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PROFESSOR: Good afternoon.

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00:00:30,170 --> 00:00:34,190
The title here is the
topic for our class today.

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00:00:34,190 --> 00:00:38,720
We want to discuss formally
the derivation Optical Bloch

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00:00:38,720 --> 00:00:43,290
Equations, but usually
when I teach you something,

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00:00:43,290 --> 00:00:46,240
I have a general concept in mind
and the concept right now here

13
00:00:46,240 --> 00:00:52,080
is, how can we get from unitary
time evolution of a quantum

14
00:00:52,080 --> 00:00:55,740
system to rate
equations anticipation.

15
00:00:55,740 --> 00:00:59,250
And this is sort of the
subject of master equation

16
00:00:59,250 --> 00:01:03,460
open system dynamics, and
this cartoon sort of tells you

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00:01:03,460 --> 00:01:05,489
what you want to do.

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00:01:05,489 --> 00:01:10,990
We have a total system, which is
one part we are interested in,

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00:01:10,990 --> 00:01:14,163
and the other one, often it has
many, many degrees of freedom,

20
00:01:14,163 --> 00:01:16,140
and we don't want to
keep track of them.

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00:01:16,140 --> 00:01:18,380
But they have one Hamiltonian.

22
00:01:18,380 --> 00:01:22,330
But we are only interested in
how our atomic system evolves,

23
00:01:22,330 --> 00:01:24,550
and we want to find
an equation, which

24
00:01:24,550 --> 00:01:26,680
is no longer a
Schrodinger equation.

25
00:01:26,680 --> 00:01:30,950
How does an initial density
matrix describing our system

26
00:01:30,950 --> 00:01:32,780
develop with time?

27
00:01:32,780 --> 00:01:37,050
And as we will see, in general,
it follows a master equation,

28
00:01:37,050 --> 00:01:39,710
and we want to discuss
what are general principles

29
00:01:39,710 --> 00:01:41,560
of such equations.

30
00:01:41,560 --> 00:01:45,080
But it's very important
here is that we are not

31
00:01:45,080 --> 00:01:47,770
keeping track of what
happens in the environment,

32
00:01:47,770 --> 00:01:51,140
and that's associated here with
a little bucket or trash can.

33
00:01:51,140 --> 00:01:54,350
Every result photons
or such are measured,

34
00:01:54,350 --> 00:01:59,040
the environment is constantly
projected on a measurement

35
00:01:59,040 --> 00:02:03,200
basis, and therefore,
re-introduce probabilistic

36
00:02:03,200 --> 00:02:07,970
element into the part
we don't observe mainly

37
00:02:07,970 --> 00:02:12,470
the part which
characterizes the atoms.

38
00:02:12,470 --> 00:02:19,690
So for that we need the
formalism of density matrix,

39
00:02:19,690 --> 00:02:26,065
and that's where
we want to start.

40
00:02:29,850 --> 00:02:34,540
At the end of last
lecture, I reminded you

41
00:02:34,540 --> 00:02:36,880
that the density
matrix can always

42
00:02:36,880 --> 00:02:39,700
be written as an
ensemble where you

43
00:02:39,700 --> 00:02:42,520
say you have a certain
probability for certain rate

44
00:02:42,520 --> 00:02:45,160
function.

45
00:02:45,160 --> 00:02:48,660
This is always possible,
but it is not unique.

46
00:02:48,660 --> 00:02:52,580
So each of you could actually
create the same density matrix

47
00:02:52,580 --> 00:02:55,460
by preparing a number
of quantum states

48
00:02:55,460 --> 00:02:57,420
with a certain
probability and saying

49
00:02:57,420 --> 00:02:58,960
this is my density matrix.

50
00:02:58,960 --> 00:03:01,700
And also each of you has
prepared different quantum

51
00:03:01,700 --> 00:03:02,630
states.

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00:03:02,630 --> 00:03:04,640
If you sum them up
in that same way,

53
00:03:04,640 --> 00:03:09,280
you get the same density matrix
and therefore, all observables

54
00:03:09,280 --> 00:03:12,280
or measurements you will do
on your ensemble in the future

55
00:03:12,280 --> 00:03:15,320
will be identical because
the density matrix is

56
00:03:15,320 --> 00:03:17,890
a full description
of the system.

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00:03:17,890 --> 00:03:20,705
So at that level, it may
look sort of very trivial

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00:03:20,705 --> 00:03:23,000
that we have
different unraveling.

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00:03:23,000 --> 00:03:25,130
Unraveling means you
look microscopically

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00:03:25,130 --> 00:03:27,140
what is behind the
density matrix,

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00:03:27,140 --> 00:03:28,640
but in your homework
assignment, you

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00:03:28,640 --> 00:03:32,970
will all show that you can have
a system which have very, very

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00:03:32,970 --> 00:03:34,990
different dissipation
mechanisms,

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00:03:34,990 --> 00:03:37,290
but they're described
by the same equation.

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00:03:37,290 --> 00:03:41,490
So therefore, it's
physically not possible

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00:03:41,490 --> 00:03:45,280
to distinguish by just measuring
the density matrix what

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00:03:45,280 --> 00:03:46,810
causes a dissipation.

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00:03:46,810 --> 00:03:49,400
Of course, if you know
what causes a dissipation,

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00:03:49,400 --> 00:03:50,990
fluctuating fields
or collisions,

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00:03:50,990 --> 00:03:53,630
you know more than the
density matrix knows.

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00:03:57,220 --> 00:04:01,900
Any questions about density
matrix or the agenda

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00:04:01,900 --> 00:04:03,240
we want to go through today?

73
00:04:07,360 --> 00:04:12,160
Just a quick reminder and
since this was covered in 8421,

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00:04:12,160 --> 00:04:18,410
and most of you know about
it, I pre-wrote the slides.

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00:04:18,410 --> 00:04:20,800
The density matrix
is a time evolution.

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00:04:20,800 --> 00:04:25,160
There is one which is sort
trivial and covered in more

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00:04:25,160 --> 00:04:29,450
elementary takes, and this
is the Hamiltonian evolution,

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00:04:29,450 --> 00:04:30,890
and I'm sure you've seen it.

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00:04:30,890 --> 00:04:33,250
The time evolution, the
unitary time evolution

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00:04:33,250 --> 00:04:35,190
involves the commutative
of the Hamiltonian

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00:04:35,190 --> 00:04:40,600
with a density matrix,
and later on, we

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00:04:40,600 --> 00:04:43,900
want to specialize including
dissipation, including

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00:04:43,900 --> 00:04:48,170
the environment to the evolution
of a tool evolutionary system

84
00:04:48,170 --> 00:04:51,120
driven by a monochromatic field

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00:04:51,120 --> 00:05:01,680
And this the famous
Jaynes-Cummings Hamiltonian,

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00:05:01,680 --> 00:05:05,920
and we can characterize this
system by a density matrix,

87
00:05:05,920 --> 00:05:08,670
but will be very important
is that we distinguish

88
00:05:08,670 --> 00:05:11,240
between populations,
the diagonal parts,

89
00:05:11,240 --> 00:05:13,350
and the coherences.

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00:05:13,350 --> 00:05:17,080
And if you simply put
this density matrix

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00:05:17,080 --> 00:05:32,980
into this equation, you find the
time evolution of the system,

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00:05:32,980 --> 00:05:36,630
and we will refer to
that result later on.

93
00:05:36,630 --> 00:05:39,470
Many of you have
seen it in 8421.

94
00:05:39,470 --> 00:05:42,470
we can parametrize
the density matrix

95
00:05:42,470 --> 00:05:44,160
for the two-level system.

96
00:05:44,160 --> 00:05:46,015
So we can parametrize
the densities

97
00:05:46,015 --> 00:05:49,080
of the two-level system
by a local vector,

98
00:05:49,080 --> 00:05:52,520
r, which is defined
by this equation.

99
00:05:52,520 --> 00:05:56,550
And then the equation
of motion is simply

100
00:05:56,550 --> 00:06:00,730
the rotation of a Bloch
vector on the Bloch sphere,

101
00:06:00,730 --> 00:06:04,770
and it has a
rotational axis, which

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00:06:04,770 --> 00:06:08,360
is given by-- the
undriven system

103
00:06:08,360 --> 00:06:10,270
rotates around the z-axis.

104
00:06:10,270 --> 00:06:12,040
This is just e to
the i omega naught

105
00:06:12,040 --> 00:06:15,280
t, the normal evolution
of the free system.

106
00:06:15,280 --> 00:06:19,440
But if you divide it with
a monochromatic field

107
00:06:19,440 --> 00:06:23,060
rotation over the x-axis
and the x-axis, of course,

108
00:06:23,060 --> 00:06:25,545
can take your two-level system,
and flip it from the ground

109
00:06:25,545 --> 00:06:26,970
to the excited state.

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00:06:26,970 --> 00:06:30,770
So it's just a reminder of the
simple unitary time evolution,

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00:06:30,770 --> 00:06:33,360
but now we want to add
dissipation on top of it.

112
00:06:36,300 --> 00:06:52,240
And what I've decided
that before I discuss

113
00:06:52,240 --> 00:06:55,500
with you Optical Bloch Equation
and master equation in general,

114
00:06:55,500 --> 00:06:58,840
I want to give you
a very simple model.

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00:06:58,840 --> 00:07:01,570
I really like
simple models which

116
00:07:01,570 --> 00:07:05,080
capture the essence of what
you are going to discuss.

117
00:07:05,080 --> 00:07:12,750
So what I want to use is I
want to use a beam splitter

118
00:07:12,750 --> 00:07:28,520
model I formulated for photons,
but it would also immediately

119
00:07:28,520 --> 00:07:30,830
apply to atoms.

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00:07:30,830 --> 00:07:33,210
And this model, what
I like about it,

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00:07:33,210 --> 00:07:35,910
it has all the ingredients
of integration of the master

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00:07:35,910 --> 00:07:38,990
equation we'll do later
on without the kind

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00:07:38,990 --> 00:07:42,640
of many indices and
summations and integration,

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00:07:42,640 --> 00:07:46,370
but it captures every single
bit of what is important.

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00:07:46,370 --> 00:07:50,640
And I usually like to present
exactly solvable, simple models

126
00:07:50,640 --> 00:07:52,780
where you get it, and
then I can go a little bit

127
00:07:52,780 --> 00:07:54,300
faster for the
general derivation

128
00:07:54,300 --> 00:07:58,097
because you know exactly what
the more complicated equations,

129
00:07:58,097 --> 00:07:58,930
what they are doing.

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00:08:02,360 --> 00:08:09,160
So in other words, what I
want to derive for you is

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00:08:09,160 --> 00:08:11,175
we want to have the
following situation.

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00:08:11,175 --> 00:08:14,260
We have a beam splitter,
and we know everything

133
00:08:14,260 --> 00:08:16,590
about beam splitters
because we talked about them

134
00:08:16,590 --> 00:08:18,860
in the first part of
the course, and we

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00:08:18,860 --> 00:08:22,680
have a wave function,
which is the input,

136
00:08:22,680 --> 00:08:25,390
and this is a photon.

137
00:08:25,390 --> 00:08:29,260
And we want to understand
after the beam splitter,

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00:08:29,260 --> 00:08:31,820
how has the system evolved.

