without lattice, and
superfluidity of fermions.

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00:01:31,270 --> 00:01:33,090
But at the same
time, I also want

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00:01:33,090 --> 00:01:36,470
to have sort of a theme for you
how we deal with interactions.

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00:01:36,470 --> 00:01:43,110
And that kind of theme
is how theoretically we

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00:01:43,110 --> 00:01:46,400
reduce unsolvable
Hamiltonians to Hamiltonians

31
00:01:46,400 --> 00:01:47,850
which can be solved.

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00:01:47,850 --> 00:01:50,170
And you will see that
actually appearing

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00:01:50,170 --> 00:01:55,240
in different situations
with similarities,

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00:01:55,240 --> 00:01:57,220
but important differences.

35
00:01:57,220 --> 00:01:59,110
So just to remind
you, we started out

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00:01:59,110 --> 00:02:03,210
with the interacting Bose
gas in a homogeneous system.

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00:02:03,210 --> 00:02:05,615
We have a very general
way to describe

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00:02:05,615 --> 00:02:09,250
scattering two particles with
initial momentum disappear

39
00:02:09,250 --> 00:02:11,760
and two particles
with momenta appear.

40
00:02:11,760 --> 00:02:14,320
This is a scattering event.

41
00:02:14,320 --> 00:02:17,750
Now this means we have products
of four operators, which

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00:02:17,750 --> 00:02:19,760
is very difficult to solve.

43
00:02:19,760 --> 00:02:23,690
And the Bogoliubov approximation
which we discussed on Monday

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00:02:23,690 --> 00:02:28,420
replaces the operator for
the condensate for the zero

45
00:02:28,420 --> 00:02:33,900
momentum state with a Z number,
saying that N 0, N 0 plus 1

46
00:02:33,900 --> 00:02:36,180
is the same-- a little
bit waving your hands.

47
00:02:36,180 --> 00:02:39,410
But it's also you can say
the macroscopic limit when

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00:02:39,410 --> 00:02:42,250
we have a photon field
with many, many photons,

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00:02:42,250 --> 00:02:44,650
we can use a Z number
in our Hamiltonian

50
00:02:44,650 --> 00:02:46,650
to describe the electric field.

51
00:02:46,650 --> 00:02:48,520
So that's the same spirit.

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00:02:48,520 --> 00:02:50,470
So maybe I should emphasize it.

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00:02:50,470 --> 00:02:52,090
This is for atoms.

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00:02:52,090 --> 00:02:56,121
What you're used to do with
photons for your whole life.

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00:02:56,121 --> 00:02:56,620
OK.

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00:02:56,620 --> 00:02:59,060
With that we have
transformed the Hamiltonian

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00:02:59,060 --> 00:03:01,740
into a bilinear expression.

58
00:03:01,740 --> 00:03:04,920
And so at the moment your
bilinear expression--

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00:03:04,920 --> 00:03:06,720
you do a Bogoliubov
transformation.

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00:03:06,720 --> 00:03:12,110
You simply diagonalize it by
finding a new set operators

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00:03:12,110 --> 00:03:16,320
where the cross term
between a and b or between a

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00:03:16,320 --> 00:03:20,240
of k and a of minus k disappear.

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00:03:20,240 --> 00:03:23,340
And then you've solved it.

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00:03:23,340 --> 00:03:28,350
And this is what we arrived at
at the end of the last lecture.

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00:03:28,350 --> 00:03:30,700
With those
approximations, we have

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00:03:30,700 --> 00:03:32,250
diagonalized the Hamiltonian.

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00:03:32,250 --> 00:03:35,650
Our Hamiltonian is now, you
can say harmonic oscillator

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00:03:35,650 --> 00:03:38,040
Hamiltonian, or
you can see it has

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00:03:38,040 --> 00:03:42,430
become a gas of non-interacting
quasi particles.

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00:03:42,430 --> 00:03:45,620
Each of those operators
creates a quasi particle,

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00:03:45,620 --> 00:03:48,360
and the quasi particle
energy is U of k.

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00:03:48,360 --> 00:03:50,880
And I explained to
you that as expected,

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00:03:50,880 --> 00:03:54,070
the quasi particle
energy is simply

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00:03:54,070 --> 00:03:57,710
particles with momentum
h-bar k for high momenta,

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00:03:57,710 --> 00:04:00,460
because in high excitation
of Bose quantums

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it is a free particle.

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00:04:02,000 --> 00:04:06,400
But a low-lying excitation
is affected in a major way

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00:04:06,400 --> 00:04:07,880
by all of these
interactions with

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00:04:07,880 --> 00:04:09,610
the Bose-Einstein condensate.

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00:04:09,610 --> 00:04:12,880
And that turns the quadratic
dispersion relation

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into a linear one,
or you can say

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00:04:15,350 --> 00:04:18,500
that turns the free particle
into a photon or something.

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00:04:21,240 --> 00:04:22,890
So that's my review
of the last class.

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Are there any questions
before we go further?

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00:04:27,050 --> 00:04:27,740
Colin.

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00:04:27,740 --> 00:04:30,073
AUDIENCE: Does this require
low density and [INAUDIBLE],

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00:04:30,073 --> 00:04:34,267
or one or the other?

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00:04:34,267 --> 00:04:36,850
PROFESSOR: We come a little bit
later to that, but in the end,

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00:04:36,850 --> 00:04:38,500
there is a small parameter.

90
00:04:38,500 --> 00:04:41,790
The small parameter will be N
a cubed-- the gas parameter--

91
00:04:41,790 --> 00:04:45,090
N, the density, a cubed,
the scattering length.

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00:04:45,090 --> 00:04:48,960
It's usually the dimensionless
combination of the two

93
00:04:48,960 --> 00:04:51,940
which decides whether we are
in the weakly or strongly

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00:04:51,940 --> 00:04:53,670
interacting limit.

95
00:04:53,670 --> 00:04:57,180
There is another assumption
which we have made here,

96
00:04:57,180 --> 00:05:00,560
but it's related to that-- as
you will see a few moments--

97
00:05:00,560 --> 00:05:03,010
that most of the are
in the condensate.

98
00:05:03,010 --> 00:05:04,620
We assumed N0 is peak.

99
00:05:04,620 --> 00:05:07,500
The condensate depletion-- the
number of atoms which are not

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00:05:07,500 --> 00:05:10,630
in the condensate,
t equals 0 is small.

101
00:05:10,630 --> 00:05:13,520
But let me just
first show you-- I

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00:05:13,520 --> 00:05:16,080
want to mix in
theory and experiment

103
00:05:16,080 --> 00:05:18,350
how sound can be observed.

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00:05:18,350 --> 00:05:21,240
I should actually say this
experiment is sort of dear

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00:05:21,240 --> 00:05:23,220
to my heart, because
at some point

106
00:05:23,220 --> 00:05:27,320
it clicked to me how the
new world of atomic physics

107
00:05:27,320 --> 00:05:29,550
connects with condensed
matter physics.

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00:05:29,550 --> 00:05:32,950
Soon after we had realized
Bose-Einstein condensates,

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00:05:32,950 --> 00:05:36,560
a famous condensed matter
theorist said hey, Wolfgang,

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00:05:36,560 --> 00:05:37,849
you should now observe sound.

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00:05:37,849 --> 00:05:38,640
Sound is important.

112
00:05:38,640 --> 00:05:41,580
Yeah, OK, but how do
you observe sound?

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00:05:41,580 --> 00:05:44,150
You know, use a piezo
and just kick the system

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00:05:44,150 --> 00:05:45,550
and create a sound wave.

115
00:05:45,550 --> 00:05:48,130
And I said, oh, gosh, this
guy doesn't know with that

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00:05:48,130 --> 00:05:51,280
if you put anything in contact
with the quantum gases--

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00:05:51,280 --> 00:05:53,540
a piezo-- the gases
will just stick to it.

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00:05:53,540 --> 00:05:55,577
He has no idea
what our system is.

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00:05:55,577 --> 00:05:57,785
But then I said wait a
moment-- we have to translate.

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00:05:57,785 --> 00:05:59,820
He says use a piezo.

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00:05:59,820 --> 00:06:01,100
Well, I have to translate.

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00:06:01,100 --> 00:06:02,840
In atomic physics
how do we kick atoms?

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00:06:02,840 --> 00:06:04,620
With a laser beam.

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00:06:04,620 --> 00:06:06,720
And in that moment,
I had the idea

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00:06:06,720 --> 00:06:09,110
that we can take a
Bose-Einstein condensate,

126
00:06:09,110 --> 00:06:12,970
suddenly switch on a blue
de-tuned repulsive laser beam.

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00:06:12,970 --> 00:06:15,980
This will do exactly as a
piezo-- create a density

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00:06:15,980 --> 00:06:18,630
perturbation, and then
the density perturbation

129
00:06:18,630 --> 00:06:21,600
will propagate with
the speed of sound.

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00:06:21,600 --> 00:06:23,720
And that worked.

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00:06:23,720 --> 00:06:28,030
That was just when Bose-Einstein
condensations were very fresh.

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00:06:28,030 --> 00:06:30,910
One of the first
scientific experiments

133
00:06:30,910 --> 00:06:33,230
done-- we switch on
the laser and you

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00:06:33,230 --> 00:06:36,520
see color-coded in red the
density perturbation which

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00:06:36,520 --> 00:06:38,080
propagates out of it.

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And the slope of this line
is the speed of sound.

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00:06:41,580 --> 00:06:44,090
And here we determined
the speed of sound

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00:06:44,090 --> 00:06:47,380
as a function of density.

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00:06:47,380 --> 00:06:50,840
So that's how phonons--
or at least how

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00:06:50,840 --> 00:06:53,180
the speed of sound
and wave packets which

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00:06:53,180 --> 00:06:56,750
propagate with the speed
of sound can be prepared.

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00:06:56,750 --> 00:06:59,880
I come back to phonons
and collective excitations

143
00:06:59,880 --> 00:07:01,830
in a few moments.

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00:07:01,830 --> 00:07:04,385
But let me first say
when we have diagonalized

145
00:07:04,385 --> 00:07:08,890
the Hamiltonian we know
everything we want to know.

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00:07:08,890 --> 00:07:11,720
I just focused on the
quasi particle energies,

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00:07:11,720 --> 00:07:15,330
but you also know the
ground-state energy.

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00:07:15,330 --> 00:07:19,300
And actually here Colin,
you see that the corrections

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00:07:19,300 --> 00:07:21,340
to the ground-state
energy scale with N

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00:07:21,340 --> 00:07:25,070
a cubed, so this is really the
small parameter in this system.

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00:07:25,070 --> 00:07:29,060
But we can also find out what is
the ground state wave function.

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00:07:29,060 --> 00:07:32,880
And let me use it to introduce
an important concept to you

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00:07:32,880 --> 00:07:35,940
called the quantum depletion.

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00:07:35,940 --> 00:07:38,890
When you have two atoms in the
condensator at zero momentum

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00:07:38,890 --> 00:07:43,510
and you switch on
the interactions,

156
00:07:43,510 --> 00:07:48,265
the delta function interaction
couples zero momentum state

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00:07:48,265 --> 00:07:50,740
to higher momentum states.

158
00:07:50,740 --> 00:07:52,850
So therefore, the
effect of interactions

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00:07:52,850 --> 00:07:56,840
is that the condensate is
not just at zero momentum.

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00:07:56,840 --> 00:08:00,010
It has some probability,
or some admixture,

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00:08:00,010 --> 00:08:02,000
of finite momentum states.

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00:08:02,000 --> 00:08:03,120
This is the ground state.

163
00:08:03,120 --> 00:08:06,400
This is how the Hamiltonian
is diagonalized.

164
00:08:06,400 --> 00:08:08,385
So from the Bogoliubov
approximation,

165
00:08:08,385 --> 00:08:12,130
where we introduced
this V and U parameter

166
00:08:12,130 --> 00:08:14,890
to transfer from one
set of Bose operators

167
00:08:14,890 --> 00:08:17,150
to another set of
Bose operators,

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00:08:17,150 --> 00:08:21,260
those coefficients
give us the population

169
00:08:21,260 --> 00:08:24,410
of those momentum states in the
[INAUDIBLE] of the condensate.

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00:08:24,410 --> 00:08:28,310
So if I now ask, what is
the condensate fraction?

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00:08:28,310 --> 00:08:31,070
What is the number of atoms
in the zero momentum states?

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00:08:31,070 --> 00:08:34,669
It's all atoms, but those
who have finite momentum.

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00:08:34,669 --> 00:08:39,330
And we find again the
small parameter N a cubed.

174
00:08:39,330 --> 00:08:42,190
Now this quantum
depletion allows me now

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00:08:42,190 --> 00:08:46,280
to make a distinction
between the cold atomic gas

176
00:08:46,280 --> 00:08:49,440
condensates and
superfluid helium form.

177
00:08:49,440 --> 00:08:54,480
In cold atomic gas condensates,
this correction is about 1%.

178
00:08:54,480 --> 00:08:59,430
So therefore, I can say
with 99% probability,

179
00:08:59,430 --> 00:09:02,410
or with 99% weight,
the condensate,

180
00:09:02,410 --> 00:09:04,790
the many-body wave
function of the condensate

181
00:09:04,790 --> 00:09:08,200
is just the zero momentum
state to the power N.

182
00:09:08,200 --> 00:09:12,030
And this very complicated
admixture of correlation

183
00:09:12,030 --> 00:09:19,800
into the ideal gas wave function
is only 1% for alkali gases.

184
00:09:19,800 --> 00:09:23,160
But for liquid helium,
the condensate fraction--

185
00:09:23,160 --> 00:09:26,030
even at zero
temperature-- is only 10%.

186
00:09:26,030 --> 00:09:27,930
So when people use
neutron scattering--

187
00:09:27,930 --> 00:09:30,870
it would be a long story in
itself how this is done--

188
00:09:30,870 --> 00:09:33,850
but when they use neutron
scattering to analyze

189
00:09:33,850 --> 00:09:36,060
the liquid helium
at low temperature,

190
00:09:36,060 --> 00:09:39,640
and figure out what is the
fraction of atoms which

191
00:09:39,640 --> 00:09:43,370
have zero momentum,
they found 10%.

192
00:09:43,370 --> 00:09:45,700
The quantum depletion is 90%.

193
00:09:45,700 --> 00:09:48,150
But that's just the
difference between a quantum

194
00:09:48,150 --> 00:09:51,310
gas and a quantum liquid.

195
00:09:51,310 --> 00:09:54,880
And in the quantum liquid--
in liquid helium-- N

196
00:09:54,880 --> 00:09:56,915
a cubed is on the
order of unity.

197
00:09:59,590 --> 00:10:04,370
OK, I mentioned this
dispersion relation.

198
00:10:04,370 --> 00:10:06,920
These are sort of
quasi particles.

199
00:10:06,920 --> 00:10:08,820
And I at least
showed you how you

200
00:10:08,820 --> 00:10:12,260
can measure the slope of the
quasi particle dispersion

201
00:10:12,260 --> 00:10:15,000
relation by propagating
sound waves.

202
00:10:15,000 --> 00:10:18,300
But let me now tell you how we
can observe quasi particles'

203
00:10:18,300 --> 00:10:22,290
elementary excitations directly.

204
00:10:22,290 --> 00:10:26,710
And this is actually simply
done by light scattering.

205
00:10:26,710 --> 00:10:32,940
If you scatter a photon,
and the scattered photon

206
00:10:32,940 --> 00:10:38,720
loses energy h-bar U. And
it is scattered at an angle.

207
00:10:38,720 --> 00:10:41,860
Therefore, it
transfers momentum q.

208
00:10:41,860 --> 00:10:44,910
These forces can
only happen if there

209
00:10:44,910 --> 00:10:52,510
is an elementary excitation
with momentum q and energy h U.

210
00:10:52,510 --> 00:10:55,280
So in other words,
on a photon basis,

211
00:10:55,280 --> 00:10:57,270
you can see photon
by photon, if you

212
00:10:57,270 --> 00:11:01,860
scatter a photon, the photon
transfers momentum and energy.

213
00:11:01,860 --> 00:11:05,290
The process can only happen if
you can form a quasiparticle

214
00:11:05,290 --> 00:11:07,680
with this momentum and energy.

215
00:11:07,680 --> 00:11:10,330
So since this is sort of the
direct way of mapping out

216
00:11:10,330 --> 00:11:12,320
whether the system
has the possibility

217
00:11:12,320 --> 00:11:19,744
to absorb momentum q and energy
h nu, this has sort of a name.

218
00:11:19,744 --> 00:11:22,160
The scattering probability is
called the dynamic structure

219
00:11:22,160 --> 00:11:22,660
factor.

220
00:11:22,660 --> 00:11:25,680
And the dynamic structure
factor is just an integral

221
00:11:25,680 --> 00:11:29,550
over all the possibilities
that a many-body system can

222
00:11:29,550 --> 00:11:32,690
absorb momentum and energy.

223
00:11:32,690 --> 00:11:38,730
Now there is one
nice feature which

224
00:11:38,730 --> 00:11:40,802
was introduced by
us at MIT, and that

225
00:11:40,802 --> 00:11:43,260
is if you're going to measure
the dynamic structure factor,

226
00:11:43,260 --> 00:11:44,926
often you do it with
neutron scattering.

