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PROFESSOR: I have
a tiring day today.

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00:00:22,870 --> 00:00:26,350
Just 10 minutes ago, I finished
chairing the first part

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00:00:26,350 --> 00:00:29,160
of a meeting of the
COA advisory committee.

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00:00:29,160 --> 00:00:30,580
And I gave a talk to them.

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00:00:30,580 --> 00:00:32,570
And right after the
lecture, I have to go back.

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00:00:32,570 --> 00:00:35,820
So while they have a lunch
break, I have to teach.

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00:00:35,820 --> 00:00:36,640
But that's OK.

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00:00:36,640 --> 00:00:40,370
That's life on the faculty.

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00:00:40,370 --> 00:00:43,130
Yes.

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00:00:43,130 --> 00:00:46,210
We're talking about applications
of the spontaneous light force.

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00:00:46,210 --> 00:00:51,230
We have used optical
block recreations and such

19
00:00:51,230 --> 00:00:55,480
to learn about the spontaneous
force, the stimulated force,

20
00:00:55,480 --> 00:00:57,100
in general.

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00:00:57,100 --> 00:01:01,850
And this is our expression for
the spontaneous light force.

22
00:01:01,850 --> 00:01:05,260
It's pretty much, the
momentum of photons

23
00:01:05,260 --> 00:01:09,100
is transferred to the atom
times to scattering rate.

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00:01:09,100 --> 00:01:14,040
And what we started last class,
and we want to finish today,

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00:01:14,040 --> 00:01:20,790
is we want to discuss
applications of this force

26
00:01:20,790 --> 00:01:25,320
to three situations which are
of experimental importance.

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00:01:25,320 --> 00:01:28,800
One can actually say that
slow beams, more or less,

28
00:01:28,800 --> 00:01:32,690
in the magneto-optical trap
are the key techniques which

29
00:01:32,690 --> 00:01:35,460
triggered a revolution
in atomic physics.

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00:01:35,460 --> 00:01:38,680
This is really the
enabling technology

31
00:01:38,680 --> 00:01:42,700
into ultracold atom science.

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00:01:42,700 --> 00:01:49,110
So I discussed with you, in
the last class, molasses.

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00:01:51,830 --> 00:01:56,040
And we discussed
what happens when

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00:01:56,040 --> 00:01:58,600
we have two
counterpropagating beams.

35
00:01:58,600 --> 00:02:03,460
Maybe the key figure to
remember is this one.

36
00:02:03,460 --> 00:02:05,890
When we have to
counterpropagating beams,

37
00:02:05,890 --> 00:02:08,860
by just adding up the
spontaneous light force

38
00:02:08,860 --> 00:02:14,820
of the two beams, we get the red
curve, force versus velocity.

39
00:02:14,820 --> 00:02:18,130
And the expert eye immediately
sees there is a linear slope.

40
00:02:18,130 --> 00:02:20,190
And that's a viscous force.

41
00:02:20,190 --> 00:02:24,782
This is the viscosity of
light interacting with atoms.

42
00:02:24,782 --> 00:02:27,990
Of course, you'd better
pick the detuning correctly.

43
00:02:27,990 --> 00:02:30,040
If you use the wrong
detuning, everything

44
00:02:30,040 --> 00:02:31,630
is flipped upside down.

45
00:02:31,630 --> 00:02:34,915
And instead of a damping force,
you have an anti-damping force.

46
00:02:39,430 --> 00:02:42,470
We then discussed
the cooling limit.

47
00:02:42,470 --> 00:02:51,335
Any cooling limit is the balance
between heating and cooling.

48
00:02:51,335 --> 00:02:54,810
And by just setting the heating
rate equal to the cooling rate,

49
00:02:54,810 --> 00:02:56,600
this describes a steady state.

50
00:02:56,600 --> 00:02:58,650
We found the famous
Doppler limit.

51
00:03:01,040 --> 00:03:01,540
OK.

52
00:03:01,540 --> 00:03:03,110
So that's where I
want to continue.

53
00:03:03,110 --> 00:03:06,710
But do you have any
questions about molasses,

54
00:03:06,710 --> 00:03:11,320
the Doppler limit, and what we
discussed in the last class?

55
00:03:15,180 --> 00:03:20,490
Well, I know we had discussions
about to take two beams which

56
00:03:20,490 --> 00:03:24,390
form a standing wave, and
take it apart, and calculate

57
00:03:24,390 --> 00:03:26,250
the force for each beam.

58
00:03:26,250 --> 00:03:28,810
Well, sorry, this is
what molasses is about.

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00:03:28,810 --> 00:03:30,590
Molasses is about two beams.

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00:03:30,590 --> 00:03:34,290
And if you want to treat
two interfering laser

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00:03:34,290 --> 00:03:38,100
beams in the full
generality, there

62
00:03:38,100 --> 00:03:39,560
are a lot of other effects.

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00:03:39,560 --> 00:03:41,310
Some of them, we
will be uncovering

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00:03:41,310 --> 00:03:43,620
in the next few lectures.

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00:03:43,620 --> 00:03:47,400
But I'm happy now, when
I discuss beam slowing,

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00:03:47,400 --> 00:03:51,870
to give you when there
is only one single laser

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00:03:51,870 --> 00:03:55,070
beam decelerating atoms.

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00:03:55,070 --> 00:03:57,590
That is the only
situation I know

69
00:03:57,590 --> 00:04:01,080
about where you have an
exact analytic solution.

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00:04:01,080 --> 00:04:03,525
So I want to present
it to you know.

71
00:04:03,525 --> 00:04:07,020
Actually, for a number of years,
I presented the exact solution

72
00:04:07,020 --> 00:04:10,720
with one laser beam before
I presented molasses,

73
00:04:10,720 --> 00:04:13,260
where I have to
make approximations.

74
00:04:13,260 --> 00:04:17,420
But I always felt that beam
slowing is more complicated,

75
00:04:17,420 --> 00:04:19,829
because you have to go
to a decelerating frame

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00:04:19,829 --> 00:04:21,690
and have to add
fictitious forces.

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00:04:21,690 --> 00:04:23,720
So some of the
intuition is lost.

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00:04:23,720 --> 00:04:27,250
So that's why I first
presented you with molasses,

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00:04:27,250 --> 00:04:30,540
with two beams, for which I
need to make approximations.

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00:04:30,540 --> 00:04:33,720
Because the interference
pattern between two laser beams

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00:04:33,720 --> 00:04:36,380
has richness and complexity.

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00:04:36,380 --> 00:04:39,590
But now I discuss
beam slowing with you.

83
00:04:39,590 --> 00:04:44,830
So this curve here
could be the result

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00:04:44,830 --> 00:04:48,110
of the very early beginnings
of laser cooling in

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00:04:48,110 --> 00:04:51,400
the late '70s, early '80s.

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00:04:51,400 --> 00:04:56,050
And this is when people had an
atomic beam with a Boltzmann

87
00:04:56,050 --> 00:04:59,960
distribution, and they had one
laser been counterpropagating.

88
00:04:59,960 --> 00:05:02,390
But the laser beam
had a fixed frequency.

89
00:05:02,390 --> 00:05:04,410
And what is this
laser beam doing?

90
00:05:04,410 --> 00:05:07,450
Well, it resonantly
interacts with the atoms

91
00:05:07,450 --> 00:05:09,920
which just have the
correct Doppler shift

92
00:05:09,920 --> 00:05:11,310
to be in resonance.

93
00:05:11,310 --> 00:05:15,770
Those atoms are pushed--
scattered photons-- are pushed

94
00:05:15,770 --> 00:05:19,110
to lower frequencies--
to lower velocities.

95
00:05:19,110 --> 00:05:21,040
But they are pushed
out of resonance.

96
00:05:21,040 --> 00:05:24,610
And eventually, the slowing
becomes slower and slower.

97
00:05:24,610 --> 00:05:27,810
And eventually, things come,
almost, to a standstill.

98
00:05:27,810 --> 00:05:31,500
So a laser beam which
has a fixed frequency

99
00:05:31,500 --> 00:05:34,790
would modify the
velocity distribution

100
00:05:34,790 --> 00:05:39,360
of an atomic beam in that way.

101
00:05:39,360 --> 00:05:40,980
People called it,
already, cooling.

102
00:05:40,980 --> 00:05:42,950
And I think some
pioneers in Russia

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00:05:42,950 --> 00:05:47,760
who did those experiments were
disappointed that they were not

104
00:05:47,760 --> 00:05:50,120
honored with a Nobel
Prize in laser cooling.

105
00:05:50,120 --> 00:05:51,260
This was the first cooling.

106
00:05:51,260 --> 00:05:54,830
Because this distribution
here is considerably narrower

107
00:05:54,830 --> 00:05:55,580
than here.

108
00:05:55,580 --> 00:05:58,605
And temperature is nothing
else than a measure

109
00:05:58,605 --> 00:06:02,640
of the reach of the
velocity distribution.

110
00:06:02,640 --> 00:06:06,060
But one can say that
those experiments did not

111
00:06:06,060 --> 00:06:07,830
trigger a revolution
in atomic physics.

112
00:06:07,830 --> 00:06:11,460
The revolution came when
people eventually figured out

113
00:06:11,460 --> 00:06:17,800
how they can narrow down this
peak from maybe a hundred

114
00:06:17,800 --> 00:06:22,410
or tens of kelvins down
to a million microkelvin.

115
00:06:22,410 --> 00:06:24,790
So it's clear what is needed.

116
00:06:24,790 --> 00:06:27,380
If you want to do
more slowing, you

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00:06:27,380 --> 00:06:30,150
have to make sure
that you are talking

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00:06:30,150 --> 00:06:32,400
to the other velocities,
and that you eventually

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00:06:32,400 --> 00:06:36,420
sweep many velocities into one
peak-- clear out the velocity

120
00:06:36,420 --> 00:06:38,670
distribution, collect
all the atoms,

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00:06:38,670 --> 00:06:41,850
and put them at a
monochromatic velocity.

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00:06:41,850 --> 00:06:47,200
So for that, you can say, well,
if you use a comb of frequency,

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00:06:47,200 --> 00:06:49,550
or a broad spectrum
of frequencies,

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00:06:49,550 --> 00:06:52,430
you could clean
out a larger area.

125
00:06:52,430 --> 00:06:54,140
And, indeed, there
are techniques

126
00:06:54,140 --> 00:06:59,640
like this-- white light slowing
or diffuse light slowing.

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00:06:59,640 --> 00:07:02,220
But since those
techniques are not

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00:07:02,220 --> 00:07:07,430
anymore of practical importance,
I don't want to discuss them.

129
00:07:07,430 --> 00:07:12,110
Let me talk about the
simplest technique-- at least,

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00:07:12,110 --> 00:07:15,950
conceptionally-- which
is chirped slowing.

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00:07:15,950 --> 00:07:18,390
And this works like follows.

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00:07:18,390 --> 00:07:20,620
You start with a laser
beam which is initially

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00:07:20,620 --> 00:07:22,750
in resonance with
the velocity group.

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00:07:22,750 --> 00:07:27,710
And as the atoms slow down,
you change the detuning

135
00:07:27,710 --> 00:07:30,810
of the laser that the
laser stays in resonance

136
00:07:30,810 --> 00:07:33,670
with the atoms.

137
00:07:33,670 --> 00:07:36,870
So it is a pulsed
technique where

138
00:07:36,870 --> 00:07:39,180
you have a continuous
atomic beam.

139
00:07:39,180 --> 00:07:42,570
You start with a laser and
say, now I start slowing.

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00:07:42,570 --> 00:07:46,055
And the laser is
following a group of atoms

141
00:07:46,055 --> 00:07:47,305
as they are decelerated.

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00:07:49,810 --> 00:07:53,940
So you can say it is like the
atoms are riding the surf.

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00:07:53,940 --> 00:07:56,140
The laser is always
following the atoms

144
00:07:56,140 --> 00:07:58,540
as they change their velocity.

145
00:07:58,540 --> 00:08:01,020
And what is, of course,
necessary for that to happen

146
00:08:01,020 --> 00:08:03,590
is that the chirp and the
deceleration of the atoms

147
00:08:03,590 --> 00:08:04,895
are synchronized.

148
00:08:04,895 --> 00:08:06,960
You will see that
in a moment, when

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00:08:06,960 --> 00:08:11,170
I put this idea into equations.

150
00:08:11,170 --> 00:08:14,440
There is another technique
which is used in many labs--

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00:08:14,440 --> 00:08:17,350
actually, in all my labs
and Martin Zwierlein's lab.

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00:08:17,350 --> 00:08:19,000
It's called Zeeman slowing.

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00:08:19,000 --> 00:08:21,690
And this is discussed
in your homework.

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00:08:21,690 --> 00:08:25,320
It is a CW version
of chirped slowing.

155
00:08:25,320 --> 00:08:29,040
And a lot of you
know what I mean.

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00:08:29,040 --> 00:08:33,080
The physics which goes on
there is almost identical.

157
00:08:33,080 --> 00:08:35,390
But for conceptional
reasons, it's

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00:08:35,390 --> 00:08:38,200
easier not to use
magnetic fields and such.

159
00:08:38,200 --> 00:08:39,710
Let's just keep it simple.

160
00:08:39,710 --> 00:08:44,720
Let's have one laser beam
talking to a two-level atom.

161
00:08:44,720 --> 00:08:49,660
And the only other ingredient
is the Doppler shift.

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00:08:49,660 --> 00:08:53,320
Any questions about--
so I've surveyed,

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00:08:53,320 --> 00:08:54,820
for you, beam slowing.

164
00:08:54,820 --> 00:08:59,160
And now I want to give
you the exact description

165
00:08:59,160 --> 00:09:02,160
for one typical example
of a slowing technique.

166
00:09:02,160 --> 00:09:04,734
And a lot of the results
we find in chirped slowing

167
00:09:04,734 --> 00:09:06,150
apply to other
slowing techniques.

168
00:09:08,730 --> 00:09:10,740
OK.

169
00:09:10,740 --> 00:09:13,870
I know I've taught
it several times.

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00:09:13,870 --> 00:09:19,770
And there is the
conceptional point

171
00:09:19,770 --> 00:09:23,630
that you can only describe
chirped slowing well

172
00:09:23,630 --> 00:09:26,920
when you go to the decelerating
frame of the atoms.

173
00:09:26,920 --> 00:09:30,600
You want to describe it in
the frame of the atoms which

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00:09:30,600 --> 00:09:33,260
are continuously
being decelerated.

175
00:09:33,260 --> 00:09:37,970
So the way I can introduce
it is the following.

176
00:09:37,970 --> 00:09:42,800
I take the maximum-- I take
the spontaneous light force.

177
00:09:42,800 --> 00:09:44,240
This is exactly
the expression we

178
00:09:44,240 --> 00:09:47,360
have discussed several times.

179
00:09:47,360 --> 00:09:53,930
And now I'm saying
we decide that we

180
00:09:53,930 --> 00:09:57,700
want to decelerate atoms
with a deceleration, a, which

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00:09:57,700 --> 00:10:00,240
is negative, because
we slow them down.

182
00:10:00,240 --> 00:10:03,810
Of course, if you pick a too
large that it would require

183
00:10:03,810 --> 00:10:05,850
more than the maximum
spontaneous light force,

184
00:10:05,850 --> 00:10:07,350
you will not find a solution.

185
00:10:07,350 --> 00:10:12,450
But let's simply assume we
want to decelerate atoms

186
00:10:12,450 --> 00:10:15,090
with a deceleration a.

187
00:10:15,090 --> 00:10:17,810
Well, what happens is
this deceleration, a,

188
00:10:17,810 --> 00:10:19,640
requires a certain force.

189
00:10:19,640 --> 00:10:23,280
If the intensity of the laser
would provide more force

190
00:10:23,280 --> 00:10:26,630
on resonance then it's
not going to work.

191
00:10:26,630 --> 00:10:29,790
Because we want to decelerate
the atoms at this deceleration.

192
00:10:29,790 --> 00:10:33,820
So therefore, the atoms,
if they are decelerated,

193
00:10:33,820 --> 00:10:38,420
have to be at a detuning,
delta prime, in such a way

194
00:10:38,420 --> 00:10:42,680
that the spontaneous
light force is exactly

195
00:10:42,680 --> 00:10:45,860
providing the
acceleration we decided

196
00:10:45,860 --> 00:10:48,700
we want to have for our atoms.