139
00:08:31,820 --> 00:08:34,870
In general, it will
be a density matrix,

140
00:08:34,870 --> 00:08:36,464
and what you want
to find out is,

141
00:08:36,464 --> 00:08:39,200
what is the equation
for the density matrix.

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00:08:39,200 --> 00:08:42,400
Maybe this density matrix goes
through the next beam splitter

143
00:08:42,400 --> 00:08:44,510
and then we want to know
what comes out of it.

144
00:08:44,510 --> 00:08:47,950
And all we have to
apply is the formalism

145
00:08:47,950 --> 00:08:52,680
we developed for the beam
splitter earlier in the course.

146
00:08:52,680 --> 00:08:55,750
Of course, the beam
splitter is not as harmless

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00:08:55,750 --> 00:08:56,970
as it looks like.

148
00:08:56,970 --> 00:09:03,710
There is another part
and another part one here

149
00:09:03,710 --> 00:09:07,530
brings in the environment,
and for the environment,

150
00:09:07,530 --> 00:09:09,300
we will use the vacuum.

151
00:09:09,300 --> 00:09:11,260
That's the simplest environment.

152
00:09:11,260 --> 00:09:12,810
It's actually
important environment

153
00:09:12,810 --> 00:09:16,090
because it is the environment
we will use all the time when

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00:09:16,090 --> 00:09:17,970
we discuss spontaneous emission.

155
00:09:17,970 --> 00:09:21,330
We send photons into the
nirvana, into the vacuum

156
00:09:21,330 --> 00:09:24,220
and they disappear, and
this is our modified.

157
00:09:24,220 --> 00:09:27,400
But the other one which
is often not so explicit.

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00:09:27,400 --> 00:09:29,860
If you send your
photons away, you're

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00:09:29,860 --> 00:09:31,800
not keeping track
of what happens,

160
00:09:31,800 --> 00:09:34,510
but you could as well
perform a measurement,

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00:09:34,510 --> 00:09:36,360
and this is what we put in here.

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00:09:36,360 --> 00:09:42,630
We say those photons
hit a bucket or detector

163
00:09:42,630 --> 00:09:44,300
and measure them,
we observe them.

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00:09:44,300 --> 00:09:46,140
There's nothing else
we do with them,

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00:09:46,140 --> 00:09:49,040
so we can as well measure them,
and this being immediately

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00:09:49,040 --> 00:09:53,281
lead to the equation
for the density matrix.

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00:09:53,281 --> 00:09:54,780
So this is what you
want to discuss,

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00:09:54,780 --> 00:09:56,980
and it will have all
the ingredients later

169
00:09:56,980 --> 00:09:59,390
on in a mathematically
simple form

170
00:09:59,390 --> 00:10:02,270
for the derivation
of master equation.

171
00:10:02,270 --> 00:10:09,170
So let's consider a
similar photon for that.

172
00:10:12,580 --> 00:10:16,630
So the wave function,
this is superposition

173
00:10:16,630 --> 00:10:20,950
of no photon and 1 photon,
and the coefficients

174
00:10:20,950 --> 00:10:23,930
are alpha and beta.

175
00:10:23,930 --> 00:10:28,630
And just for
simplifying notation,

176
00:10:28,630 --> 00:10:31,820
I pick alpha and
beta to be real.

177
00:10:31,820 --> 00:10:44,640
So what do we expect to
happen at the beam splitter?

178
00:10:44,640 --> 00:10:48,175
Well, there's a probability
that the photon gets reflected.

179
00:10:54,100 --> 00:10:58,760
Probability to
reflect the photon

180
00:10:58,760 --> 00:11:00,500
and therefore to
observe the photon.

181
00:11:06,470 --> 00:11:10,940
This probability is, of course,
beta square-- the probability

182
00:11:10,940 --> 00:11:15,280
that we have a photon to
begin with-- and then the beam

183
00:11:15,280 --> 00:11:18,560
splitter, remember we
categorized the beam

184
00:11:18,560 --> 00:11:21,280
splitter with angle
sine theta cosine theta.

185
00:11:21,280 --> 00:11:23,720
Sine theta was the
reflection amplitude,

186
00:11:23,720 --> 00:11:26,470
cosine theta, the
transmission amplitude.

187
00:11:26,470 --> 00:11:30,940
So this probability,
which I call P1

188
00:11:30,940 --> 00:11:37,660
is the probability for
reflection and for measurement.

189
00:11:37,660 --> 00:11:42,460
And now naively you
would think what

190
00:11:42,460 --> 00:11:45,270
happens after the system
has passed through the beam

191
00:11:45,270 --> 00:11:49,170
splitter, with a probability of
P1, we've measured the photon.

192
00:11:49,170 --> 00:11:51,980
We know for sure there
is no photon left.

193
00:11:51,980 --> 00:11:56,930
The system is in
the vacuum state,

194
00:11:56,930 --> 00:11:59,980
but then you would say,
well, maybe with probability

195
00:11:59,980 --> 00:12:05,030
1 minus P1, we have
not measured anything.

196
00:12:05,030 --> 00:12:07,480
Nothing has happened
to the wave function,

197
00:12:07,480 --> 00:12:09,760
and that means the wave
function just continuous.

198
00:12:12,570 --> 00:12:17,750
Well as we will
see, this is wrong.

199
00:12:17,750 --> 00:12:19,640
We are missing something.

200
00:12:19,640 --> 00:12:24,730
What we are actually missing
is that if you measure nothing,

201
00:12:24,730 --> 00:12:26,460
the wave function is not sine.

202
00:12:26,460 --> 00:12:29,540
The possibility that we could
have measured something,

203
00:12:29,540 --> 00:12:31,930
changes the wave function.

204
00:12:31,930 --> 00:12:36,260
I will comment on that in much
more detail in a few lectures

205
00:12:36,260 --> 00:12:38,290
down the road when
I derive for you

206
00:12:38,290 --> 00:12:40,040
quantum Monte Carlo
wave function.

207
00:12:40,040 --> 00:12:42,450
I will have a wonderful
discussions with you

208
00:12:42,450 --> 00:12:46,530
about how does non-observation
change a wave function.

209
00:12:46,530 --> 00:12:49,190
So we will talk about
the physics behind it

210
00:12:49,190 --> 00:12:50,580
in some more detail.

211
00:12:50,580 --> 00:12:54,150
Right now, I don't want to
get into this discussion.

212
00:12:54,150 --> 00:12:56,850
I simply want to use our
beam splitter equation,

213
00:12:56,850 --> 00:13:02,310
so we can just take the beam
splitter equation and apply it.

214
00:13:02,310 --> 00:13:19,800
So our output state
is obtained by taking

215
00:13:19,800 --> 00:13:23,510
the operator for
our beam splitter,

216
00:13:23,510 --> 00:13:29,230
and maybe you remember that the
propagation for beam splitter

217
00:13:29,230 --> 00:13:31,580
was discovered by
an operator, which

218
00:13:31,580 --> 00:13:37,020
had a dagger b dagger
a in the exponent.

219
00:13:37,020 --> 00:13:39,195
a and b are the two input nodes.

220
00:13:42,240 --> 00:13:46,540
And the angle of
the beam splitter,

221
00:13:46,540 --> 00:13:49,700
which interpolates between
0% and 100% reflection

222
00:13:49,700 --> 00:13:52,180
transmission is theta.

223
00:13:52,180 --> 00:13:55,090
And we're now looking
for the output

224
00:13:55,090 --> 00:13:58,010
state of the total system.

225
00:13:58,010 --> 00:14:00,190
We're not performing
the measurement yet,

226
00:14:00,190 --> 00:14:08,330
and this is now acting on
the total system, which

227
00:14:08,330 --> 00:14:11,110
is the cross product
of our photon system,

228
00:14:11,110 --> 00:14:13,010
of our system of interest.

229
00:14:13,010 --> 00:14:15,940
And the other
input, which we call

230
00:14:15,940 --> 00:14:19,430
the environment or
the vacuum, is 0.

231
00:14:19,430 --> 00:14:25,096
Well, look a few weeks
back, we have done that all.

232
00:14:25,096 --> 00:14:32,360
The output state is, well,
there was a probability, alpha,

233
00:14:32,360 --> 00:14:38,410
that we had no photon
in the state psi.

234
00:14:38,410 --> 00:14:40,340
And if we have no
photon in the state psi

235
00:14:40,340 --> 00:14:43,500
and no photon in the vacuum,
this is the state 0, 0.

236
00:14:46,770 --> 00:14:50,240
What I denote here
with this second place

237
00:14:50,240 --> 00:14:51,120
is the environment.

238
00:14:53,750 --> 00:14:57,510
And now we have one photon.

239
00:14:57,510 --> 00:15:00,920
We have exactly one photon
with the amplitude beta,

240
00:15:00,920 --> 00:15:06,810
and this photon is split
with cosine theta transmitted

241
00:15:06,810 --> 00:15:10,850
and with phi theta reflected.

242
00:15:10,850 --> 00:15:15,350
If you transmit
it, we have 1, 0.

243
00:15:15,350 --> 00:15:19,210
If we reflect it, we have 0, 1.

244
00:15:19,210 --> 00:15:23,399
And again, this is
the environment,

245
00:15:23,399 --> 00:15:25,065
and here is a photon
in the environment.

246
00:15:30,240 --> 00:15:33,330
So let me just be
clear that this

247
00:15:33,330 --> 00:15:35,300
is where the
environment comes in.

248
00:15:35,300 --> 00:15:43,710
It is a vacuum
state, and here, this

249
00:15:43,710 --> 00:15:46,030
is the output part
for the environment.

250
00:15:46,030 --> 00:15:49,110
This is where we
do the measurement.

251
00:15:49,110 --> 00:15:53,880
And I don't think it matters.

252
00:15:53,880 --> 00:15:56,010
I haven't really told you
which is mode A, which

253
00:15:56,010 --> 00:15:58,770
is mode B. It doesn't
matter, but one, let's say

254
00:15:58,770 --> 00:16:01,650
the environment is
mode B, and the system

255
00:16:01,650 --> 00:16:18,260
evolves in mode
A. As you can see,

256
00:16:18,260 --> 00:16:20,110
I'm using a new
program, which has

257
00:16:20,110 --> 00:16:22,620
some nicer features in
terms of handwriting,

258
00:16:22,620 --> 00:16:26,770
but it is a little bit
rough in scrolling,

259
00:16:26,770 --> 00:16:30,850
so I sometimes have to
scroll back and forth.

260
00:16:30,850 --> 00:16:34,500
So what is our output?

261
00:16:34,500 --> 00:16:36,770
Now, we have two possibilities.

262
00:16:36,770 --> 00:16:40,382
The environment is 0,
or the environment is 1,

263
00:16:40,382 --> 00:16:41,590
and we perform a measurement.

264
00:16:45,610 --> 00:16:50,120
So we have to now go into a
probabilistic description.

265
00:16:50,120 --> 00:16:55,570
So with probability P1, we
have done a measurement,

266
00:16:55,570 --> 00:17:04,700
and our output state is
now the vacuum state.