227
00:11:44,926 --> 00:11:48,230
You scatter neutrons or x-rays
and they scatter spontaneously

228
00:11:48,230 --> 00:11:50,870
at an angle, and
you need a detector

229
00:11:50,870 --> 00:11:54,430
to detect the
scattered particles.

230
00:11:54,430 --> 00:11:58,470
But in our case, because
the gas is so dilute,

231
00:11:58,470 --> 00:12:00,670
the distance between
atoms is on the order

232
00:12:00,670 --> 00:12:02,240
of the optical wavelengths.

233
00:12:02,240 --> 00:12:03,350
We don't want neutrons.

234
00:12:03,350 --> 00:12:04,570
We don't want x-rays.

235
00:12:04,570 --> 00:12:07,690
We want photons, because
the wavelengths of photons

236
00:12:07,690 --> 00:12:09,850
is perfectly matched
to the properties,

237
00:12:09,850 --> 00:12:13,260
to momentum transfer and
such we need for our system.

238
00:12:13,260 --> 00:12:16,200
But photons-- we
have photon lasers.

239
00:12:16,200 --> 00:12:20,920
So instead of in a painstakingly
way analyzing the frequency

240
00:12:20,920 --> 00:12:23,140
and the momentum of
scattered photons,

241
00:12:23,140 --> 00:12:26,360
we don't do the spontaneous
scattering process.

242
00:12:26,360 --> 00:12:28,180
We do a stimulated process.

243
00:12:28,180 --> 00:12:31,425
We use two laser beams,
and we stimulate a photon

244
00:12:31,425 --> 00:12:34,320
to be scattered into
the other laser beam.

245
00:12:34,320 --> 00:12:37,490
And by having the two laser
beams at frequency difference

246
00:12:37,490 --> 00:12:40,690
delta omega, we're
really asking the system,

247
00:12:40,690 --> 00:12:43,580
are you ready to absorb
delta omega energy?

248
00:12:43,580 --> 00:12:46,330
If yes, then you
have a quasiparticle.

249
00:12:46,330 --> 00:12:50,090
So this is how we do
quasiparticle spectroscopy.

250
00:12:50,090 --> 00:12:52,820
And a few years
later, this method

251
00:12:52,820 --> 00:12:58,310
was defined by the
[INAUDIBLE] Institute.

252
00:12:58,310 --> 00:13:00,630
And what you see
here is they varied

253
00:13:00,630 --> 00:13:03,380
the angles between the
two laser beams, realized

254
00:13:03,380 --> 00:13:05,370
different momentum
transfer, and what

255
00:13:05,370 --> 00:13:09,190
you see is the low dispersion,
the linear dispersion

256
00:13:09,190 --> 00:13:11,200
relation for low
momenta, and then

257
00:13:11,200 --> 00:13:14,970
the quadratic part
at high momentum.

258
00:13:14,970 --> 00:13:19,340
So this is called
BEC spectroscopy.

259
00:13:19,340 --> 00:13:23,060
It's a variant, you can see,
of Raman spectroscopy where

260
00:13:23,060 --> 00:13:27,530
you go from a zero quasiparticle
state to a one quasiparticle

261
00:13:27,530 --> 00:13:30,830
state through a
simulated Raman process,

262
00:13:30,830 --> 00:13:33,875
and this is how the dispersion
relation is determined.

263
00:13:38,920 --> 00:13:43,820
OK so I've so far
dealt with aspects

264
00:13:43,820 --> 00:13:47,200
of a homogeneous Bose-Einstein
condensate where,

265
00:13:47,200 --> 00:13:49,250
of course, if you're
a homogeneous system,

266
00:13:49,250 --> 00:13:52,590
you formulate everything
in momentum space.

267
00:13:52,590 --> 00:13:55,090
But now we want to
deal with the situation

268
00:13:55,090 --> 00:13:58,640
that our condensates
are finite in size.

269
00:13:58,640 --> 00:14:02,990
And in addition, they're
usually in a harmonic oscillator

270
00:14:02,990 --> 00:14:08,020
potential, and therefore their
density is inhomogeneous.

271
00:14:08,020 --> 00:14:13,350
Let me start the discussion with
the inhomogeneous Bose-Einstein

272
00:14:13,350 --> 00:14:16,440
condensate by showing
again a picture which

273
00:14:16,440 --> 00:14:19,900
brings in some
memories of '90, '96.

274
00:14:19,900 --> 00:14:22,130
It was the first
time that we could

275
00:14:22,130 --> 00:14:24,850
see the Bose-Einstein
condensate in the trap.

276
00:14:24,850 --> 00:14:27,820
Before people have just
seen it by time of flight

277
00:14:27,820 --> 00:14:29,900
when it was pretty
much already destroyed

278
00:14:29,900 --> 00:14:32,040
or it was just flying out.

279
00:14:32,040 --> 00:14:35,980
But here we see the trapped
Bose-Einstein condensate

280
00:14:35,980 --> 00:14:37,870
and we could even
take multiple pictures

281
00:14:37,870 --> 00:14:39,280
of the same condensate.

282
00:14:39,280 --> 00:14:41,380
So you can see that
this was the first time

283
00:14:41,380 --> 00:14:43,080
that a condensate
was seen alive.

284
00:14:46,550 --> 00:14:49,260
What you see here is actually
the bimodal distribution.

285
00:14:49,260 --> 00:14:52,400
You see the condensate and
then due to speckled pattern,

286
00:14:52,400 --> 00:14:54,640
the Fermi cloud
looks fragmented,

287
00:14:54,640 --> 00:14:56,395
but this is just a
speckled pattern due

288
00:14:56,395 --> 00:14:58,400
to the signal to noise.

289
00:14:58,400 --> 00:15:01,740
But here you really see now--
not in ballistic expansion

290
00:15:01,740 --> 00:15:06,030
as I showed you on Monday--
but really insight to the size

291
00:15:06,030 --> 00:15:08,860
and shape of the condensate.

292
00:15:08,860 --> 00:15:11,380
OK, how do we describe it?

293
00:15:11,380 --> 00:15:15,920
Well, the message is pretty
much exactly the same way

294
00:15:15,920 --> 00:15:19,010
as we did with the Bogoliubov
approximation and such,

295
00:15:19,010 --> 00:15:20,970
but instead of in
momentum space,

296
00:15:20,970 --> 00:15:23,960
we do it now in position space.

297
00:15:23,960 --> 00:15:25,720
Actually, I have
to say this week,

298
00:15:25,720 --> 00:15:28,370
I go through a lot of material.

299
00:15:28,370 --> 00:15:30,960
But what I'm trying
is to give you

300
00:15:30,960 --> 00:15:33,780
sort of a spirited
and animated overview

301
00:15:33,780 --> 00:15:35,730
that you really know
what is important.

302
00:15:35,730 --> 00:15:38,260
Where is the same
concept appearing again

303
00:15:38,260 --> 00:15:40,300
in a different way.

304
00:15:40,300 --> 00:15:42,590
The details-- I've
posted, actually,

305
00:15:42,590 --> 00:15:44,687
the original
articles, references.

306
00:15:44,687 --> 00:15:46,270
Some of are school
notes, some of them

307
00:15:46,270 --> 00:15:48,490
written by myself
on the backside.

308
00:15:48,490 --> 00:15:51,180
So yes, it's a little
bit different character

309
00:15:51,180 --> 00:15:52,680
than other lectures.

310
00:15:52,680 --> 00:15:54,750
I want to show you
a lot of things,

311
00:15:54,750 --> 00:15:59,160
and put a special
emphasis on the ideas.

312
00:15:59,160 --> 00:16:01,850
OK, so in second
quantization, we

313
00:16:01,850 --> 00:16:05,270
are now using field operators
which create and annihilate

314
00:16:05,270 --> 00:16:09,460
particles at position R.
This is the single particle

315
00:16:09,460 --> 00:16:12,400
Hamiltonian-- kinetic
energy and potential energy.

316
00:16:12,400 --> 00:16:17,720
And our interaction term has
now, again, four operators--

317
00:16:17,720 --> 00:16:20,740
two interrelation operators,
two creation operators,

318
00:16:20,740 --> 00:16:24,460
but they're now formulated
in position space.

319
00:16:24,460 --> 00:16:27,730
We do exactly the same as we
did in the homogeneous gas.

320
00:16:27,730 --> 00:16:30,590
We assume the potential
is short range.

321
00:16:30,590 --> 00:16:33,130
That means delta function.

322
00:16:33,130 --> 00:16:35,040
And that means we
can get rid of one

323
00:16:35,040 --> 00:16:38,670
of the integrations
in r and r prime.

324
00:16:38,670 --> 00:16:44,430
But we still have product
of four operators.

325
00:16:44,430 --> 00:16:48,490
We can formally solve the
equation by writing down

326
00:16:48,490 --> 00:16:53,090
Heisenberg's situation for the
equation of motion for those

327
00:16:53,090 --> 00:16:56,620
field operators, but of
course this cannot be solved.

328
00:16:56,620 --> 00:17:00,090
Remember products are
four operators are hard.

329
00:17:00,090 --> 00:17:03,280
We have to reduce
it to two operators.

330
00:17:03,280 --> 00:17:07,829
And this is actually done by the
Bogoliubov approximation again.

331
00:17:07,829 --> 00:17:10,740
But it's done here
in the following way.

332
00:17:10,740 --> 00:17:16,390
Remember, in the momentum space
we replaced a0 and a0 dagger

333
00:17:16,390 --> 00:17:18,750
by the square root of a0.

334
00:17:18,750 --> 00:17:24,160
What we do here is we say this
is sort of a quantum field.

335
00:17:24,160 --> 00:17:31,010
And when we have a superfluid,
this field operator

336
00:17:31,010 --> 00:17:32,790
has an average value.

337
00:17:32,790 --> 00:17:36,580
It's actually the macroscopic
superfluid order parameter.

338
00:17:36,580 --> 00:17:40,720
So we replace the operator
by an average value

339
00:17:40,720 --> 00:17:43,860
which we assume is large,
because many, many particles

340
00:17:43,860 --> 00:17:47,110
are in the superfluid state.

341
00:17:47,110 --> 00:17:50,220
And then we have
fluctuations which are small.

342
00:17:50,220 --> 00:17:53,430
And what we will do is-- it's
pretty clear-- when it comes

343
00:17:53,430 --> 00:17:57,000
to the fluctuations, we will
neglect higher products--

344
00:17:57,000 --> 00:17:59,060
products of four
fluctuation operators.

345
00:17:59,060 --> 00:18:02,605
And we are down to C
numbers and, as you

346
00:18:02,605 --> 00:18:04,260
will see, two operators.

347
00:18:08,010 --> 00:18:09,960
We can-- and this is
even more dramatic--

348
00:18:09,960 --> 00:18:11,850
it wouldn't make sense
in momentum space,

349
00:18:11,850 --> 00:18:13,870
but it does make sense
in position space--

350
00:18:13,870 --> 00:18:16,340
we can even do a
first approximation

351
00:18:16,340 --> 00:18:19,470
where we completely
neglect those fluctuations.

352
00:18:19,470 --> 00:18:22,190
And then what we have
is when we simply

353
00:18:22,190 --> 00:18:28,600
insert a [? Z ?] number
into this Hamiltonian.

354
00:18:28,600 --> 00:18:33,310
We have then an equation
for this number,

355
00:18:33,310 --> 00:18:34,870
for this function psi.

356
00:18:34,870 --> 00:18:37,600
And this is the nonlinear
Schrodinger equation,

357
00:18:37,600 --> 00:18:40,730
or it's also called
Gross-Pitaevskii equation,

358
00:18:40,730 --> 00:18:44,990
which is now the analogy
to Schrodinger's equation,

359
00:18:44,990 --> 00:18:48,000
but now for the macroscopic
wave function which

360
00:18:48,000 --> 00:18:51,250
is occupied by many,
many particles.

361
00:18:51,250 --> 00:18:53,470
And in addition to
the kinetic energy

362
00:18:53,470 --> 00:18:57,760
and the trapping potential
in the external potential,

363
00:18:57,760 --> 00:19:01,740
it has a term which is
proportional to the density.

364
00:19:01,740 --> 00:19:05,620
And this is just
mean field repulsion

365
00:19:05,620 --> 00:19:11,060
which one atom feels
exerted by the other atoms.

366
00:19:11,060 --> 00:19:15,080
I should say--
just to connect it

367
00:19:15,080 --> 00:19:18,250
to what I've said earlier
about the quantum depletion--

368
00:19:18,250 --> 00:19:22,740
that you can regard the
function psi as the best

369
00:19:22,740 --> 00:19:26,340
approximation to an ideal
Bose-Einstein condensate.

370
00:19:26,340 --> 00:19:28,470
So if you want to write
your many-body wave

371
00:19:28,470 --> 00:19:32,300
function as a function
psi to the power N,

372
00:19:32,300 --> 00:19:35,350
all particles are in
the same quantum state,

373
00:19:35,350 --> 00:19:38,720
then a variational
calculation would

374
00:19:38,720 --> 00:19:43,960
say that this function psi
should obey this equation.

375
00:19:43,960 --> 00:19:45,560
If you want to read
up what I've said,

376
00:19:45,560 --> 00:19:49,150
I've posted a recent paper,
which I found very pedagogical,

377
00:19:49,150 --> 00:19:53,510
where they derive the
Gross-Pitaevskii equation

378
00:19:53,510 --> 00:19:56,570
without even using an
operator psi dagger,

379
00:19:56,570 --> 00:19:58,940
without using any
second quantization.

380
00:19:58,940 --> 00:20:02,810
They just say we have an
Hamiltonian for an interaction

381
00:20:02,810 --> 00:20:08,780
system, and we try to write
the complicated, many-body wave

382
00:20:08,780 --> 00:20:13,810
function as a wave
function to the power N.

383
00:20:13,810 --> 00:20:17,250
And then you put this
ansatz into your Hamiltonian

384
00:20:17,250 --> 00:20:21,330
and do a variation optimization
which single-particle wave

385
00:20:21,330 --> 00:20:25,020
function psi, if
taken to the power N,

386
00:20:25,020 --> 00:20:28,220
minimizes the total
energy of N particles,

387
00:20:28,220 --> 00:20:29,970
including interactions.

388
00:20:29,970 --> 00:20:33,500
And the answer is this equation.

389
00:20:33,500 --> 00:20:35,160
So there are many
ways this equation

390
00:20:35,160 --> 00:20:37,200
is sort of very natural.

391
00:20:40,120 --> 00:20:45,510
Now you all know about
single-particle physics--

392
00:20:45,510 --> 00:20:47,640
kinetic energy in an
external potential.

393
00:20:47,640 --> 00:20:51,310
Let's now learn what this
density-dependent term does.

394
00:20:51,310 --> 00:20:53,220
Well, it's pretty clear.

395
00:20:53,220 --> 00:20:56,650
There is a price to
be paid for density.

396
00:20:56,650 --> 00:21:00,330
And while the total
integrated density, of course,

397
00:21:00,330 --> 00:21:02,600
is given by the total
number of particles.

398
00:21:02,600 --> 00:21:05,460
That means if you lower
the density somewhere,

399
00:21:05,460 --> 00:21:07,890
you have to increase
the density.

400
00:21:07,890 --> 00:21:11,350
But because this is
a non-linear term,

401
00:21:11,350 --> 00:21:14,090
if you have an average
density and you lower it

402
00:21:14,090 --> 00:21:18,340
here and increase it here, you
have actually more repulsive

403
00:21:18,340 --> 00:21:21,170
energy than if the
density is flat.

404
00:21:21,170 --> 00:21:23,780
So this term has only one goal.

405
00:21:23,780 --> 00:21:27,660
It wants to flatten
out densities.

406
00:21:27,660 --> 00:21:31,530
So therefore if we
have a box potential

407
00:21:31,530 --> 00:21:33,240
and you know the
ground state in a box

408
00:21:33,240 --> 00:21:37,180
is just half a period
of a sine function.

409
00:21:39,730 --> 00:21:41,964
This is a non-interacting
condensate.

410
00:21:41,964 --> 00:21:43,380
This would be the
macroscopic wave

411
00:21:43,380 --> 00:21:44,810
function in a box potential.

412
00:21:44,810 --> 00:21:47,360
But if you now put in
strong interactions,

413
00:21:47,360 --> 00:21:51,290
the strong interaction's going
to flatten out the potential.

414
00:21:51,290 --> 00:21:54,190
They flatten out the
potential, and only at the very

415
00:21:54,190 --> 00:21:56,570
last moment-- all
right, now it's

416
00:21:56,570 --> 00:22:00,550
time to go down because we have
to meet our boundary condition.

417
00:22:00,550 --> 00:22:04,520
And the length scale where
you eventually go down-- it

418
00:22:04,520 --> 00:22:07,100
has a famous name,
the healing length--

419
00:22:07,100 --> 00:22:10,720
is the length scale
where the kinetic energy

420
00:22:10,720 --> 00:22:13,080
due to the curvature
of the wave function

421
00:22:13,080 --> 00:22:17,330
is now comparable to
the interaction energy.