197
00:10:48,700 --> 00:10:53,580
In other words, this detuning--
in the frame of the atom,

198
00:10:53,580 --> 00:10:58,140
the atoms are on resonance
when the light force is exactly

199
00:10:58,140 --> 00:11:01,330
providing the force to give
you the acceleration a.

200
00:11:01,330 --> 00:11:05,520
But if you have more laser
power than absolutely necessary,

201
00:11:05,520 --> 00:11:07,770
in the frame of the
atoms, the atoms

202
00:11:07,770 --> 00:11:12,670
have to be detuned in
such a way that the force,

203
00:11:12,670 --> 00:11:15,260
with the detuning,
is exactly providing

204
00:11:15,260 --> 00:11:18,930
the acceleration we want.

205
00:11:18,930 --> 00:11:22,650
So that's the easiest
way to explain it.

206
00:11:22,650 --> 00:11:25,110
We have to set the stage.

207
00:11:25,110 --> 00:11:29,310
We have to introduce
this nominal detuning.

208
00:11:29,310 --> 00:11:29,810
OK.

209
00:11:29,810 --> 00:11:37,150
So you can say that this
detuning is just a definition.

210
00:11:37,150 --> 00:11:41,850
And now we want to describe
the slowing process.

211
00:11:41,850 --> 00:11:47,210
So we have said we want
to start with atoms

212
00:11:47,210 --> 00:11:50,290
at an initial velocity-- which
we'll actually cancel out

213
00:11:50,290 --> 00:11:51,490
in a moment.

214
00:11:51,490 --> 00:11:54,450
But let's do it in
a systematic way--

215
00:11:54,450 --> 00:11:58,640
and afterwards, we want the
velocity to decrease, linearly,

216
00:11:58,640 --> 00:11:59,440
in time.

217
00:12:01,990 --> 00:12:05,180
I simply defined this
parameter, delta prime,

218
00:12:05,180 --> 00:12:06,940
to be a solution
of this equation.

219
00:12:06,940 --> 00:12:09,430
So this is a well
defined quantity.

220
00:12:09,430 --> 00:12:13,740
And the laser detuning--
which, I've broken my laser

221
00:12:13,740 --> 00:12:16,470
to change its detuning
as a function of time--

222
00:12:16,470 --> 00:12:20,790
is now this nominal detuning
minus the Doppler shift

223
00:12:20,790 --> 00:12:24,400
due to the velocity
of the atoms.

224
00:12:24,400 --> 00:12:27,150
I'm really solving the
problem backwards for you.

225
00:12:27,150 --> 00:12:32,040
I first decide which should
be the velocity trajectory

226
00:12:32,040 --> 00:12:34,650
of the atoms-- how should
the atoms slow down.

227
00:12:34,650 --> 00:12:39,090
And then, I define what
is the detuning which

228
00:12:39,090 --> 00:12:40,910
will result in that.

229
00:12:40,910 --> 00:12:43,290
That's the easier
way to understand it.

230
00:12:43,290 --> 00:12:44,890
Of course, if you
pick this detuning,

231
00:12:44,890 --> 00:12:46,790
the atoms will exactly do that.

232
00:12:49,380 --> 00:12:49,900
OK.

233
00:12:49,900 --> 00:12:54,660
But now what we're
doing is the following.

234
00:12:54,660 --> 00:13:01,200
This is sort of the
acceleration and the velocity

235
00:13:01,200 --> 00:13:03,810
we want the atom to have.

236
00:13:03,810 --> 00:13:07,820
But there may be some
atoms in the beam which

237
00:13:07,820 --> 00:13:09,760
do not have this velocity.

238
00:13:09,760 --> 00:13:13,210
They deviate from this
velocity with v prime.

239
00:13:13,210 --> 00:13:15,810
So the goal of
cooling would now be

240
00:13:15,810 --> 00:13:20,750
to reduce v prime to 0 to
make all the atoms follow

241
00:13:20,750 --> 00:13:22,700
the trajectory we have
designed for them.

242
00:13:25,551 --> 00:13:26,050
OK.

243
00:13:26,050 --> 00:13:29,730
Many words, many definitions.

244
00:13:29,730 --> 00:13:31,267
Now we can simply
substitute what

245
00:13:31,267 --> 00:13:32,870
we have defined
into the equation.

246
00:13:32,870 --> 00:13:34,120
And we get a wonderful result.

247
00:13:39,580 --> 00:13:44,040
So we take equation
one, which was

248
00:13:44,040 --> 00:13:48,270
the equation for
the light force.

249
00:13:48,270 --> 00:13:50,210
And we put all
those definitions,

250
00:13:50,210 --> 00:13:56,290
which I went to great lengths
to explain to you, into it.

251
00:13:56,290 --> 00:14:01,040
And we get this result.

252
00:14:01,040 --> 00:14:04,690
It's just that the detuning has
been expressed by delta prime,

253
00:14:04,690 --> 00:14:07,970
and the velocity by this
difference velocity v prime.

254
00:14:07,970 --> 00:14:10,426
It's pure mathematical
substitution.

255
00:14:12,970 --> 00:14:16,962
But now comes the
important point

256
00:14:16,962 --> 00:14:20,410
that we want to describe
everything, now,

257
00:14:20,410 --> 00:14:23,170
in the decelerating frame.

258
00:14:23,170 --> 00:14:26,550
The atoms are meant to
decelerate with a deceleration

259
00:14:26,550 --> 00:14:27,446
a.

260
00:14:27,446 --> 00:14:29,070
And you know, from
classical mechanics,

261
00:14:29,070 --> 00:14:31,340
if you describe something
in a decelerating frame,

262
00:14:31,340 --> 00:14:34,890
you have to add a
fictitious force.

263
00:14:34,890 --> 00:14:39,250
This fictitious force, since
the acceleration is constant,

264
00:14:39,250 --> 00:14:40,335
is a constant force.

265
00:14:43,330 --> 00:14:47,440
But I'm expressing it, now,
by the other parameters.

266
00:14:47,440 --> 00:14:49,490
We've chosen our acceleration.

267
00:14:49,490 --> 00:14:51,520
We've chosen the
detuning, delta prime,

268
00:14:51,520 --> 00:14:54,180
that-- everything is
sort of connected.

269
00:14:54,180 --> 00:14:59,170
And this is just a way to
use the correct-- the most

270
00:14:59,170 --> 00:15:03,600
easiest units, or the
correct parametrization,

271
00:15:03,600 --> 00:15:05,010
for this constant force.

272
00:15:07,530 --> 00:15:10,660
So I have done-- and the
important thing for you

273
00:15:10,660 --> 00:15:13,160
is, this equation is exact.

274
00:15:13,160 --> 00:15:18,000
It's exact for an arbitrary
velocity v prime of the atoms.

275
00:15:18,000 --> 00:15:19,310
So let me just summarize.

276
00:15:19,310 --> 00:15:21,440
I've taken the
spontaneous light force.

277
00:15:21,440 --> 00:15:23,650
I've given you a
bunch of definitions,

278
00:15:23,650 --> 00:15:25,200
what delta prime is.

279
00:15:25,200 --> 00:15:28,780
That's how you
program your laser.

280
00:15:28,780 --> 00:15:32,090
But then, in the
decelerating frame,

281
00:15:32,090 --> 00:15:33,900
we need a fictitious force.

282
00:15:33,900 --> 00:15:39,150
And this is nothing else than
a mathematically exact rewrite

283
00:15:39,150 --> 00:15:41,840
of the spontaneous light force
in the decelerating frame.

284
00:15:49,070 --> 00:15:55,620
But now what happens is--
we had a decelerating--

285
00:15:55,620 --> 00:15:58,990
we had a light force where,
because we assume the laser is

286
00:15:58,990 --> 00:16:01,180
counterpropagating,
there's a minus sign.

287
00:16:01,180 --> 00:16:04,960
The fictitious force
has a plus sign.

288
00:16:04,960 --> 00:16:11,480
And that means that we
can-- and if v prime is 0,

289
00:16:11,480 --> 00:16:13,060
the force is 0.

290
00:16:13,060 --> 00:16:17,970
So we can now make a Taylor
expansion for small v prime.

291
00:16:17,970 --> 00:16:22,980
So we have, again, a viscous
force, a friction force.

292
00:16:22,980 --> 00:16:25,990
And it turns out that
this friction coefficient

293
00:16:25,990 --> 00:16:30,810
for beam deceleration is
exactly 1/2 what we got before,

294
00:16:30,810 --> 00:16:32,270
for molasses.

295
00:16:32,270 --> 00:16:34,320
But it's clear we
have one laser beam.

296
00:16:34,320 --> 00:16:37,800
Whereas in molasses,
we had two laser beams.

297
00:16:37,800 --> 00:16:40,740
If you ask, what is
the heating, well,

298
00:16:40,740 --> 00:16:43,340
if you have one laser beam
and not two laser beams,

299
00:16:43,340 --> 00:16:46,940
the heating described by the
momentum diffusion coefficient

300
00:16:46,940 --> 00:16:48,280
is also just 1/2.

301
00:16:48,280 --> 00:16:53,090
One beam heats exactly half
of what we got for two beams.

302
00:16:53,090 --> 00:16:56,060
And therefore, the
temperature, which

303
00:16:56,060 --> 00:16:59,660
characterizes the width of
the velocity distribution

304
00:16:59,660 --> 00:17:02,900
in the frame of the
decelerating atoms

305
00:17:02,900 --> 00:17:05,849
is exactly the Doppler
temperature we had before.

306
00:17:05,849 --> 00:17:08,089
Because the two
factors of 1/2 cancel.

307
00:17:16,140 --> 00:17:18,420
So now I have given
you-- and I hope

308
00:17:18,420 --> 00:17:22,390
you take some pleasure
in it-- an example where

309
00:17:22,390 --> 00:17:24,670
the laser cooling for
a two-level system

310
00:17:24,670 --> 00:17:26,210
is exactly described.

311
00:17:26,210 --> 00:17:27,829
There's only a single beam.

312
00:17:27,829 --> 00:17:30,250
And the Doppler limit
is the exact solution.

313
00:17:30,250 --> 00:17:32,720
I've not made any
approximation other

314
00:17:32,720 --> 00:17:35,860
than assuming I have a
two-level system for which

315
00:17:35,860 --> 00:17:38,900
the spontaneous light force has
the expression we've derived.

316
00:17:41,551 --> 00:17:42,050
OK.

317
00:17:42,050 --> 00:17:45,520
I know, even with
my explanation,

318
00:17:45,520 --> 00:17:47,680
you have to read through
it once or twice.

319
00:17:47,680 --> 00:17:50,720
Because I made a
few substitutions.

320
00:17:50,720 --> 00:17:53,560
And you really have to
digest them at your own pace.

321
00:17:53,560 --> 00:17:57,860
So let me give you a summary of
what we have done graphically.

322
00:17:57,860 --> 00:18:01,080
One laser beam scatters light.

323
00:18:01,080 --> 00:18:04,280
And it is this Lorentzian.

324
00:18:04,280 --> 00:18:06,499
And it's a negative
Lorentzian because we

325
00:18:06,499 --> 00:18:07,790
have a counterpropagating beam.

326
00:18:07,790 --> 00:18:10,000
We push the atoms into
the negative direction,

327
00:18:10,000 --> 00:18:12,810
they fly in the
positive direction.

328
00:18:12,810 --> 00:18:16,550
But by going-- by
chirping the laser,

329
00:18:16,550 --> 00:18:23,040
by going into the
decelerating frame,

330
00:18:23,040 --> 00:18:26,960
the force doesn't go to 0.

331
00:18:26,960 --> 00:18:29,260
I had to add an offset.

332
00:18:29,260 --> 00:18:32,240
And this offset is
the fictitious force

333
00:18:32,240 --> 00:18:35,540
due to the transformation
of the decelerating frame.

334
00:18:35,540 --> 00:18:40,500
And now you realize
that this force-- and it

335
00:18:40,500 --> 00:18:44,230
is the correct force in the
decelerating frame-- has two 0

336
00:18:44,230 --> 00:18:45,700
crossings.

337
00:18:45,700 --> 00:18:50,330
One is a stable point, and
one is an unstable point.

338
00:18:50,330 --> 00:18:54,790
And the stable point says, if
an atom happens to be too fast,

339
00:18:54,790 --> 00:18:56,400
it gets a correcting force.

340
00:18:56,400 --> 00:18:58,410
And this is like a lock point.

341
00:18:58,410 --> 00:19:00,230
It has a stable lock point.

342
00:19:00,230 --> 00:19:06,790
All the atoms which did not have
the correct velocity we wanted

343
00:19:06,790 --> 00:19:10,660
them to have, they are sort of
sucked into this lock point,

344
00:19:10,660 --> 00:19:14,130
and eventually, will
pile up at a huge peak

345
00:19:14,130 --> 00:19:15,940
at this nominal velocity.

346
00:19:15,940 --> 00:19:17,950
And the width of
this peak is simply

347
00:19:17,950 --> 00:19:21,560
described by the Doppler limit.

348
00:19:21,560 --> 00:19:25,310
Before I take your
questions, let

349
00:19:25,310 --> 00:19:31,175
me show you what happens
in the two frames.

350
00:19:36,720 --> 00:19:39,350
This is v prime.

351
00:19:39,350 --> 00:19:46,340
This is the velocity in
the decelerating frame.

352
00:19:46,340 --> 00:19:52,190
And let's say our
initial velocity is this.

353
00:19:52,190 --> 00:19:58,460
So what happens now is all
the atoms within the atoms

354
00:19:58,460 --> 00:20:04,810
are decelerated towards
v prime equals 0.

355
00:20:04,810 --> 00:20:09,030
The friction force puts
them to v prime equals 0.

356
00:20:09,030 --> 00:20:11,950
But since we are in
a decelerating frame,

357
00:20:11,950 --> 00:20:15,310
in the absence of the laser,
the whole Maxwell-Boltzmann

358
00:20:15,310 --> 00:20:17,960
distribution is
just shifted-- not

359
00:20:17,960 --> 00:20:19,800
because the velocity
of the atoms

360
00:20:19,800 --> 00:20:22,460
changed-- because we are
in a decelerating frame

361
00:20:22,460 --> 00:20:25,230
and we are accelerating
away from the atoms.

362
00:20:25,230 --> 00:20:28,660
And what I just described for
you is the following-- that,

363
00:20:28,660 --> 00:20:32,650
at v prime equals 0, we
have our friction force.

364
00:20:32,650 --> 00:20:36,460
And the atomic
distribution in this frame

365
00:20:36,460 --> 00:20:38,210
is now pushed through.

366
00:20:38,210 --> 00:20:41,420
And everything which goes
through this resonant region

367
00:20:41,420 --> 00:20:43,130
is sort of collected.

368
00:20:43,130 --> 00:20:50,230
And you have a peak which is
piling up at v prime equals 0.

369
00:20:50,230 --> 00:20:55,630
Probably, it's easier--
but mathematically,

370
00:20:55,630 --> 00:20:57,490
I needed the
decelerating frame--

371
00:20:57,490 --> 00:20:59,750
if you use the
following picture.

372
00:20:59,750 --> 00:21:03,710
And just think of this beam
slowing I explained to you.

373
00:21:03,710 --> 00:21:05,740
If you just switched
on a laser here,

374
00:21:05,740 --> 00:21:08,315
you would just burn a hole
into the velocity distribution

375
00:21:08,315 --> 00:21:10,160
and pile up the atoms.

376
00:21:10,160 --> 00:21:11,700
But now you sweep.

377
00:21:11,700 --> 00:21:14,820
And when you sweep, you take
the peak, push it further.

378
00:21:14,820 --> 00:21:16,180
It gets a bigger peak here.

379
00:21:16,180 --> 00:21:17,380
Push it further.

380
00:21:17,380 --> 00:21:19,750
So eventually, what
you're doing is,

381
00:21:19,750 --> 00:21:22,550
by chirping the laser
in the correct way

382
00:21:22,550 --> 00:21:25,060
with the correct laser
power and all that,

383
00:21:25,060 --> 00:21:29,020
you're just pushing
the atoms all out.

384
00:21:29,020 --> 00:21:31,690
And eventually,
they are piled up

385
00:21:31,690 --> 00:21:35,840
in a narrow distribution which
is defined by the time when you

386
00:21:35,840 --> 00:21:38,280
just switch off your laser beam.