267
00:17:04,700 --> 00:17:20,050
With probability 1
minus P1, we have not

268
00:17:20,050 --> 00:17:24,609
detected anything in the
vacuum, and therefore, our state

269
00:17:24,609 --> 00:17:29,200
of the system is alpha 0
plus beta cosine theta 1.

270
00:17:36,530 --> 00:17:46,350
Is alpha 0, so it is not beta 1
as naively would have assumed.

271
00:17:46,350 --> 00:17:48,080
It's not the original state.

272
00:17:48,080 --> 00:17:52,060
There is a cosine theta
factor, which we got exactly

273
00:17:52,060 --> 00:17:55,940
from the beam splitter from
the unitary evolution provided

274
00:17:55,940 --> 00:17:58,130
by the beam splitter.

275
00:17:58,130 --> 00:18:01,660
And since these state
is no longer normalized,

276
00:18:01,660 --> 00:18:07,210
I have to normalize
it by alpha squared

277
00:18:07,210 --> 00:18:13,250
plus beta square
cosine square theta.

278
00:18:13,250 --> 00:18:17,150
So now we have done our
measurement probability P1

279
00:18:17,150 --> 00:18:18,430
to detect the photon.

280
00:18:18,430 --> 00:18:22,020
This projects the system
into the vacuum state

281
00:18:22,020 --> 00:18:24,150
with the probability 1 minus P1.

282
00:18:24,150 --> 00:18:25,500
We have that state.

283
00:18:25,500 --> 00:18:26,370
Just one second.

284
00:18:26,370 --> 00:18:27,570
Scroll in the pictures.

285
00:18:30,330 --> 00:18:32,040
Write that down.

286
00:18:32,040 --> 00:18:36,140
In fact, millions
that our system is now

287
00:18:36,140 --> 00:18:43,200
described by a density
matrix with probability P1

288
00:18:43,200 --> 00:18:44,890
and 1 minus 1 minus P1.

289
00:18:48,810 --> 00:18:57,100
With probability P1, we
are in the vacuum state,

290
00:18:57,100 --> 00:18:59,920
and with probability
1 minus P1, we

291
00:18:59,920 --> 00:19:04,640
are in that state, the
denormalized state psi

292
00:19:04,640 --> 00:19:07,360
naught, which I just hold down.

293
00:19:07,360 --> 00:19:08,870
Question?

294
00:19:08,870 --> 00:19:10,680
AUDIENCE: Have you
considered theta

295
00:19:10,680 --> 00:19:15,784
to be some like a dynamical
phase evolution system.

296
00:19:15,784 --> 00:19:18,760
It's very low order
like when you expand it

297
00:19:18,760 --> 00:19:21,240
the first time it looks almost
identical to [INAUDIBLE]

298
00:19:21,240 --> 00:19:23,224
quantum effect maybe.

299
00:19:23,224 --> 00:19:28,860
The environment is measuring the
state in some way, and I mean,

300
00:19:28,860 --> 00:19:30,120
it's the lowest order now.

301
00:19:30,120 --> 00:19:32,286
PROFESSOR: Yeah, I Quite
agree that random 0 is just

302
00:19:32,286 --> 00:19:33,110
an example of it.

303
00:19:35,809 --> 00:19:37,100
Pretty much, it's all the same.

304
00:19:37,100 --> 00:19:38,550
Yeah.

305
00:19:38,550 --> 00:19:41,840
What we do here is, I
like the beam splitter

306
00:19:41,840 --> 00:19:44,695
because the beam splitter
provides an exact formulation

307
00:19:44,695 --> 00:19:46,190
of the measurement process.

308
00:19:46,190 --> 00:19:47,780
You really can use
a beam splitter

309
00:19:47,780 --> 00:19:50,440
to discuss what
happens fundamentally

310
00:19:50,440 --> 00:19:52,450
when you perform a measurement.

311
00:19:52,450 --> 00:19:56,570
And the beams splitter is one
typical implementation of that,

312
00:19:56,570 --> 00:19:58,000
but it has all the
features you'll

313
00:19:58,000 --> 00:19:59,291
find in any measurement system.

314
00:20:02,120 --> 00:20:04,350
And especially what
you observe here,

315
00:20:04,350 --> 00:20:07,470
let me just emphasizes is,
the fact that we you do not

316
00:20:07,470 --> 00:20:09,970
make a measurement
is changing the way

317
00:20:09,970 --> 00:20:14,910
function from the initial wave
function psi to psi naught,

318
00:20:14,910 --> 00:20:18,760
we have a factor of
cosine theta here.

319
00:20:18,760 --> 00:20:21,170
And that's also very general.

320
00:20:21,170 --> 00:20:27,289
A measurement perturbs,
modifies your wave function

321
00:20:27,289 --> 00:20:29,330
no matter what the outcome
of the measurement is.

322
00:20:38,330 --> 00:20:41,460
So let me write
it down because we

323
00:20:41,460 --> 00:20:45,400
want to take it
to the next level.

324
00:20:45,400 --> 00:20:52,830
So we have now found in terms of
the beam splitter, angle theta,

325
00:20:52,830 --> 00:20:59,070
and the parameters of the
initial state, alpha beta.

326
00:20:59,070 --> 00:21:01,610
We found the density matrix
after the beam splitter.

327
00:21:10,760 --> 00:21:12,920
Yes.

328
00:21:12,920 --> 00:21:15,000
So what is the next step?

329
00:21:15,000 --> 00:21:21,250
Our goal is to derive the
master equation for the density

330
00:21:21,250 --> 00:21:24,100
matrix, the time evolution
of the density matrix.

331
00:21:24,100 --> 00:21:27,950
So since we want to
discuss the time evolution,

332
00:21:27,950 --> 00:21:30,620
we want to find a
differential equation.

333
00:21:30,620 --> 00:21:32,260
So what we want
to figure out is,

334
00:21:32,260 --> 00:21:36,710
what is the difference
between the output density

335
00:21:36,710 --> 00:21:39,720
matrix and the input
density matrix.

336
00:21:39,720 --> 00:21:42,730
The input was, of
course, pure state

337
00:21:42,730 --> 00:21:48,130
characterized by the matrix
population alpha squared

338
00:21:48,130 --> 00:21:54,170
and beta squared of diagonal
matrix element of alpha beta.

339
00:22:03,810 --> 00:22:06,030
The difference between
the density matrix

340
00:22:06,030 --> 00:22:08,660
is can just calculate
the difference.

341
00:22:12,920 --> 00:22:19,640
You can simplify
things by applying

342
00:22:19,640 --> 00:22:23,760
some trigonometric
identities, so this

343
00:22:23,760 --> 00:22:40,710
is an exact result,
cosine theta minus 1.

344
00:22:40,710 --> 00:22:48,450
Here we have alpha beta
cosine theta minus 1,

345
00:22:48,450 --> 00:22:53,850
and on the diagonal,
we have cosine

346
00:22:53,850 --> 00:22:58,870
2 theta minus 1 divided by 2.

347
00:22:58,870 --> 00:23:02,530
Anyway this is an
intermediate result.

348
00:23:02,530 --> 00:23:04,760
We're interested in the
differential equation.

349
00:23:04,760 --> 00:23:07,610
We want to sort of find out
what happens when we observe,

350
00:23:07,610 --> 00:23:11,190
when we have the density
matrix interacting

351
00:23:11,190 --> 00:23:13,600
with the environment
all the time.

352
00:23:13,600 --> 00:23:17,850
And this can be simulated
by beam splitters

353
00:23:17,850 --> 00:23:22,220
by using many beam splitters
with a small degree

354
00:23:22,220 --> 00:23:24,650
of reflection.

355
00:23:24,650 --> 00:23:28,810
So we want to simplify
this result now

356
00:23:28,810 --> 00:23:33,550
for the case of many beam
splitters, and each of them

357
00:23:33,550 --> 00:23:44,040
has a small tipping
angle, theta,

358
00:23:44,040 --> 00:23:53,620
and for later convenience, I
defined theta to be gamma times

359
00:23:53,620 --> 00:23:55,460
delta t over 2.

360
00:23:55,460 --> 00:23:59,390
That's just my
definition of theta.

361
00:23:59,390 --> 00:24:07,510
So what we have in mind now is
that we start with the system

362
00:24:07,510 --> 00:24:15,230
psi, and we have many such beam
splitters with an infinitesimal

363
00:24:15,230 --> 00:24:15,840
tipping angle.

364
00:24:21,250 --> 00:24:29,160
Each beam splitter has the
vacuum at its input state.

365
00:24:32,620 --> 00:24:38,023
And we always perform
the measurement.

366
00:24:48,390 --> 00:24:51,320
If I take the
equation above, which

367
00:24:51,320 --> 00:24:55,740
I know you can't see anymore,
we find a differential equation

368
00:24:55,740 --> 00:25:05,740
for the density matrix, which is
we find an infinitesimal change

369
00:25:05,740 --> 00:25:11,390
delta over the density
matrix, which looks like this.

370
00:25:19,650 --> 00:25:24,480
So all I've done is, I've
used the equation above,

371
00:25:24,480 --> 00:25:30,120
and I've done a Taylor expansion
in the small angle theta.

372
00:25:30,120 --> 00:25:32,820
And the reason why I brought
in the square root, well,

373
00:25:32,820 --> 00:25:34,630
we get cosine theta.

374
00:25:34,630 --> 00:25:37,430
The first order Taylor expansion
or the lowest order Taylor

375
00:25:37,430 --> 00:25:41,020
expansion from cosine is
1 minus theta squared.

376
00:25:41,020 --> 00:25:44,630
So I get the square
root squared.

377
00:25:44,630 --> 00:25:46,610
So I get gamma,
which appears here,

378
00:25:46,610 --> 00:25:49,240
and then I divide by delta t.

379
00:25:49,240 --> 00:25:52,800
So this is just an exact
mathematical expression,

380
00:25:52,800 --> 00:26:00,350
and the next step is to form
a differential equation.

381
00:26:00,350 --> 00:26:04,830
But before I do that, I want
to emphasize the two features

382
00:26:04,830 --> 00:26:06,520
we are using here.

383
00:26:06,520 --> 00:26:08,140
They sort of enter
automatically,

384
00:26:08,140 --> 00:26:11,280
but these are the
two big assumptions

385
00:26:11,280 --> 00:26:14,360
we make when we derive
a master equation.

386
00:26:14,360 --> 00:26:25,850
The first one is that we
always have a vacuum state

387
00:26:25,850 --> 00:26:27,820
as the input.

388
00:26:27,820 --> 00:26:35,460
So in other words,
the environment

389
00:26:35,460 --> 00:26:43,900
is always in the same state,
which is a vacuum state,

390
00:26:43,900 --> 00:26:48,860
and this is sort of called
a Bohr approximation.

391
00:26:48,860 --> 00:26:53,700
What it means is that we
do a measurement here,

392
00:26:53,700 --> 00:26:55,860
but the vacuum is not changing.

393
00:26:55,860 --> 00:26:58,600
In other words, we are
not overloading the vacuum

394
00:26:58,600 --> 00:27:02,830
with so many photons
that suddenly the vacuum

395
00:27:02,830 --> 00:27:04,760
Is no longer in
the vacuum state.

396
00:27:04,760 --> 00:27:06,770
Or in the case of
spontaneous emission,

397
00:27:06,770 --> 00:27:10,500
the vacuum can just take as many
photons as you dumping into it.