422
00:22:17,330 --> 00:22:19,560
If the system would
curve down earlier,

423
00:22:19,560 --> 00:22:23,130
it would cause too
much repulsive energy,

424
00:22:23,130 --> 00:22:25,150
because the density
is not kept flat.

425
00:22:25,150 --> 00:22:28,280
If it would curve down too
late, the enormous curvature

426
00:22:28,280 --> 00:22:30,020
would mean a lot
of kinetic energy.

427
00:22:30,020 --> 00:22:31,520
And this is just
the best compromise

428
00:22:31,520 --> 00:22:33,570
between those two criteria.

429
00:22:33,570 --> 00:22:36,150
So that's how you derive
the healing length.

430
00:22:36,150 --> 00:22:38,580
The healing length is
now the length scale

431
00:22:38,580 --> 00:22:45,010
over which the system is willing
to meet boundary conditions,

432
00:22:45,010 --> 00:22:49,080
and not stay flat,
as flat as possible.

433
00:22:49,080 --> 00:22:49,845
Yes.

434
00:22:49,845 --> 00:22:51,594
AUDIENCE: Just curious--
in that equation,

435
00:22:51,594 --> 00:22:57,180
the atom has the mass
of each individual--

436
00:22:57,180 --> 00:22:59,476
PROFESSOR: The mass is
the mass of a single atom.

437
00:22:59,476 --> 00:23:02,260
AUDIENCE: Single atom,
it's not-- sorry,

438
00:23:02,260 --> 00:23:03,260
I wasn't here last time.

439
00:23:03,260 --> 00:23:06,580
But the quasiparticles are
the same mass [INAUDIBLE].

440
00:23:06,580 --> 00:23:08,880
PROFESSOR: In a Bose-Einstein
condensate they are.

441
00:23:08,880 --> 00:23:11,070
We haven't changed the mass.

442
00:23:11,070 --> 00:23:13,350
But we're not talking
actually about quasiparticle.

443
00:23:13,350 --> 00:23:18,810
We're actually really talking
about here qualitative features

444
00:23:18,810 --> 00:23:22,020
of the solution of the
Gross-Pitaevskii equation.

445
00:23:22,020 --> 00:23:23,670
And the Gross-Pitaevskii
equation

446
00:23:23,670 --> 00:23:27,990
is sort of a single particle
equation for particles of mass,

447
00:23:27,990 --> 00:23:32,650
of the original bare mass
M. It's a macroscopic wave

448
00:23:32,650 --> 00:23:35,520
function, but I sometimes say
it describes the wave function

449
00:23:35,520 --> 00:23:38,520
of a single particle where
all the other particles are

450
00:23:38,520 --> 00:23:40,610
included at the
mean field level.

451
00:23:40,610 --> 00:23:43,120
So therefore, it is really
this, the atomic mass,

452
00:23:43,120 --> 00:23:46,460
and not any form of
collective mass here.

453
00:23:46,460 --> 00:23:49,640
Well, if you would now
ask how do interactions

454
00:23:49,640 --> 00:23:56,510
transform the first excited
state in the box potential?

455
00:23:56,510 --> 00:24:00,410
Well, then it's again flat
this as much as possible.

456
00:24:00,410 --> 00:24:03,760
But then if you want to
maintain the parity of the wave

457
00:24:03,760 --> 00:24:06,430
function, then it's
only close to the zero

458
00:24:06,430 --> 00:24:10,570
crossing within a healing length
that the system says OK, now I

459
00:24:10,570 --> 00:24:12,360
change sign.

460
00:24:12,360 --> 00:24:18,080
So that's sort of what is inside
the Gross-Pitaevskii equation.

461
00:24:18,080 --> 00:24:23,440
Now once we realize that, we
can take it to the next level

462
00:24:23,440 --> 00:24:26,781
and say, well, if you
neglect-- let's say

463
00:24:26,781 --> 00:24:28,280
we're interested
in the ground state

464
00:24:28,280 --> 00:24:30,530
and we neglect this
boundary region where

465
00:24:30,530 --> 00:24:33,130
the kinetic energy
becomes important-- maybe

466
00:24:33,130 --> 00:24:35,920
we can simplify the
Gross-Pitaevskii equation

467
00:24:35,920 --> 00:24:38,846
by neglecting the
kinetic energy.

468
00:24:38,846 --> 00:24:43,580
If potential energy dominates
by far, we can neglect that.

469
00:24:43,580 --> 00:24:45,670
And then we should
get a good description

470
00:24:45,670 --> 00:24:49,510
which will not be valid in
the wings of the wave function

471
00:24:49,510 --> 00:24:53,210
but in most of space.

472
00:24:53,210 --> 00:24:54,941
But now if you look
at this equation,

473
00:24:54,941 --> 00:24:56,565
it's no longer a
differential equation.

474
00:24:56,565 --> 00:24:58,120
It has no derivatives.

475
00:24:58,120 --> 00:25:01,530
It's just something
which applies

476
00:25:01,530 --> 00:25:04,500
to the wave function psi itself.

477
00:25:04,500 --> 00:25:09,140
And we can simply solve that.

478
00:25:09,140 --> 00:25:11,580
The solution of that
is that psi squared--

479
00:25:11,580 --> 00:25:15,370
the density-- is
nothing else than there

480
00:25:15,370 --> 00:25:20,000
is a constant minus the
inverted trapping potential.

481
00:25:20,000 --> 00:25:24,550
So therefore in this
Thomas-Fermi approximation

482
00:25:24,550 --> 00:25:27,390
where we neglect the
kinetic energy completely,

483
00:25:27,390 --> 00:25:32,010
you just take your trapping
potential, turn it upside down,

484
00:25:32,010 --> 00:25:34,790
and then you fill
it up with density

485
00:25:34,790 --> 00:25:37,500
until you have accommodated
the number of atoms

486
00:25:37,500 --> 00:25:38,890
you want to accommodate.

487
00:25:38,890 --> 00:25:41,084
And this is, of
course-- the constraint

488
00:25:41,084 --> 00:25:43,250
in the number of atoms is
determined by the chemical

489
00:25:43,250 --> 00:25:44,490
potential.

490
00:25:44,490 --> 00:25:48,100
Or if you have a more
complicated W-shaped potential,

491
00:25:48,100 --> 00:25:49,420
the same construction.

492
00:25:49,420 --> 00:25:54,070
Flip it over, fill it up,
gives you the condensate wave

493
00:25:54,070 --> 00:25:56,180
function, the
density distribution

494
00:25:56,180 --> 00:26:00,150
of the condensate
in this potential.

495
00:26:00,150 --> 00:26:02,940
I don't want to belabor
it, but coming back

496
00:26:02,940 --> 00:26:05,680
to the question of
the small parameter,

497
00:26:05,680 --> 00:26:08,240
if you look at
those equations, you

498
00:26:08,240 --> 00:26:13,290
can identify a parameter-- this
is now the small parameter,

499
00:26:13,290 --> 00:26:16,060
or the parameter
in the system which

500
00:26:16,060 --> 00:26:19,030
is the important
dimensionless parameter.

501
00:26:19,030 --> 00:26:20,950
It depends on the
number of atoms,

502
00:26:20,950 --> 00:26:24,470
and it depends now on the ratio
of the scattering length--

503
00:26:24,470 --> 00:26:26,590
which characterizes
the interaction--

504
00:26:26,590 --> 00:26:29,860
and the harmonic
oscillator length.

505
00:26:29,860 --> 00:26:31,520
You can say the
harmonic oscillator

506
00:26:31,520 --> 00:26:36,720
length is the ideal wave
function harmonic oscillator.

507
00:26:36,720 --> 00:26:39,790
So this parametrizes the
importance of kinetic energy,

508
00:26:39,790 --> 00:26:42,180
whereas the scattering
length parametrizes

509
00:26:42,180 --> 00:26:44,160
the importance of interactions.

510
00:26:44,160 --> 00:26:48,930
And all those solutions
can be nicely written

511
00:26:48,930 --> 00:26:53,750
as what you would
have in an ideal gas,

512
00:26:53,750 --> 00:26:59,600
and then this parameter X. So
in typical experiments, N a,

513
00:26:59,600 --> 00:27:00,871
N is a million.

514
00:27:00,871 --> 00:27:03,370
The scattering length is smaller
than the oscillator length,

515
00:27:03,370 --> 00:27:08,440
but N is a million, and this
parameter X is usually large.

516
00:27:08,440 --> 00:27:11,370
So therefore-- and I'm
simply just talking

517
00:27:11,370 --> 00:27:14,840
about this solution-- we have
the situation that the chemical

518
00:27:14,840 --> 00:27:17,720
potential is larger
by a power of X

519
00:27:17,720 --> 00:27:22,400
than the ideal gas solution,
which would just be the ground

520
00:27:22,400 --> 00:27:25,910
state with its zero point energy
in the harmonic oscillator

521
00:27:25,910 --> 00:27:31,620
potential, or the width-- the
size-- of the wave function

522
00:27:31,620 --> 00:27:35,690
is larger than the ground state
of the harmonic oscillator,

523
00:27:35,690 --> 00:27:40,100
but only with an exponent,
which is one-fifth.

524
00:27:40,100 --> 00:27:42,290
Well, we can see that.

525
00:27:42,290 --> 00:27:45,580
These are now, again,
somewhat improved pictures

526
00:27:45,580 --> 00:27:47,670
of condensates inside the trap.

527
00:27:47,670 --> 00:27:49,860
For the expert, the
previous picture

528
00:27:49,860 --> 00:27:51,160
was dark-ground imaging.

529
00:27:51,160 --> 00:27:53,960
This is now
phase-contrast imaging.

530
00:27:53,960 --> 00:27:56,510
And if you take a
profile, we clearly

531
00:27:56,510 --> 00:28:02,370
see the condensate wave function
and we see the thermal wings.

532
00:28:02,370 --> 00:28:07,450
When we look at the size of
the condensate wave function,

533
00:28:07,450 --> 00:28:11,594
you realize what I just meant--
that the ground-state wave

534
00:28:11,594 --> 00:28:13,260
function, the harmonic
oscillator length

535
00:28:13,260 --> 00:28:15,980
is 7 micron in the
axial direction,

536
00:28:15,980 --> 00:28:18,650
but here you have 300 micron.

537
00:28:18,650 --> 00:28:23,850
So this condensate is completely
dominated by interactions.

538
00:28:23,850 --> 00:28:30,720
And it fulfils very nice, and
pretty much the whole shape,

539
00:28:30,720 --> 00:28:33,150
except maybe some
details in the wings,

540
00:28:33,150 --> 00:28:40,030
are quantitatively described
by the simple approximation

541
00:28:40,030 --> 00:28:42,270
I have explained to you.

542
00:28:42,270 --> 00:28:44,020
So just a little
bit show and tell.

543
00:28:44,020 --> 00:28:47,660
We have this
Gross-Pitaevskii equation.

544
00:28:47,660 --> 00:28:50,680
The Gross-Pitaevskii equation
has, as Schrodinger's equation,

545
00:28:50,680 --> 00:28:53,320
a time independent form
to get the ground state.

546
00:28:53,320 --> 00:28:56,240
It has also a time
dependent form.

547
00:28:56,240 --> 00:28:58,840
You simply replace the
energy by the derivative

548
00:28:58,840 --> 00:28:59,750
of the wave function.

549
00:28:59,750 --> 00:29:01,580
This is the time-dependent form.

550
00:29:01,580 --> 00:29:06,170
Everything is very simple, and
you can do many-body physics,

551
00:29:06,170 --> 00:29:07,991
but on your computer,
you pretty much look

552
00:29:07,991 --> 00:29:09,365
for single-particle
wave function

553
00:29:09,365 --> 00:29:11,690
and take them to the power N.

554
00:29:11,690 --> 00:29:14,280
So some areas where the
Gross-Pitaevskii equation

555
00:29:14,280 --> 00:29:17,820
has quantitatively
explained experiments-- one

556
00:29:17,820 --> 00:29:21,590
is the expansion of a
Bose-Einstein condensate.

557
00:29:21,590 --> 00:29:23,540
It's this famous
situation when you

558
00:29:23,540 --> 00:29:26,570
have an elongated condensate
and let it expand.

559
00:29:26,570 --> 00:29:31,760
It expands faster in
the radial direction.

560
00:29:31,760 --> 00:29:33,970
One simple argument
is the pressure

561
00:29:33,970 --> 00:29:37,900
of the mean field is larger and
leads to faster acceleration.

562
00:29:37,900 --> 00:29:40,150
So therefore, if you have
a cigar-shaped condensate

563
00:29:40,150 --> 00:29:45,230
and release it, it
turns into a disk.

564
00:29:45,230 --> 00:29:48,790
It inverts the aspect ratio,
going from a cigar shape

565
00:29:48,790 --> 00:29:50,030
to a disk shape.

566
00:29:50,030 --> 00:29:53,270
And that has been beautifully
and quantitatively described

567
00:29:53,270 --> 00:29:55,960
by the Gross-Pitaevskii
equation.

568
00:29:55,960 --> 00:29:58,550
Here we have measured
the interaction energy

569
00:29:58,550 --> 00:30:00,740
as a number of condensed atoms.

570
00:30:00,740 --> 00:30:05,370
And I mentioned to you that
this X parameter comes often

571
00:30:05,370 --> 00:30:07,750
with power 1/5 and 2/5.

572
00:30:07,750 --> 00:30:10,154
And this here is a
fit to the power 2/5,

573
00:30:10,154 --> 00:30:11,570
beautifully
confirming the theory.

574
00:30:17,390 --> 00:30:19,400
So the Gross-Pitaevskii
equation was

575
00:30:19,400 --> 00:30:26,680
invented in 1962, about 50 years
ago by Gross and Pitaevskii

576
00:30:26,680 --> 00:30:29,260
to describe vortices.

577
00:30:29,260 --> 00:30:34,450
Actually, Lev Pitaevskii
is still alive,

578
00:30:34,450 --> 00:30:38,200
going strong, and
publishing papers.

579
00:30:38,200 --> 00:30:42,210
One of his latest predictions
was solitons in Fermi gases,

580
00:30:42,210 --> 00:30:46,000
and I know in Professor
Zwierlein's group,

581
00:30:46,000 --> 00:30:49,760
one of his lab just looked
at the same physics,

582
00:30:49,760 --> 00:30:52,440
and compared to the
theory of Lev Pitaevskii.

583
00:30:52,440 --> 00:30:55,610
And this year we
are celebrating--

584
00:30:55,610 --> 00:30:58,200
I forgot, the 85th or
90th birthday of him.

585
00:30:58,200 --> 00:31:00,190
AUDIENCE: [INAUDIBLE]

586
00:31:00,190 --> 00:31:00,866
PROFESSOR: 90?

587
00:31:00,866 --> 00:31:01,990
AUDIENCE: There's a poster.

588
00:31:01,990 --> 00:31:04,180
PROFESSOR: I know there's
a poster next to my door,

589
00:31:04,180 --> 00:31:06,600
but I forgot which anniversary.

590
00:31:06,600 --> 00:31:10,430
So I mean, he's an legend.

591
00:31:10,430 --> 00:31:12,110
But he's still walking.

592
00:31:12,110 --> 00:31:13,750
He's still doing science.

593
00:31:13,750 --> 00:31:18,840
So if you meet him, you go
back to 50 years in history.

594
00:31:18,840 --> 00:31:21,700
Anyway, it is this
Lev Pitaevskii,

595
00:31:21,700 --> 00:31:27,410
and he invented his equations
to describe vortices.

596
00:31:27,410 --> 00:31:32,640
So anyway so we have this
nonlinear Schrodinger equation.

597
00:31:32,640 --> 00:31:34,660
Let me just show you
what vortices are

598
00:31:34,660 --> 00:31:37,340
and how they are formed.

599
00:31:37,340 --> 00:31:42,710
Vortices come-- if you solve
the Gross-Pitaevskii equation--

600
00:31:42,710 --> 00:31:45,090
if you ever any quantum
fluid or quantum gas,

601
00:31:45,090 --> 00:31:49,410
and you add angular momentum,
the angular momentum cannot

602
00:31:49,410 --> 00:31:51,820
lead to rigid body rotation.

603
00:31:51,820 --> 00:31:55,500
This would violate the fact
that the velocity field has

604
00:31:55,500 --> 00:31:58,640
to be irrotational to make
sure that the phase of the wave

605
00:31:58,640 --> 00:32:01,320
function is well defined.

606
00:32:01,320 --> 00:32:06,620
So if you rotate the system,
it can absorb angular momentum

607
00:32:06,620 --> 00:32:09,050
only by forming vortices.

608
00:32:09,050 --> 00:32:13,390
And vortices are singular points
of the wave function-- singular

609
00:32:13,390 --> 00:32:16,770
points where the density is 0.

610
00:32:16,770 --> 00:32:21,100
In other words-- I don't have
time to be exhaustive here--

611
00:32:21,100 --> 00:32:23,090
but when something
rotates, there

612
00:32:23,090 --> 00:32:26,397
is a dynamic phase you
can say, because there

613
00:32:26,397 --> 00:32:28,230
are matter waves
going in circles.