387
00:21:38,280 --> 00:21:41,360
So one chirp of the laser,
from an initial detuning

388
00:21:41,360 --> 00:21:45,430
in the lab frame to a final
detuning in the lab frame,

389
00:21:45,430 --> 00:21:49,950
will just sweep out the whole
Maxwell-Boltzmann distribution.

390
00:21:49,950 --> 00:21:52,400
And if you suddenly
switch off the laser,

391
00:21:52,400 --> 00:21:56,580
you have frozen in a velocity
distribution which is exactly

392
00:21:56,580 --> 00:21:58,535
described by the Doppler limit.

393
00:22:04,180 --> 00:22:06,290
So what is easier to
understand-- molasses

394
00:22:06,290 --> 00:22:12,050
with two beams, or beam
slowing with a single beam?

395
00:22:12,050 --> 00:22:14,140
I think molasses is
easier, because everything

396
00:22:14,140 --> 00:22:15,330
happens in the lab frame.

397
00:22:15,330 --> 00:22:16,670
But think about it.

398
00:22:16,670 --> 00:22:23,275
It's a nice example for which no
approximations have to be made.

399
00:22:23,275 --> 00:22:23,775
Questions?

400
00:22:28,520 --> 00:22:30,310
Yes, Nancy.

401
00:22:30,310 --> 00:22:31,980
AUDIENCE: It's not
really obvious to me

402
00:22:31,980 --> 00:22:36,450
how, in this mathematics,
we are choosing

403
00:22:36,450 --> 00:22:39,324
a time-dependent
detuning as opposed

404
00:22:39,324 --> 00:22:41,719
to a time-dependent
deceleration.

405
00:22:47,470 --> 00:22:49,900
PROFESSOR: The way I
presented it logically, we

406
00:22:49,900 --> 00:22:52,950
said we want a certain
acceleration to happen.

407
00:22:52,950 --> 00:23:00,250
And then, we provide our laser
that k/kv is the Doppler shift.

408
00:23:00,250 --> 00:23:04,620
ka is how the Doppler shift
changes as a function of time.

409
00:23:04,620 --> 00:23:09,330
And we chirp our laser
exactly in frequency

410
00:23:09,330 --> 00:23:12,840
with a chirp which is ka.

411
00:23:12,840 --> 00:23:17,950
So in other words, I started
to define a constant a.

412
00:23:17,950 --> 00:23:20,890
And that would require
that the laser is now

413
00:23:20,890 --> 00:23:23,100
chirped in a linear way.

414
00:23:23,100 --> 00:23:25,500
I could have started the
other around and said, hey,

415
00:23:25,500 --> 00:23:28,220
let's assume we chirp the
laser in a linear way.

416
00:23:28,220 --> 00:23:30,740
And then, we do
corresponding substitutions.

417
00:23:30,740 --> 00:23:33,260
And the result
would be the same.

418
00:23:33,260 --> 00:23:35,715
AUDIENCE: No, actually,
like if we did not

419
00:23:35,715 --> 00:23:39,760
want to have a detuning
which was time-dependent,

420
00:23:39,760 --> 00:23:42,670
would we have gotten--
we still have a force.

421
00:23:42,670 --> 00:23:44,657
And that force would
have been just changing.

422
00:23:44,657 --> 00:23:46,740
PROFESSOR: Well then, it's
more complicated, yeah.

423
00:23:46,740 --> 00:23:49,480
Then, you have to integrate
a differential equation.

424
00:23:49,480 --> 00:23:53,200
But what happened
is the following.

425
00:23:53,200 --> 00:23:57,300
What I've done is, by defining
the acceleration and then

426
00:23:57,300 --> 00:24:02,570
the chirp, I have the situation
that, in the reference

427
00:24:02,570 --> 00:24:06,800
frame of the decelerating
atoms, the detuning is constant.

428
00:24:06,800 --> 00:24:10,020
So usually, you go to
a decelerating frame

429
00:24:10,020 --> 00:24:12,100
if something else simplifies.

430
00:24:12,100 --> 00:24:16,670
And what simplifies is that,
in the frame of the atom,

431
00:24:16,670 --> 00:24:18,910
the laser detuning is constant.

432
00:24:18,910 --> 00:24:22,770
So you can also say the frame
in which the atoms decelerate is

433
00:24:22,770 --> 00:24:27,900
exactly the frame in which
the chirp has disappeared.

434
00:24:27,900 --> 00:24:30,870
Because the decelerating
frame compensates

435
00:24:30,870 --> 00:24:33,810
the chirp with a
linearly varying,

436
00:24:33,810 --> 00:24:35,600
time varying Doppler shift.

437
00:24:39,490 --> 00:24:42,020
OK.

438
00:24:42,020 --> 00:24:45,860
But the physics, I
think, is really--

439
00:24:45,860 --> 00:24:49,160
the moment you have a force
which has a 0 crossing,

440
00:24:49,160 --> 00:24:50,380
it's a lock point.

441
00:24:50,380 --> 00:24:52,020
You determine the slope.

442
00:24:52,020 --> 00:24:54,260
And alpha, together with
the momentum diffusion,

443
00:24:54,260 --> 00:24:55,650
gives you the final temperature.

444
00:24:55,650 --> 00:24:58,520
So I hope you enjoy that it's
actually the same physics--

445
00:24:58,520 --> 00:25:00,910
exactly the same
physics-- we discussed,

446
00:25:00,910 --> 00:25:03,700
with some approximation
for molasses.

447
00:25:09,060 --> 00:25:09,560
OK.

448
00:25:09,560 --> 00:25:14,230
We had a discussion last time
about, we have the force,

449
00:25:14,230 --> 00:25:16,720
we find exactly what
happens to the atoms.

450
00:25:16,720 --> 00:25:20,850
But we may have to think
about it in different ways

451
00:25:20,850 --> 00:25:26,020
to figure out where the
kinetic energy goes from.

452
00:25:26,020 --> 00:25:29,700
And I think we mentioned it
already, in the last class,

453
00:25:29,700 --> 00:25:33,960
that what happens is you have
a red-detuned laser, which is,

454
00:25:33,960 --> 00:25:36,130
the photons are absorbed.

455
00:25:36,130 --> 00:25:40,200
But the atom is scattering
light in all directions.

456
00:25:40,200 --> 00:25:43,000
And so if you go
at 90 degrees, you

457
00:25:43,000 --> 00:25:44,920
don't have a linear
Doppler shift.

458
00:25:44,920 --> 00:25:48,080
If you go forward-backward
with the same probability--

459
00:25:48,080 --> 00:25:51,890
a symmetric pattern-- the
Doppler shift averages to 0.

460
00:25:51,890 --> 00:25:56,810
So on average, you emit
photons which are on resonance.

461
00:25:56,810 --> 00:26:00,880
And therefore, every time
you scatter a photon,

462
00:26:00,880 --> 00:26:05,270
you radiate away
the energy which

463
00:26:05,270 --> 00:26:06,530
is equal to the Doppler shift.

464
00:26:11,300 --> 00:26:14,000
That means, initially, when
the Doppler shift is huge,

465
00:26:14,000 --> 00:26:17,480
each photon transports
away a lot of energy.

466
00:26:17,480 --> 00:26:20,910
And the slower the atoms
are, the less a photon

467
00:26:20,910 --> 00:26:22,570
transports away.

468
00:26:22,570 --> 00:26:23,870
But that makes sense.

469
00:26:23,870 --> 00:26:25,800
If you have a huge
velocity and you

470
00:26:25,800 --> 00:26:29,570
subtract h bar k, the
momentum of the photon,

471
00:26:29,570 --> 00:26:33,380
the kinetic energy is
quadratic in velocity.

472
00:26:33,380 --> 00:26:37,300
You know, h bar k, the
momentum transfer of a photon,

473
00:26:37,300 --> 00:26:40,260
takes out more kinetic
energy the faster

474
00:26:40,260 --> 00:26:42,550
the velocity of the atoms is.

475
00:26:42,550 --> 00:26:44,445
So everything makes
sense and is consistent.

476
00:26:52,450 --> 00:26:53,140
OK.

477
00:26:53,140 --> 00:26:58,570
Let's go from the-- we
started with two beams.

478
00:26:58,570 --> 00:27:02,080
We just discussed one
beam, chirped slowing.

479
00:27:02,080 --> 00:27:06,360
And let's go now to what most
of you use in the laboratory--

480
00:27:06,360 --> 00:27:08,440
namely, six beams.

481
00:27:08,440 --> 00:27:11,210
You want to cool in x, y, z.

482
00:27:11,210 --> 00:27:13,670
So you have
one-dimensional molasses

483
00:27:13,670 --> 00:27:17,130
with two beams times 3.

484
00:27:17,130 --> 00:27:22,610
So in the limit that
the intensity is low,

485
00:27:22,610 --> 00:27:26,140
you just add up all the forces.

486
00:27:26,140 --> 00:27:28,710
You say each beam has a
spontaneous light force.

487
00:27:28,710 --> 00:27:31,618
And you add the six
spontaneous light forces up.

488
00:27:34,610 --> 00:27:38,110
At low intensity-- and you
have to take my word for it--

489
00:27:38,110 --> 00:27:41,150
you can ignore all the
interference effect

490
00:27:41,150 --> 00:27:43,280
between the beams.

491
00:27:43,280 --> 00:27:47,760
They can actually lead to
very interesting effects.

492
00:27:47,760 --> 00:27:51,530
I will give you at least a taste
of what will happen-- actually,

493
00:27:51,530 --> 00:27:55,610
a pretty good taste-- what
happens at higher power.

494
00:27:55,610 --> 00:27:57,495
We need the traced atom picture.

495
00:27:57,495 --> 00:28:01,155
And we'll discuss how
the stimulated force

496
00:28:01,155 --> 00:28:02,155
can be used for cooling.

497
00:28:04,750 --> 00:28:07,180
But if you don't have
a two-level system,

498
00:28:07,180 --> 00:28:10,450
you have sigma plus
sigma minus transitions

499
00:28:10,450 --> 00:28:17,270
and the beams interfere, you get
spatially varying polarizations

500
00:28:17,270 --> 00:28:19,350
of your interference
patterns, and then you

501
00:28:19,350 --> 00:28:22,240
have much [INAUDIBLE] physics.

502
00:28:22,240 --> 00:28:26,120
But in the simplest case
of a two-level system,

503
00:28:26,120 --> 00:28:28,150
there is a regime
at low intensity

504
00:28:28,150 --> 00:28:32,510
where you can simply take the
equations we got for molasses

505
00:28:32,510 --> 00:28:33,715
and apply them to XYZ.

506
00:28:36,530 --> 00:28:41,660
I put, on the website, a nice
handout, a review paper by Bill

507
00:28:41,660 --> 00:28:46,560
Phillips, who is discussing 1D
molasses versus 3D molasses,

508
00:28:46,560 --> 00:28:48,480
and if you enjoy
it, you can read

509
00:28:48,480 --> 00:28:52,400
how numerical factors
of 1 or 2 or 3

510
00:28:52,400 --> 00:28:55,920
now appear in the
friction coefficient,

511
00:28:55,920 --> 00:28:57,650
in the heating
coefficient, and how

512
00:28:57,650 --> 00:28:59,860
they affect the
final temperature.

513
00:28:59,860 --> 00:29:03,510
But you got the full [INAUDIBLE]
concepts and the correct idea

514
00:29:03,510 --> 00:29:04,930
for one dimension molasses.

515
00:29:11,290 --> 00:29:22,310
Now, let me give you
an outlook for what

516
00:29:22,310 --> 00:29:24,250
we will be doing next week.

517
00:29:24,250 --> 00:29:28,790
Next week, we want to
understand the following-- what

518
00:29:28,790 --> 00:29:31,170
happens to the
friction coefficient

519
00:29:31,170 --> 00:29:34,070
when you increase the power?

520
00:29:34,070 --> 00:29:38,190
Well, you remember when
we have red-detuned light,

521
00:29:38,190 --> 00:29:43,670
then we had one Lorentzian,
another Lorentzian.

522
00:29:43,670 --> 00:29:45,770
The slope was
negative, and negative

523
00:29:45,770 --> 00:29:49,480
means positive
friction, we cool.

524
00:29:49,480 --> 00:29:51,840
And if you put
more power into it,

525
00:29:51,840 --> 00:29:55,140
the linear slope increases,
and for low power,

526
00:29:55,140 --> 00:29:57,620
the friction coefficient is
proportional to the laser

527
00:29:57,620 --> 00:29:59,540
intensity.

528
00:29:59,540 --> 00:30:01,850
But now look what happens.

529
00:30:01,850 --> 00:30:10,330
If you approach saturation,
then the friction

530
00:30:10,330 --> 00:30:16,680
is not only saturating, it
suddenly changes the sign,

531
00:30:16,680 --> 00:30:19,230
and this is qualitatively new.

532
00:30:19,230 --> 00:30:22,050
And you should a little
bit wonder about it.

533
00:30:22,050 --> 00:30:25,280
Wasn't the reason why we
have cooling that you have

534
00:30:25,280 --> 00:30:28,140
a laser which is red-detuned?

535
00:30:28,140 --> 00:30:30,440
And the photon which is
spontaneously emitted

536
00:30:30,440 --> 00:30:33,800
is on resonance, and this
is the energy difference.

537
00:30:33,800 --> 00:30:37,470
But now I'm telling
you, at high intensity,

538
00:30:37,470 --> 00:30:40,210
it's not the red-detuned
laser which cools,

539
00:30:40,210 --> 00:30:44,020
it's the blue-detuned laser.

540
00:30:44,020 --> 00:30:45,660
So you should wonder about it.

541
00:30:45,660 --> 00:30:49,040
There is really something
new to be learned.

542
00:30:49,040 --> 00:30:53,440
So I want to motivate you to
follow me today and next week

543
00:30:53,440 --> 00:30:57,940
through an alternative
description

544
00:30:57,940 --> 00:31:01,700
of what light does to atoms
using the dressed atom picture,

545
00:31:01,700 --> 00:31:03,550
and in the dressed
atom picture, we

546
00:31:03,550 --> 00:31:06,040
will very naturally
understand why

547
00:31:06,040 --> 00:31:09,180
now blue-detuned light
provides cooling.

548
00:31:09,180 --> 00:31:11,770
But right now, for you,
this should be a mystery,

549
00:31:11,770 --> 00:31:14,260
but it should be something
where you say, hey,

550
00:31:14,260 --> 00:31:17,000
still something
qualitative is missing

551
00:31:17,000 --> 00:31:19,175
in our understanding
of light forces.

552
00:31:21,860 --> 00:31:25,240
OK, I will not be
able to explain

553
00:31:25,240 --> 00:31:28,200
it [? physically ?] to you.

554
00:31:28,200 --> 00:31:30,350
I first have to introduce
the dressed atom picture,

555
00:31:30,350 --> 00:31:33,780
and I will do that it
in 10 minutes or so.

556
00:31:33,780 --> 00:31:37,190
But what I want to do
here is if I'm telling you

557
00:31:37,190 --> 00:31:40,350
something goes terribly
wrong and we completely

558
00:31:40,350 --> 00:31:43,690
miss even the sign of the
force, blue-detuned light is now

559
00:31:43,690 --> 00:31:45,710
cooling, not red-detuned
light, I at least

560
00:31:45,710 --> 00:31:48,370
want to tell you
what went wrong.

561
00:31:48,370 --> 00:31:52,720
Because haven't we done
an almost exact discussion

562
00:31:52,720 --> 00:31:54,380
of light forces?

563
00:31:54,380 --> 00:31:57,484
We used the gradient of d dot e.

564
00:31:57,484 --> 00:32:01,090
We used for d the steady state
solution of the optical Bloch

565
00:32:01,090 --> 00:32:01,660
equations.

566
00:32:01,660 --> 00:32:05,170
Haven't we done
everything right?

567
00:32:05,170 --> 00:32:09,790
Well, we did approximations.

568
00:32:09,790 --> 00:32:13,910
And what happens is,
and this is the reason

569
00:32:13,910 --> 00:32:18,240
for those qualitatively
different things, that we

570
00:32:18,240 --> 00:32:23,980
should, in order to get
those effects, not use,

571
00:32:23,980 --> 00:32:27,960
for the dipole moment, for the
off-diagonal matrix elements

572
00:32:27,960 --> 00:32:30,930
of the density matrix,
the steady state solution

573
00:32:30,930 --> 00:32:33,070
for optical Bloch vector.