398
00:27:10,500 --> 00:27:14,120
They disappear so quickly that
for all practical purposes,

399
00:27:14,120 --> 00:27:18,380
the environment stays
in the vacuum state.

400
00:27:18,380 --> 00:27:20,300
So this is called the
Bohr approximation.

401
00:27:20,300 --> 00:27:22,390
The environment is not changing.

402
00:27:22,390 --> 00:27:25,851
It has enough capacity you to
be modified by the measurement

403
00:27:25,851 --> 00:27:26,350
process.

404
00:27:29,820 --> 00:27:42,260
And the second thing
which is related

405
00:27:42,260 --> 00:27:49,540
is, the vacuum is always
in the same state,

406
00:27:49,540 --> 00:27:56,120
and there are no correlations
from here to here to here,

407
00:27:56,120 --> 00:27:57,990
there is no memory effect.

408
00:27:57,990 --> 00:27:59,690
Everything is
completely uncorrelated.

409
00:28:07,340 --> 00:28:10,240
So the environment
is uncorrelated.

410
00:28:13,990 --> 00:28:16,380
It has no memory.

411
00:28:16,380 --> 00:28:20,100
It's correlation function
is a delta function,

412
00:28:20,100 --> 00:28:21,985
and this is called
Markov approximation.

413
00:28:28,390 --> 00:28:31,040
So these are the two
effects which are important.

414
00:28:31,040 --> 00:28:33,820
One is no memory
for the environment,

415
00:28:33,820 --> 00:28:37,020
Delta function correlation,
Markov approximation,

416
00:28:37,020 --> 00:28:40,290
and the two, of
course, are related

417
00:28:40,290 --> 00:28:42,680
in the environment is all
of this in the same state.

418
00:28:46,920 --> 00:28:59,100
So remember this is a change
for the density matrix,

419
00:28:59,100 --> 00:29:03,480
and alpha beta where
the original parameters

420
00:29:03,480 --> 00:29:06,460
of the density matrix
for the input state.

421
00:29:06,460 --> 00:29:10,256
So I can now rewrite everything
as a differential equation.

422
00:29:14,270 --> 00:29:19,130
The density matrix
has a derivative

423
00:29:19,130 --> 00:29:26,370
for the diagonal matrix
elements and for the coherences.

424
00:29:29,700 --> 00:29:33,700
Here we have plus gamma.

425
00:29:33,700 --> 00:29:36,950
Here we have minus
gamma 0, 1, 1,

426
00:29:36,950 --> 00:29:40,640
makes sense because
we conserve the trace.

427
00:29:40,640 --> 00:29:44,290
We have unity probability
that we have a stellar system.

428
00:29:44,290 --> 00:29:47,830
And therefore the two
diagonal matrix elements,

429
00:29:47,830 --> 00:29:52,160
the population, the sum of
them cannot change with time.

430
00:29:54,970 --> 00:30:00,120
And for the coherence,
we have gamma over 2.

431
00:30:03,340 --> 00:30:10,010
And if you're familiar with
Optical Bloch Equations, which

432
00:30:10,010 --> 00:30:13,460
we derive next, we
can say that these

433
00:30:13,460 --> 00:30:20,970
means if 0 is the ground state
that the ground state changes

434
00:30:20,970 --> 00:30:25,050
because-- call it spontaneous
emission from the excited

435
00:30:25,050 --> 00:30:26,480
state.

436
00:30:26,480 --> 00:30:30,940
This equation would say that
the excited state decays

437
00:30:30,940 --> 00:30:35,620
with the rate gamma,
and sometimes you

438
00:30:35,620 --> 00:30:38,430
may have wondered about
that there are factors

439
00:30:38,430 --> 00:30:41,800
of two appearing,
which also appears here

440
00:30:41,800 --> 00:30:46,790
that when the excited state
decays with a rate gamma,

441
00:30:46,790 --> 00:30:49,310
we have a factor here
for the coherences, which

442
00:30:49,310 --> 00:30:50,380
is gamma over 2.

443
00:30:53,800 --> 00:30:56,950
So what we have
accomplished in contrast

444
00:30:56,950 --> 00:31:00,430
to let's say Einstein's
equation with the Einstein

445
00:31:00,430 --> 00:31:03,500
a and b coefficient,
which lead to rate

446
00:31:03,500 --> 00:31:10,080
equations for the population,
we have now a new feature.

447
00:31:10,080 --> 00:31:14,450
We have an the equation
for the coherences,

448
00:31:14,450 --> 00:31:17,685
and we find a decay
of the coherences

449
00:31:17,685 --> 00:31:20,535
with half the rate as a
decay of the population.

450
00:31:24,921 --> 00:31:25,420
Questions?

451
00:31:36,190 --> 00:31:41,400
So if you want you could rewrite
this model for photon, which

452
00:31:41,400 --> 00:31:44,180
goes through beam splitters,
undergoes measurement.

453
00:31:44,180 --> 00:31:46,840
You can rewrite it from
atomic wave function

454
00:31:46,840 --> 00:31:48,620
and you measure
whether the atomics

455
00:31:48,620 --> 00:31:52,390
in the excited and ground
state and the equation

456
00:31:52,390 --> 00:31:54,490
for the measurement
performed on the atom

457
00:31:54,490 --> 00:32:01,060
is exactly as the equation by
which the beam splitter acts

458
00:32:01,060 --> 00:32:02,460
on the photon state.

459
00:32:02,460 --> 00:32:04,150
So what I've shown
here it's very

460
00:32:04,150 --> 00:32:06,450
specific for a single
photon because I

461
00:32:06,450 --> 00:32:09,000
could use simple
equations, but everything

462
00:32:09,000 --> 00:32:14,535
is what you find in a much
more general situation.

463
00:32:17,180 --> 00:32:23,100
So before I give you the
general derivation of the master

464
00:32:23,100 --> 00:32:33,490
equation, let me talk
about what we have learned

465
00:32:33,490 --> 00:32:41,080
from this example and what
the general procedure is.

466
00:32:46,620 --> 00:32:53,650
The first thing is our goal
is to find a differential

467
00:32:53,650 --> 00:32:58,760
equation for the density
matrix of the system.

468
00:32:58,760 --> 00:32:59,690
I just remember.

469
00:32:59,690 --> 00:33:02,940
There was one thing
I wanted to mention.

470
00:33:02,940 --> 00:33:05,370
In the previous derivation
with the beam splitter,

471
00:33:05,370 --> 00:33:09,250
I started with a purer
state, and the purer state

472
00:33:09,250 --> 00:33:11,930
developed into a
statistical mixture,

473
00:33:11,930 --> 00:33:14,350
and this statistical
mixture would then

474
00:33:14,350 --> 00:33:16,490
transform the next
beam splitter into

475
00:33:16,490 --> 00:33:18,610
another statistical mixture.

476
00:33:18,610 --> 00:33:21,550
I derived the differential
equation for you

477
00:33:21,550 --> 00:33:23,530
for the first state
from the pure state

478
00:33:23,530 --> 00:33:25,340
to the statistical mixture.

479
00:33:25,340 --> 00:33:27,210
But if you would
spend a few minutes,

480
00:33:27,210 --> 00:33:29,100
you could immediately
show that you

481
00:33:29,100 --> 00:33:31,950
can start with an
elementary density matrix,

482
00:33:31,950 --> 00:33:34,300
look how it evolves
through the beam splitter,

483
00:33:34,300 --> 00:33:38,750
and you get exactly the
same differential equation.

484
00:33:46,380 --> 00:33:52,260
So the general procedure is,
we want a differential equation

485
00:33:52,260 --> 00:33:57,490
how the density matrix
evolves with time.

486
00:33:57,490 --> 00:34:03,360
And this will be obtained
by finding an operator which

487
00:34:03,360 --> 00:34:05,925
acts on the initial
density matrix.

488
00:34:08,730 --> 00:34:11,949
This operator is not
a unitary operator

489
00:34:11,949 --> 00:34:14,100
because we are
performing measurements

490
00:34:14,100 --> 00:34:16,530
through the environment,
even if you don't actually

491
00:34:16,530 --> 00:34:17,750
perform them.

492
00:34:17,750 --> 00:34:20,429
Once we dump something
into the environment,

493
00:34:20,429 --> 00:34:22,750
it's out of our
control and anybody

494
00:34:22,750 --> 00:34:25,260
could go and perform
a measurement,

495
00:34:25,260 --> 00:34:27,330
and so we should assume
that this measurement has

496
00:34:27,330 --> 00:34:27,790
been taken.

497
00:34:27,790 --> 00:34:29,581
It's one of those
quantum mechanical things

498
00:34:29,581 --> 00:34:32,380
that you don't even have to
care whether somebody does it.

499
00:34:32,380 --> 00:34:35,790
The environment does it for you.

500
00:34:35,790 --> 00:34:42,699
So this operator is called
a Liouvillian operator.

501
00:34:45,850 --> 00:34:48,120
It's sometimes called--
and I haven't really

502
00:34:48,120 --> 00:34:51,389
traced down why-- it's called
sometimes super operator.

503
00:34:55,830 --> 00:34:57,730
I know what
superconductivity is,

504
00:34:57,730 --> 00:35:01,070
but I don't know what the super
powers of this operator are,

505
00:35:01,070 --> 00:35:06,080
but that's just a name
which you will find.

506
00:35:06,080 --> 00:35:08,010
The second thing
which we have used

507
00:35:08,010 --> 00:35:14,390
is that the evolution
of this system

508
00:35:14,390 --> 00:35:23,050
can be obtained from the time
evolution of the total system

509
00:35:23,050 --> 00:35:28,430
by performing a trace over
the degrees of freedom

510
00:35:28,430 --> 00:35:29,870
of the environment.

511
00:35:29,870 --> 00:35:31,750
This was exactly
what we actually

512
00:35:31,750 --> 00:35:36,070
did when we said the system
continues with probability P

513
00:35:36,070 --> 00:35:38,480
naught in one state
and probability

514
00:35:38,480 --> 00:35:39,810
P1 in the other state.

515
00:35:39,810 --> 00:35:42,620
The operation which lead
to this density matrix

516
00:35:42,620 --> 00:35:44,060
was exactly the partial trace.

517
00:35:49,820 --> 00:35:57,385
So this is the second
general feature

518
00:35:57,385 --> 00:36:00,630
which we have to implement.

519
00:36:00,630 --> 00:36:05,510
Thirdly, if we could
do one and two exactly,

520
00:36:05,510 --> 00:36:07,100
we would have an
exact formulation

521
00:36:07,100 --> 00:36:10,220
for a small part
of a quantum system

522
00:36:10,220 --> 00:36:13,030
no matter how complicated
the environment is.

523
00:36:13,030 --> 00:36:17,800
In practice, we can
solve the equations only

524
00:36:17,800 --> 00:36:22,350
when we make
simplifying assumptions

525
00:36:22,350 --> 00:36:23,740
about the environment.

526
00:36:27,870 --> 00:36:34,350
One is, it is large,
and more important,

527
00:36:34,350 --> 00:36:42,010
therefore, it's unchanging, and
this is the Bohr approximation.