614
00:32:28,230 --> 00:32:30,320
And the integral
of the phase has

615
00:32:30,320 --> 00:32:33,525
to be an integer number
of 2 pi, otherwise

616
00:32:33,525 --> 00:32:35,650
you would not have a
well-defined phase of the wave

617
00:32:35,650 --> 00:32:36,790
function.

618
00:32:36,790 --> 00:32:39,920
And if you now say you make
a circle, which is 2 pi.

619
00:32:39,920 --> 00:32:42,170
You make the circle
smaller, smaller, smaller.

620
00:32:42,170 --> 00:32:44,410
In the middle of the
circle, you go around.

621
00:32:44,410 --> 00:32:46,620
The wave function
changes by 2 pi,

622
00:32:46,620 --> 00:32:49,200
but what should the wave
function do on one point?

623
00:32:49,200 --> 00:32:52,990
Which number between 0 and 2 pi
should the wave function peak?

624
00:32:52,990 --> 00:32:55,010
Well, the wave function
says I can't peak.

625
00:32:55,010 --> 00:32:56,500
I just go to 0.

626
00:32:56,500 --> 00:32:59,360
And a 0 value has no
phase and I'm fine.

627
00:32:59,360 --> 00:33:01,640
That's exactly what the
wave function tells you.

628
00:33:01,640 --> 00:33:04,290
So therefore if you want
to describe vortices,

629
00:33:04,290 --> 00:33:08,000
you want to now describe the
Gross-Pitaevskii equation,

630
00:33:08,000 --> 00:33:11,140
but with the boundary condition
that the wave function

631
00:33:11,140 --> 00:33:12,520
goes to 0.

632
00:33:12,520 --> 00:33:14,530
And if you want to
describe one vortex,

633
00:33:14,530 --> 00:33:20,130
you want the wave function to
go 0 in the center of the cloud.

634
00:33:20,130 --> 00:33:23,010
And so you make a
corresponding Ansatz.

635
00:33:23,010 --> 00:33:26,310
You allow the phase to
wrap around by 2 pi,

636
00:33:26,310 --> 00:33:28,530
when the angle phi is varied.

637
00:33:28,530 --> 00:33:32,210
And then when you solve
it, you have, of course,

638
00:33:32,210 --> 00:33:37,280
put in that there should be a
0 of the density at the center.

639
00:33:37,280 --> 00:33:41,230
And it's now nice for me to--
there's an important review

640
00:33:41,230 --> 00:33:43,120
paper which describes
all that and more--

641
00:33:43,120 --> 00:33:45,610
but it's nice to
show you now the two

642
00:33:45,610 --> 00:33:49,680
extreme cases of the ideal
condensate without interaction,

643
00:33:49,680 --> 00:33:52,190
and the strong
interacting condensate.

644
00:33:52,190 --> 00:33:57,090
The ideal condensate-- well, we
are in the first excited state

645
00:33:57,090 --> 00:33:59,480
of the harmonic
oscillator potential.

646
00:33:59,480 --> 00:34:01,580
And this is the dashed line.

647
00:34:01,580 --> 00:34:06,120
And of course, the size
of the whole, the 0,

648
00:34:06,120 --> 00:34:09,090
is pretty much on the order
of the oscillator length,

649
00:34:09,090 --> 00:34:10,960
because there is no
other length scale

650
00:34:10,960 --> 00:34:13,440
in the ideal
harmonic oscillator.

651
00:34:13,440 --> 00:34:15,949
But if you have the
interacting system,

652
00:34:15,949 --> 00:34:18,360
remember what
interactions are doing.

653
00:34:18,360 --> 00:34:21,630
They want to keep the density
as constant as possible.

654
00:34:21,630 --> 00:34:23,300
Well, we are not
in a box potential.

655
00:34:23,300 --> 00:34:26,050
We're in an inverted
parabola potential.

656
00:34:26,050 --> 00:34:28,489
And remember, our
Thomas-Fermi solution,

657
00:34:28,489 --> 00:34:32,650
which neglects kinetic energy,
is the inverted parabola

658
00:34:32,650 --> 00:34:33,780
up to here.

659
00:34:33,780 --> 00:34:36,285
And then the tail is
when we can no longer

660
00:34:36,285 --> 00:34:38,429
neglect the kinetic energy.

661
00:34:38,429 --> 00:34:41,720
But if we now say, OK, fine,
but now in addition, we

662
00:34:41,720 --> 00:34:43,679
want a vortex in
the center, the wave

663
00:34:43,679 --> 00:34:45,650
function out there says no.

664
00:34:45,650 --> 00:34:48,040
We do what minimizes
kinetic energy,

665
00:34:48,040 --> 00:34:51,850
and what minimizes
repulsive energy.

666
00:34:51,850 --> 00:34:54,520
We follow the inverted
parabola, and only

667
00:34:54,520 --> 00:34:59,970
at the very last moment-- on
the scale of the healing length,

668
00:34:59,970 --> 00:35:02,700
the system meets the
required boundary condition

669
00:35:02,700 --> 00:35:05,400
that the density goes to 0.

670
00:35:05,400 --> 00:35:07,510
Anyway, with this
qualitative understanding,

671
00:35:07,510 --> 00:35:08,930
you can get a lot out of those.

672
00:35:08,930 --> 00:35:11,300
You can immediately
understand the salient feature

673
00:35:11,300 --> 00:35:12,100
of the solution.

674
00:35:15,274 --> 00:35:16,242
Question?

675
00:35:16,242 --> 00:35:18,178
AUDIENCE: I know we're
not talking details,

676
00:35:18,178 --> 00:35:20,350
but is there a simple,
maybe clear reason why

677
00:35:20,350 --> 00:35:22,802
when you spin a
Bose-Einstein condensate

678
00:35:22,802 --> 00:35:25,085
you get many vortices, but
when you spin your coffee

679
00:35:25,085 --> 00:35:28,285
in your cup, you
just get one big one?

680
00:35:28,285 --> 00:35:28,910
PROFESSOR: Yes.

681
00:35:33,430 --> 00:35:39,720
So the question is, if I have
a wrap-around of 4 pi in phase,

682
00:35:39,720 --> 00:35:48,575
whether the system should have
a doubly charged vortex, or two

683
00:35:48,575 --> 00:35:50,740
singly charged vortices?

684
00:35:50,740 --> 00:35:53,660
What is the
difference in energy?

685
00:35:53,660 --> 00:35:58,760
Well, what happens is doubly
charged vortices are unstable.

686
00:35:58,760 --> 00:36:00,842
My group, at some point,
were the first to create

687
00:36:00,842 --> 00:36:02,300
doubly charged
vortices, but we saw

688
00:36:02,300 --> 00:36:04,815
that they immediately decayed.

689
00:36:04,815 --> 00:36:05,690
AUDIENCE: [INAUDIBLE]

690
00:36:05,690 --> 00:36:06,370
PROFESSOR: Pardon?

691
00:36:06,370 --> 00:36:07,286
AUDIENCE: [INAUDIBLE]?

692
00:36:10,310 --> 00:36:12,060
PROFESSOR: Initially,
we couldn't observe.

693
00:36:12,060 --> 00:36:14,810
We just saw that
it was unstable.

694
00:36:14,810 --> 00:36:16,560
But it's a decay
into two vortices.

695
00:36:16,560 --> 00:36:18,220
But it's easy to understand.

696
00:36:18,220 --> 00:36:21,280
If you put two vortices
on top of each other,

697
00:36:21,280 --> 00:36:25,110
what is the energy of the
system compared to one vortex?

698
00:36:25,110 --> 00:36:29,550
Well, the energy of a vortex is
the rotational field around it,

699
00:36:29,550 --> 00:36:32,420
and if you superimpose
two vortices,

700
00:36:32,420 --> 00:36:35,450
the velocity around the
doubly charged vortices

701
00:36:35,450 --> 00:36:38,500
is twice the velocity
of one vortex.

702
00:36:38,500 --> 00:36:42,300
And therefore, the kinetic
energy is four times.

703
00:36:42,300 --> 00:36:44,210
If you have two
vortices which are

704
00:36:44,210 --> 00:36:47,620
far separated that it has
its own velocity field.

705
00:36:47,620 --> 00:36:49,580
It has its own
velocity field here.

706
00:36:49,580 --> 00:36:51,970
And when the two velocity
fields come together,

707
00:36:51,970 --> 00:36:55,150
the velocity is
already so low that you

708
00:36:55,150 --> 00:36:58,400
don't have to consider that,
then those two vortices have

709
00:36:58,400 --> 00:37:02,550
an energy which is two times
the energy of a single vortex.

710
00:37:02,550 --> 00:37:04,980
So therefore, when
the two vortices,

711
00:37:04,980 --> 00:37:08,260
they start out with four times
the energy of a single vortex,

712
00:37:08,260 --> 00:37:11,170
and when they dissociate
and repel each other,

713
00:37:11,170 --> 00:37:13,850
they have shed half
of their energy.

714
00:37:13,850 --> 00:37:16,230
So this argument
tells you immediately

715
00:37:16,230 --> 00:37:19,850
that vortices are
in effect repulsive.

716
00:37:19,850 --> 00:37:23,530
And therefore, any
multiple-charged vortex

717
00:37:23,530 --> 00:37:27,170
will spontaneously decay.

718
00:37:27,170 --> 00:37:30,110
It's also this net
repulsion between vortices

719
00:37:30,110 --> 00:37:33,800
which makes the vortices
arrange in a regular lattice.

720
00:37:33,800 --> 00:37:36,090
The regular lattice
which you saw before

721
00:37:36,090 --> 00:37:39,120
is Nature's answer to
how can we minimize

722
00:37:39,120 --> 00:37:41,270
the energy of all
those vortices?

723
00:37:41,270 --> 00:37:44,990
And the idea is let's keep the
average distance between them

724
00:37:44,990 --> 00:37:49,230
as large as possible, and the
answer is a hexagonal lattice.

725
00:37:49,230 --> 00:37:51,870
Yeah.

726
00:37:51,870 --> 00:37:55,060
AUDIENCE: So a few
slides back, when

727
00:37:55,060 --> 00:38:01,120
you write the G-P equation, when
you plotted sort of for the box

728
00:38:01,120 --> 00:38:04,920
potential, the ground state
and the excited state comparing

729
00:38:04,920 --> 00:38:08,510
the G-P equation to the typical
single-particle [INAUDIBLE].

730
00:38:08,510 --> 00:38:11,472
And so I guess I'm a
little bit confused

731
00:38:11,472 --> 00:38:15,040
about how excitations manifest
themselves in the system.

732
00:38:15,040 --> 00:38:18,730
Because in some sense, if you
were to solve the G-P equation,

733
00:38:18,730 --> 00:38:25,180
would you arrive at a
spectrum of solutions?

734
00:38:25,180 --> 00:38:28,150
Then I guess I'd be confused
at whether excitations manifest

735
00:38:28,150 --> 00:38:30,490
themselves into that
spectrum of solutions,

736
00:38:30,490 --> 00:38:35,990
or rather than they become
excitations, sort of deviations

737
00:38:35,990 --> 00:38:39,706
from the mean field as
including the [INAUDIBLE].

738
00:38:39,706 --> 00:38:41,580
PROFESSOR: You are now
asking about something

739
00:38:41,580 --> 00:38:42,288
more complicated.

740
00:38:42,288 --> 00:38:43,740
You're asking what
are the excited

741
00:38:43,740 --> 00:38:45,690
states of the many-body system?

742
00:38:45,690 --> 00:38:48,900
And actually, we have
already found one answer.

743
00:38:48,900 --> 00:38:53,200
And this is if you want
these small excitations,

744
00:38:53,200 --> 00:38:56,020
it's one particle
becomes a phonon,

745
00:38:56,020 --> 00:38:58,990
or one particle becomes
a quasiparticle.

746
00:38:58,990 --> 00:39:01,300
What we are talking
about here is

747
00:39:01,300 --> 00:39:04,840
what happens if the
whole macroscopic wave

748
00:39:04,840 --> 00:39:07,020
function is in an excited state.

749
00:39:07,020 --> 00:39:09,600
So we are asking here in
the box potential what

750
00:39:09,600 --> 00:39:15,030
happens if you force all N atoms
to have one node in the wave

751
00:39:15,030 --> 00:39:15,620
function.

752
00:39:15,620 --> 00:39:17,590
And this is what I'm
talking about here.

753
00:39:17,590 --> 00:39:20,400
So the excitation
energy of this state

754
00:39:20,400 --> 00:39:24,300
here is much, much higher
than of single quasiparticle

755
00:39:24,300 --> 00:39:26,010
excitations.

756
00:39:26,010 --> 00:39:29,580
I'm not sure if I'm addressing
your question here, but--

757
00:39:29,580 --> 00:39:32,585
AUDIENCE: So with
single quasiparticle

758
00:39:32,585 --> 00:39:34,209
excitations, are
those essentially sort

759
00:39:34,209 --> 00:39:35,682
of deviations from
the mean field?

760
00:39:40,570 --> 00:39:43,280
PROFESSOR: No, they
are-- you really

761
00:39:43,280 --> 00:39:52,390
look for many-body physics--
main field deviations--

762
00:39:52,390 --> 00:39:53,550
probably.

763
00:39:53,550 --> 00:40:00,500
I mean, what you do
is in that sense, yes.

764
00:40:00,500 --> 00:40:04,070
You're allowing in this
Ansatz that psi operator

765
00:40:04,070 --> 00:40:07,040
is psi average
plus fluctuations.

766
00:40:07,040 --> 00:40:09,710
You now look for
fluctuations, and you're

767
00:40:09,710 --> 00:40:12,200
looking for the
energy eigenspectrum

768
00:40:12,200 --> 00:40:14,090
of those fluctuations.

769
00:40:14,090 --> 00:40:16,840
And the answer are
quasiparticles.

770
00:40:16,840 --> 00:40:21,000
So it is, actually,
the Bogoliubov solution

771
00:40:21,000 --> 00:40:26,090
for the spectrum of
the fluctuations.

772
00:40:26,090 --> 00:40:30,440
And the answer is the dispersion
relation I presented to you.

773
00:40:30,440 --> 00:40:33,740
Whereas here we
are asking what are

774
00:40:33,740 --> 00:40:36,600
excited states of the
macroscopic wave function?

775
00:40:36,600 --> 00:40:38,330
It's a very, very
different question

776
00:40:38,330 --> 00:40:40,660
which we've addressed here.

777
00:40:40,660 --> 00:40:41,510
Yes.

778
00:40:41,510 --> 00:40:43,259
AUDIENCE: So when you
stir the condensate,

779
00:40:43,259 --> 00:40:48,310
do you view it as-- do you
stir, sort of, at the trap

780
00:40:48,310 --> 00:40:51,312
frequency, so I do sort
of N single excitations?

781
00:40:51,312 --> 00:40:55,866
Or do you stir at N
times the trap frequency,

782
00:40:55,866 --> 00:40:59,608
so you get one sort of
N particle excitation?

783
00:40:59,608 --> 00:41:01,590
PROFESSOR: Well, the
experimental answer

784
00:41:01,590 --> 00:41:04,140
is you want to stir at the
quadrupole frequency, which

785
00:41:04,140 --> 00:41:07,100
is square root 2 times
the trap frequency.

786
00:41:07,100 --> 00:41:10,080
Then you create
quadrupolar excitations.

787
00:41:10,080 --> 00:41:14,660
The quadrupolar excitation can
be regarded as a standing sound

788
00:41:14,660 --> 00:41:17,990
wave, but you create
a macroscopic number

789
00:41:17,990 --> 00:41:19,730
of those excitations.

790
00:41:19,730 --> 00:41:22,450
So you make, actually, the
whole condensate wave function

791
00:41:22,450 --> 00:41:25,680
oscillate in a
quadrupolar pattern,

792
00:41:25,680 --> 00:41:28,390
but it's a rotating
quadrupole, and that

793
00:41:28,390 --> 00:41:33,620
eventually then rearranges
itself and leads to vortices.

794
00:41:33,620 --> 00:41:40,070
So I think the correct answer
is the most efficient way

795
00:41:40,070 --> 00:41:45,350
to create vortices would be to
excite quasiparticles, but then

796
00:41:45,350 --> 00:41:49,150
create so many-- and this may
relate to Matt's question--

797
00:41:49,150 --> 00:41:52,490
create so many quasiparticles
that you have really

798
00:41:52,490 --> 00:41:55,570
a time-dependent and oscillating
macroscopic wave function.

799
00:42:00,300 --> 00:42:02,990
So eventually, you have
a coherent excitation

800
00:42:02,990 --> 00:42:04,200
of quasiparticles.

801
00:42:04,200 --> 00:42:06,780
And that eventually means
the condensate moves

802
00:42:06,780 --> 00:42:10,310
in a quadrupolar pattern, but
it's a quadrupolar pattern

803
00:42:10,310 --> 00:42:12,290
with rotation, and
that eventually

804
00:42:12,290 --> 00:42:15,940
turns into many vortices.