574
00:32:33,070 --> 00:32:37,910
We have to acknowledge that
it always takes a finite time

575
00:32:37,910 --> 00:32:41,420
to reach steady state.

576
00:32:41,420 --> 00:32:46,090
For instance, if he takes, let's
say, a spontaneous emission

577
00:32:46,090 --> 00:32:49,730
time, if the atom goes
through a standing wave,

578
00:32:49,730 --> 00:32:52,830
it goes from high
intensity to low intensity,

579
00:32:52,830 --> 00:32:55,630
it takes about one
spontaneous emission time.

580
00:32:55,630 --> 00:32:57,850
When you are highly
excited, but now you

581
00:32:57,850 --> 00:33:00,120
go through a node of
the standing wave,

582
00:33:00,120 --> 00:33:02,030
the steady state
solution is [INAUDIBLE].

583
00:33:02,030 --> 00:33:04,450
At the node of the standing
wave, there is no light.

584
00:33:04,450 --> 00:33:06,090
The steady state
solution with no light

585
00:33:06,090 --> 00:33:07,960
is you better go back
to the ground state,

586
00:33:07,960 --> 00:33:10,200
but that takes
something on the order

587
00:33:10,200 --> 00:33:12,480
of the spontaneous
emission time.

588
00:33:12,480 --> 00:33:15,680
So therefore, when I
am excited because I'm

589
00:33:15,680 --> 00:33:18,110
in the middle of the
[? pied ?] laser beam

590
00:33:18,110 --> 00:33:20,820
and I go fast into
the dark region,

591
00:33:20,820 --> 00:33:23,060
I'm still excited because
I didn't have time

592
00:33:23,060 --> 00:33:25,310
to get rid of my excitation.

593
00:33:25,310 --> 00:33:29,340
In other words, the
internal degree of freedom,

594
00:33:29,340 --> 00:33:32,040
the population of
grounded excited state,

595
00:33:32,040 --> 00:33:35,580
will lag behind the
steady state solution

596
00:33:35,580 --> 00:33:39,940
by a lag time which is a
spontaneous emission time.

597
00:33:39,940 --> 00:33:43,730
And that would mean if I
multiply with the velocity,

598
00:33:43,730 --> 00:33:46,820
it's pretty much that there
is a spacial displacement.

599
00:33:53,360 --> 00:33:59,770
I'm always a certain distance
away from the steady state

600
00:33:59,770 --> 00:34:01,200
solution.

601
00:34:01,200 --> 00:34:09,880
And actually, this point,
to introduce the lag time,

602
00:34:09,880 --> 00:34:13,130
we with do it in a
wonderful, physical way when

603
00:34:13,130 --> 00:34:14,830
we have the dressed
atom picture,

604
00:34:14,830 --> 00:34:18,620
but it can be done with
optical Bloch equation.

605
00:34:18,620 --> 00:34:21,770
It's just a real pain because
the optical Bloch equation

606
00:34:21,770 --> 00:34:24,469
is sort of like a
matrix the equation.

607
00:34:24,469 --> 00:34:27,239
It's completely
obscure, how it is done,

608
00:34:27,239 --> 00:34:29,480
but in order that you
convince yourself,

609
00:34:29,480 --> 00:34:33,580
I posted the original paper
from Gordon and Ashkin

610
00:34:33,580 --> 00:34:34,900
where this was done.

611
00:34:34,900 --> 00:34:36,080
You read the math.

612
00:34:36,080 --> 00:34:37,710
You understand the
result, but you

613
00:34:37,710 --> 00:34:40,710
don't understand the derivation,
the physical picture behind it.

614
00:34:40,710 --> 00:34:44,300
The dressed atom picture will
give us a lucid explanation

615
00:34:44,300 --> 00:34:47,340
for exactly what a
lag time does and why

616
00:34:47,340 --> 00:34:50,159
this results in colling.

617
00:34:50,159 --> 00:34:53,645
But anyway, this is the
physics we have done wrong.

618
00:34:53,645 --> 00:34:55,570
At low power, it's OK.

619
00:34:55,570 --> 00:34:58,620
At high power, we really
have to take care of effect.

620
00:35:01,790 --> 00:35:05,210
And if we would take care
of effect-- And as I said,

621
00:35:05,210 --> 00:35:07,320
we don't need the
dressed atom picture,

622
00:35:07,320 --> 00:35:09,960
we can fully do it within the
framework of the optical Bloch

623
00:35:09,960 --> 00:35:12,120
equation, but it's messy.

624
00:35:12,120 --> 00:35:15,710
--we would actually get out, and
you see that in the references,

625
00:35:15,710 --> 00:35:19,230
the result that the friction
coefficient in a standing wave

626
00:35:19,230 --> 00:35:23,820
at weak power is exactly two
times the friction coefficient

627
00:35:23,820 --> 00:35:29,880
for a travelling wave, and
that justifies that we describe

628
00:35:29,880 --> 00:35:33,120
molasses by the spontaneous
light force of two travelling

629
00:35:33,120 --> 00:35:36,070
waves, completely ignoring
the standing wave.

630
00:35:36,070 --> 00:35:39,250
But the same treatment of
the optical Bloch equation

631
00:35:39,250 --> 00:35:41,690
will then, eventually,
give us the result

632
00:35:41,690 --> 00:35:45,810
that alpha changes
sign at high intensity.

633
00:35:45,810 --> 00:35:48,400
But as I said, I
want to describe

634
00:35:48,400 --> 00:35:51,460
that after we have introduced
the dressed atom picture.

635
00:35:54,640 --> 00:35:57,180
Questions?

636
00:35:57,180 --> 00:36:00,040
OK, we have done molasses
in one dimension.

637
00:36:00,040 --> 00:36:01,840
We have done beam slowing.

638
00:36:01,840 --> 00:36:04,160
We just did a short
discussion about molasses

639
00:36:04,160 --> 00:36:06,140
in arbitrary dimensions.

640
00:36:06,140 --> 00:36:08,010
The one technique
which is missing now

641
00:36:08,010 --> 00:36:09,890
is the magneto-optic trap.

642
00:36:12,920 --> 00:36:22,430
Let me just spend five
minutes to discuss

643
00:36:22,430 --> 00:36:25,650
the background behind
the magneto-optic trap

644
00:36:25,650 --> 00:36:29,610
because for many of you,
I think you were just

645
00:36:29,610 --> 00:36:32,910
born when the magneto-optic
trap was invented.

646
00:36:32,910 --> 00:36:39,880
Well, I joined
the field in 1990,

647
00:36:39,880 --> 00:36:43,620
and that was three years
after the magneto-optic trap

648
00:36:43,620 --> 00:36:45,520
has been invented.

649
00:36:45,520 --> 00:36:52,490
And many people say if there is
one most singular development

650
00:36:52,490 --> 00:36:54,420
in laser cooling and
such, it's molasses

651
00:36:54,420 --> 00:36:56,860
and the magneto-optical trap.

652
00:36:56,860 --> 00:37:00,200
So let me tell you why,
the magneto-optical trap,

653
00:37:00,200 --> 00:37:03,760
you shouldn't take
it for granted.

654
00:37:03,760 --> 00:37:11,610
It was actually in
1983 that Ashkin--

655
00:37:11,610 --> 00:37:12,610
Was it Gordon or Ashkin?

656
00:37:12,610 --> 00:37:14,130
I think Ashkin, Art Ashkin.

657
00:37:14,130 --> 00:37:17,670
--that he presented an
Optical Earnshaw theorem,

658
00:37:17,670 --> 00:37:20,970
and it is nicely described
in Bill Phillips' Varenna

659
00:37:20,970 --> 00:37:23,670
notes which I have
posted on the website.

660
00:37:23,670 --> 00:37:30,530
And it's pretty much a proof
that you cannot build a trap

661
00:37:30,530 --> 00:37:32,720
using the spontaneous
light force.

662
00:37:32,720 --> 00:37:35,110
So in other words,
this is a proof

663
00:37:35,110 --> 00:37:38,881
which people took
as a correct proof

664
00:37:38,881 --> 00:37:40,630
that something like
the magneto-optic trap

665
00:37:40,630 --> 00:37:42,750
could not exist.

666
00:37:42,750 --> 00:37:45,200
So what this is proof like?

667
00:37:45,200 --> 00:37:47,870
Well, and that
sort of teaches you

668
00:37:47,870 --> 00:37:52,115
what was the mental concept
or the mental barrier people

669
00:37:52,115 --> 00:37:54,180
had to break through
before they invented

670
00:37:54,180 --> 00:37:55,580
the magneto-optic trap.

671
00:37:55,580 --> 00:37:57,280
The proof is the following.

672
00:37:57,280 --> 00:38:02,400
If you think that you
have beams-- I mean,

673
00:38:02,400 --> 00:38:06,600
a model for the spontaneous
light force is you have photons

674
00:38:06,600 --> 00:38:09,900
and the photons are
sand blasting the atom.

675
00:38:09,900 --> 00:38:12,580
Each photon is like
a grain of sand,

676
00:38:12,580 --> 00:38:16,040
and the atom is in scattering
descent in all directions.

677
00:38:16,040 --> 00:38:18,470
You can really make this
completely classical picture

678
00:38:18,470 --> 00:38:19,890
of the spontaneous light force.

679
00:38:22,570 --> 00:38:23,070
Almost.

680
00:38:27,650 --> 00:38:31,310
If you do with that you would
say the spontaneous light force

681
00:38:31,310 --> 00:38:35,840
is proportional to
the stream of photons

682
00:38:35,840 --> 00:38:39,480
which is described by
the Poynting vector.

683
00:38:39,480 --> 00:38:42,480
But now, you have
in electromagnetism

684
00:38:42,480 --> 00:38:47,460
a continuity equation which says
the divergence of the Poynting

685
00:38:47,460 --> 00:38:52,360
vector plus the change of energy
density in a certain volume

686
00:38:52,360 --> 00:38:55,160
has to be 0.

687
00:38:55,160 --> 00:38:57,060
So if you are in
steady state-- If you

688
00:38:57,060 --> 00:38:58,790
have an arrangement
of laser beams

689
00:38:58,790 --> 00:39:03,500
and the energy density of
the electromagnetic field

690
00:39:03,500 --> 00:39:06,260
is at steady state,
that tells you

691
00:39:06,260 --> 00:39:10,620
that the divergence of the
Poynting vector has to be 0.

692
00:39:10,620 --> 00:39:13,340
Well, if the divergence
of the Poynting vector

693
00:39:13,340 --> 00:39:15,806
is 0, using this
equation that says

694
00:39:15,806 --> 00:39:17,680
the divergence of the
spontaneous light force

695
00:39:17,680 --> 00:39:19,750
has to be 0.

696
00:39:19,750 --> 00:39:21,830
But if the divergence
of the force

697
00:39:21,830 --> 00:39:25,610
has to be 0, that means you
can not have a force which

698
00:39:25,610 --> 00:39:29,540
is inward from all directions
because such a force,

699
00:39:29,540 --> 00:39:31,670
a trap needs a force
which is inward.

700
00:39:31,670 --> 00:39:34,255
A trap requires a
negative divergence.

701
00:39:38,506 --> 00:39:43,520
Do you see the wrong
assumption in this proof,

702
00:39:43,520 --> 00:39:46,190
or did I convince you that
the magneto-optic trap is not

703
00:39:46,190 --> 00:39:52,750
possible because this
proof would say you can not

704
00:39:52,750 --> 00:39:55,270
make a trap out of the
spontaneous light force?

705
00:40:00,080 --> 00:40:02,806
AUDIENCE: Well, you through out
the change in energy density,

706
00:40:02,806 --> 00:40:04,090
so that's--

707
00:40:04,090 --> 00:40:07,200
PROFESSOR: But that's
OK because if you switch

708
00:40:07,200 --> 00:40:12,470
on a laser beam, light
travels at the speed of light,

709
00:40:12,470 --> 00:40:16,380
and in 10 to the minus 10
seconds, in picoseconds,

710
00:40:16,380 --> 00:40:18,200
you reach steady state.

711
00:40:18,200 --> 00:40:20,180
And you know when
you operate the MOT,

712
00:40:20,180 --> 00:40:24,400
you're not switching on and
off your lasers at picoseconds.

713
00:40:24,400 --> 00:40:27,072
It's not that.

714
00:40:27,072 --> 00:40:27,780
Yes, [INAUDIBLE].

715
00:40:27,780 --> 00:40:30,636
AUDIENCE: --the
spacial variation of c.

716
00:40:30,636 --> 00:40:34,450
PROFESSOR: Yes, exactly.

717
00:40:34,450 --> 00:40:39,220
The force is not just
proportional to the Poynting

718
00:40:39,220 --> 00:40:39,930
vector.

719
00:40:39,930 --> 00:40:45,020
This would be true if
I had grains of sand,

720
00:40:45,020 --> 00:40:49,000
but photons have to fill
a resonance condition.

721
00:40:49,000 --> 00:40:54,860
You can maybe
assume I'm an atom,

722
00:40:54,860 --> 00:40:59,100
but I have an spatially varying
magnetic field around me.

723
00:40:59,100 --> 00:41:02,990
At the one magnetic
field, I feel the force

724
00:41:02,990 --> 00:41:05,060
of the laser beam
because I am resonant.

725
00:41:05,060 --> 00:41:07,250
At another magnetic
field, the laser beam

726
00:41:07,250 --> 00:41:10,950
just goes through me because
I'm not in resonance.

727
00:41:10,950 --> 00:41:12,880
So therefore, the
proportionality

728
00:41:12,880 --> 00:41:14,950
between the Poynting
vector of the laser

729
00:41:14,950 --> 00:41:18,430
beam and the light force
ignores that the light force

730
00:41:18,430 --> 00:41:20,360
is resonant, that the
light force depends

731
00:41:20,360 --> 00:41:21,401
on polarization and such.

732
00:41:31,170 --> 00:41:34,590
So the solution
is the divergence

733
00:41:34,590 --> 00:41:39,390
of-- If the c parameter
is spatially dependent,

734
00:41:39,390 --> 00:41:43,260
then we can have a divergence,
and then, we can do trapping.

735
00:41:43,260 --> 00:41:49,250
And it was actually Dave
Pritchard in '85 or in '86

736
00:41:49,250 --> 00:41:51,820
who wrote a really
influential paper.

737
00:41:51,820 --> 00:41:55,850
He said the Optical Earnshaw
theorem can be circumvented.

738
00:41:55,850 --> 00:41:58,800
We can have a spatially
dependent factor,

739
00:41:58,800 --> 00:42:01,840
and what can be spatially
dependent is saturation.

740
00:42:01,840 --> 00:42:05,230
If you focus a laser beam,
you have a spatially varying

741
00:42:05,230 --> 00:42:06,850
saturation parameter.

742
00:42:06,850 --> 00:42:08,700
There can be some
optical pumping.

743
00:42:08,700 --> 00:42:11,710
There can be light--

744
00:42:11,710 --> 00:42:13,900
So he discussed
several possibilities

745
00:42:13,900 --> 00:42:17,750
how the Optical Earnshaw
theorem was circumvented.

746
00:42:17,750 --> 00:42:21,010
At this point in 1986,
he didn't realize

747
00:42:21,010 --> 00:42:24,350
that it can be done with
a magnetic field gradient.

748
00:42:24,350 --> 00:42:28,570
When he talked about possible
ways of circumventing

749
00:42:28,570 --> 00:42:32,630
the Optical Earnshaw theorem,
Jean Dalibard, who many of who

750
00:42:32,630 --> 00:42:34,830
know, who's a famous
researcher in our field,

751
00:42:34,830 --> 00:42:37,530
had the idea that applying
a magnetic field gradient

752
00:42:37,530 --> 00:42:38,730
could work.

753
00:42:38,730 --> 00:42:40,760
But nobody thought,
at this time,

754
00:42:40,760 --> 00:42:43,110
that it would work
in three dimensions,

755
00:42:43,110 --> 00:42:45,040
so the fact that it
works in three dimensions

756
00:42:45,040 --> 00:42:47,180
is almost a miracle.

757
00:42:47,180 --> 00:42:51,190
So when the idea was born,
all those schemes, and also

758
00:42:51,190 --> 00:42:54,710
the original idea of applying
a magnetic field gradient,

759
00:42:54,710 --> 00:42:57,970
were conceived as
one dimension scheme.