528
00:36:42,010 --> 00:36:47,080
And the second feature is,
it has a short correlation

529
00:36:47,080 --> 00:36:55,690
time, tau c.

530
00:36:55,690 --> 00:36:58,160
In the beam splitter, I
have made the assumption

531
00:36:58,160 --> 00:37:00,490
that there is no correlation
between different beam

532
00:37:00,490 --> 00:37:02,460
splitters in the
derivation, which

533
00:37:02,460 --> 00:37:04,240
I want to walk you through.

534
00:37:04,240 --> 00:37:09,130
Right now, you
will see explicitly

535
00:37:09,130 --> 00:37:11,170
where the correlation
times enter.

536
00:37:11,170 --> 00:37:13,355
And this is called the
Markov approximation.

537
00:37:18,910 --> 00:37:26,920
And finally, this is number
four, the whole possibility

538
00:37:26,920 --> 00:37:33,850
to derive a master
equation hinges on the fact

539
00:37:33,850 --> 00:37:37,780
that we have different
time scales which

540
00:37:37,780 --> 00:37:39,120
are very different.

541
00:37:48,490 --> 00:37:58,100
We are interested in the
evolution of our system.

542
00:37:58,100 --> 00:38:02,680
We want to know how it relaxes,
and this is on a time scale

543
00:38:02,680 --> 00:38:03,700
1 over gamma.

544
00:38:06,240 --> 00:38:08,670
So we call this slow.

545
00:38:08,670 --> 00:38:13,130
We are interested
in the variation

546
00:38:13,130 --> 00:38:18,100
of our system, the atomic
system or the photon state,

547
00:38:18,100 --> 00:38:21,510
which passes through
the beam splitter,

548
00:38:21,510 --> 00:38:25,270
and this time scale
has to be much slower

549
00:38:25,270 --> 00:38:27,600
than the fluctuations
of the environment.

550
00:38:35,550 --> 00:38:45,440
So therefore, if the
environment has fluctuations,

551
00:38:45,440 --> 00:38:47,110
which in the beam
splitter model where

552
00:38:47,110 --> 00:38:50,510
assumed to be 0, it was
delta function of time,

553
00:38:50,510 --> 00:38:55,880
if that correlation time is
much smaller than the time

554
00:38:55,880 --> 00:39:00,480
it takes for the system
to relax and to evolve,

555
00:39:00,480 --> 00:39:06,300
that opens a window
delta t, and this

556
00:39:06,300 --> 00:39:09,370
is the time scale of
the master equation.

557
00:39:16,610 --> 00:39:19,650
Just to give you one example
for the spontaneous emission,

558
00:39:19,650 --> 00:39:22,910
the correlation
time, tau c, would

559
00:39:22,910 --> 00:39:27,920
be the time it takes the photon
to disappear from the atom.

560
00:39:27,920 --> 00:39:30,610
And the photon has
disappeared from the atom

561
00:39:30,610 --> 00:39:33,140
when it is one wavelengths away.

562
00:39:33,140 --> 00:39:35,490
So typically, the
correlation time

563
00:39:35,490 --> 00:39:38,330
f the vacuum for
spontaneous emission

564
00:39:38,330 --> 00:39:44,070
is one cycle of the
optical frequency.

565
00:39:44,070 --> 00:39:45,840
It's very, very fast.

566
00:39:45,840 --> 00:39:50,490
Whereas typical decay times
of the excited states,

567
00:39:50,490 --> 00:39:51,180
a nano second.

568
00:39:51,180 --> 00:39:53,400
It's six orders of
magnitude slower,

569
00:39:53,400 --> 00:39:56,160
and this is what we describe.

570
00:39:56,160 --> 00:39:58,892
But on a time scale
of a femtosecond

571
00:39:58,892 --> 00:40:04,460
of one optical cycle, the photon
has not detached from the atom

572
00:40:04,460 --> 00:40:06,980
and it could go actually
back to the atom.

573
00:40:06,980 --> 00:40:11,440
During that time, we talked
a little bit about it

574
00:40:11,440 --> 00:40:14,770
when we did this diagrammatic
discussions of resonance

575
00:40:14,770 --> 00:40:17,090
scattering for very,
very early times.

576
00:40:17,090 --> 00:40:21,130
You don't have exponential
decay because you cannot do

577
00:40:21,130 --> 00:40:24,130
the approximations where
we approximated the kernel

578
00:40:24,130 --> 00:40:26,860
by something which was
completely energy dependent.

579
00:40:26,860 --> 00:40:30,030
And so what happens
at such short times,

580
00:40:30,030 --> 00:40:32,994
we encounter here again
in such short times,

581
00:40:32,994 --> 00:40:35,160
we will not have a simple
description of the system.

582
00:40:47,200 --> 00:40:50,120
So the last point,
let me summarize.

583
00:40:50,120 --> 00:40:57,310
Our goal is that we describe the
density matrix of the system,

584
00:40:57,310 --> 00:41:00,420
and we want to find
the Liouville operator

585
00:41:00,420 --> 00:41:04,470
or some matrix which acts on it.

586
00:41:04,470 --> 00:41:11,490
And because we will integrate
over time steps, which

587
00:41:11,490 --> 00:41:14,910
are larger than the
correlation time of the system,

588
00:41:14,910 --> 00:41:21,520
we can also call it, it will
be a coarse-grained evolution.

589
00:41:28,810 --> 00:41:30,400
Any questions about that?

590
00:41:44,640 --> 00:41:48,550
I really like the discussion,
the derivation of the master

591
00:41:48,550 --> 00:41:53,490
equation, how it is presented
in atom-photon indirection.

592
00:41:53,490 --> 00:41:57,230
But it is presented
on more than 50 pages

593
00:41:57,230 --> 00:41:59,470
with many, many equations.

594
00:41:59,470 --> 00:42:02,640
So after giving you all
of the principles, all

595
00:42:02,640 --> 00:42:06,030
of the concepts, I
want to go with you now

596
00:42:06,030 --> 00:42:09,820
over those equations
and point out how

597
00:42:09,820 --> 00:42:13,430
the principles, which we
encountered with the beam

598
00:42:13,430 --> 00:42:16,120
splitter, how they
are now implemented

599
00:42:16,120 --> 00:42:18,520
in a very general context.

600
00:42:18,520 --> 00:42:23,890
I will not be able to give you
all of the mathematical aspects

601
00:42:23,890 --> 00:42:27,920
of it, but I think by now you
know that the book atom-photon

602
00:42:27,920 --> 00:42:29,660
interactions are
actually wonderful.

603
00:42:29,660 --> 00:42:33,400
You can get a lot of conception
information out of it

604
00:42:33,400 --> 00:42:37,150
by looking at the equations
without understanding

605
00:42:37,150 --> 00:42:38,830
every technical detail.

606
00:42:38,830 --> 00:42:42,670
So I would really encourage
you, if there is something which

607
00:42:42,670 --> 00:42:45,800
piques your interest and I
hope there will be things which

608
00:42:45,800 --> 00:42:48,840
you'll find very interesting,
that you go to the book

609
00:42:48,840 --> 00:42:50,070
and read it.

610
00:42:50,070 --> 00:42:51,950
So I'm exactly following
that actually I

611
00:42:51,950 --> 00:42:53,960
used copies of the book.

612
00:42:53,960 --> 00:42:58,550
So we have a
Hamiltonian, which is

613
00:42:58,550 --> 00:43:01,730
describing the atomic system.

614
00:43:01,730 --> 00:43:06,521
It describes the reservoir and
then there is an interaction.

615
00:43:06,521 --> 00:43:09,020
We keep it very general here,
but you may always think well,

616
00:43:09,020 --> 00:43:13,340
the atom is your favorite
two-level system.

617
00:43:13,340 --> 00:43:16,350
The environment is
maybe the vacuum

618
00:43:16,350 --> 00:43:19,030
with all its possible
modes, and the interaction

619
00:43:19,030 --> 00:43:22,670
is the dipole interaction
or the a dot p interaction.

620
00:43:33,290 --> 00:43:41,060
So we start out
with an equation,

621
00:43:41,060 --> 00:43:46,630
which is nothing else
than Schrodinger's

622
00:43:46,630 --> 00:43:52,400
equation for the density matrix.

623
00:43:52,400 --> 00:43:54,360
The time derivation
of the density matrix

624
00:43:54,360 --> 00:43:57,670
is commutative
with the Hamilton.

625
00:43:57,670 --> 00:44:01,320
But it is often useful and
you've seen it many times,

626
00:44:01,320 --> 00:44:03,880
to go to the interaction
representation,

627
00:44:03,880 --> 00:44:08,370
that the time dependence
due to the unperturbed part

628
00:44:08,370 --> 00:44:12,810
of the operator is absorbed
in a unitary transformation,

629
00:44:12,810 --> 00:44:16,850
so therefore, this density
matrix in the interaction

630
00:44:16,850 --> 00:44:22,030
representation evolves not with,
h, because h naught is taken

631
00:44:22,030 --> 00:44:25,396
care of, it only involves due
to the coupling between the two

632
00:44:25,396 --> 00:44:27,395
systems, between the
system and the environment.

633
00:44:35,220 --> 00:44:38,580
So now this equation, we are
interested in a time step delta

634
00:44:38,580 --> 00:44:40,000
t.

635
00:44:40,000 --> 00:44:42,700
And this time step,
delta t remember,

636
00:44:42,700 --> 00:44:45,040
we want to coarse
grain, will be larger

637
00:44:45,040 --> 00:44:47,230
than the correlation
time of the reservoir,

638
00:44:47,230 --> 00:44:49,660
and you will see exactly
where it comes about.

639
00:44:49,660 --> 00:44:53,390
So we want to now do one of
those coarse-grain steps.

640
00:44:53,390 --> 00:44:57,180
We take this equation and
we integrate from time t

641
00:44:57,180 --> 00:44:58,580
to time t plus delta t.

642
00:45:01,260 --> 00:45:04,560
So this is exact here.

643
00:45:07,750 --> 00:45:19,815
But now we want to iterate,
and that means the following.

644
00:45:22,640 --> 00:45:26,750
We have expressed the time
step in the density matrix

645
00:45:26,750 --> 00:45:29,570
by having the
density matrix there.

646
00:45:29,570 --> 00:45:36,080
But now we can do in a
first-order perturbation

647
00:45:36,080 --> 00:45:40,260
theory, we can do one step and
we get the second order result

648
00:45:40,260 --> 00:45:43,640
by plugging the first order
result into this equation.

649
00:45:43,640 --> 00:45:46,490
It's the same we have seen
with our diagrams and such.

650
00:45:46,490 --> 00:45:49,950
We have an exact equation.

651
00:45:49,950 --> 00:45:53,090
It's useless unless we do
something, and what we do is,

652
00:45:53,090 --> 00:45:56,630
we realize that
we can iterate it

653
00:45:56,630 --> 00:46:00,630
because the part we don't
know involves one more

654
00:46:00,630 --> 00:46:02,840
occurrence of the
interaction potential.

655
00:46:02,840 --> 00:46:05,740
And when you plug the nth
order solution in here,

656
00:46:05,740 --> 00:46:07,647
you get the n plus
first order solution.

657
00:46:07,647 --> 00:46:09,230
And this is exactly
what is done here.