805
00:42:15,940 --> 00:42:19,500
But that's really a very rich
question which you're asking,

806
00:42:19,500 --> 00:42:24,210
which has been studied-- a lot
of different aspects of that

807
00:42:24,210 --> 00:42:24,960
have been studied.

808
00:42:30,220 --> 00:42:35,760
OK, so we've talked about
Bose-Einstein condensates

809
00:42:35,760 --> 00:42:39,350
in a homogeneous system, just
sort of to lay the groundwork.

810
00:42:39,350 --> 00:42:41,570
We've talked about
Bose-Einstein condensates

811
00:42:41,570 --> 00:42:44,150
in traps in
inhomogeneous system.

812
00:42:44,150 --> 00:42:47,070
Now we want to talk about
Bose-Einstein condensates

813
00:42:47,070 --> 00:42:48,850
in optical lattices.

814
00:42:48,850 --> 00:42:52,050
Well, there are two reasons
why we want to do that.

815
00:42:52,050 --> 00:42:56,390
One is we want to use the
Bose-Einstein condensate

816
00:42:56,390 --> 00:43:00,490
to obtain deeper insight into
the properties of matter.

817
00:43:00,490 --> 00:43:05,640
And a lot of forms of matter
appear in periodic lattices.

818
00:43:05,640 --> 00:43:07,990
So if you put a
Bose-Einstein condensate

819
00:43:07,990 --> 00:43:11,580
into a periodic
potential, we can at least

820
00:43:11,580 --> 00:43:14,700
understand some
of the properties

821
00:43:14,700 --> 00:43:17,870
of crystalline
matter, or electrons

822
00:43:17,870 --> 00:43:22,200
which are block waves
in a periodic potential.

823
00:43:22,200 --> 00:43:24,470
So this is one reason why.

824
00:43:24,470 --> 00:43:27,630
The other reason why we want
to go to optical lattices

825
00:43:27,630 --> 00:43:31,530
is the following-- ideal
Bose-Einstein condensates

826
00:43:31,530 --> 00:43:33,580
are trivial.

827
00:43:33,580 --> 00:43:39,020
Weakly interacting Bose-Einstein
condensates are entertaining,

828
00:43:39,020 --> 00:43:41,570
and you can write a lot of
papers, have a lot of fun

829
00:43:41,570 --> 00:43:45,800
with it, develop your methods--
also mildly intellectually

830
00:43:45,800 --> 00:43:49,460
interesting because how those
weak interactions manifest

831
00:43:49,460 --> 00:43:51,690
itself in vortices and all.

832
00:43:51,690 --> 00:43:53,580
It's really rich
and interesting.

833
00:43:53,580 --> 00:43:56,760
But the conceptional
problems appear

834
00:43:56,760 --> 00:43:59,560
when you go to
strong interactions.

835
00:43:59,560 --> 00:44:03,420
Strongly correlated matter is
where mean field descriptions

836
00:44:03,420 --> 00:44:04,430
no longer work.

837
00:44:04,430 --> 00:44:07,340
This is really the frontier
of our understanding

838
00:44:07,340 --> 00:44:08,440
of many-body systems.

839
00:44:08,440 --> 00:44:13,917
And when you want
to be there, you

840
00:44:13,917 --> 00:44:15,750
want to create a strongly
correlated system.

841
00:44:15,750 --> 00:44:19,020
And strongly correlated systems
means that the interaction

842
00:44:19,020 --> 00:44:23,330
energy dominates
over kinetic energy.

843
00:44:23,330 --> 00:44:25,300
In Bose-Einstein
condensates you can only

844
00:44:25,300 --> 00:44:28,360
go so far with
increasing interactions--

845
00:44:28,360 --> 00:44:30,210
why are Feshbach resonances?

846
00:44:30,210 --> 00:44:32,770
Because you get into some
inelastic collisions.

847
00:44:32,770 --> 00:44:34,080
So that's one knob to turn.

848
00:44:34,080 --> 00:44:36,490
You go to larger and larger
of scattering lengths,

849
00:44:36,490 --> 00:44:38,530
and crank up the interactions.

850
00:44:38,530 --> 00:44:42,600
But if atoms strongly interact,
they start to do bad chemistry.

851
00:44:42,600 --> 00:44:43,790
They start to spin flip.

852
00:44:43,790 --> 00:44:45,020
They do other things.

853
00:44:45,020 --> 00:44:47,410
And in some cases you
keep it under control.

854
00:44:47,410 --> 00:44:51,310
In others, you just can't
keep it under control.

855
00:44:51,310 --> 00:44:54,270
But another way to get
to strong interactions

856
00:44:54,270 --> 00:44:56,570
means you reduce
the kinetic energy.

857
00:44:56,570 --> 00:44:59,030
It's the ratio of the
two which matters.

858
00:44:59,030 --> 00:45:02,390
And when you put particles
in a lattice, well,

859
00:45:02,390 --> 00:45:05,410
the lattice actually
reduces the kinetic energy.

860
00:45:05,410 --> 00:45:07,200
You may know from
condensed matter physics

861
00:45:07,200 --> 00:45:09,500
that in a lattice there is
an effective mass, which

862
00:45:09,500 --> 00:45:11,520
is higher than the
[INAUDIBLE] mass.

863
00:45:11,520 --> 00:45:16,220
Therefore you've reduced
the kinetic energy.

864
00:45:16,220 --> 00:45:19,020
Or if you want another
hand-waving approximation,

865
00:45:19,020 --> 00:45:21,630
the kinetic energy is
given by the bandwidths.

866
00:45:21,630 --> 00:45:24,390
And if the tunneling
becomes slower and slower,

867
00:45:24,390 --> 00:45:26,580
the width of your
band becomes narrower,

868
00:45:26,580 --> 00:45:28,880
and your kinetic energy is less.

869
00:45:28,880 --> 00:45:31,800
So anyway, I can give you many
hand-waving approximations

870
00:45:31,800 --> 00:45:35,810
why in a lattice,
kinetic energy is reduced

871
00:45:35,810 --> 00:45:39,120
and repulsive energy
is probably enhanced,

872
00:45:39,120 --> 00:45:42,100
because instead of having
your atoms spread out,

873
00:45:42,100 --> 00:45:47,160
they're bunched up
at each lattice site

874
00:45:47,160 --> 00:45:49,960
into a higher density cloud.

875
00:45:49,960 --> 00:45:52,190
So anyway, maybe you're
interested in parity

876
00:45:52,190 --> 00:45:55,510
potentials, or you're
interested in quantum systems

877
00:45:55,510 --> 00:45:57,070
with strong correlations.

878
00:45:57,070 --> 00:46:01,590
For whatever reasons, you
want optical lattices.

879
00:46:01,590 --> 00:46:05,270
Now I have here a
few slides which

880
00:46:05,270 --> 00:46:09,280
introduce periodic
potentials, but it is really

881
00:46:09,280 --> 00:46:13,010
just the single particle
physics in a periodic potential.

882
00:46:13,010 --> 00:46:15,030
It has nothing do
with quantum gases.

883
00:46:15,030 --> 00:46:16,780
This could be the
first lecture of how

884
00:46:16,780 --> 00:46:20,230
to describe
electrons in a metal.

885
00:46:20,230 --> 00:46:22,330
So let me just quickly
go through assuming

886
00:46:22,330 --> 00:46:25,350
that almost all of
you are familiar.

887
00:46:25,350 --> 00:46:29,480
But I use those slides also just
to give you a few definitions

888
00:46:29,480 --> 00:46:31,850
and introduce a few symbols.

889
00:46:31,850 --> 00:46:36,050
So this is simple, boring,
exactly understood physics.

890
00:46:36,050 --> 00:46:39,090
We have a Hamiltonian
which has kinetic energy,

891
00:46:39,090 --> 00:46:43,170
and a periodic potential which
is our lattice potential.

892
00:46:43,170 --> 00:46:48,090
It's rather trivial to solve,
but exactly for your wave

893
00:46:48,090 --> 00:46:50,690
function, you use
Bloch's theorem

894
00:46:50,690 --> 00:46:54,260
and divide it by an exponential
factor with quasi momentum

895
00:46:54,260 --> 00:46:56,870
times a periodic function.

896
00:46:56,870 --> 00:47:00,180
And if you now solve
Schrodinger's equation,

897
00:47:00,180 --> 00:47:04,210
you want to use Fourier space.

898
00:47:04,210 --> 00:47:06,900
You Fourier analyze
the wave function.

899
00:47:06,900 --> 00:47:09,010
You Fourier analyze
the potential.

900
00:47:09,010 --> 00:47:12,730
And since the
potential is periodic,

901
00:47:12,730 --> 00:47:16,550
sine square potential has
only three Fourier components

902
00:47:16,550 --> 00:47:23,240
at 0 plus/minus 1 times the
periodicity of the lattice.

903
00:47:23,240 --> 00:47:28,800
You do a Fourier expansion for
your periodic wave function.

904
00:47:28,800 --> 00:47:32,480
And if you insert that into
Schrodinger's equation,

905
00:47:32,480 --> 00:47:35,600
well, the Fourier transform
has turned the differential

906
00:47:35,600 --> 00:47:39,120
equation into an
algebraic equation,

907
00:47:39,120 --> 00:47:41,380
because the second
derivative simply

908
00:47:41,380 --> 00:47:46,970
becomes now k
squared or q squared.

909
00:47:46,970 --> 00:47:50,245
So in other words, you have
a set of linear equations.

910
00:47:53,350 --> 00:47:55,630
There is an index
which is the band

911
00:47:55,630 --> 00:47:58,340
index-- how high
do you want to go?

912
00:47:58,340 --> 00:48:00,170
And usually you truncate it.

913
00:48:00,170 --> 00:48:01,360
But it's the same.

914
00:48:01,360 --> 00:48:03,730
It is that same
trivial story which

915
00:48:03,730 --> 00:48:07,620
is told in all textbooks of
condensed matter physics.

916
00:48:07,620 --> 00:48:11,810
If you have no band structure,
well, you have the parabola,

917
00:48:11,810 --> 00:48:13,570
but to prepare for
band structure,

918
00:48:13,570 --> 00:48:16,850
I've opened it down here in
the first Brillouin zone.

919
00:48:16,850 --> 00:48:19,250
And if you now
introduce a lattice,

920
00:48:19,250 --> 00:48:23,810
you introduce band gaps, and you
go from the left to the middle

921
00:48:23,810 --> 00:48:28,190
to the right for stronger
and stronger lattices.

922
00:48:28,190 --> 00:48:30,500
So the case which
I will focus on,

923
00:48:30,500 --> 00:48:35,610
because it is the most extreme
case away from free space,

924
00:48:35,610 --> 00:48:41,580
is this case which is called
the tight binding limit, where

925
00:48:41,580 --> 00:48:46,220
the potential energy of
the lattice is large.

926
00:48:46,220 --> 00:48:51,880
And large means compared
with the kinetic energy

927
00:48:51,880 --> 00:48:53,980
at the Brillouin
Zone, which is k

928
00:48:53,980 --> 00:48:57,170
squared-- k is the lattice
momentum-- k squared over 2 m.

929
00:48:57,170 --> 00:48:59,260
And that's the recoil energy.

930
00:48:59,260 --> 00:49:01,350
So that our dimensionless
parameter here

931
00:49:01,350 --> 00:49:04,880
is the depth of the
lattice measured

932
00:49:04,880 --> 00:49:09,000
in recoil energies
of the photon,

933
00:49:09,000 --> 00:49:10,650
because that is
the kinetic energy

934
00:49:10,650 --> 00:49:14,290
of the free gas at
the Brillouin zone.

935
00:49:14,290 --> 00:49:19,190
OK there are few things which
immediately simplify that.

936
00:49:19,190 --> 00:49:20,910
Once we're in the
tight binding limit,

937
00:49:20,910 --> 00:49:24,610
our lattice is really
deep, and each side

938
00:49:24,610 --> 00:49:26,710
forms a harmonic oscillator.

939
00:49:26,710 --> 00:49:28,530
And the harmonic
oscillator frequency

940
00:49:28,530 --> 00:49:32,650
is analytically given by
the depth of the lattice.

941
00:49:32,650 --> 00:49:37,150
And the solution here
for the lowest band

942
00:49:37,150 --> 00:49:40,190
is that the lowest
band-- the energy

943
00:49:40,190 --> 00:49:43,780
in the lowest band, the
dispersion relation is-- well,

944
00:49:43,780 --> 00:49:47,170
we have a harmonic oscillator
in each site at x y z.

945
00:49:47,170 --> 00:49:49,810
So the average energy
in the lowest band

946
00:49:49,810 --> 00:49:54,940
is the 0 point energy in x, y,
and z-- three half h-bar omega

947
00:49:54,940 --> 00:49:56,000
0.

948
00:49:56,000 --> 00:49:59,780
And then we have a
cosinusoidal band structure

949
00:49:59,780 --> 00:50:02,570
where q is the quasimomentum.

950
00:50:02,570 --> 00:50:05,930
And what appears here as the
only interesting parameter

951
00:50:05,930 --> 00:50:07,600
is j.

952
00:50:07,600 --> 00:50:15,990
And j is I think this 4
should be 4j is the bandwidth.

953
00:50:19,850 --> 00:50:23,310
So what appears here now
for the first time is j.

954
00:50:23,310 --> 00:50:26,030
It appears here
as the bandwidth.

955
00:50:26,030 --> 00:50:30,250
But let me immediately give
it another interpretation

956
00:50:30,250 --> 00:50:34,940
as a tunneling matrix element in
the following way-- right now,

957
00:50:34,940 --> 00:50:39,030
we have formulated the physics
in Bloch wave functions

958
00:50:39,030 --> 00:50:41,150
which are infinitely extended.

959
00:50:41,150 --> 00:50:43,580
The Bloch wave functions
are for the lattice

960
00:50:43,580 --> 00:50:46,770
what plain waves
are for free space.

961
00:50:46,770 --> 00:50:50,780
But if you have
tight binding limit,

962
00:50:50,780 --> 00:50:52,860
there is another limit
which is important.

963
00:50:52,860 --> 00:50:58,340
Namely, a particle is localized
and hops around in the lattice.

964
00:50:58,340 --> 00:51:02,130
The localized particle, of
course, is in free space.

965
00:51:02,130 --> 00:51:06,540
It would be a wave packet-- a
superposition of plain waves.

966
00:51:06,540 --> 00:51:08,490
So let's do the
same in the lattice.

967
00:51:08,490 --> 00:51:12,060
Let's construct
superpositions of Bloch waves.

968
00:51:12,060 --> 00:51:13,910
And these are our wave packets.

969
00:51:13,910 --> 00:51:17,420
And the wave packet is now
called the Wannier function.

970
00:51:17,420 --> 00:51:20,510
And there is a
mathematical procedure

971
00:51:20,510 --> 00:51:22,530
how you should pick
the phases here

972
00:51:22,530 --> 00:51:24,710
to get the maximal
normalization.

973
00:51:24,710 --> 00:51:28,210
But the simple picture is
those Wannier functions are

974
00:51:28,210 --> 00:51:31,730
very, very close to the
Gaussian ground state

975
00:51:31,730 --> 00:51:35,050
solution of the harmonic
oscillator on each side.

976
00:51:35,050 --> 00:51:39,080
The wings are different, but
I don't want to go into that.

977
00:51:39,080 --> 00:51:43,740
You transform from an
orthonormal basis of Bloch wave

978
00:51:43,740 --> 00:51:48,630
function to another orthonormal
basis Wannier function.

979
00:51:48,630 --> 00:51:53,760
And the Wannier functions are
as well localized as possible.

980
00:51:53,760 --> 00:51:55,800
That's the procedure.

981
00:51:55,800 --> 00:51:59,970
And now we can simply
rewrite our total Hamiltonian

982
00:51:59,970 --> 00:52:03,385
or everything we're interested
in, not in Bloch wave functions

983
00:52:03,385 --> 00:52:06,960
but in Wannier wave functions.

984
00:52:06,960 --> 00:52:12,250
And what comes out now is,
is that the bandwidth j

985
00:52:12,250 --> 00:52:17,860
is nothing else than our
Hamiltonian with kinetic energy

986
00:52:17,860 --> 00:52:19,540
in the periodic potential.

987
00:52:19,540 --> 00:52:22,600
But j becomes now
a matrix element

988
00:52:22,600 --> 00:52:24,040
between two Wannier functions.

989
00:52:27,690 --> 00:52:31,990
But the Wannier
functions have now-- I'll

990
00:52:31,990 --> 00:52:34,010
put in some indices
in a moment--

991
00:52:34,010 --> 00:52:36,890
connect now two different sides.

992
00:52:36,890 --> 00:52:39,540
So it is-- you have
Wannier function.

993
00:52:39,540 --> 00:52:41,760
You have the Hamiltonian
and connect it

994
00:52:41,760 --> 00:52:43,690
to another Wannier function.

995
00:52:43,690 --> 00:52:49,050
So it is the amplitude
that, with the Hamiltonian,

996
00:52:49,050 --> 00:52:52,890
the particle can hop from
one side to another side.