760
00:42:57,970 --> 00:43:00,950
Most of the schemes can
not be easily generalized

761
00:43:00,950 --> 00:43:05,100
to three dimension,
but the MOT scheme

762
00:43:05,100 --> 00:43:08,830
works as well in three dimension
as it works in one dimension.

763
00:43:08,830 --> 00:43:12,370
Anyway, so this is
sort of the background.

764
00:43:12,370 --> 00:43:17,170
Since I was a postdoc
when there were only two

765
00:43:17,170 --> 00:43:20,400
or three groups who were
operating a magneto-optic trap,

766
00:43:20,400 --> 00:43:22,140
I put together,
with my own hands,

767
00:43:22,140 --> 00:43:24,525
the first magneto-optic trap
which was ever used at MIT.

768
00:43:24,525 --> 00:43:28,480
I mean, it's ancient
history, so I

769
00:43:28,480 --> 00:43:31,500
wanted to share a little
bit this history to you.

770
00:43:31,500 --> 00:43:33,890
But let me now simply
describe to you

771
00:43:33,890 --> 00:43:35,710
how the magneto-optic
trap works.

772
00:43:40,060 --> 00:43:45,524
Let me describe it
for an atom which

773
00:43:45,524 --> 00:43:46,690
we are not using in the lab.

774
00:43:46,690 --> 00:43:49,030
We all usually use atoms which
have hyperfine structure.

775
00:43:49,030 --> 00:43:51,700
But let me just for
pedagogical simplicity

776
00:43:51,700 --> 00:43:55,930
assume that we have an atom
which has an s to p transition,

777
00:43:55,930 --> 00:44:01,390
and in the p transition,
we have the three different

778
00:44:01,390 --> 00:44:02,980
magnetic [? quantum ?] numbers.

779
00:44:02,980 --> 00:44:05,610
So it's a simplified model.

780
00:44:05,610 --> 00:44:08,990
And what we're
doing right now is

781
00:44:08,990 --> 00:44:13,650
we are applying molasses, two
beams which act as molasses,

782
00:44:13,650 --> 00:44:15,470
but there is one difference now.

783
00:44:15,470 --> 00:44:17,750
They have different
polarizations, sigma plus

784
00:44:17,750 --> 00:44:21,080
and sigma minus,
and in addition, we

785
00:44:21,080 --> 00:44:24,530
apply a linear magnetic
field gradient.

786
00:44:24,530 --> 00:44:26,670
So what happens now
is the following.

787
00:44:26,670 --> 00:44:28,820
The linear magnetic
field gradient

788
00:44:28,820 --> 00:44:32,030
does Zeeman shifts in
the opposite direction

789
00:44:32,030 --> 00:44:35,400
to the n plus 1 and
n minus 1 level.

790
00:44:35,400 --> 00:44:40,290
Now, the transition to m minus
1 is done by sigma plus light.

791
00:44:40,290 --> 00:44:44,470
The transition to m plus 1
is done by sigma minus light.

792
00:44:44,470 --> 00:44:48,730
So what happens now
is if you have an atom

793
00:44:48,730 --> 00:44:52,050
and it moves away
from the origin,

794
00:44:52,050 --> 00:44:55,350
it will eventually come
closer and closer in resonance

795
00:44:55,350 --> 00:44:58,410
with the sigma plus light,
but the sigma plus light

796
00:44:58,410 --> 00:45:02,550
comes from the left side so
the atom will be pushed back.

797
00:45:02,550 --> 00:45:06,580
If the atom moves out hear, it
moves further and further away,

798
00:45:06,580 --> 00:45:08,670
in resonance, from
the sigma plus light,

799
00:45:08,670 --> 00:45:11,760
but it moves now, in resonance,
over the sigma minus light,

800
00:45:11,760 --> 00:45:15,920
and the sigma minus light
is pushing the atoms back.

801
00:45:15,920 --> 00:45:21,230
So this geometry has an inward
force from both directions

802
00:45:21,230 --> 00:45:23,298
and therefore, works as a trap.

803
00:45:31,914 --> 00:45:33,580
So let me now describe
it mathematically

804
00:45:33,580 --> 00:45:36,400
which is very, very simple.

805
00:45:36,400 --> 00:45:39,940
What we have to do is we
just have, like in molasses,

806
00:45:39,940 --> 00:45:43,090
two laser beams two
times the expression

807
00:45:43,090 --> 00:45:44,960
for the spontaneous light force.

808
00:45:44,960 --> 00:45:48,860
And you'll remember we
got the molasses equation

809
00:45:48,860 --> 00:45:51,820
by taking the two laser beams
and putting in the Doppler

810
00:45:51,820 --> 00:45:54,710
shift which is a
velocity dependent shift.

811
00:45:54,710 --> 00:45:58,580
Well, all we have to do is,
for those two laser beams,

812
00:45:58,580 --> 00:46:03,260
we have to put now a spatially
dependent Zeeman shift which

813
00:46:03,260 --> 00:46:06,960
varies linearly with position.

814
00:46:06,960 --> 00:46:14,620
So therefore, all we have to
do is we sum up the total force

815
00:46:14,620 --> 00:46:16,720
as the spontaneous
force for the right

816
00:46:16,720 --> 00:46:19,350
and for the left laser
beam, and in addition

817
00:46:19,350 --> 00:46:24,570
to the molasses effect, we put
now in the spatial dependence.

818
00:46:24,570 --> 00:46:26,840
So we don't have to
do the math again.

819
00:46:26,840 --> 00:46:29,840
We did already a
Taylor expansion in v,

820
00:46:29,840 --> 00:46:31,565
but let's now just
do a Taylor expansion

821
00:46:31,565 --> 00:46:33,390
in the blue expression.

822
00:46:33,390 --> 00:46:36,830
And then, we find that there
is a linearly restoring force

823
00:46:36,830 --> 00:46:42,360
both in velocity space and
in configuration space which

824
00:46:42,360 --> 00:46:45,180
actually tells you
something interesting.

825
00:46:45,180 --> 00:46:50,410
The more friction you have, the
larger is the restoring force.

826
00:46:50,410 --> 00:46:54,670
So the trapping and the
cooling are really connected,

827
00:46:54,670 --> 00:47:00,730
at least in the simple model
of the spontaneous light force.

828
00:47:00,730 --> 00:47:04,800
OK, so if we have
this kind of force,

829
00:47:04,800 --> 00:47:09,130
a linear force in velocity,
a linear force in position,

830
00:47:09,130 --> 00:47:11,380
this is the equation of
motion of a damped harmonic

831
00:47:11,380 --> 00:47:12,470
oscillator.

832
00:47:12,470 --> 00:47:15,600
You have a spatial
force constant, k,

833
00:47:15,600 --> 00:47:18,440
this linear force which
is linear in displacement.

834
00:47:18,440 --> 00:47:22,330
So this is a damped
harmonic oscillator.

835
00:47:22,330 --> 00:47:24,560
Usually, you don't
see atoms sloshing

836
00:47:24,560 --> 00:47:27,820
in your magneto-optic trap
like in an harmonic oscillator

837
00:47:27,820 --> 00:47:30,720
because if you look through
the [? typical ?] numbers

838
00:47:30,720 --> 00:47:32,830
and by just doing
the Taylor expansion

839
00:47:32,830 --> 00:47:35,040
of the spontaneous light
force, you can immediately

840
00:47:35,040 --> 00:47:38,540
get an alluded expressions
for those numbers.

841
00:47:38,540 --> 00:47:41,270
In almost all cases,
the harmonic oscillator

842
00:47:41,270 --> 00:47:42,040
is overdamped.

843
00:47:45,240 --> 00:47:48,050
So that's how the
magneto-optic trap works.

844
00:47:54,510 --> 00:48:00,820
However, the magneto-optic
trap is not so simple.

845
00:48:00,820 --> 00:48:02,520
I would actually
say nobody fully

846
00:48:02,520 --> 00:48:04,600
understands the
magneto-optic trap

847
00:48:04,600 --> 00:48:07,230
because we have
multi-level structure.

848
00:48:07,230 --> 00:48:09,510
We don't have just
one count level.

849
00:48:09,510 --> 00:48:17,360
A sodium and a
rubidium atom has--

850
00:48:17,360 --> 00:48:21,450
f equals one and f equals two--
has eight different hyperfine

851
00:48:21,450 --> 00:48:22,220
levels.

852
00:48:22,220 --> 00:48:25,650
So there is not one level
and we can excite the atoms.

853
00:48:25,650 --> 00:48:27,260
There are eight
different levels.

854
00:48:27,260 --> 00:48:30,290
And those laser beams to optical
pumping among those eight

855
00:48:30,290 --> 00:48:32,640
levels, and you have to
solve a rate equation

856
00:48:32,640 --> 00:48:34,560
for eight different levels.

857
00:48:34,560 --> 00:48:37,280
And the amazing thing
is it still works.

858
00:48:37,280 --> 00:48:39,470
It still works
pretty much, at least

859
00:48:39,470 --> 00:48:43,060
qualitatively, as well as
I just described to you.

860
00:48:43,060 --> 00:48:47,160
So it's a combination of optical
pumping and Zeeman shifts.

861
00:48:47,160 --> 00:48:49,640
It even works better
than advertised

862
00:48:49,640 --> 00:48:52,520
because as I will tell
you in the next few weeks,

863
00:48:52,520 --> 00:48:55,170
you can reach
temperatures-- Because

864
00:48:55,170 --> 00:48:57,260
of the multi-level
structure, you

865
00:48:57,260 --> 00:49:01,030
reach temperatures which are
much colder than the Doppler

866
00:49:01,030 --> 00:49:02,080
limit.

867
00:49:02,080 --> 00:49:03,890
It's called polarization
gradient cooling.

868
00:49:03,890 --> 00:49:05,740
It's just if you
have laser beams

869
00:49:05,740 --> 00:49:07,573
and you have atoms with
hyperfine structure,

870
00:49:07,573 --> 00:49:09,620
they are colder than
the Doppler limit.

871
00:49:09,620 --> 00:49:12,270
First, nobody understood
it, and then, Bill Phillips

872
00:49:12,270 --> 00:49:14,180
figured it out.

873
00:49:14,180 --> 00:49:17,450
And that he figured
out why atoms

874
00:49:17,450 --> 00:49:24,380
get colder that
was the main point

875
00:49:24,380 --> 00:49:25,840
emphasized by the
Nobel Committee

876
00:49:25,840 --> 00:49:27,990
when he got the Nobel
Prize for laser cooling.

877
00:49:27,990 --> 00:49:30,510
Of course, he had already
invented Zeeman slowing before,

878
00:49:30,510 --> 00:49:33,330
so he had a few other
things under his belt.

879
00:49:33,330 --> 00:49:36,820
But anyway, the fact
that the MOT works

880
00:49:36,820 --> 00:49:40,220
even better than advertised
is another miracle,

881
00:49:40,220 --> 00:49:43,200
and this is related to
the hyperfine structure.

882
00:49:43,200 --> 00:49:50,490
And finally, the fact that
you can simply extended it

883
00:49:50,490 --> 00:49:54,570
to three dimension, is-- I said
the first, the second --maybe

884
00:49:54,570 --> 00:49:55,680
the third miracle.

885
00:49:55,680 --> 00:49:57,030
It also works.

886
00:49:57,030 --> 00:49:59,110
Because when you
look at the field

887
00:49:59,110 --> 00:50:04,390
of two coils, a so-called
Anti-Helmholtz field,

888
00:50:04,390 --> 00:50:11,110
there are some directions
where the magnetic field is not

889
00:50:11,110 --> 00:50:12,250
radially outward.

890
00:50:12,250 --> 00:50:14,650
This is a much more
complicated magnetic field.

891
00:50:14,650 --> 00:50:16,720
It goes outward
here, outward here,

892
00:50:16,720 --> 00:50:21,090
but in between the quadrupolar
field, it sort of curves.

893
00:50:21,090 --> 00:50:25,060
There are some directions
where the magnetic field is not

894
00:50:25,060 --> 00:50:28,404
radial, and you would say, hey,
the one dimensional scheme is

895
00:50:28,404 --> 00:50:29,320
[INAUDIBLE] [? set. ?]

896
00:50:32,290 --> 00:50:34,710
Actually, before the
magneto-optic trap

897
00:50:34,710 --> 00:50:36,590
was demonstrated, Dave
Pritchard's student,

898
00:50:36,590 --> 00:50:38,650
Eric [? Rob ?] did
calculations what

899
00:50:38,650 --> 00:50:42,770
happens in these
diagonal directions which

900
00:50:42,770 --> 00:50:45,630
have no resemblance to
the one dimension scheme.

901
00:50:45,630 --> 00:50:47,992
The solution is in
those directions,

902
00:50:47,992 --> 00:50:49,700
the trapping form is
a little bit weaker,

903
00:50:49,700 --> 00:50:53,200
but the big picture
is it doesn't matter.

904
00:50:53,200 --> 00:50:55,940
So there was an idea
conceived in one dimension

905
00:50:55,940 --> 00:51:00,180
for a simple atom structure,
and this idea worked even better

906
00:51:00,180 --> 00:51:02,250
for real atoms and
in three dimensions.

907
00:51:06,920 --> 00:51:09,260
OK, in your homework
assignment, you

908
00:51:09,260 --> 00:51:13,860
will put in some
numbers which show

909
00:51:13,860 --> 00:51:19,740
what happens if you take
a vapor, an alkali vapor

910
00:51:19,740 --> 00:51:23,420
at a very low density of 10
to the 8 per cubic centimeter,

911
00:51:23,420 --> 00:51:25,840
pressure of, typically, 10
to the minus 8 [? Tor, ?]

912
00:51:25,840 --> 00:51:29,740
and you arrange six
laser beams around it.

913
00:51:29,740 --> 00:51:35,140
Well, those laser beams have
a restoring force in velocity.

914
00:51:35,140 --> 00:51:39,250
That means they cool and
increase the phase space

915
00:51:39,250 --> 00:51:43,010
density over your sample
by 8 orders of magnitude.

916
00:51:43,010 --> 00:51:45,900
Those laser beams have
also a restoring force

917
00:51:45,900 --> 00:51:47,730
in the spatial dimension.

918
00:51:47,730 --> 00:51:50,540
Therefore, you get
a cloud which has

919
00:51:50,540 --> 00:51:52,440
3 orders of magnitude
higher density.

920
00:51:57,670 --> 00:52:01,130
So the temperature goes down
by 8 orders of magnitude

921
00:52:01,130 --> 00:52:03,600
which means the phase
space density is enhanced

922
00:52:03,600 --> 00:52:05,910
by 12 orders of magnitude.

923
00:52:05,910 --> 00:52:08,550
So if you're talking
about the goal of reaching

924
00:52:08,550 --> 00:52:12,010
Bose-Einstein condensation
which requires a phase space

925
00:52:12,010 --> 00:52:16,450
density of 1, just
having a dilute vapor

926
00:52:16,450 --> 00:52:20,530
and these six laser beams with
a little bit of magnetic field,

927
00:52:20,530 --> 00:52:23,190
increases your
phase space density

928
00:52:23,190 --> 00:52:26,250
by 15 orders of magnitude.

929
00:52:26,250 --> 00:52:32,720
So this is why molasses and
the MOT were a real milestone.

930
00:52:32,720 --> 00:52:35,570
Now, with 15 orders
of magnitude,

931
00:52:35,570 --> 00:52:38,400
you are just 4 or 5
orders of magnitude

932
00:52:38,400 --> 00:52:41,520
from Bose-Einstein
condensation, and we

933
00:52:41,520 --> 00:52:44,400
will discuss, in
a week and a half,

934
00:52:44,400 --> 00:52:47,100
what techniques were
needed to bridge

935
00:52:47,100 --> 00:52:51,570
the gap between those
temperatures and densities

936
00:52:51,570 --> 00:52:53,226
and quantum
degeneracy of gasses.

937
00:53:01,020 --> 00:53:04,710
So that's the story of
the magneto-optic trap.

938
00:53:04,710 --> 00:53:07,010
I'm also telling
you that because I'm

939
00:53:07,010 --> 00:53:10,310
proud that a lot of the
pioneering work in this regard

940
00:53:10,310 --> 00:53:11,145
happened at MIT.