658
00:46:15,810 --> 00:46:18,370
And I skipped a
few equations here.

659
00:46:18,370 --> 00:46:20,990
This is what he's
done here, number one.

660
00:46:20,990 --> 00:46:24,160
And number two is, we are
interested in the system,

661
00:46:24,160 --> 00:46:26,380
not in the reservoir,
so therefore we

662
00:46:26,380 --> 00:46:29,910
perform the trace
over the reservoir.

663
00:46:29,910 --> 00:46:32,920
And the trace of the reservoir
for the photon beam splitter

664
00:46:32,920 --> 00:46:38,370
mend, we say we have
two possible states,

665
00:46:38,370 --> 00:46:40,240
we detect a photon or not.

666
00:46:40,240 --> 00:46:43,540
And for the system, we
have now a density matrix

667
00:46:43,540 --> 00:46:47,680
which is probability P naught
in one state, probability

668
00:46:47,680 --> 00:46:49,750
P1 in the other state.

669
00:46:49,750 --> 00:46:54,590
And this is exactly what the
operator partial trace does.

670
00:46:54,590 --> 00:46:57,030
Remember also I want
to really make sure

671
00:46:57,030 --> 00:46:59,190
that you recognize
all the structures.

672
00:46:59,190 --> 00:47:04,550
The time evolution of a density
matrix was a commutator with h,

673
00:47:04,550 --> 00:47:08,350
but in the interaction picture,
it's a commutator with v.

674
00:47:08,350 --> 00:47:13,010
But since we are putting the
first order result in here,

675
00:47:13,010 --> 00:47:16,610
the second order result
is now the commutator

676
00:47:16,610 --> 00:47:19,910
of v with the
commutator of v and o.

677
00:47:19,910 --> 00:47:22,430
It's just we have
iterated one more time.

678
00:47:30,240 --> 00:47:40,290
So the sigma tilde, the
density matrix for our system,

679
00:47:40,290 --> 00:47:43,470
tilde means in the
interaction picture

680
00:47:43,470 --> 00:47:48,180
is now the partial
trace over the reservoir

681
00:47:48,180 --> 00:47:50,570
of the total density matrix.

682
00:47:50,570 --> 00:47:52,630
Tilde means in the
interaction picture.

683
00:47:55,380 --> 00:48:01,840
And the important part
here is that it is exact.

684
00:48:01,840 --> 00:48:06,982
We have not done any
approximation here.

685
00:48:06,982 --> 00:48:07,565
Any questions?

686
00:48:24,840 --> 00:48:28,510
Of course, now we have to
make approximations because we

687
00:48:28,510 --> 00:48:31,870
cannot solve an interactive
problem exactly.

688
00:48:31,870 --> 00:48:37,080
The first one is-- and what
do we want to do in the end?

689
00:48:37,080 --> 00:48:41,790
We want to keep the
first non-trivial term

690
00:48:41,790 --> 00:48:44,910
but to the extent possible, we
want to factorize everything.

691
00:48:44,910 --> 00:48:47,490
We want to get rid of the
entanglement of the environment

692
00:48:47,490 --> 00:48:51,260
in the system and only get
sort of the minimum which

693
00:48:51,260 --> 00:48:53,370
is provided by the coupling.

694
00:48:53,370 --> 00:48:55,790
So this evolves as follows.

695
00:48:55,790 --> 00:49:01,640
The interaction we assume is
a product of two operators.

696
00:49:01,640 --> 00:49:04,380
One operator acts on
the system, one operator

697
00:49:04,380 --> 00:49:06,390
acts on the environment.

698
00:49:06,390 --> 00:49:10,115
So this could be the dipole
acting on the atom, the vacuum

699
00:49:10,115 --> 00:49:13,730
field, e, interacting
with the environment

700
00:49:13,730 --> 00:49:16,160
or it could be p dot a.

701
00:49:16,160 --> 00:49:20,440
Or maybe your system has
a magnetic moment, m,

702
00:49:20,440 --> 00:49:25,870
and the environment consists
of fluctuating magnetic fields.

703
00:49:25,870 --> 00:49:29,382
So we'll pretty much find
in every kind of measurement

704
00:49:29,382 --> 00:49:33,460
that a measurement involves
the product of two operators.

705
00:49:33,460 --> 00:49:36,700
One is an operator
for your system

706
00:49:36,700 --> 00:49:39,200
and one is an operator for the
reservoir of the environment.

707
00:49:44,280 --> 00:49:50,600
And so this is one
thing we want to use,

708
00:49:50,600 --> 00:49:53,700
and now there is one
thing which the moment we

709
00:49:53,700 --> 00:49:57,390
will set in our equations,
there is one thing

710
00:49:57,390 --> 00:50:01,560
which will naturally appear.

711
00:50:01,560 --> 00:50:03,414
Let me scroll back.

712
00:50:07,130 --> 00:50:10,520
What we have here is the
interaction operator, v,

713
00:50:10,520 --> 00:50:13,240
at two different times.

714
00:50:13,240 --> 00:50:16,410
So this means if something
happens at different times

715
00:50:16,410 --> 00:50:18,150
and we integrate over times.

716
00:50:18,150 --> 00:50:21,000
This is a correlation function,
a correlation function

717
00:50:21,000 --> 00:50:26,360
between v at the time t prime
and the time t double prime

718
00:50:26,360 --> 00:50:33,450
and since the reservoir
part of this interaction

719
00:50:33,450 --> 00:50:37,690
is the operator, r, so
what we have here now is,

720
00:50:37,690 --> 00:50:41,650
we have a correlation
between the operator, r,

721
00:50:41,650 --> 00:50:46,220
at two times, which
characterizes the environment.

722
00:50:46,220 --> 00:50:48,690
And now comes an
important approximation.

723
00:50:48,690 --> 00:50:54,630
You remember I said we want to
assume that the environment has

724
00:50:54,630 --> 00:50:56,600
a very short correlation time.

725
00:50:56,600 --> 00:51:01,150
Whenever a photon is emitted,
it appears dramatically fast.

726
00:51:01,150 --> 00:51:03,470
It disappears in
one optical cycle,

727
00:51:03,470 --> 00:51:05,220
and the environment
is sort of reset,

728
00:51:05,220 --> 00:51:07,500
it's back in the vacuum state.

729
00:51:07,500 --> 00:51:11,100
So this is now expressed here
that this product over which

730
00:51:11,100 --> 00:51:17,120
we take the partial trace has
a very short coherence time.

731
00:51:17,120 --> 00:51:20,900
And the fact is
now the following.

732
00:51:20,900 --> 00:51:29,380
We are integrating over a
coarse-grained step delta t,

733
00:51:29,380 --> 00:51:33,910
but this correlation function
goes to 0 in a very short time.

734
00:51:33,910 --> 00:51:39,810
So therefore it will
not contribute a lot.

735
00:51:39,810 --> 00:51:40,900
Let me write that down.

736
00:51:54,000 --> 00:52:00,080
What we are going to approximate
is our total density matrix

737
00:52:00,080 --> 00:52:08,870
is now approximately
factorizing in a density matrix

738
00:52:08,870 --> 00:52:11,420
describing the atomic system.

739
00:52:11,420 --> 00:52:14,500
Well, we describe
the atomic system

740
00:52:14,500 --> 00:52:17,570
when we trace out
the environment.

741
00:52:17,570 --> 00:52:24,190
We describe the environment when
we trace out the atomic part.

742
00:52:24,190 --> 00:52:26,820
And if we now form
the direct product,

743
00:52:26,820 --> 00:52:28,990
we are back to the
total system, but we

744
00:52:28,990 --> 00:52:32,500
have factorized the total
density matrix into two parts.

745
00:52:35,360 --> 00:52:42,140
What we neglect here is a part
which cannot be factorized

746
00:52:42,140 --> 00:52:44,700
which is the
correlated part of it.

747
00:52:47,520 --> 00:52:52,390
But what happens is, since we
are integrating over time steps

748
00:52:52,390 --> 00:52:55,790
delta t and the
correlation decay

749
00:52:55,790 --> 00:53:00,920
in a very, very short
time, the result is

750
00:53:00,920 --> 00:53:05,200
that this complicated part,
which we could never calculate,

751
00:53:05,200 --> 00:53:12,230
is smaller than the first
part by the ratio of the time

752
00:53:12,230 --> 00:53:16,590
where the correlations
contribute over the time delta

753
00:53:16,590 --> 00:53:20,100
t, the time step we
are going to take.

754
00:53:20,100 --> 00:53:23,720
So this is a very
critical assumption.

755
00:53:23,720 --> 00:53:26,900
There is a whole page
or two in the book where

756
00:53:26,900 --> 00:53:29,250
an photon-atom
interaction they discuss

757
00:53:29,250 --> 00:53:34,270
the validity of this
assumption, but I've given you

758
00:53:34,270 --> 00:53:38,810
the physical motivation that we
indicate over much larger time,

759
00:53:38,810 --> 00:53:41,330
and if this time is large
and the correlation is

760
00:53:41,330 --> 00:53:48,284
lost for short time,
they only contribute

761
00:53:48,284 --> 00:53:49,950
with this small
parameter to the result.

762
00:53:52,470 --> 00:53:56,950
So in other words,
this means after-- we

763
00:53:56,950 --> 00:53:59,750
have an interaction between
the environment and the system.

764
00:53:59,750 --> 00:54:01,690
We write it down
in second order,

765
00:54:01,690 --> 00:54:07,080
but the second
order result is now

766
00:54:07,080 --> 00:54:11,810
we evaluated by
factorizing the density

767
00:54:11,810 --> 00:54:17,180
matrix into our system
in the reservoir.

768
00:54:17,180 --> 00:54:19,810
So that means in that sense
if you factorize something,

769
00:54:19,810 --> 00:54:21,550
it looks as if it's
not interacting,

770
00:54:21,550 --> 00:54:23,050
but the trick is the same.

771
00:54:23,050 --> 00:54:25,870
You write down something
to first and second order,

772
00:54:25,870 --> 00:54:29,190
and once you have factored
out the important physics,

773
00:54:29,190 --> 00:54:31,820
now you can evaluate
the expression

774
00:54:31,820 --> 00:54:36,760
by using an approximation,
which is now the approximation

775
00:54:36,760 --> 00:54:38,910
that the density
matrix factorizes.

776
00:54:51,850 --> 00:55:00,430
So with that, this
is the approximation

777
00:55:00,430 --> 00:55:05,080
that we have made that the
correlation time is very short.

778
00:55:05,080 --> 00:55:09,170
And now we have a
differential equation

779
00:55:09,170 --> 00:55:11,870
for the density matrix
sigma, which describes

780
00:55:11,870 --> 00:55:12,960
or atomic system.

781
00:55:18,612 --> 00:55:21,170
We have traced out the
degrees of the reservoir.

782
00:55:24,120 --> 00:55:28,900
And now we want to insert B.17.

783
00:55:28,900 --> 00:55:31,000
You probably don't
remember what B.17 is.

784
00:55:31,000 --> 00:55:32,950
It says that the
interaction operator

785
00:55:32,950 --> 00:55:36,940
is a product of a, the
operator a for the atoms

786
00:55:36,940 --> 00:55:40,130
and r for the reservoir.