997
00:52:52,890 --> 00:52:56,390
So therefore j, which
was the bandwidth,

998
00:52:56,390 --> 00:52:59,890
is now the tunneling
energy divided

999
00:52:59,890 --> 00:53:03,020
by h bar-- the tunneling
rate from on one side

1000
00:53:03,020 --> 00:53:05,910
to the other side.

1001
00:53:05,910 --> 00:53:08,100
For very deep lattice,
everything is analytic

1002
00:53:08,100 --> 00:53:11,460
and it can easily be solved.

1003
00:53:11,460 --> 00:53:14,210
And I mentioned already in the
tight binding approximation,

1004
00:53:14,210 --> 00:53:16,060
you should think about
your Wannier function

1005
00:53:16,060 --> 00:53:19,310
as just localized
Gaussian eigensolutions

1006
00:53:19,310 --> 00:53:22,070
of the harmonic oscillator.

1007
00:53:22,070 --> 00:53:24,190
Yes?

1008
00:53:24,190 --> 00:53:28,660
AUDIENCE: So qualitatively, this
question sounds sort of silly,

1009
00:53:28,660 --> 00:53:32,140
but normally when we write
down the solution for j,

1010
00:53:32,140 --> 00:53:35,810
we're only considering
nearest neighbor.

1011
00:53:35,810 --> 00:53:39,275
But from the math,
I don't immediately

1012
00:53:39,275 --> 00:53:44,230
see why we wouldn't
include i equal to j.

1013
00:53:44,230 --> 00:53:46,790
PROFESSOR: Give
me one more slide.

1014
00:53:46,790 --> 00:53:53,870
So I should've
actually-- what I should

1015
00:53:53,870 --> 00:53:57,920
have done is that--
j has an index here.

1016
00:53:57,920 --> 00:54:02,580
I just didn't want to
overload you with indices.

1017
00:54:02,580 --> 00:54:04,330
I mean, this is sort
of just telling you

1018
00:54:04,330 --> 00:54:07,010
what j is in its simplest form.

1019
00:54:07,010 --> 00:54:10,610
It is loaded with indices and
I well show you in a moment

1020
00:54:10,610 --> 00:54:12,780
where those indices come in.

1021
00:54:12,780 --> 00:54:14,920
So I just wanted to
give you the idea

1022
00:54:14,920 --> 00:54:17,750
if we want to have
Wannier function.

1023
00:54:17,750 --> 00:54:20,850
We hop from one Wannier
function to the other one,

1024
00:54:20,850 --> 00:54:23,060
and the operator
is the Hamiltonian.

1025
00:54:23,060 --> 00:54:28,840
And therefore, there
should be an index j and l.

1026
00:54:28,840 --> 00:54:34,280
And maybe-- let me just do the
next step first and then come

1027
00:54:34,280 --> 00:54:36,030
back to it, but I
wanted to tell you here

1028
00:54:36,030 --> 00:54:39,040
what I'm aiming at--
namely interpretation of j,

1029
00:54:39,040 --> 00:54:41,460
of tunneling from
one side to the next.

1030
00:54:41,460 --> 00:54:47,400
But right now I haven't really
told you which j I really mean.

1031
00:54:47,400 --> 00:54:49,930
There should be-- based
on the right-hand side,

1032
00:54:49,930 --> 00:54:53,040
there is a j which
has two indices.

1033
00:54:53,040 --> 00:54:56,480
And I will make the indices
disappear in a moment.

1034
00:54:56,480 --> 00:54:59,730
But before I make those
indices disappear,

1035
00:54:59,730 --> 00:55:03,350
let me introduce the other
relevant parameter, which

1036
00:55:03,350 --> 00:55:05,850
will also have
indices, and that is we

1037
00:55:05,850 --> 00:55:08,540
have to bring in
the interactions.

1038
00:55:08,540 --> 00:55:11,150
We want to describe
an interacting system.

1039
00:55:11,150 --> 00:55:13,680
We describe an
interacting system

1040
00:55:13,680 --> 00:55:15,490
using the short-range
approximation

1041
00:55:15,490 --> 00:55:20,690
by assuming that two particles
interact with a delta function.

1042
00:55:20,690 --> 00:55:26,360
And if you have two particles
on site, each of them

1043
00:55:26,360 --> 00:55:30,370
has a density which is the
Wannier function squared.

1044
00:55:30,370 --> 00:55:33,950
And the product of the
two densities integrated

1045
00:55:33,950 --> 00:55:38,910
gives us the expectation value
for the repulsive energy.

1046
00:55:38,910 --> 00:55:41,440
And this is given here.

1047
00:55:41,440 --> 00:55:44,000
So the moment we
introduce interactions,

1048
00:55:44,000 --> 00:55:50,820
we are now interested in
the interaction energy

1049
00:55:50,820 --> 00:55:57,150
between two particles which in
this case occupy the same side.

1050
00:55:57,150 --> 00:56:01,010
OK so I've tried to introduce
was what sort of j is.

1051
00:56:01,010 --> 00:56:03,730
j is a matrix element
between two Wannier functions

1052
00:56:03,730 --> 00:56:05,750
with a Hamiltonian in between.

1053
00:56:05,750 --> 00:56:10,430
And u is the matrix element
of two Wannier functions

1054
00:56:10,430 --> 00:56:13,730
with the interaction
operator in between.

1055
00:56:13,730 --> 00:56:15,860
And now I want to
use that concept

1056
00:56:15,860 --> 00:56:21,200
to take my full
Hamiltonian and transform

1057
00:56:21,200 --> 00:56:26,570
from field operators localized
at x to Wannier functions.

1058
00:56:26,570 --> 00:56:29,650
So they B operators are
now creation operators.

1059
00:56:29,650 --> 00:56:31,820
They create an atom
in a Wannier function.

1060
00:56:31,820 --> 00:56:34,920
In other words, B
dagger means you

1061
00:56:34,920 --> 00:56:38,370
have a particle in a
Wannier function at site i.

1062
00:56:38,370 --> 00:56:40,380
And I just use that as
a basis transformation

1063
00:56:40,380 --> 00:56:42,380
and the exact transformation
of this Hamiltonian

1064
00:56:42,380 --> 00:56:47,250
is now into this form.

1065
00:56:47,250 --> 00:56:47,750
So

1066
00:56:47,750 --> 00:56:52,850
What I have right now is, I have
the tunneling matrix element

1067
00:56:52,850 --> 00:56:54,570
between particles
at site i and j,

1068
00:56:54,570 --> 00:56:57,140
and I have to sum
over all of them.

1069
00:56:57,140 --> 00:57:15,060
And in terms of interaction,
I can calculate this matrix

1070
00:57:15,060 --> 00:57:17,620
element by using Wannier
function with four

1071
00:57:17,620 --> 00:57:21,540
different indices
and ask what happens.

1072
00:57:21,540 --> 00:57:24,880
And this is simply an
exact way of rewriting it.

1073
00:57:24,880 --> 00:57:27,190
And here I've given
you the definition.

1074
00:57:27,190 --> 00:57:28,720
So if you want to
forget everything

1075
00:57:28,720 --> 00:57:34,800
I told you about j and u, I've
done an exact transformation

1076
00:57:34,800 --> 00:57:38,490
from field operators to Wannier
function creation operators.

1077
00:57:38,490 --> 00:57:45,050
And this introduces tunneling
terms like this from site i

1078
00:57:45,050 --> 00:57:47,070
to site j.

1079
00:57:47,070 --> 00:57:51,940
And these here includes products
of four Wannier functions

1080
00:57:51,940 --> 00:57:55,370
which are responsible
for the interaction term.

1081
00:57:55,370 --> 00:57:57,960
Actually, if this is
not complicated enough,

1082
00:57:57,960 --> 00:57:59,670
I've suppressed
band indices here.

1083
00:57:59,670 --> 00:58:03,050
I should also now sum
over all possible bands.

1084
00:58:03,050 --> 00:58:07,540
But OK, I want to come now
to the leading approximation

1085
00:58:07,540 --> 00:58:09,020
in a tight binding model.

1086
00:58:09,020 --> 00:58:11,210
And that is where--
I mean those Wannier

1087
00:58:11,210 --> 00:58:12,910
functions are overlapped.

1088
00:58:12,910 --> 00:58:15,770
Two neighboring Wannier
functions barely overlap.

1089
00:58:15,770 --> 00:58:19,050
If I go further away, the
overlap becomes even smaller.

1090
00:58:19,050 --> 00:58:22,950
So the most dominant terms are
nearest neighbor interactions.

1091
00:58:22,950 --> 00:58:24,450
And the nearest
neighbor interaction

1092
00:58:24,450 --> 00:58:27,230
is where i and j differ by 1.

1093
00:58:27,230 --> 00:58:30,440
And this is what I
call j without indices.

1094
00:58:30,440 --> 00:58:35,970
And similarly, when it comes
to the interaction term

1095
00:58:35,970 --> 00:58:40,620
where we have products
of four Wannier functions

1096
00:58:40,620 --> 00:58:44,020
and we want to get the
overlap of all four,

1097
00:58:44,020 --> 00:58:47,650
and then multiply it with g,
the prefactor of our delta

1098
00:58:47,650 --> 00:58:50,260
function to get an
interaction term,

1099
00:58:50,260 --> 00:58:54,530
well the best overlap is if
all indices are the same.

1100
00:58:54,530 --> 00:58:56,780
And this is what I call U.

1101
00:58:56,780 --> 00:59:00,750
So in that limit, in
that tight binding limit,

1102
00:59:00,750 --> 00:59:03,300
my Hamiltonian is
now very simple.

1103
00:59:03,300 --> 00:59:08,530
It consists of a tunneling
term parametrized with j,

1104
00:59:08,530 --> 00:59:16,470
and an on site interaction
term parametrized with U. Yes.

1105
00:59:16,470 --> 00:59:19,830
AUDIENCE: So you're saying
i, j and k are all the same,

1106
00:59:19,830 --> 00:59:22,710
so the interaction
is with itself?

1107
00:59:26,550 --> 00:59:29,790
PROFESSOR: No, two
particles per site.

1108
00:59:29,790 --> 00:59:32,460
When the four
particles are the same,

1109
00:59:32,460 --> 00:59:40,590
it turns into-- this
Hamiltonian here has

1110
00:59:40,590 --> 00:59:42,510
all indices are the same.

1111
00:59:42,510 --> 00:59:49,680
And if you calculate that, it
turns into the product of 2 b.

1112
00:59:49,680 --> 00:59:53,520
b dagger is the occupation
number at each site.

1113
00:59:53,520 --> 00:59:55,690
But if you're careful
with commutators,

1114
00:59:55,690 --> 00:59:58,980
it becomes occupation number
times occupation number

1115
00:59:58,980 --> 01:00:00,140
minus 1.

1116
01:00:00,140 --> 01:00:03,700
If you have only one particle
per site, this term is 0.

1117
01:00:03,700 --> 01:00:09,930
So technically U is, if
you put in 2 times 1,

1118
01:00:09,930 --> 01:00:11,980
if you have two
particles per site,

1119
01:00:11,980 --> 01:00:15,460
U is the interaction energy
between two particles.

1120
01:00:15,460 --> 01:00:17,240
Just use this
expression to figure out

1121
01:00:17,240 --> 01:00:18,740
what it is for three or four.

1122
01:00:18,740 --> 01:00:21,280
But for one particle, you get 0.

1123
01:00:21,280 --> 01:00:25,230
The self interaction of a
particle is absolutely 0.

1124
01:00:25,230 --> 01:00:27,235
One particle does not
interact with itself.

1125
01:00:31,029 --> 01:00:33,570
AUDIENCE: That gets back to my
question with the [INAUDIBLE].

1126
01:00:33,570 --> 01:00:40,120
So when you have i equals j, the
[INAUDIBLE] term, do you just--

1127
01:00:40,120 --> 01:00:43,670
PROFESSOR: No, I assume,
i j is nearest neighbor.

1128
01:00:43,670 --> 01:00:46,966
I assume that the index
i and j differ by 1.

1129
01:00:46,966 --> 01:00:49,215
AUDIENCE: Yeah, I mean your
argument was that the best

1130
01:00:49,215 --> 01:00:50,915
overlap is between
nearest neighbors.

1131
01:00:50,915 --> 01:00:51,540
PROFESSOR: Yes.

1132
01:00:54,100 --> 01:00:57,390
AUDIENCE: So an overlap
with the Wannier functions

1133
01:00:57,390 --> 01:01:01,145
at the same site is sort
of the kinetic energy term.

1134
01:01:01,145 --> 01:01:04,070
Is it approximately 0,
or you just absorb that

1135
01:01:04,070 --> 01:01:06,310
into the chemical potential?

1136
01:01:06,310 --> 01:01:07,430
PROFESSOR: This is a 0.

1137
01:01:07,430 --> 01:01:14,970
AUDIENCE: [INAUDIBLE]
So we're only

1138
01:01:14,970 --> 01:01:18,635
looking at nearest neighbor,
so are we essentially saying

1139
01:01:18,635 --> 01:01:23,254
the case where i equals j is
approximately 0 [INAUDIBLE].

1140
01:01:23,254 --> 01:01:25,045
AUDIENCE 2: Because
it's the matrix element

1141
01:01:25,045 --> 01:01:25,870
of a Hamiltonian.

1142
01:01:25,870 --> 01:01:29,830
So p, the candidate, plus
the lattice operator.

1143
01:01:29,830 --> 01:01:32,800
So if I take my
Wannier function,

1144
01:01:32,800 --> 01:01:35,770
which is built out of
eigenvalues for that equation,

1145
01:01:35,770 --> 01:01:38,245
[INAUDIBLE] bunch
of energy terms.

1146
01:01:38,245 --> 01:01:40,720
On every site [INAUDIBLE]
they're the same.

1147
01:01:40,720 --> 01:01:43,690
Because you have a common
[INAUDIBLE] energy that you can

1148
01:01:43,690 --> 01:01:44,720
[INAUDIBLE].

1149
01:01:44,720 --> 01:01:46,886
AUDIENCE: Well then just
multiple it by [INAUDIBLE].

1150
01:01:51,120 --> 01:01:53,340
PROFESSOR: My gut
feeling is-- and this

1151
01:01:53,340 --> 01:01:55,580
is why nobody considers
it-- it's just

1152
01:01:55,580 --> 01:01:59,660
a constant energy, which is
probably something like the 0

1153
01:01:59,660 --> 01:02:02,150
point energy times the number
of particles, which is not

1154
01:02:02,150 --> 01:02:03,600
affecting any dynamics.

1155
01:02:03,600 --> 01:02:06,510
It's pretty much a constant
which can be simply dropped.

1156
01:02:11,814 --> 01:02:13,230
Let me just go
back to that slide.

1157
01:02:13,230 --> 01:02:19,570
I hope-- let me just
look up the reference.

1158
01:02:19,570 --> 01:02:21,820
I haven't looked at it recently.

1159
01:02:21,820 --> 01:02:24,930
When we do this
exact transformation,

1160
01:02:24,930 --> 01:02:26,760
there should be a
reference whether i

1161
01:02:26,760 --> 01:02:30,010
and j, what happens
when i equals j.

1162
01:02:30,010 --> 01:02:32,869
I think it's just
a constant term.

1163
01:02:32,869 --> 01:02:34,327
AUDIENCE: Because
if you're summing

1164
01:02:34,327 --> 01:02:35,857
over every single
lattice site, so

1165
01:02:35,857 --> 01:02:38,100
you get the number of
particles times 0 [INAUDIBLE].

1166
01:02:38,100 --> 01:02:39,766
PROFESSOR: Yeah, but
even mathematically

1167
01:02:39,766 --> 01:02:43,160
it gives just a constant term,
here, which is [INAUDIBLE].

1168
01:02:43,160 --> 01:02:43,660
Thanks.

1169
01:02:43,660 --> 01:02:46,530
It was good to clarify it.

1170
01:02:46,530 --> 01:02:50,680
OK, I think within
the next 20 minutes,

1171
01:02:50,680 --> 01:02:53,986
I can step you through
the superfluid to Mott

1172
01:02:53,986 --> 01:02:54,860
insulator transition.

1173
01:02:57,490 --> 01:03:03,100
First, references for what
I've just said are given here.

1174
01:03:05,670 --> 01:03:11,940
But let me now come take
this Bose-Hubbard model

1175
01:03:11,940 --> 01:03:15,930
and discuss its
two limiting cases.

1176
01:03:15,930 --> 01:03:19,560
One case is where U
is much larger than j.

1177
01:03:19,560 --> 01:03:22,500
The other case is where
j is much larger than U.

1178
01:03:22,500 --> 01:03:24,740
These are the two
limiting cases,

1179
01:03:24,740 --> 01:03:27,820
and it will turn out
that one is an insulator

1180
01:03:27,820 --> 01:03:29,980
and one is a superfluid.

1181
01:03:29,980 --> 01:03:32,150
And that makes perfect
sense, of course.

1182
01:03:32,150 --> 01:03:37,040
If U is much larger than
j, I can set j equals to 0

1183
01:03:37,040 --> 01:03:38,410
if I can neglect it.