941
00:53:11,145 --> 00:53:13,940
It was Dave Pritchard who
realized at the Optical

942
00:53:13,940 --> 00:53:17,530
Earnshaw theorem
can be circumvented,

943
00:53:17,530 --> 00:53:21,340
and it was him, in a
collaboration with Steve Chu,

944
00:53:21,340 --> 00:53:26,210
that they together realized
the first magneto-optic trap,

945
00:53:26,210 --> 00:53:29,151
and this was in 1987.

946
00:53:29,151 --> 00:53:29,650
Questions?

947
00:53:40,965 --> 00:53:44,670
OK, I think simple applications
of the spontaneous light force.

948
00:53:47,820 --> 00:53:54,150
So we have time to talk
about the dressed atom.

949
00:53:54,150 --> 00:53:56,700
Now, I was waiting
until this point

950
00:53:56,700 --> 00:53:59,570
to introduce the dressed
atom picture for you

951
00:53:59,570 --> 00:54:07,760
because the dressed atom picture
is the natural picture you want

952
00:54:07,760 --> 00:54:10,500
to think about, for instance,
for the Mollow triplet,

953
00:54:10,500 --> 00:54:12,630
and we had already some
discussion about it.

954
00:54:12,630 --> 00:54:14,310
But it is also the
natural picture

955
00:54:14,310 --> 00:54:17,140
you want to think about
it when you describe

956
00:54:17,140 --> 00:54:19,690
the stimulated light force.

957
00:54:19,690 --> 00:54:23,660
And I mentioned already, it is
the dressed atom picture which

958
00:54:23,660 --> 00:54:29,240
will help me to give you the
physical picture behind if you

959
00:54:29,240 --> 00:54:32,580
have strong laser beams,
why blue-detuned light cools

960
00:54:32,580 --> 00:54:34,360
and not red-detuned light.

961
00:54:34,360 --> 00:54:37,230
So you should be motivated now
to learn about the dressed atom

962
00:54:37,230 --> 00:54:39,040
picture.

963
00:54:39,040 --> 00:54:41,910
But first, I have
to say it's not

964
00:54:41,910 --> 00:54:44,530
anything new, what
I'm teaching you.

965
00:54:47,060 --> 00:54:50,180
It has exactly the
same physics in it

966
00:54:50,180 --> 00:54:52,940
as the optical Bloch equations.

967
00:54:52,940 --> 00:54:56,620
It's just that the optical
Bloch equations are

968
00:54:56,620 --> 00:54:59,070
often more complicated,
and they don't

969
00:54:59,070 --> 00:55:00,880
provide the clear insight.

970
00:55:00,880 --> 00:55:03,305
And the reason is the following.

971
00:55:03,305 --> 00:55:05,690
Often when you describe
a physical phenomenon,

972
00:55:05,690 --> 00:55:09,850
you can choose one basis set, or
you can use another basis set.

973
00:55:09,850 --> 00:55:11,950
But if you're in
the wrong basis,

974
00:55:11,950 --> 00:55:15,180
you get the correct results,
but in the wrong basis,

975
00:55:15,180 --> 00:55:17,900
you just can't describe
the physics easily.

976
00:55:17,900 --> 00:55:21,010
So in other words, with
the dressed atom picture,

977
00:55:21,010 --> 00:55:25,430
I want to introduce dressed
atom states for you which

978
00:55:25,430 --> 00:55:29,160
are no longer the ground
state and the excited state.

979
00:55:29,160 --> 00:55:33,310
I want to treat the
atoms dressed up

980
00:55:33,310 --> 00:55:36,360
by the laser beam,
which is one mode

981
00:55:36,360 --> 00:55:38,370
of the electromagnetic field.

982
00:55:38,370 --> 00:55:41,690
So the atoms, together
with this one laser beam,

983
00:55:41,690 --> 00:55:44,980
will be partially excited,
partially in the ground state.

984
00:55:44,980 --> 00:55:47,730
And this partially
excited atoms, this

985
00:55:47,730 --> 00:55:49,980
is the dressed atom basis.

986
00:55:49,980 --> 00:55:54,990
But if you think in
terms of those states,

987
00:55:54,990 --> 00:55:57,180
you have a much easier
way to formulate

988
00:55:57,180 --> 00:56:00,560
certain physical processes,
exactly processes

989
00:56:00,560 --> 00:56:02,080
where you have high laser power.

990
00:56:02,080 --> 00:56:04,120
At low laser power,
the dress states

991
00:56:04,120 --> 00:56:06,280
are identical to
the naked states

992
00:56:06,280 --> 00:56:07,730
to the ground and
excited states.

993
00:56:07,730 --> 00:56:11,810
But at high laser power,
at infinite laser power,

994
00:56:11,810 --> 00:56:16,240
the two dressed states are the
symmetric and anti-symmetric

995
00:56:16,240 --> 00:56:19,330
equal superposition of
grounded and excited state.

996
00:56:19,330 --> 00:56:22,600
So that's what we
want to describe now.

997
00:56:22,600 --> 00:56:27,180
So the difference to the
Optical Bloch Equation

998
00:56:27,180 --> 00:56:29,290
is that we are
now really looking

999
00:56:29,290 --> 00:56:31,360
for the combined
state of the atom

1000
00:56:31,360 --> 00:56:34,530
in the electromagnetic field.

1001
00:56:34,530 --> 00:56:43,130
So what we want to do is--
this is sort of our picture

1002
00:56:43,130 --> 00:56:44,310
that we have.

1003
00:56:44,310 --> 00:56:46,680
This is our Hamiltonian.

1004
00:56:46,680 --> 00:56:50,310
We have the atom
interacting with the vacuum

1005
00:56:50,310 --> 00:56:52,560
through spontaneous
emission with a reservoir

1006
00:56:52,560 --> 00:56:54,340
of empty modes.

1007
00:56:54,340 --> 00:56:58,110
But now, it also interacts
with the laser beam.

1008
00:56:58,110 --> 00:57:00,820
And you remember when we
did the master equation,

1009
00:57:00,820 --> 00:57:02,530
we looked sort of at this.

1010
00:57:02,530 --> 00:57:05,070
We derived a master
equation for this.

1011
00:57:05,070 --> 00:57:07,740
And then we added
in the unitary time

1012
00:57:07,740 --> 00:57:09,830
evolution due to laser beam.

1013
00:57:09,830 --> 00:57:12,030
But what we want to
do now is we want

1014
00:57:12,030 --> 00:57:16,790
to first take care of the
atom and the laser beam

1015
00:57:16,790 --> 00:57:19,820
almost exactly in an
exact diagonalization.

1016
00:57:19,820 --> 00:57:22,170
And once we have
re-diagonalized that,

1017
00:57:22,170 --> 00:57:26,030
we have now eigenstates which
we call the dressed states.

1018
00:57:26,030 --> 00:57:29,640
Then we allow not the
ground and excited state.

1019
00:57:29,640 --> 00:57:33,457
We then allow the dressed
states to emit into the vacuum.

1020
00:57:33,457 --> 00:57:35,540
And this is the formulation
we want to obtain now.

1021
00:57:47,130 --> 00:57:50,430
Let me make one
comment before I go

1022
00:57:50,430 --> 00:57:54,930
through a few simple equation,
and that's the following.

1023
00:57:54,930 --> 00:57:59,190
If I want to treat this
system almost exactly,

1024
00:57:59,190 --> 00:58:02,290
I have two choices.

1025
00:58:02,290 --> 00:58:04,650
I use a strong laser beam.

1026
00:58:04,650 --> 00:58:08,190
So I could, as we have
discussed several times,

1027
00:58:08,190 --> 00:58:11,990
simply use the time dependent
external electric field.

1028
00:58:11,990 --> 00:58:15,550
The laser beam is E0
times cosine omega t.

1029
00:58:15,550 --> 00:58:18,780
And I just introduced
that in the Hamiltonian.

1030
00:58:18,780 --> 00:58:21,870
And you can get
everything I describe now

1031
00:58:21,870 --> 00:58:23,500
for you out of this Hamiltonian.

1032
00:58:26,690 --> 00:58:28,860
However, there is
an alternative way.

1033
00:58:28,860 --> 00:58:34,980
And this one is I simply
assume that the laser beam is

1034
00:58:34,980 --> 00:58:37,240
described by the
fully quantized field.

1035
00:58:37,240 --> 00:58:39,070
And I use the fog state.

1036
00:58:39,070 --> 00:58:42,530
I just assume N, the number
of photons, is very big.

1037
00:58:47,570 --> 00:58:49,720
And this is, at least
for that purpose,

1038
00:58:49,720 --> 00:58:52,880
a valid description
of the laser beam.

1039
00:58:52,880 --> 00:58:55,810
So what happens is you really
want to look through that.

1040
00:58:55,810 --> 00:58:58,380
I've really learned
a lot by comparing

1041
00:58:58,380 --> 00:59:00,860
the purely classical
treatment of the laser beam

1042
00:59:00,860 --> 00:59:03,590
and the fully quantized
description of the laser beam.

1043
00:59:03,590 --> 00:59:04,700
The results are the same.

1044
00:59:04,700 --> 00:59:07,150
They are Absolutely identical.

1045
00:59:07,150 --> 00:59:10,570
And what you will see is, by
a bunch of approximations,

1046
00:59:10,570 --> 00:59:13,880
we reduce the total
Hamiltonian to-- I've

1047
00:59:13,880 --> 00:59:16,260
seen it many times--
a two by two matrix.

1048
00:59:16,260 --> 00:59:18,330
And whether you start
with the classical field

1049
00:59:18,330 --> 00:59:20,540
or whether you start with
the fully quantized field,

1050
00:59:20,540 --> 00:59:22,520
if you make the
right approximation,

1051
00:59:22,520 --> 00:59:25,010
you get the same
two by two matrix.

1052
00:59:25,010 --> 00:59:26,920
But there's a difference.

1053
00:59:26,920 --> 00:59:29,720
This Hamiltonian is
time independent.

1054
00:59:29,720 --> 00:59:33,870
a dagger a, time
independent Hamiltonian.

1055
00:59:33,870 --> 00:59:38,430
If you take a classical field,
which is cosine omega lt,

1056
00:59:38,430 --> 00:59:41,130
you don't have a time
independent Hamiltonian.

1057
00:59:41,130 --> 00:59:43,520
You have a driven system,
and your wave function

1058
00:59:43,520 --> 00:59:45,830
has an external time dependence.

1059
00:59:45,830 --> 00:59:49,040
The laser beam is driving
the wave function.

1060
00:59:49,040 --> 00:59:51,390
So in other words,
you have two choices

1061
00:59:51,390 --> 00:59:53,870
when you want to describe
how an atom interacts

1062
00:59:53,870 --> 00:59:55,180
with a strong laser beam.

1063
00:59:55,180 --> 00:59:58,545
You can use the fully quantized
field, which is conceptionally

1064
00:59:58,545 --> 01:00:00,610
a little bit more
complicated, because we've

1065
01:00:00,610 --> 01:00:02,290
quantized the
electromagnetic field.

1066
01:00:02,290 --> 01:00:04,780
But then everything
is time independent,

1067
01:00:04,780 --> 01:00:07,880
and everything I do
here is really easy.

1068
01:00:07,880 --> 01:00:09,600
I can go over it
now in five minutes,

1069
01:00:09,600 --> 01:00:12,120
and you understand everything.

1070
01:00:12,120 --> 01:00:14,330
Well, in your
homework, I've already

1071
01:00:14,330 --> 01:00:18,340
asked you at an early homework
to look at simply the two level

1072
01:00:18,340 --> 01:00:20,630
system driven by
classical field.

1073
01:00:20,630 --> 01:00:25,300
And in your next homework,
number 10, your last homework,

1074
01:00:25,300 --> 01:00:28,820
I'm asking you now to
interpret the solution

1075
01:00:28,820 --> 01:00:30,620
with the classical field.

1076
01:00:30,620 --> 01:00:32,810
And it's actually a
little bit a challenge

1077
01:00:32,810 --> 01:00:35,350
to take the time
dependent wave function

1078
01:00:35,350 --> 01:00:38,647
and read the physics out of this
time dependent wave function,

1079
01:00:38,647 --> 01:00:41,230
which corresponds to the physics
we derive in the dressed atom

1080
01:00:41,230 --> 01:00:43,620
picture.

1081
01:00:43,620 --> 01:00:44,760
But you have two ways.

1082
01:00:44,760 --> 01:00:46,550
One is the classical field.

1083
01:00:46,550 --> 01:00:49,214
Then the wave function, the
Hamiltonian, is time dependent.

1084
01:00:49,214 --> 01:00:50,880
And the wave function
is time dependent.

1085
01:00:50,880 --> 01:00:53,220
Or here, I start with
the quantum field

1086
01:00:53,220 --> 01:00:55,250
and everything is
time independent.

1087
01:00:55,250 --> 01:00:58,100
But the solutions are identical.

1088
01:00:58,100 --> 01:01:00,470
Well, one is a time
dependent wave function.

1089
01:01:00,470 --> 01:01:01,610
One is time independent.

1090
01:01:01,610 --> 01:01:03,570
But if you interpret
the solution

1091
01:01:03,570 --> 01:01:06,030
in terms of what happens
to the physical system,

1092
01:01:06,030 --> 01:01:09,120
you will find exactly the same.

1093
01:01:09,120 --> 01:01:11,990
So in other words, I decided
I do the easy part here,

1094
01:01:11,990 --> 01:01:13,980
which is a fully quantized part.

1095
01:01:13,980 --> 01:01:16,460
And this is probably what
will get stuck in your head.

1096
01:01:16,460 --> 01:01:19,592
But since I think it's an
important pedagogical exercise,

1097
01:01:19,592 --> 01:01:21,050
you have to go
through the homework

1098
01:01:21,050 --> 01:01:23,960
and do the time dependent part.

1099
01:01:23,960 --> 01:01:30,240
OK, so we want to
know exactly-- well

1100
01:01:30,240 --> 01:01:33,230
not-- we want to do the
correct approximations.

1101
01:01:33,230 --> 01:01:35,450
And after some
approximations, exactly

1102
01:01:35,450 --> 01:01:38,390
solve what the atom does
with one laser beam.

1103
01:01:41,890 --> 01:01:45,530
So the laser beam is described
by the standard Hamiltonian

1104
01:01:45,530 --> 01:01:46,880
for single mode light.

1105
01:01:46,880 --> 01:01:51,150
The atom is described by, well,
excitation energy omega 0.

1106
01:01:51,150 --> 01:01:53,860
And b is the excited state.

1107
01:01:53,860 --> 01:01:57,990
So if you don't have any
light atom interaction,

1108
01:01:57,990 --> 01:02:02,740
the eigenstates are just product
states, an atom in the excited

1109
01:02:02,740 --> 01:02:04,160
or the ground state.

1110
01:02:04,160 --> 01:02:06,520
And N is the photon number.

1111
01:02:06,520 --> 01:02:07,810
[? Collin? ?]

1112
01:02:07,810 --> 01:02:10,320
AUDIENCE: You said that
it's usually easier

1113
01:02:10,320 --> 01:02:12,480
to understand physics
in a dressed atom

1114
01:02:12,480 --> 01:02:14,454
picture versus optical
block equations.

1115
01:02:14,454 --> 01:02:15,120
PROFESSOR: Yeah.

1116
01:02:15,120 --> 01:02:20,165
AUDIENCE: Can you think
of-- is that a general rule?

1117
01:02:20,165 --> 01:02:24,595
Or is this just like it's only
applicable in certain areas?

1118
01:02:24,595 --> 01:02:25,095
[INAUDIBLE]?

1119
01:02:36,460 --> 01:02:38,750
PROFESSOR: You're really
asking an expert question.

1120
01:02:38,750 --> 01:02:42,800
And I talked to Jean Dalibard
and Claude Cohen-Tannoudji,

1121
01:02:42,800 --> 01:02:45,210
who invented the dressed
atom picture, when

1122
01:02:45,210 --> 01:02:47,510
I wanted to understand
what happens when

1123
01:02:47,510 --> 01:02:49,030
we have a blue detuned lattice.

1124
01:02:49,030 --> 01:02:52,300
And I used the dressed atom
picture and I got confused.

1125
01:02:52,300 --> 01:02:54,540
So what happened
is the following.

1126
01:02:54,540 --> 01:02:59,340
If you pick your right bases,
the description is simple.