787
00:55:40,130 --> 00:55:44,280
So the reservoir part, tau
prime tau double prime,

788
00:55:44,280 --> 00:55:46,340
gives a correlation function.

789
00:55:46,340 --> 00:55:49,690
This is the correlation
function between the operator,

790
00:55:49,690 --> 00:55:54,290
r, at two different
times and the part

791
00:55:54,290 --> 00:55:59,009
which acts on our system, the a
part, is explicitly kept here.

792
00:56:03,320 --> 00:56:07,510
So this is now a
general master equation.

793
00:56:07,510 --> 00:56:10,200
It tells us the time
evolution of the density

794
00:56:10,200 --> 00:56:16,270
matrix in this form.

795
00:56:16,270 --> 00:56:18,370
It looks very
complicated, but this

796
00:56:18,370 --> 00:56:20,990
is because it's very general.

797
00:56:20,990 --> 00:56:31,600
In order to bring it
into an easier form,

798
00:56:31,600 --> 00:56:38,590
we want to now introduce a basis
of states, energy eigenstates

799
00:56:38,590 --> 00:56:41,960
of the unperturbed
system, and write down

800
00:56:41,960 --> 00:56:45,920
all of these operator into
such a basis of states.

801
00:56:48,960 --> 00:56:53,080
But anyway you saw here how
we had an exact equation,

802
00:56:53,080 --> 00:56:57,120
and the main
approximation we made

803
00:56:57,120 --> 00:57:01,680
is that the operator
acting on the environment

804
00:57:01,680 --> 00:57:03,460
has a very short
correlation time.

805
00:57:07,907 --> 00:57:08,490
Any questions?

806
00:57:11,676 --> 00:57:13,850
Well, you're only
a few minutes away

807
00:57:13,850 --> 00:57:17,040
from producing this result
to Fermi's golden rule, which

808
00:57:17,040 --> 00:57:18,810
you have known for
a long, long time.

809
00:57:18,810 --> 00:57:21,350
It's just we have made
very general assumptions.

810
00:57:21,350 --> 00:57:25,770
You see sort of how the
assumptions propagate,

811
00:57:25,770 --> 00:57:31,730
but now if you write it down
for an energy eigenbasis,

812
00:57:31,730 --> 00:57:35,550
you will immediately see
results you have probably

813
00:57:35,550 --> 00:57:39,310
known since your childhood.

814
00:57:39,310 --> 00:57:41,330
So we want to have
energy eigenstates

815
00:57:41,330 --> 00:57:42,940
of the atomic
operators, so this is

816
00:57:42,940 --> 00:57:44,650
sort of ground and
excited state if you

817
00:57:44,650 --> 00:57:48,060
think about a two-level system.

818
00:57:48,060 --> 00:57:53,100
The previous equation,
I have to go back to it.

819
00:58:00,520 --> 00:58:04,130
Our previous equation is
a differential equation

820
00:58:04,130 --> 00:58:08,730
for the density matrix here,
and here is the density matrix.

821
00:58:08,730 --> 00:58:12,340
So now we formulate
this equation

822
00:58:12,340 --> 00:58:16,230
into an energy eigenbasis,
and what do we get?

823
00:58:16,230 --> 00:58:18,850
Well, we get an equation
for the matrix elements,

824
00:58:18,850 --> 00:58:21,290
and what matrix
elements are important?

825
00:58:21,290 --> 00:58:24,160
diagonal matrix elements
which are population

826
00:58:24,160 --> 00:58:26,870
of diagonal matrix elements
which are coherences.

827
00:58:26,870 --> 00:58:29,190
So we pretty much
take this equation,

828
00:58:29,190 --> 00:58:31,800
use the energy
eigenbasis and look,

829
00:58:31,800 --> 00:58:34,327
what do we get for
the populations

830
00:58:34,327 --> 00:58:35,910
and what do we get
for the coherences.

831
00:58:50,450 --> 00:58:57,140
So the structure is
now the following.

832
00:58:57,140 --> 00:58:59,860
That we have our
matrix elements ab.

833
00:59:03,780 --> 00:59:09,030
There is one part which looks
like a unitary time evolution.

834
00:59:09,030 --> 00:59:11,180
This is what comes from
the Hamilton operator.

835
00:59:11,180 --> 00:59:15,420
This is sort of the-- we'll
see that in a moment--

836
00:59:15,420 --> 00:59:19,020
but this is the time
evolution without relaxation

837
00:59:19,020 --> 00:59:21,330
and now we have
something here which

838
00:59:21,330 --> 00:59:24,650
are generalized
relaxation coefficients.

839
00:59:24,650 --> 00:59:26,960
And you will find if
you go further above

840
00:59:26,960 --> 00:59:29,450
that those relaxation
coefficients are directly

841
00:59:29,450 --> 00:59:34,440
related to the correlation
function of the reservoir.

842
00:59:34,440 --> 00:59:42,260
So we can now specify what
happens between populations.

843
00:59:42,260 --> 00:59:45,580
Population means that we
have a differential equation,

844
00:59:45,580 --> 00:59:48,770
let's say between
sigma aa the puller,

845
00:59:48,770 --> 01:00:01,410
and sigma cc, so we
have a rate coefficient

846
01:00:01,410 --> 01:00:04,900
which connects the
population in state A

847
01:00:04,900 --> 01:00:10,370
with the population in state C.

848
01:00:10,370 --> 01:00:14,910
And if you take this expression,
you find several things.

849
01:00:14,910 --> 01:00:18,180
Well, you find
Fermi's golden rule,

850
01:00:18,180 --> 01:00:21,510
in a generalized
way-- that's always

851
01:00:21,510 --> 01:00:25,590
nice-- you find
Fermi's golden rule.

852
01:00:25,590 --> 01:00:29,530
When you integrate over time,
you often get data function,

853
01:00:29,530 --> 01:00:32,200
and you expect to get a
delta function because

854
01:00:32,200 --> 01:00:34,040
of energy conservation.

855
01:00:34,040 --> 01:00:36,710
So you get that, of
course, naturally.

856
01:00:36,710 --> 01:00:42,040
Secondly, we have second
order matrix element,

857
01:00:42,040 --> 01:00:44,920
which you know from
Fermi's golden rule,

858
01:00:44,920 --> 01:00:47,710
but now we have the
following situation

859
01:00:47,710 --> 01:00:51,770
that the matrix element
in Fermi's golden rule

860
01:00:51,770 --> 01:00:57,190
may actually depend on the
state, mu, of the environment.

861
01:00:57,190 --> 01:00:59,360
So you have maybe 10
different possibilities

862
01:00:59,360 --> 01:01:01,550
for the environment,
and Fermi's golden rule

863
01:01:01,550 --> 01:01:03,610
gives you spontaneous
emission, which

864
01:01:03,610 --> 01:01:05,810
is different for
those 10 states.

865
01:01:05,810 --> 01:01:08,940
And naturally, since we have
performed the partial trace

866
01:01:08,940 --> 01:01:12,000
over the environment,
we have all those rates

867
01:01:12,000 --> 01:01:14,460
weighted with the probability
that the environment

868
01:01:14,460 --> 01:01:16,850
is in one of those states.

869
01:01:16,850 --> 01:01:20,560
So what you find here is
a simple generalization

870
01:01:20,560 --> 01:01:21,950
of Fermi's golden rule.

871
01:01:26,330 --> 01:01:36,980
And if you look at the
off-diagonal matrix elements,

872
01:01:36,980 --> 01:01:39,260
for instance, you
want to know what

873
01:01:39,260 --> 01:01:44,540
is this rate, what is the rate
coefficient, which gives you

874
01:01:44,540 --> 01:01:48,130
the time derivative
of the coherence,

875
01:01:48,130 --> 01:01:51,360
and it's multiplied
with the coherence.

876
01:01:51,360 --> 01:01:56,640
You'll find now that in general,
this rate coefficient has

877
01:01:56,640 --> 01:02:01,640
a damping term, but it may
also have an imaginary term.

878
01:02:01,640 --> 01:02:05,320
And I hope you remember when we
played with diagrams, that we

879
01:02:05,320 --> 01:02:07,040
had something similar.

880
01:02:07,040 --> 01:02:11,300
There was something which we
called the radiative shift.

881
01:02:11,300 --> 01:02:14,800
I called it the AC stock
effect of a single photon,

882
01:02:14,800 --> 01:02:17,170
and here it is a
level shift which

883
01:02:17,170 --> 01:02:19,940
comes because the environment
interacts with your system

884
01:02:19,940 --> 01:02:21,960
and it shifts the
levels a little bit.

885
01:02:21,960 --> 01:02:23,940
So in addition to
just relaxation,

886
01:02:23,940 --> 01:02:25,950
spontaneous emission,
and damping,

887
01:02:25,950 --> 01:02:29,940
there is also a dispersive
part, a level shift,

888
01:02:29,940 --> 01:02:35,950
and it has exactly
the same structure.

889
01:02:35,950 --> 01:02:40,240
Let me add that delta
ab is the difference

890
01:02:40,240 --> 01:02:46,470
between the shift of state
b and the shift of state a.

891
01:02:46,470 --> 01:02:51,100
And those shifts have
exactly the same structure.

892
01:02:51,100 --> 01:02:55,010
You have to take the principle
part of something which

893
01:02:55,010 --> 01:02:58,070
has 1 over the
difference of energies,

894
01:02:58,070 --> 01:03:02,390
and we discussed that
this has to be understood

895
01:03:02,390 --> 01:03:07,930
by somewhere adding an
infinitesimal imaginary part

896
01:03:07,930 --> 01:03:10,120
and doing the right thing
with complex function.

897
01:03:10,120 --> 01:03:13,360
It's actually related to
Laplace time difference

898
01:03:13,360 --> 01:03:17,590
between Laplace transformation
and Fourier transformation.

899
01:03:17,590 --> 01:03:21,110
So anyway what I find
sort of beautiful

900
01:03:21,110 --> 01:03:23,780
is that we started with
a most general situation.

901
01:03:23,780 --> 01:03:25,400
We perform the partial trace.

902
01:03:25,400 --> 01:03:28,190
We made one assumption of
short correlation times,

903
01:03:28,190 --> 01:03:31,880
and a lot of things we have
known about quantum system

904
01:03:31,880 --> 01:03:34,310
just pops out in a
very general form here.

905
01:03:37,060 --> 01:03:39,130
Any questions about so far?

906
01:03:49,280 --> 01:03:55,160
Well, the coherences are,
of course, more interesting

907
01:03:55,160 --> 01:03:56,140
than the population.

908
01:03:56,140 --> 01:03:59,780
Coherence is always something
physicists get excited about it

909
01:03:59,780 --> 01:04:03,000
because it captures
something which goes often

910
01:04:03,000 --> 01:04:08,200
beyond classical
system that we have

911
01:04:08,200 --> 01:04:10,170
quantum mechanical coherences.

912
01:04:10,170 --> 01:04:14,610
And what happens is the
coefficient here, which

913
01:04:14,610 --> 01:04:18,700
provides the damping
of the coherence,

914
01:04:18,700 --> 01:04:22,460
just comes out of the
formalism, has two parts.

915
01:04:22,460 --> 01:04:34,040
And one part is
an adiabatic part

916
01:04:34,040 --> 01:04:38,090
and the other one is
a non-adiabatic part.