1184
01:03:38,410 --> 01:03:42,330
No tunneling means no transport,
and that means an insulator.

1185
01:03:42,330 --> 01:03:47,250
It's also clear that when
there is no tunneling,

1186
01:03:47,250 --> 01:03:50,720
that the system is
really described

1187
01:03:50,720 --> 01:03:55,690
by a product of so and so
many particles per site.

1188
01:03:55,690 --> 01:03:58,260
So it could be one
particle per site.

1189
01:03:58,260 --> 01:04:00,020
And I have a product
over all sites,

1190
01:04:00,020 --> 01:04:03,270
or two particles per site.

1191
01:04:03,270 --> 01:04:05,010
So this will be
the ground state.

1192
01:04:05,010 --> 01:04:08,610
And it's called the ground
state of the Mott insulator.

1193
01:04:08,610 --> 01:04:10,610
It's also trivial
to discuss what

1194
01:04:10,610 --> 01:04:13,050
happens if j is much,
much larger than U,

1195
01:04:13,050 --> 01:04:17,620
because then I simply neglect
U, and I have a free gas.

1196
01:04:17,620 --> 01:04:20,300
Well, a free gas of
Bloch wave function,

1197
01:04:20,300 --> 01:04:26,910
but that's the same as a free
gas of planar wave function.

1198
01:04:26,910 --> 01:04:30,600
It's just that quasimomentum
replaces momentum.

1199
01:04:30,600 --> 01:04:32,950
And if you have just
an ideal Bose gas

1200
01:04:32,950 --> 01:04:36,490
in a periodic lattice instead
of Bose-Einstein condensation

1201
01:04:36,490 --> 01:04:38,520
in the lowest
momentum state, you

1202
01:04:38,520 --> 01:04:41,490
have Bose-Einstein condensation
in the lowest quasimomentum

1203
01:04:41,490 --> 01:04:43,480
state.

1204
01:04:43,480 --> 01:04:46,400
The lowest quasimomentum
state is a superposition

1205
01:04:46,400 --> 01:04:49,170
of all Wannier
function-- I mean,

1206
01:04:49,170 --> 01:04:51,580
the 0 momentum
state in free space

1207
01:04:51,580 --> 01:04:57,830
is a superposition of all
position delta functions.

1208
01:04:57,830 --> 01:04:59,870
The plane wave is delocalized.

1209
01:04:59,870 --> 01:05:01,740
And the lowest
quasimomentum state

1210
01:05:01,740 --> 01:05:06,290
is just completely delocalized
over all Wannier functions.

1211
01:05:06,290 --> 01:05:07,910
But this is nothing
else than the q

1212
01:05:07,910 --> 01:05:11,660
equals 0 quasimomentum
Bloch wave.

1213
01:05:11,660 --> 01:05:13,410
The interesting
question is-- and this

1214
01:05:13,410 --> 01:05:17,270
has led to hundreds if not
more papers in the literature--

1215
01:05:17,270 --> 01:05:21,230
how do we go from one
limit to the other limit?

1216
01:05:21,230 --> 01:05:23,690
And now you would
say, well, maybe we

1217
01:05:23,690 --> 01:05:25,850
should use what
has worked so well.

1218
01:05:25,850 --> 01:05:28,490
We should use the
Bogoliubov approximation.

1219
01:05:28,490 --> 01:05:30,330
Just assume we
have a condensate,

1220
01:05:30,330 --> 01:05:36,580
replace all those kind of a 0
operators for the lowest Bloch

1221
01:05:36,580 --> 01:05:37,610
waves.

1222
01:05:37,610 --> 01:05:41,390
However, this doesn't work,
because the interesting thing

1223
01:05:41,390 --> 01:05:44,750
here is to find the
transition from the superfluid

1224
01:05:44,750 --> 01:05:46,125
from the Bose-Einstein
condensate

1225
01:05:46,125 --> 01:05:48,710
to an insulating state.

1226
01:05:48,710 --> 01:05:50,800
The nature of this
approximation is

1227
01:05:50,800 --> 01:05:52,960
that you need N 0 to be large.

1228
01:05:52,960 --> 01:05:58,200
But we're interested when the
condensate wave function turns

1229
01:05:58,200 --> 01:06:01,670
to 0, and in the
insulator it becomes 0.

1230
01:06:01,670 --> 01:06:04,970
So we are actually interested
in the opposite limit.

1231
01:06:04,970 --> 01:06:06,950
And indeed, if you
would ignore everything

1232
01:06:06,950 --> 01:06:08,590
I've just said that
doesn't make sense

1233
01:06:08,590 --> 01:06:11,260
to make this approximation
and make it nevertheless,

1234
01:06:11,260 --> 01:06:14,500
you will find that you never
get the insulating state

1235
01:06:14,500 --> 01:06:17,200
because you've pretty much
eliminated the possibility

1236
01:06:17,200 --> 01:06:21,316
to describe an insulating state
by doing this approximation.

1237
01:06:21,316 --> 01:06:22,690
So I want to show
you now that we

1238
01:06:22,690 --> 01:06:25,680
have to do another mean
field approximation, which

1239
01:06:25,680 --> 01:06:27,800
is actually nice.

1240
01:06:27,800 --> 01:06:32,890
It's very different from this
Bogoliubov approximation,

1241
01:06:32,890 --> 01:06:34,850
but it's also a mean
field approximation

1242
01:06:34,850 --> 01:06:37,830
which will describe our system.

1243
01:06:37,830 --> 01:06:43,010
So the goal is now
that I want to find

1244
01:06:43,010 --> 01:06:47,540
an effective Hamiltonian
which describes

1245
01:06:47,540 --> 01:06:50,090
the transition
from here to there.

1246
01:06:50,090 --> 01:06:53,190
And the important
approximation I will use

1247
01:06:53,190 --> 01:06:58,300
was is the following-- again,
I have to get rid of operators.

1248
01:06:58,300 --> 01:07:01,500
Products of two many
operators cannot be solved.

1249
01:07:01,500 --> 01:07:05,290
And so what I will do is I
will use products of operators,

1250
01:07:05,290 --> 01:07:08,960
write them as average
value plus fluctuations.

1251
01:07:08,960 --> 01:07:11,730
And then when I
multiply that out,

1252
01:07:11,730 --> 01:07:15,240
I take the product of
the average values,

1253
01:07:15,240 --> 01:07:19,050
and I include the
fluctuations in leading order.

1254
01:07:19,050 --> 01:07:23,370
So I take delta A times B
and delta B times A average,

1255
01:07:23,370 --> 01:07:27,360
but I neglect the product
of those fluctuations.

1256
01:07:27,360 --> 01:07:31,070
You can say I neglect the
correlation of fluctuations

1257
01:07:31,070 --> 01:07:33,480
here.

1258
01:07:33,480 --> 01:07:36,240
So this is spelled out here.

1259
01:07:36,240 --> 01:07:40,870
But the sign is important.

1260
01:07:40,870 --> 01:07:45,390
Just look at this equation--
A delta B plus delta B with B

1261
01:07:45,390 --> 01:07:48,440
plus A average times B average.

1262
01:07:48,440 --> 01:07:55,920
If I absorb A times B by
upgrading delta B to B,

1263
01:07:55,920 --> 01:07:59,170
but I do the same here,
I have to subtract

1264
01:07:59,170 --> 01:08:01,780
1 product of the
two average values.

1265
01:08:01,780 --> 01:08:05,510
It's actually this minus sign
which will play a role later,

1266
01:08:05,510 --> 01:08:09,110
but here you see already that
I will make this decoupling

1267
01:08:09,110 --> 01:08:11,600
approximation-- that I
decouple the fluctuations

1268
01:08:11,600 --> 01:08:14,160
from each other and I
write it in this way,

1269
01:08:14,160 --> 01:08:16,240
there is an
important minus sign.

1270
01:08:18,859 --> 01:08:24,000
OK, so we want to start
in the insulating state.

1271
01:08:24,000 --> 01:08:27,620
And we're going to figure out
how does the system develop

1272
01:08:27,620 --> 01:08:30,569
superfluidity out of
the insulation state?

1273
01:08:30,569 --> 01:08:33,110
And the perturbation
operator which takes me out

1274
01:08:33,110 --> 01:08:37,290
of the insulating
state is tunneling.

1275
01:08:37,290 --> 01:08:41,470
So therefore, the operator which
is responsible for breaking out

1276
01:08:41,470 --> 01:08:44,890
of the insulating
state is the operator

1277
01:08:44,890 --> 01:08:49,060
which induces tunneling
between neighboring sites.

1278
01:08:49,060 --> 01:08:51,890
And the tunneling operator
in this Hamiltonian--

1279
01:08:51,890 --> 01:08:56,580
remember, the Bose-Hubbard
model had j times B dagger B--

1280
01:08:56,580 --> 01:09:00,550
this was our tunneling
term-- this involves now

1281
01:09:00,550 --> 01:09:02,240
products of operators.

1282
01:09:02,240 --> 01:09:05,330
And I told you we want to get
rid of products of operators

1283
01:09:05,330 --> 01:09:07,470
to get something we
can easily solve.

1284
01:09:07,470 --> 01:09:10,370
So we use now this
product of operators

1285
01:09:10,370 --> 01:09:14,310
on two neighboring sites.

1286
01:09:14,310 --> 01:09:20,660
And we use exactly this
decoupling approximation.

1287
01:09:20,660 --> 01:09:26,210
So therefore, we replace each
operator by an average value,

1288
01:09:26,210 --> 01:09:28,880
and we neglect the product
of the fluctuations.

1289
01:09:28,880 --> 01:09:30,899
And then we obtain this.

1290
01:09:30,899 --> 01:09:34,750
And I explained to you where
the minus sign came from.

1291
01:09:34,750 --> 01:09:45,659
So now I call this average
value of the operator B l,

1292
01:09:45,659 --> 01:09:48,529
I call the superfluid
order parameter psi.

1293
01:09:54,380 --> 01:09:56,820
I think I could have
chosen psi to be complex,

1294
01:09:56,820 --> 01:10:01,410
and then B l dagger B l
would have complex conjugate.

1295
01:10:01,410 --> 01:10:05,450
But it's sufficient here
to restrict the discussion

1296
01:10:05,450 --> 01:10:08,690
on real numbers, and it
makes the notation simpler.

1297
01:10:08,690 --> 01:10:12,040
So anyway, I introduce that.

1298
01:10:12,040 --> 01:10:15,650
And what happens now
is the following--

1299
01:10:15,650 --> 01:10:19,770
that I have my Hamiltonian.

1300
01:10:19,770 --> 01:10:23,510
The Hamiltonian had
the interaction energy.

1301
01:10:28,460 --> 01:10:33,110
But the tunneling term
is now very simplified,

1302
01:10:33,110 --> 01:10:35,520
because instead of
tunneling from one site

1303
01:10:35,520 --> 01:10:39,660
to a neighboring
site, the other side

1304
01:10:39,660 --> 01:10:42,640
is sort of absorbed
by the mean field,

1305
01:10:42,640 --> 01:10:45,730
by the superfluid
order parameter psi.

1306
01:10:45,730 --> 01:10:51,470
And therefore, and this
here gives me a psi

1307
01:10:51,470 --> 01:10:55,270
squared term which appears here.

1308
01:10:55,270 --> 01:10:58,720
Trust me, this is just
an identical rewrite

1309
01:10:58,720 --> 01:11:02,850
of the previous Hamiltonian
by using this decoupling

1310
01:11:02,850 --> 01:11:05,160
approximation.

1311
01:11:05,160 --> 01:11:09,860
So what we have gained now
is something really dramatic.

1312
01:11:09,860 --> 01:11:13,540
We had many sites and
tunneling from site to site.

1313
01:11:13,540 --> 01:11:20,020
But if you look at
it now, we simply sum

1314
01:11:20,020 --> 01:11:23,850
the effective Hamiltonian
over site index l.

1315
01:11:23,850 --> 01:11:26,460
So our effective
Hamiltonian is now

1316
01:11:26,460 --> 01:11:29,950
the sum of an identical
Hamiltonian per site.

1317
01:11:29,950 --> 01:11:32,760
The sites no longer
interact with each other.

1318
01:11:32,760 --> 01:11:37,930
Each site interacts with all
the other sites described

1319
01:11:37,930 --> 01:11:41,530
by the mean field by the
superfluid order parameter psi.

1320
01:11:41,530 --> 01:11:43,800
So therefore, our
many-body problem,

1321
01:11:43,800 --> 01:11:45,380
which is still a
many-body problem,

1322
01:11:45,380 --> 01:11:49,720
but has turned into an effective
Hamiltonian for each site,

1323
01:11:49,720 --> 01:11:52,910
because each side has
the same Hamiltonian.

1324
01:11:52,910 --> 01:11:53,410
Colin.

1325
01:11:53,410 --> 01:11:57,730
AUDIENCE: What happened
to the j psi squared term?

1326
01:11:57,730 --> 01:12:00,132
PROFESSOR: This here?

1327
01:12:00,132 --> 01:12:00,840
AUDIENCE: Oh, OK.

1328
01:12:04,640 --> 01:12:08,740
PROFESSOR: OK so I can say
instead of solving for the sum,

1329
01:12:08,740 --> 01:12:11,850
I can just solve for
each site individually.

1330
01:12:11,850 --> 01:12:17,780
And this is now my effective
Hamiltonian for each site.

1331
01:12:17,780 --> 01:12:23,510
Now I want to catch the
onset of superfluidity.

1332
01:12:23,510 --> 01:12:27,240
So I want to get the
system when psi is small.

1333
01:12:27,240 --> 01:12:30,690
And therefore, I can just
ask-- I don't know what psi is.

1334
01:12:30,690 --> 01:12:32,060
It's part of my solution.

1335
01:12:32,060 --> 01:12:34,290
But I'm interested
in the moment when

1336
01:12:34,290 --> 01:12:39,720
psi begins to take off from
0, when superfluidity emerges.

1337
01:12:39,720 --> 01:12:45,280
So what I can therefore do is,
I can regard psi as an epsilon

1338
01:12:45,280 --> 01:12:47,930
parameter, as a small parameter.

1339
01:12:47,930 --> 01:12:52,310
And the psi parameter
comes with an operator V.

1340
01:12:52,310 --> 01:12:57,700
And this operator V is nothing
else than B l dagger plus B l.

1341
01:12:57,700 --> 01:13:00,030
So in other words,
what I'm doing is,

1342
01:13:00,030 --> 01:13:04,370
I'm separating my Hamiltonian
into a Hamiltonian which

1343
01:13:04,370 --> 01:13:06,960
is diagonal in the
quantum numbers

1344
01:13:06,960 --> 01:13:09,680
of the isolating state--
just one, two, three

1345
01:13:09,680 --> 01:13:11,600
particles per site.

1346
01:13:11,600 --> 01:13:13,340
And psi squared is the Z number.

1347
01:13:13,340 --> 01:13:17,170
Psi squared is also
diagonal in that.

1348
01:13:17,170 --> 01:13:19,560
And now the possibility
of tunneling,

1349
01:13:19,560 --> 01:13:22,720
the possibility of
superfluidity is now

1350
01:13:22,720 --> 01:13:29,610
perturbative in this
term psi times V.

1351
01:13:29,610 --> 01:13:33,460
OK, I don't want to explain,
actually, this expression.

1352
01:13:33,460 --> 01:13:37,020
It just formalizes
[INAUDIBLE] intermediate step.

1353
01:13:37,020 --> 01:13:38,970
When we have the
chemical potential,

1354
01:13:38,970 --> 01:13:41,380
and we raise the
chemical potential--

1355
01:13:41,380 --> 01:13:44,330
we go from zero to
one particle per site

1356
01:13:44,330 --> 01:13:45,950
to two particles per site.

1357
01:13:45,950 --> 01:13:48,180
And whatever the
chemical potential is

1358
01:13:48,180 --> 01:13:51,410
determines whether we have
one or two particles per site.

1359
01:13:51,410 --> 01:13:56,350
This is just telling me as a
function of chemical potential,

1360
01:13:56,350 --> 01:13:59,550
what is the ground
state of the insulator?

1361
01:13:59,550 --> 01:14:01,960
So now we take this ground
state-- it's actually

1362
01:14:01,960 --> 01:14:04,240
much easier described
in words then

1363
01:14:04,240 --> 01:14:07,210
by this formula-- we
take this ground state

1364
01:14:07,210 --> 01:14:14,900
and do perturbation theory in
our term psi times V. Remember,

1365
01:14:14,900 --> 01:14:20,620
the operator is B dagger
B. And the epsilon is psi.

1366
01:14:20,620 --> 01:14:26,220
So in second-order perturbation
theory, we get psi squared.

1367
01:14:26,220 --> 01:14:30,480
And then V, because B B
dagger, is very simple.

1368
01:14:30,480 --> 01:14:37,490
It only couples one occupation
number N to N plus 1

1369
01:14:37,490 --> 01:14:40,440
and N minus 1,
because B and B dagger

1370
01:14:40,440 --> 01:14:43,120
destroy or create a
particle per site.