1127
01:02:59,340 --> 01:03:02,770
But simple means
if all the physics

1128
01:03:02,770 --> 01:03:04,605
is your atom is in
one dressed state

1129
01:03:04,605 --> 01:03:06,730
or the other dressed state,
and it does transitions

1130
01:03:06,730 --> 01:03:09,270
between the dressed
states and that is easy,

1131
01:03:09,270 --> 01:03:12,280
and this happens in the
limit of large detuning.

1132
01:03:12,280 --> 01:03:14,190
But if you're in
the regime where

1133
01:03:14,190 --> 01:03:17,130
coherences between the
dressed states matter.

1134
01:03:17,130 --> 01:03:20,300
In other words, the density
matrix in the dressed state

1135
01:03:20,300 --> 01:03:22,780
basis has now off
diagonal matrix element.

1136
01:03:22,780 --> 01:03:25,340
The physics becomes
complicated again.

1137
01:03:25,340 --> 01:03:28,140
So in other words, what
I'm telling you is you

1138
01:03:28,140 --> 01:03:33,170
are the best description
happens when we have a two level

1139
01:03:33,170 --> 01:03:35,410
system, we always need
a two by two matrix.

1140
01:03:35,410 --> 01:03:39,020
But if you pick the description
where the solution is diagonal,

1141
01:03:39,020 --> 01:03:41,300
if you have to deal with
coherence, using coherence

1142
01:03:41,300 --> 01:03:43,830
is coupled to population,
it's complicated.

1143
01:03:43,830 --> 01:03:47,210
And there's some cases-- one is
the blue detuned lattice, which

1144
01:03:47,210 --> 01:03:50,190
I encounter where you have to
consider the coherences to get

1145
01:03:50,190 --> 01:03:51,530
the correct answer.

1146
01:03:51,530 --> 01:03:55,730
And usually, also when
the laser beams are weak

1147
01:03:55,730 --> 01:03:58,530
and on resonance, the
dressed atom picture

1148
01:03:58,530 --> 01:04:01,340
is not providing any advantages.

1149
01:04:01,340 --> 01:04:05,640
So you want to use the bare
picture for weak excitations.

1150
01:04:05,640 --> 01:04:09,990
And yes, in the resonance
case, the dressed atom picture,

1151
01:04:09,990 --> 01:04:11,900
if you're very
close to resonance,

1152
01:04:11,900 --> 01:04:15,380
the dressed atom picture
gets more complicated.

1153
01:04:15,380 --> 01:04:18,420
But to give you the answer in
something we have discussed,

1154
01:04:18,420 --> 01:04:21,330
you know already that when
you drive an atom with a laser

1155
01:04:21,330 --> 01:04:24,350
beam, you get the
Mollow triplet.

1156
01:04:24,350 --> 01:04:28,120
And the Mollow
triplet shows you--

1157
01:04:28,120 --> 01:04:32,010
I hope we can finish that
today-- shows you the energy

1158
01:04:32,010 --> 01:04:33,980
splitting between the
dressed energy levels,

1159
01:04:33,980 --> 01:04:35,520
between the dressed
energy levels.

1160
01:04:35,520 --> 01:04:38,500
But if the Mollow
triplet is not resolved

1161
01:04:38,500 --> 01:04:41,940
because the generalized
Rabi frequency is small,

1162
01:04:41,940 --> 01:04:45,280
then you're not resolving
your dressed atom levels.

1163
01:04:45,280 --> 01:04:46,882
And then the dressed
atom level is not

1164
01:04:46,882 --> 01:04:47,965
providing a big advantage.

1165
01:04:52,520 --> 01:04:54,470
For certain problems,
the dressed atom picture

1166
01:04:54,470 --> 01:04:57,450
will not help you.

1167
01:04:57,450 --> 01:04:59,167
But I can reassure
you, and this is

1168
01:04:59,167 --> 01:05:01,000
what I learned from
Jean Dalibard and Claude

1169
01:05:01,000 --> 01:05:03,330
Cohen-Tannoudji,
if you really look

1170
01:05:03,330 --> 01:05:06,060
at the full matrix with the
off diagonal matrix append

1171
01:05:06,060 --> 01:05:09,590
and everything, you will always
find exactly the same solution.

1172
01:05:16,100 --> 01:05:17,870
Anyway, we were
just writing down

1173
01:05:17,870 --> 01:05:22,720
product states of the atoms
and the eigenfunctions

1174
01:05:22,720 --> 01:05:24,900
of the Hamiltonian
for the light, which

1175
01:05:24,900 --> 01:05:28,600
are fog states, number
states with N, L N plus 1.

1176
01:05:28,600 --> 01:05:31,540
But the important thing
is now the following.

1177
01:05:31,540 --> 01:05:35,535
In the limit of small
detuning, the ground state

1178
01:05:35,535 --> 01:05:39,720
with N plus 1 photon is almost
degenerate with the excited

1179
01:05:39,720 --> 01:05:41,190
state with N photons.

1180
01:05:41,190 --> 01:05:44,720
And we call these two
states a manifold.

1181
01:05:44,720 --> 01:05:48,550
The manifold E of N. And
N is the photon number

1182
01:05:48,550 --> 01:05:49,630
in the excited state.

1183
01:05:53,620 --> 01:05:58,140
So what happens is, if you
look at all those states,

1184
01:05:58,140 --> 01:06:01,740
they are grouped into manifolds.

1185
01:06:01,740 --> 01:06:05,080
And the energy separation
between two manifolds

1186
01:06:05,080 --> 01:06:12,320
is, of course, I go from N to N
plus 1, the energy of a photon.

1187
01:06:12,320 --> 01:06:16,550
So now I introduce the
atom laser coupling

1188
01:06:16,550 --> 01:06:19,050
in the usual dipole
approximation.

1189
01:06:19,050 --> 01:06:23,600
And what happens is, this
dipole approximation,

1190
01:06:23,600 --> 01:06:28,530
this dipole light atom coupling
will couple, of course,

1191
01:06:28,530 --> 01:06:31,020
ground and excited state.

1192
01:06:31,020 --> 01:06:34,400
And it will, of course,
have a big effect

1193
01:06:34,400 --> 01:06:37,220
when the energy
splitting is small.

1194
01:06:37,220 --> 01:06:41,150
Well, you will see
that in the next line.

1195
01:06:41,150 --> 01:06:45,230
But we have the co and
counter rotating terms.

1196
01:06:45,230 --> 01:06:49,910
This dipole Hamiltonian will
also couple the ground state

1197
01:06:49,910 --> 01:06:53,040
to the excited state
in the other manifold.

1198
01:06:53,040 --> 01:06:58,150
This is the off resonant
term, which will be neglect.

1199
01:06:58,150 --> 01:06:59,760
So here are the
equations for that.

1200
01:07:02,830 --> 01:07:06,530
Our electric field is, of
course, a plus a dagger.

1201
01:07:06,530 --> 01:07:09,530
The dipole operator is that.

1202
01:07:09,530 --> 01:07:15,210
And if you take what is in
front of the electric field

1203
01:07:15,210 --> 01:07:19,390
and define this as the
coupling constant, g,

1204
01:07:19,390 --> 01:07:24,950
we have what I just told
you, that within a manifold,

1205
01:07:24,950 --> 01:07:26,980
the ground state
with N plus 1 photons

1206
01:07:26,980 --> 01:07:31,800
is coupled to the excited
state with N photons.

1207
01:07:31,800 --> 01:07:33,560
g is simply the
coupling constant,

1208
01:07:33,560 --> 01:07:37,770
which all parameters have
been included in that.

1209
01:07:37,770 --> 01:07:41,000
But then, because of the matrix
element of a or a dagger,

1210
01:07:41,000 --> 01:07:42,980
we have the square
root of N plus 1.

1211
01:07:45,500 --> 01:07:50,440
OK, so what we are now doing is
we are going with this coupling

1212
01:07:50,440 --> 01:07:53,020
within a manifold, which
is strong because the two

1213
01:07:53,020 --> 01:07:54,760
levels are energetically close.

1214
01:07:54,760 --> 01:07:58,270
And we will neglect the
non-resonant couplings

1215
01:07:58,270 --> 01:07:59,810
to the other manifolds.

1216
01:07:59,810 --> 01:08:02,027
And that's just another
way of doing, again,

1217
01:08:02,027 --> 01:08:03,360
the rotating wave approximation.

1218
01:08:10,640 --> 01:08:13,920
Now, when we describe an atom
in the laser field and the atom

1219
01:08:13,920 --> 01:08:16,420
scatters light, or
we have a laser beam,

1220
01:08:16,420 --> 01:08:19,340
and I have to tell you,
when we have a laser beam,

1221
01:08:19,340 --> 01:08:23,069
we don't provide exactly
N atom, N photons.

1222
01:08:23,069 --> 01:08:25,279
The laser beam is
in a coherent state.

1223
01:08:25,279 --> 01:08:27,890
And there are
Poissonian fluctuations.

1224
01:08:27,890 --> 01:08:31,600
But now, we do the next
approximation, namely we

1225
01:08:31,600 --> 01:08:34,670
say when N is
large, then delta N

1226
01:08:34,670 --> 01:08:38,450
can be neglected
with regard to N.

1227
01:08:38,450 --> 01:08:41,770
Then at least for the
range of important photon

1228
01:08:41,770 --> 01:08:45,819
numbers for the problem, this
coupling becomes independent

1229
01:08:45,819 --> 01:08:50,550
of N. And we simply take the
average number of photons

1230
01:08:50,550 --> 01:08:58,090
in our laser beam and define
now an electric field, which

1231
01:08:58,090 --> 01:09:01,859
is independent of N.

1232
01:09:01,859 --> 01:09:03,920
And the product with
the dipole operator

1233
01:09:03,920 --> 01:09:07,930
of the atom, that's what
we call the Rabi frequency.

1234
01:09:07,930 --> 01:09:12,380
So at that point, we introduce
a classical electric field

1235
01:09:12,380 --> 01:09:15,210
through this definition, and
therefore, the dressed atom

1236
01:09:15,210 --> 01:09:15,779
picture.

1237
01:09:15,779 --> 01:09:19,660
Also, we start with the fully
quantized Hamiltonian is now

1238
01:09:19,660 --> 01:09:24,160
identical to the classical
field with this electric field

1239
01:09:24,160 --> 01:09:24,660
amplitude.

1240
01:09:29,600 --> 01:09:32,960
But what we have gained is
we could discuss everything

1241
01:09:32,960 --> 01:09:35,640
in a time independent picture.

1242
01:09:35,640 --> 01:09:39,670
And now, I promised you,
we'll go back to a two

1243
01:09:39,670 --> 01:09:42,120
by two matrix, which you
have seen many times.

1244
01:09:42,120 --> 01:09:44,569
It's a two level system.

1245
01:09:44,569 --> 01:09:47,720
We have the detuning.

1246
01:09:47,720 --> 01:09:50,450
We are in the rotating frame.

1247
01:09:50,450 --> 01:09:55,400
So that means the ground and
excited state with N and N

1248
01:09:55,400 --> 01:09:59,070
minus 1 photons have
the same energy.

1249
01:09:59,070 --> 01:10:02,410
But if we have a detuning,
there is a difference

1250
01:10:02,410 --> 01:10:04,400
whether we excite the
atom or whether we

1251
01:10:04,400 --> 01:10:06,060
put a photon in the laser field.

1252
01:10:06,060 --> 01:10:08,210
But you have seen
that many, many times.

1253
01:10:08,210 --> 01:10:10,240
And the coupling
is now described

1254
01:10:10,240 --> 01:10:13,130
by the Rabi frequency.

1255
01:10:13,130 --> 01:10:17,590
So that's kind of
our typical two level

1256
01:10:17,590 --> 01:10:20,680
system where the two levels
[INAUDIBLE] the rotating frame,

1257
01:10:20,680 --> 01:10:23,410
everything is time independent.

1258
01:10:23,410 --> 01:10:28,860
The two levels are split,
and they are coupled.

1259
01:10:28,860 --> 01:10:32,760
And well, how many times have
you seen the solution of two

1260
01:10:32,760 --> 01:10:35,210
by two matrix?

1261
01:10:35,210 --> 01:10:38,820
The solution is that
the coupling is now

1262
01:10:38,820 --> 01:10:40,450
increasing the splitting.

1263
01:10:40,450 --> 01:10:42,100
The splitting between
the two levels

1264
01:10:42,100 --> 01:10:45,000
is, again, the generalized
Rabi frequency,

1265
01:10:45,000 --> 01:10:48,830
which you have seen
many, many times.

1266
01:10:48,830 --> 01:10:51,410
But now we want to really
get the wave function.

1267
01:10:51,410 --> 01:10:54,920
We want to get something out
of the dressed atom picture.

1268
01:10:54,920 --> 01:10:57,830
And that's the following.

1269
01:10:57,830 --> 01:11:01,760
We started out with the
bare states-- ground

1270
01:11:01,760 --> 01:11:04,700
state with N plus 1, excited
state with N photons.

1271
01:11:04,700 --> 01:11:09,880
And the two by two matrix
after the diagonalization

1272
01:11:09,880 --> 01:11:12,260
is now a linear
combination of those.

1273
01:11:12,260 --> 01:11:14,870
And it's convenient to use
sine theta, cosine theta,

1274
01:11:14,870 --> 01:11:18,920
because then the states are
automatically normalized.

1275
01:11:18,920 --> 01:11:21,780
If I introduce the
solution is now

1276
01:11:21,780 --> 01:11:24,790
that the angle theta is
given by this equation,

1277
01:11:24,790 --> 01:11:28,290
that's just the solution
for the two by two matrix.

1278
01:11:28,290 --> 01:11:32,530
And the physical picture,
which we have obtained,

1279
01:11:32,530 --> 01:11:33,285
is the following.

1280
01:11:38,420 --> 01:11:43,190
If this here is the energy
of the excited state with N

1281
01:11:43,190 --> 01:11:49,720
photons, I've just sort of drawn
this as a reference parallel.

1282
01:11:49,720 --> 01:11:52,800
But now I want to draw it
as a function of the laser

1283
01:11:52,800 --> 01:11:54,660
frequency.

1284
01:11:54,660 --> 01:11:57,560
Well, the ground state
with N plus 1 photons

1285
01:11:57,560 --> 01:12:01,050
has exactly the same
energy on resonance.

1286
01:12:01,050 --> 01:12:04,980
But if the photon is more
energetic then this state,

1287
01:12:04,980 --> 01:12:07,870
because it has N plus 1
photons versus n photons

1288
01:12:07,870 --> 01:12:11,480
if the photon is more energetic
than a resonant photon,

1289
01:12:11,480 --> 01:12:12,850
this state has higher energy.

1290
01:12:12,850 --> 01:12:15,160
And here, it has lower energy.

1291
01:12:15,160 --> 01:12:17,610
In other words, we
have now a level

1292
01:12:17,610 --> 01:12:21,300
crossing between
the two bare states.

1293
01:12:21,300 --> 01:12:23,500
But a level crossing
with interaction

1294
01:12:23,500 --> 01:12:25,510
turns into an avoided crossing.

1295
01:12:25,510 --> 01:12:27,450
And the solution which
we have just derived

1296
01:12:27,450 --> 01:12:29,850
are the blue curves.

1297
01:12:29,850 --> 01:12:32,830
And if I go far
away from resonance,

1298
01:12:32,830 --> 01:12:37,240
this behavior here,
which is perturbative,

1299
01:12:37,240 --> 01:12:40,970
is nothing else than
the AC Stark shift.

1300
01:12:40,970 --> 01:12:43,820
So this is exactly what we
have discussed at this point.

1301
01:12:47,970 --> 01:12:51,380
Now, let's get some
mileage out of it.

1302
01:12:51,380 --> 01:12:54,862
And we will talk
about light forces.

1303
01:12:54,862 --> 01:12:55,945
That's really interesting.

1304
01:12:55,945 --> 01:12:58,820
And I hope it's another
highlight of this course

1305
01:12:58,820 --> 01:13:01,650
to describe light forces in
the dressed atom picture.

1306
01:13:01,650 --> 01:13:03,840
But what I want to
do here is already

1307
01:13:03,840 --> 01:13:05,480
in preparation for that.

1308
01:13:05,480 --> 01:13:10,080
I want to not yet
introduce forces, simply

1309
01:13:10,080 --> 01:13:14,690
figure out what happens in
the dressed atom picture

1310
01:13:14,690 --> 01:13:16,590
when the atoms scatter light.