1371
01:14:43,120 --> 01:14:47,520
So therefore, I can
immediately write down

1372
01:14:47,520 --> 01:14:50,830
what this matrix element is
in second-order perturbation

1373
01:14:50,830 --> 01:14:52,570
theory.

1374
01:14:52,570 --> 01:14:54,930
I mean, these are all--
sorry, it's all defined here.

1375
01:14:54,930 --> 01:14:57,020
I know I'm losing you now.

1376
01:14:57,020 --> 01:15:00,970
Nobody will tell me what is the
difference between U bar and U,

1377
01:15:00,970 --> 01:15:03,950
but it's trivially defined.

1378
01:15:03,950 --> 01:15:07,330
So the idea is we have
the isolating system.

1379
01:15:07,330 --> 01:15:10,020
We do perturbation
theory in tunneling.

1380
01:15:10,020 --> 01:15:17,180
And the perturbative operator
is psi times B plus B dagger.

1381
01:15:17,180 --> 01:15:19,160
The B plus B dagger
matrix elements

1382
01:15:19,160 --> 01:15:22,555
are trivial, because
they admix to N particles

1383
01:15:22,555 --> 01:15:25,450
per site-- N plus
1 and N minus 1.

1384
01:15:25,450 --> 01:15:27,794
And this is what
we've done here.

1385
01:15:27,794 --> 01:15:29,710
AUDIENCE: And the j is
just occupation number?

1386
01:15:29,710 --> 01:15:34,259
PROFESSOR: The j is the
occupation number of our site.

1387
01:15:34,259 --> 01:15:35,550
AUDIENCE: I have two questions.

1388
01:15:35,550 --> 01:15:41,044
So in the bottom equation, what
happened to the psi squared?

1389
01:15:41,044 --> 01:15:42,460
PROFESSOR: Sorry,
this is the sum.

1390
01:15:42,460 --> 01:15:44,394
The psi square is missing.

1391
01:15:44,394 --> 01:15:46,804
AUDIENCE: I guess I
had the same question

1392
01:15:46,804 --> 01:15:50,184
for the equation next
to the green thing.

1393
01:15:50,184 --> 01:15:52,860
Is there supposed to be
a psi squared in there?

1394
01:15:52,860 --> 01:15:56,360
Because originally, there's
a psi squared and an h0.

1395
01:15:56,360 --> 01:15:59,090
PROFESSOR: No, this is the
ground state where psi is 0.

1396
01:15:59,090 --> 01:16:00,600
And now we do
perturbation theory

1397
01:16:00,600 --> 01:16:05,500
in psi V. These are the
unperturbed energies which

1398
01:16:05,500 --> 01:16:07,640
appear in the
energy denominator.

1399
01:16:07,640 --> 01:16:09,530
The wave functions
we are using are

1400
01:16:09,530 --> 01:16:12,280
Fock states-- number
states-- per site.

1401
01:16:12,280 --> 01:16:17,070
And here we couple occupation
number j to all possible N's.

1402
01:16:17,070 --> 01:16:21,050
But because of B and B dagger,
N has only two values--

1403
01:16:21,050 --> 01:16:22,741
j plus 1 and j minus 1.

1404
01:16:22,741 --> 01:16:23,990
And this is what's given here.

1405
01:16:26,680 --> 01:16:28,270
It's mathematically trivial.

1406
01:16:28,270 --> 01:16:31,820
The notation is more complicated
than the physics behind it.

1407
01:16:31,820 --> 01:16:36,110
But now comes again-- I think I
need five minutes and I'm done.

1408
01:16:36,110 --> 01:16:37,920
But now comes the
interesting physics.

1409
01:16:37,920 --> 01:16:41,080
We have to ask what
are we doing here?

1410
01:16:41,080 --> 01:16:44,720
It's all the mathematical, and
the math is really simple here.

1411
01:16:44,720 --> 01:16:48,860
What we have done is we have
looked at the isolating state,

1412
01:16:48,860 --> 01:16:53,700
and we have done perturbation
theory in psi times B plus B

1413
01:16:53,700 --> 01:16:54,770
dagger.

1414
01:16:54,770 --> 01:16:56,710
And now we get an
energy correction

1415
01:16:56,710 --> 01:16:59,760
which is psi squared.

1416
01:16:59,760 --> 01:17:03,300
And what are we
really doing here?

1417
01:17:03,300 --> 01:17:05,440
Well, you can say
the following--

1418
01:17:05,440 --> 01:17:07,330
we started with a
hypothesis that we

1419
01:17:07,330 --> 01:17:11,720
have a superfluid state
characterized by psi and psi,

1420
01:17:11,720 --> 01:17:14,300
at the onset of
superfluidity-- is small.

1421
01:17:14,300 --> 01:17:16,210
But now we have
done the calculation

1422
01:17:16,210 --> 01:17:18,590
assuming that there is a psi.

1423
01:17:18,590 --> 01:17:21,370
But now we are turning around
and said have we really

1424
01:17:21,370 --> 01:17:24,450
done the system a favor by
introducing superfluidity?

1425
01:17:24,450 --> 01:17:27,810
In other words, has our
perturbation theory in psi

1426
01:17:27,810 --> 01:17:30,250
lowered the energy of
the state or raised

1427
01:17:30,250 --> 01:17:32,000
the energy of the state.

1428
01:17:32,000 --> 01:17:35,920
So in other words, we've done
a hypothetical calculation.

1429
01:17:35,920 --> 01:17:38,920
Hey, what would the
system feel like if there

1430
01:17:38,920 --> 01:17:40,630
were a little bit
of superfluidity?

1431
01:17:40,630 --> 01:17:43,730
If the system says great,
I've lowered my energy,

1432
01:17:43,730 --> 01:17:45,970
then we know we are in
the superfluid state.

1433
01:17:45,970 --> 01:17:48,830
When the system says no,
I raised my energy because

1434
01:17:48,830 --> 01:17:52,090
of the psi, then the system
has rejected our idea

1435
01:17:52,090 --> 01:17:54,120
to introduce superfluidity.

1436
01:17:54,120 --> 01:17:56,520
So therefore, the question
we are raising now

1437
01:17:56,520 --> 01:18:00,830
is after we have done the
calculation, for what values

1438
01:18:00,830 --> 01:18:07,100
of U and j is it favorable
to introduce a psi or not?

1439
01:18:07,100 --> 01:18:09,390
Now I was expecting the
question of some of you

1440
01:18:09,390 --> 01:18:12,030
that in second-order
perturbation theory,

1441
01:18:12,030 --> 01:18:15,690
second-order perturbation
theory always lowers the energy.

1442
01:18:15,690 --> 01:18:18,850
But remember, this is why I
emphasized the minus sign--

1443
01:18:18,850 --> 01:18:24,910
we had a psi squared term
which came from that, which

1444
01:18:24,910 --> 01:18:28,450
came from the last term of the
decoupling approximation, which

1445
01:18:28,450 --> 01:18:30,090
had a minus sign.

1446
01:18:30,090 --> 01:18:34,120
And therefore, we
have in psi squared

1447
01:18:34,120 --> 01:18:37,755
one term which came
from this special psi--

1448
01:18:37,755 --> 01:18:40,530
and I emphasized in the
decoupling approximation.

1449
01:18:40,530 --> 01:18:42,500
And we have a contribution
of psi squared

1450
01:18:42,500 --> 01:18:44,480
which comes from
perturbation theory.

1451
01:18:44,480 --> 01:18:49,030
And the two together can
actually change their sign.

1452
01:18:49,030 --> 01:18:51,695
So what we have right now is
if you describe the ground

1453
01:18:51,695 --> 01:18:55,910
state as a function of psi,
we have our unperturbed energy

1454
01:18:55,910 --> 01:18:58,500
of the Mott insulating
states, and then we

1455
01:18:58,500 --> 01:19:01,390
have a term in psi squared.

1456
01:19:01,390 --> 01:19:06,620
And we should-- and we could,
but we don't-- calculate

1457
01:19:06,620 --> 01:19:09,740
the next order in
psi to the four.

1458
01:19:09,740 --> 01:19:12,290
And it turns out in fourth-order
perturbation theory,

1459
01:19:12,290 --> 01:19:14,490
this term is always positive.

1460
01:19:14,490 --> 01:19:17,300
So what happens now
to the total energy

1461
01:19:17,300 --> 01:19:22,120
when this term A2, which we
have exactly analytically

1462
01:19:22,120 --> 01:19:27,610
calculated-- if this term A2
is larger or smaller than 0.

1463
01:19:27,610 --> 01:19:33,090
Well, if you have a parabolic
term and a quartic term,

1464
01:19:33,090 --> 01:19:38,390
in this case, both the
quadratic and the quartic term

1465
01:19:38,390 --> 01:19:40,850
are opening up
like in a U shape,

1466
01:19:40,850 --> 01:19:44,510
but here the total energy
turns into W shape.

1467
01:19:44,510 --> 01:19:47,240
And the interpretation
is fairly simple.

1468
01:19:47,240 --> 01:19:49,390
Under those
conditions, the system

1469
01:19:49,390 --> 01:19:54,010
prefers to have psi equal
0, whereas here, it's

1470
01:19:54,010 --> 01:19:58,120
like the system wants to
have a psi parameter which

1471
01:19:58,120 --> 01:20:00,960
is finite, either
positive or negative.

1472
01:20:00,960 --> 01:20:03,110
And whether it's
positive or negative

1473
01:20:03,110 --> 01:20:05,520
is sometimes called
spontaneous symmetry breaking

1474
01:20:05,520 --> 01:20:08,190
between two
degenerate solutions.

1475
01:20:08,190 --> 01:20:10,160
What I'm showing
here is, of course,

1476
01:20:10,160 --> 01:20:12,680
very similar to the
Ginzburg-Landau theory

1477
01:20:12,680 --> 01:20:15,980
of phase transitions, where
you have an effective potential

1478
01:20:15,980 --> 01:20:19,370
and the phase transition
takes place when

1479
01:20:19,370 --> 01:20:21,980
the second-order
coefficient changes sign,

1480
01:20:21,980 --> 01:20:24,890
and the effective potential
turns from U-shaped

1481
01:20:24,890 --> 01:20:27,020
into W-shaped.

1482
01:20:27,020 --> 01:20:28,260
So that's what we have done.

1483
01:20:31,190 --> 01:20:34,240
So anyway, I think this
is a nice problem where

1484
01:20:34,240 --> 01:20:36,980
the interpretation
is, I think, much more

1485
01:20:36,980 --> 01:20:39,120
subtle than the
calculation itself.

1486
01:20:39,120 --> 01:20:42,410
But we've calculated
the phase transition.

1487
01:20:42,410 --> 01:20:44,200
Everything is analytic.

1488
01:20:44,200 --> 01:20:46,090
Here is the analytic
result, which

1489
01:20:46,090 --> 01:20:48,515
I know with very
indigestible notation.

1490
01:20:51,570 --> 01:20:56,070
I followed exactly the paper
by Vanderstraeten, [? Stouff ?]

1491
01:20:56,070 --> 01:20:58,860
and collaborators, which
is posted on the website.

1492
01:20:58,860 --> 01:21:02,060
You can plot this phase
diagram in this way.

1493
01:21:02,060 --> 01:21:04,200
But if you use what
is more common,

1494
01:21:04,200 --> 01:21:09,260
normalize the chemical potential
by U and normalize j by U,

1495
01:21:09,260 --> 01:21:13,470
you get these wonderful
lobes of the Mott insulator

1496
01:21:13,470 --> 01:21:15,940
where you see that,
if you increase

1497
01:21:15,940 --> 01:21:19,790
the chemical potential, you
have N equals 1, N equals 2,

1498
01:21:19,790 --> 01:21:23,192
N equals 3, Mott insulator,
and in between you

1499
01:21:23,192 --> 01:21:25,530
go through superfluid regions.

1500
01:21:25,530 --> 01:21:29,290
However if your
tunneling is larger,

1501
01:21:29,290 --> 01:21:32,920
if your tunneling is too
large, you're only superfluid.

1502
01:21:32,920 --> 01:21:36,330
So this is sort
of the way how you

1503
01:21:36,330 --> 01:21:39,890
derive this rather
rich phase diagram

1504
01:21:39,890 --> 01:21:43,190
of bosonic atoms in
an optical lattice.

1505
01:21:43,190 --> 01:21:45,260
Green is an insulating state.

1506
01:21:45,260 --> 01:21:49,990
And white is the
superfluid state.

1507
01:21:49,990 --> 01:21:52,350
Let me just conclude by
showing a few slides how

1508
01:21:52,350 --> 01:21:54,090
this can be observed.

1509
01:21:54,090 --> 01:21:57,410
So in one case, we have
an insulating state

1510
01:21:57,410 --> 01:22:00,410
with a definite number
of particles per site.

1511
01:22:00,410 --> 01:22:02,680
And here we have
superfluid state,

1512
01:22:02,680 --> 01:22:06,790
which has the normal
fluctuations in number.

1513
01:22:10,260 --> 01:22:13,830
Here's another cartoon picture
of an condensate in the lowest

1514
01:22:13,830 --> 01:22:15,130
Bloch wave function.

1515
01:22:15,130 --> 01:22:19,430
And here you have
localized number states.

1516
01:22:19,430 --> 01:22:22,810
Often, you observe the
transition by taking the system

1517
01:22:22,810 --> 01:22:25,430
and releasing it
in time of flight

1518
01:22:25,430 --> 01:22:27,470
into ballistic expansion.

1519
01:22:27,470 --> 01:22:29,850
If the superfluidity
is coherent,

1520
01:22:29,850 --> 01:22:33,760
the wave function on the
different sites is coherent.

1521
01:22:33,760 --> 01:22:38,310
And what you get is a
multi-slit interference pattern,

1522
01:22:38,310 --> 01:22:40,320
which is a diffraction pattern.

1523
01:22:40,320 --> 01:22:43,830
And you see these characteristic
diffraction patterns.

1524
01:22:43,830 --> 01:22:46,320
It's an in-slit
diffraction pattern

1525
01:22:46,320 --> 01:22:50,360
which characterizes the
coherence in the initial state.

1526
01:22:50,360 --> 01:22:53,870
However if you have a
completely isolating state,

1527
01:22:53,870 --> 01:22:56,950
you have a Gaussian wave
packet on each side, which

1528
01:22:56,950 --> 01:22:59,900
has nothing to do with
the next neighbor.

1529
01:22:59,900 --> 01:23:02,420
And then the Gaussian
wave function simply

1530
01:23:02,420 --> 01:23:05,550
expands in a structureless way.

1531
01:23:05,550 --> 01:23:07,600
And so the transition
from here to there

1532
01:23:07,600 --> 01:23:12,670
happens exactly when
the ratio of U over j

1533
01:23:12,670 --> 01:23:14,900
crosses the value
we have derived.

1534
01:23:17,530 --> 01:23:19,140
There's another
technique of observing

1535
01:23:19,140 --> 01:23:21,240
that which is more recent.

1536
01:23:21,240 --> 01:23:23,800
This is a quantum
gas microscopes

1537
01:23:23,800 --> 01:23:26,240
where you do the same
physics in two dimensions,

1538
01:23:26,240 --> 01:23:31,210
so you have only one plane
where this physics takes place.

1539
01:23:31,210 --> 01:23:34,860
And you can observe atoms
now with a microscope.

1540
01:23:34,860 --> 01:23:38,710
So each green dot
here is an atom.

1541
01:23:38,710 --> 01:23:41,720
If you sort of zoom in.

1542
01:23:41,720 --> 01:23:45,780
And if you now study
the region where

1543
01:23:45,780 --> 01:23:49,050
you go through the superfluid
to the Mott insulator,

1544
01:23:49,050 --> 01:23:56,030
you see that you have
exactly one atom per site.

1545
01:23:56,030 --> 01:24:00,530
If you increase the atom
number, you have an insulator

1546
01:24:00,530 --> 01:24:02,800
with one atom per site.

1547
01:24:02,800 --> 01:24:06,450
And in the middle,
you have state

1548
01:24:06,450 --> 01:24:08,530
with two atoms per site,
which-- for reasons

1549
01:24:08,530 --> 01:24:11,800
I don't want to discuss-- are
color-coded here in black.

1550
01:24:11,800 --> 01:24:14,070
And that sort of
goes on and goes on.

1551
01:24:14,070 --> 01:24:18,430
So you can really
resolve site per site

1552
01:24:18,430 --> 01:24:21,430
the occupation of the
number of atoms per site.

1553
01:24:24,120 --> 01:24:27,580
So I think I'll stop
here with the Bose gases.

1554
01:24:27,580 --> 01:24:29,440
When you think the
Mott insulator is

1555
01:24:29,440 --> 01:24:32,410
the end of the story, all
motion has been frozen out

1556
01:24:32,410 --> 01:24:34,920
and you have one
particle per site, well,

1557
01:24:34,920 --> 01:24:37,170
it could be the
beginning of a new story.

1558
01:24:37,170 --> 01:24:40,640
Because if you use two
different hyperfine states,

1559
01:24:40,640 --> 01:24:44,640
spin up and spin down, we
can talk about spin ordering.

1560
01:24:44,640 --> 01:24:48,410
But this would be a
whole different lecture.