1311
01:13:16,590 --> 01:13:18,520
And you know already
the solution.

1312
01:13:18,520 --> 01:13:21,740
The solution is
the Mollow triplet.

1313
01:13:21,740 --> 01:13:25,710
And we have discussed with
the Optical Bloch Equations

1314
01:13:25,710 --> 01:13:28,260
that at low laser power,
the Mollow triplet

1315
01:13:28,260 --> 01:13:34,160
has three curves.

1316
01:13:34,160 --> 01:13:36,930
It has three broadened curves.

1317
01:13:36,930 --> 01:13:39,010
And they have
different broadening.

1318
01:13:39,010 --> 01:13:41,860
And I didn't prove it to
you, but I mentioned to you,

1319
01:13:41,860 --> 01:13:47,112
at low laser power there is--
perturbative laser power,

1320
01:13:47,112 --> 01:13:49,570
we discussed at great length
that there's a delta function.

1321
01:13:49,570 --> 01:13:52,870
And at low laser power,
the delta function remains.

1322
01:13:52,870 --> 01:13:55,720
So this is sort of the
spectrum we get out of it.

1323
01:13:55,720 --> 01:13:57,670
And what I want
to show you now is

1324
01:13:57,670 --> 01:14:04,350
how this spectrum is described
in the dressed atom picture.

1325
01:14:04,350 --> 01:14:07,210
Atom photon interaction
has a wonderful discussion

1326
01:14:07,210 --> 01:14:08,580
of all the details.

1327
01:14:08,580 --> 01:14:11,260
I will share with
you a few highlights.

1328
01:14:11,260 --> 01:14:14,990
But we will go much further in
the description of the Mollow

1329
01:14:14,990 --> 01:14:17,760
triplet then we've gone before.

1330
01:14:17,760 --> 01:14:22,460
But I will not go into
all excruciating details.

1331
01:14:22,460 --> 01:14:27,310
So let's discuss first.

1332
01:14:27,310 --> 01:14:29,100
That's something we
hadn't done so far.

1333
01:14:29,100 --> 01:14:32,050
What are the intensities
in this Mollow triplet?

1334
01:14:35,340 --> 01:14:45,430
So the way how we obtain
it is the following.

1335
01:14:45,430 --> 01:14:51,080
We have the manifold, the
manifold with N minus 1

1336
01:14:51,080 --> 01:14:53,200
and the manifold with N photons.

1337
01:14:53,200 --> 01:14:56,140
And this is the detuning.

1338
01:14:56,140 --> 01:14:59,160
And if we allow for the
atom laser coupling,

1339
01:14:59,160 --> 01:15:02,520
the splitting here is a
generalized Rabi frequency.

1340
01:15:02,520 --> 01:15:06,160
And well, you immediately see
that in this picture, if you

1341
01:15:06,160 --> 01:15:09,250
allow spontaneous emission from
the upper manifold to the lower

1342
01:15:09,250 --> 01:15:13,620
manifold, you have four
different combinations.

1343
01:15:13,620 --> 01:15:16,520
One is the long photon
is the blue sideband.

1344
01:15:16,520 --> 01:15:18,367
This is the red sideband.

1345
01:15:18,367 --> 01:15:19,450
This is the blue sideband.

1346
01:15:19,450 --> 01:15:20,500
This is the red sideband.

1347
01:15:20,500 --> 01:15:23,210
And those two are the carriers.

1348
01:15:23,210 --> 01:15:25,480
But now, we can
do the following.

1349
01:15:25,480 --> 01:15:28,610
I've just written down for
you the solution of the wave

1350
01:15:28,610 --> 01:15:30,530
function with sine
and cosine theta.

1351
01:15:37,300 --> 01:15:41,950
If we want to emit the blue
sideband, we go from the top

1352
01:15:41,950 --> 01:15:43,930
to the bottom.

1353
01:15:43,930 --> 01:15:47,320
But that means, since a
photon in the rotating wave

1354
01:15:47,320 --> 01:15:53,980
approximation can only be
emitted when we go from b to a,

1355
01:15:53,980 --> 01:15:57,890
we find that the
matrix element for that

1356
01:15:57,890 --> 01:16:02,240
involves cosine theta
times sine theta.

1357
01:16:02,240 --> 01:16:04,480
However, if you want
to emit the carrier,

1358
01:16:04,480 --> 01:16:07,300
you go from here to there.

1359
01:16:07,300 --> 01:16:11,970
Well, we always have to find
the excited state to begin with.

1360
01:16:11,970 --> 01:16:21,000
And we go-- I must
have made a mistake.

1361
01:16:24,340 --> 01:16:24,850
Sorry.

1362
01:16:24,850 --> 01:16:26,660
When I went to emit
the blue side, when

1363
01:16:26,660 --> 01:16:28,880
I go from all the way
up to all the way down,

1364
01:16:28,880 --> 01:16:31,650
and I always have to find
an excited state which

1365
01:16:31,650 --> 01:16:33,070
connects to the ground state.

1366
01:16:33,070 --> 01:16:37,865
And this gives me a term,
which is cosine square.

1367
01:16:37,865 --> 01:16:41,290
The red detuned sideband
gives me sine squared.

1368
01:16:41,290 --> 01:16:46,600
And if I want to get
the carrier, this one,

1369
01:16:46,600 --> 01:16:48,410
I go from here to here.

1370
01:16:48,410 --> 01:16:51,090
I get cosine theta sine theta.

1371
01:16:51,090 --> 01:16:53,570
So in other words,
we can immediately

1372
01:16:53,570 --> 01:16:56,640
read off the matrix elements.

1373
01:16:56,640 --> 01:16:59,240
And since the intensity,
Fermi's Golden Rule,

1374
01:16:59,240 --> 01:17:01,290
is the matrix
element squared, we

1375
01:17:01,290 --> 01:17:05,410
find immediately that the upper
lower sideband and the carrier

1376
01:17:05,410 --> 01:17:09,340
involve terms cosine to the
4 theta, sine to the 4 theta,

1377
01:17:09,340 --> 01:17:11,510
cosine square sine square.

1378
01:17:11,510 --> 01:17:13,670
So we got the matrix element.

1379
01:17:13,670 --> 01:17:15,560
And the only thing
we have to know now

1380
01:17:15,560 --> 01:17:22,530
is that the width or the rate
is given by the matrix element.

1381
01:17:27,370 --> 01:17:30,290
The rate is given by the
matrix element square.

1382
01:17:30,290 --> 01:17:34,330
But now, what is the
intensity of the line?

1383
01:17:34,330 --> 01:17:38,720
Well, the upper
sideband is emitted

1384
01:17:38,720 --> 01:17:40,580
by the upper dressed
state, because you

1385
01:17:40,580 --> 01:17:42,270
have to go from the
upper dressed state

1386
01:17:42,270 --> 01:17:45,430
to the lower dressed state to
get the blue detuned photon.

1387
01:17:45,430 --> 01:17:48,050
So therefore, the
intensity for this sideband

1388
01:17:48,050 --> 01:17:51,380
is this rate times
the population

1389
01:17:51,380 --> 01:17:54,260
in the upper dressed state.

1390
01:17:54,260 --> 01:17:56,560
So what is missing
at this point,

1391
01:17:56,560 --> 01:18:00,100
and we need that urgently
for next week, what

1392
01:18:00,100 --> 01:18:04,740
is the population in the
upper and the lower sideband?

1393
01:18:04,740 --> 01:18:08,320
Well, we know the
transition rates

1394
01:18:08,320 --> 01:18:10,510
between the upper and
the lower sidebands--

1395
01:18:10,510 --> 01:18:13,840
well, between the upper
and lower dressed state.

1396
01:18:13,840 --> 01:18:18,572
We know we've just
found here the rate.

1397
01:18:18,572 --> 01:18:19,280
What is the rate?

1398
01:18:19,280 --> 01:18:23,970
What is the matrix element
squared when you go from one

1399
01:18:23,970 --> 01:18:27,820
is the upper, two is the
lower state of the manifold.

1400
01:18:27,820 --> 01:18:30,140
So we have just
gotten those rates.

1401
01:18:30,140 --> 01:18:34,110
And rate equation says,
the change of population

1402
01:18:34,110 --> 01:18:38,370
in the upper dressed
state is given

1403
01:18:38,370 --> 01:18:41,040
by the departure
from the upper state

1404
01:18:41,040 --> 01:18:44,970
and what arrives
from the lower state.

1405
01:18:44,970 --> 01:18:49,110
So since we are interested
in the steady state solution,

1406
01:18:49,110 --> 01:18:52,940
we can just set the
left side to zero.

1407
01:18:52,940 --> 01:18:56,190
And then, the rate
equations simply

1408
01:18:56,190 --> 01:19:00,250
give us the population
in terms of the rates.

1409
01:19:00,250 --> 01:19:03,750
So in other words, if you
ask, what is the steady state

1410
01:19:03,750 --> 01:19:06,370
solution in the upper
or lower dressed state,

1411
01:19:06,370 --> 01:19:10,510
it's given by a ratio of
those rate coefficients.

1412
01:19:10,510 --> 01:19:12,450
But we just got those
rate coefficients.

1413
01:19:12,450 --> 01:19:18,030
And this gives us another bunch
of sine and cosine functions.

1414
01:19:18,030 --> 01:19:20,350
So in other words,
we know now what

1415
01:19:20,350 --> 01:19:23,450
is the population of
the dressed states.

1416
01:19:23,450 --> 01:19:26,700
I mean, everything is expressed
by one non-trivial parameter

1417
01:19:26,700 --> 01:19:27,520
theta.

1418
01:19:27,520 --> 01:19:31,260
And all we get is sine square,
cosine square, sine to the 4th,

1419
01:19:31,260 --> 01:19:33,110
we just get those
trigonometric functions.

1420
01:19:38,180 --> 01:19:38,720
Yes.

1421
01:19:38,720 --> 01:19:42,230
I think we can finish that.

1422
01:19:42,230 --> 01:19:42,730
Good.

1423
01:19:48,240 --> 01:19:50,550
So this is the steady
state solution.

1424
01:19:50,550 --> 01:19:53,190
But now, we can go back
to the rate equation

1425
01:19:53,190 --> 01:19:58,940
and say the time derivative
of the rate equation

1426
01:19:58,940 --> 01:20:03,380
has to relax to
the steady state.

1427
01:20:03,380 --> 01:20:07,410
This is nothing else than
rewriting the equation above.

1428
01:20:07,410 --> 01:20:12,480
But by rewriting the
equation above in that way,

1429
01:20:12,480 --> 01:20:17,210
we find now the
prefactor in front of-- I

1430
01:20:17,210 --> 01:20:20,470
mean, here, what you have is
the time derivative is given

1431
01:20:20,470 --> 01:20:22,440
by the difference
from steady state.

1432
01:20:22,440 --> 01:20:24,630
And this is the
relaxation coefficient.

1433
01:20:24,630 --> 01:20:28,100
And the relaxation coefficient
is-- that's the solution.

1434
01:20:28,100 --> 01:20:31,240
You just have to do the
substitutions yourself.

1435
01:20:31,240 --> 01:20:33,790
The relaxation coefficient,
how populations

1436
01:20:33,790 --> 01:20:36,530
relax to steady
state, is now given

1437
01:20:36,530 --> 01:20:38,960
by the sum of the two rates.

1438
01:20:38,960 --> 01:20:42,230
One is going from the
upper to the lower

1439
01:20:42,230 --> 01:20:44,260
and from the lower
to the upper state.

1440
01:20:44,260 --> 01:20:46,330
And each of those rates
has been expressed

1441
01:20:46,330 --> 01:20:47,860
by sine square and
cosine squares.

1442
01:20:50,430 --> 01:20:52,990
I'm not doing it here,
because it's a little bit more

1443
01:20:52,990 --> 01:20:53,920
complicated.

1444
01:20:53,920 --> 01:20:57,620
But you can also rewrite for
the dressed atom picture what

1445
01:20:57,620 --> 01:21:01,745
happens-- you can rewrite those
equations in a way what happens

1446
01:21:01,745 --> 01:21:06,140
not to the population, but
what happens to the coherences,

1447
01:21:06,140 --> 01:21:09,100
to the off diagram
matrix element formulated

1448
01:21:09,100 --> 01:21:11,530
in the dressed state basis.

1449
01:21:11,530 --> 01:21:15,630
And what you find is, you find
a correlation time, a relaxation

1450
01:21:15,630 --> 01:21:18,880
time, for the coherences.

1451
01:21:18,880 --> 01:21:22,090
Remember, for the naked
state, population's

1452
01:21:22,090 --> 01:21:23,900
relaxed with gamma.

1453
01:21:23,900 --> 01:21:26,110
Coherence is relaxed
with gamma over 2.

1454
01:21:26,110 --> 01:21:27,880
This is for the Optical
Bloch Equations.

1455
01:21:27,880 --> 01:21:31,960
But now we have reformulated
the physics in the dressed atom

1456
01:21:31,960 --> 01:21:33,640
picture.

1457
01:21:33,640 --> 01:21:38,110
And what the solution
is the following.

1458
01:21:41,120 --> 01:21:43,000
This is sort of a
weak laser power.

1459
01:21:43,000 --> 01:21:46,790
But in the most general
case, what you will find

1460
01:21:46,790 --> 01:21:58,180
is that the width
of the central peak

1461
01:21:58,180 --> 01:22:01,790
reflects the relaxation
time for the population.

1462
01:22:01,790 --> 01:22:04,890
And the width for the sidebands
is the relaxation time

1463
01:22:04,890 --> 01:22:08,880
for the coherences in
the dressed atom picture.

1464
01:22:08,880 --> 01:22:13,440
So we had before with the
Optical Bloch Equations,

1465
01:22:13,440 --> 01:22:15,420
I showed you different cases.

1466
01:22:15,420 --> 01:22:17,780
We had a three quarter gamma.

1467
01:22:17,780 --> 01:22:21,890
I need two more minutes and
I'm finished with my unit.

1468
01:22:21,890 --> 01:22:24,930
So we had weird factors
of three quarters.

1469
01:22:24,930 --> 01:22:28,520
And I showed you how they
come from the eigenvalues

1470
01:22:28,520 --> 01:22:31,670
of the matrix of the
Optical Bloch Equation.

1471
01:22:31,670 --> 01:22:34,080
But now we understand
them physically,

1472
01:22:34,080 --> 01:22:37,260
namely the central
feature has a peak,

1473
01:22:37,260 --> 01:22:39,410
which is the relaxation
time for the populations

1474
01:22:39,410 --> 01:22:40,820
in the dressed atom picture.

1475
01:22:40,820 --> 01:22:42,740
And the sidebands
have a width, which

1476
01:22:42,740 --> 01:22:44,750
is the relaxation time
for the coherences.

1477
01:22:47,930 --> 01:22:50,960
They no longer differ
by a factor of two,

1478
01:22:50,960 --> 01:22:55,375
as in the naked bases in which
we formulated the Optical Bloch

1479
01:22:55,375 --> 01:22:55,875
Equations.

1480
01:23:02,110 --> 01:23:04,120
By the way, I know
a lot of you may

1481
01:23:04,120 --> 01:23:07,170
be curious about
the coherent peak.

1482
01:23:07,170 --> 01:23:09,220
You can read about it in API.

1483
01:23:09,220 --> 01:23:12,860
It's 10 pages about
correlation function.

1484
01:23:12,860 --> 01:23:19,810
But in the end, the rate
of the delta function

1485
01:23:19,810 --> 01:23:23,700
can be, again, described by
values we have just calculated.

1486
01:23:23,700 --> 01:23:26,340
We have just calculated
steady state populations

1487
01:23:26,340 --> 01:23:28,370
and transition rates.

1488
01:23:28,370 --> 01:23:32,160
So this rate is nothing else
than another combination

1489
01:23:32,160 --> 01:23:34,820
of sine square and cosine
squares of this angle theta.

1490
01:23:34,820 --> 01:23:38,910
So everything is wonderfully
and simply described

1491
01:23:38,910 --> 01:23:41,790
the full physics of
that by just this angle

1492
01:23:41,790 --> 01:23:44,970
theta in the dressed
atom picture.

1493
01:23:44,970 --> 01:23:56,890
OK So with that, we are done.

1494
01:23:56,890 --> 01:24:02,750
And I think I'll discuss the
last page with you on Monday.