917
01:04:38,090 --> 01:04:39,940
Well, and that makes sense.

918
01:04:39,940 --> 01:04:42,710
If you have two quantum
states and there

919
01:04:42,710 --> 01:04:45,886
is a coherence, some
phase between the two,

920
01:04:45,886 --> 01:04:49,750
the phase can get lost if you
do a transition between the two

921
01:04:49,750 --> 01:04:53,890
states or one state undergoes
a collision and is quenched.

922
01:04:53,890 --> 01:04:58,480
So you definitely have one
part which is due to the fact

923
01:04:58,480 --> 01:05:03,760
that the quantum states
or the population

924
01:05:03,760 --> 01:05:08,660
changes, and you find that there
is this state-changing part,

925
01:05:08,660 --> 01:05:11,760
which is pretty much
the sum of all the rate

926
01:05:11,760 --> 01:05:14,030
coefficient leading
out of state,

927
01:05:14,030 --> 01:05:16,310
leading to the decay
of state a and leading

928
01:05:16,310 --> 01:05:18,810
to the decay of state b.

929
01:05:18,810 --> 01:05:21,560
In other words, if you have
a two-level system, which

930
01:05:21,560 --> 01:05:29,510
has a coherence and you
have decay of the excited

931
01:05:29,510 --> 01:05:31,640
state and decay of
the ground state,

932
01:05:31,640 --> 01:05:36,430
you would expect that those
decay terms appear also

933
01:05:36,430 --> 01:05:39,950
in the decay of the coherence
between the two levels

934
01:05:39,950 --> 01:05:46,070
and they do, and they appear
with the correct factor of 1/2.

935
01:05:46,070 --> 01:05:49,580
But there is another possibility
and this is the following.

936
01:05:49,580 --> 01:05:54,540
You can have no [INAUDIBLE] of
the population of the state,

937
01:05:54,540 --> 01:05:57,450
but you can still
lose the coherence.

938
01:05:57,450 --> 01:05:59,360
The model you
should maybe make is

939
01:05:59,360 --> 01:06:01,500
that you have spin
up, spin down.

940
01:06:01,500 --> 01:06:03,920
You are not perturbing
the populations

941
01:06:03,920 --> 01:06:06,650
in spin up and spin
down, but the environment

942
01:06:06,650 --> 01:06:09,930
provides fluctuating
magnetic field.

943
01:06:09,930 --> 01:06:13,680
Then due to the
fluctuating magnetic field,

944
01:06:13,680 --> 01:06:16,240
you no longer can keep
track of the phase,

945
01:06:16,240 --> 01:06:18,430
and that means in
your identity matrix

946
01:06:18,430 --> 01:06:21,150
the off-diagonal
matrix elements decay.

947
01:06:21,150 --> 01:06:25,670
And we find that here this
is the second part which

948
01:06:25,670 --> 01:06:28,840
in this book is called
the adiabatic part,

949
01:06:28,840 --> 01:06:32,090
and the physics behind it
is now pure de-phasing.

950
01:06:35,210 --> 01:06:37,880
So it's an independent
way for coherences

951
01:06:37,880 --> 01:06:42,275
to decay independent of the
decay of the population.

952
01:06:56,374 --> 01:06:56,873
Questions?

953
01:06:59,900 --> 01:07:00,400
Collin.

954
01:07:00,400 --> 01:07:03,700
AUDIENCE: Where does the
Markov approximation come in?

955
01:07:03,700 --> 01:07:06,730
PROFESSOR: The Markov
approximation is, so to speak,

956
01:07:06,730 --> 01:07:10,040
the delta function
approximation,

957
01:07:10,040 --> 01:07:14,740
which would say that-- I mean,
I introduced the correlation

958
01:07:14,740 --> 01:07:16,710
function between the
reservoir operator

959
01:07:16,710 --> 01:07:20,320
and said the correlation time
tau c is very, very short.

960
01:07:20,320 --> 01:07:22,050
The Markov approximation
would actually

961
01:07:22,050 --> 01:07:25,670
state that it would actually
say in a more radical way

962
01:07:25,670 --> 01:07:29,360
the correlation time is 0.

963
01:07:29,360 --> 01:07:32,440
And the Bohr
approximation, the fact

964
01:07:32,440 --> 01:07:35,940
that the reservoir
is unchanged came

965
01:07:35,940 --> 01:07:39,190
in when we said
the total density

966
01:07:39,190 --> 01:07:43,320
matrix for the second order
expression just factorizes.

967
01:07:43,320 --> 01:07:45,690
It factorizes into
the environment,

968
01:07:45,690 --> 01:07:48,850
which is just this density
matrix of the environment.

969
01:07:48,850 --> 01:07:52,100
It's not changed by the
interaction with the system.

970
01:07:52,100 --> 01:07:53,820
And this is the
Bohr approximation.

971
01:07:53,820 --> 01:07:58,650
We just use the same expression
for the reservoir independent

972
01:07:58,650 --> 01:08:00,610
of the measurements
the reservoir has done.

973
01:08:04,340 --> 01:08:06,320
Other questions?

974
01:08:06,320 --> 01:08:06,834
Yes, Nancy.

975
01:08:06,834 --> 01:08:07,750
AUDIENCE: [INAUDIBLE].

976
01:08:12,760 --> 01:08:15,020
PROFESSOR: This is
something very general.

977
01:08:15,020 --> 01:08:16,990
Thank you actually
for the question.

978
01:08:16,990 --> 01:08:30,140
Whenever we have some
damping of the population,

979
01:08:30,140 --> 01:08:33,170
the coherence is only
damped with a factor of 1/2.

980
01:08:37,740 --> 01:08:40,479
One way to explain it
in a very simple way

981
01:08:40,479 --> 01:08:57,750
is, that if you have an
amplitude alpha excited

982
01:08:57,750 --> 01:09:06,260
and alpha ground, the
population in the excited state

983
01:09:06,260 --> 01:09:09,180
is this squared.

984
01:09:09,180 --> 01:09:13,470
So you sometimes make the
model that the amplitude decays

985
01:09:13,470 --> 01:09:17,319
with gamma over 2,
but the population

986
01:09:17,319 --> 01:09:22,290
is-- because you take the
product, decays with gamma.

987
01:09:22,290 --> 01:09:27,580
So you would say alpha
e and alpha g both decay

988
01:09:27,580 --> 01:09:30,200
with half the rate,
but the probability

989
01:09:30,200 --> 01:09:33,689
is for the total rate,
and the coherence

990
01:09:33,689 --> 01:09:37,109
is the product of
the amplitudes.

991
01:09:37,109 --> 01:09:39,750
So therefore when you
look at the coherence,

992
01:09:39,750 --> 01:09:42,680
this decays with 1/2 gamma e.

993
01:09:42,680 --> 01:09:46,350
This decays with 1/2 gamma g,
and this is what you get here,

994
01:09:46,350 --> 01:09:51,120
1/2 gamma in state a,
1/2 gamma in state b.

995
01:09:51,120 --> 01:09:54,160
Whereas the probability
to be in this state

996
01:09:54,160 --> 01:09:59,370
decays with twice that because
the probability is squared.

997
01:09:59,370 --> 01:10:03,190
So you find that pretty much
in any quantum mechanically

998
01:10:03,190 --> 01:10:05,420
corrects derivation,
which you do

999
01:10:05,420 --> 01:10:07,515
about the decay of
population coherences.

1000
01:10:26,654 --> 01:10:27,320
Other questions?

1001
01:10:32,230 --> 01:10:33,590
So we have done two things.

1002
01:10:33,590 --> 01:10:36,580
We have done the very,
very simple derivation

1003
01:10:36,580 --> 01:10:39,440
using the beam splitter
model where you may not even

1004
01:10:39,440 --> 01:10:42,680
notice where I did the Markov
approximation because I jumped

1005
01:10:42,680 --> 01:10:44,320
from beam splitter
to beam splitter

1006
01:10:44,320 --> 01:10:46,890
and left all the
correlations behind.

1007
01:10:46,890 --> 01:10:49,940
Here in the most
general calculation,

1008
01:10:49,940 --> 01:10:52,720
you have seen exactly
where it enters,

1009
01:10:52,720 --> 01:10:56,140
but maybe now you have the
full forest in front of you,

1010
01:10:56,140 --> 01:10:58,480
and you don't recognize
the trees anymore.

1011
01:10:58,480 --> 01:11:02,910
So let me wrap up this
lecture by now focusing

1012
01:11:02,910 --> 01:11:06,550
on the system we want
to discuss further on,

1013
01:11:06,550 --> 01:11:13,930
namely a two-level system
interacting with a vacuum

1014
01:11:13,930 --> 01:11:15,690
through spontaneous emission.

1015
01:11:15,690 --> 01:11:19,590
But I also want to make
some generalizations.

1016
01:11:19,590 --> 01:11:27,040
I want to give you some
generalizations about what

1017
01:11:27,040 --> 01:11:30,385
kind of environments are
possible in quantum physics.

1018
01:11:34,150 --> 01:11:38,480
Let me just see how I do that.

1019
01:11:41,020 --> 01:11:45,960
So this part I actually owe to
Professor [? Ikschuan ?] who

1020
01:11:45,960 --> 01:11:48,430
wonderfully compiled that.

1021
01:11:48,430 --> 01:11:50,740
So what I want to
do now is, I want

1022
01:11:50,740 --> 01:11:53,640
to call your attention
to the operator

1023
01:11:53,640 --> 01:11:56,550
form which, is rather unique.

1024
01:11:56,550 --> 01:12:01,680
Remember when we did second
order perturbation theory,

1025
01:12:01,680 --> 01:12:08,350
we had sort of the
commutator of v

1026
01:12:08,350 --> 01:12:11,020
with the commutator
of v and rho.

1027
01:12:11,020 --> 01:12:15,670
This came from iterating
the exact equation of motion

1028
01:12:15,670 --> 01:12:17,890
for the density matrix.

1029
01:12:17,890 --> 01:12:21,567
And you want to specialize that
now to Jaynes-Cummings model.

1030
01:12:21,567 --> 01:12:23,400
I mean, in the end, at
least in this course,

1031
01:12:23,400 --> 01:12:25,400
we always come back to
the Jaynes-Cummings model

1032
01:12:25,400 --> 01:12:29,360
because it captures a lot
of what we want to explore.

1033
01:12:29,360 --> 01:12:33,120
So the Jaynes-Cummings model in
the rotating wave approximation

1034
01:12:33,120 --> 01:12:34,360
is very simple.

1035
01:12:34,360 --> 01:12:38,160
It raises the atom from
ground to excited state

1036
01:12:38,160 --> 01:12:41,680
and destroys a photon,
or it does the opposite.

1037
01:12:41,680 --> 01:12:45,280
So this is our
simple interaction

1038
01:12:45,280 --> 01:12:48,390
between our system,
the two-level atom

1039
01:12:48,390 --> 01:12:53,210
and our reservoir which is just
the vacuum of all the modes.

1040
01:12:53,210 --> 01:12:56,680
And you, again, recognize
what I said in general.

1041
01:12:56,680 --> 01:12:59,380
You usually always
find it by linear form,

1042
01:12:59,380 --> 01:13:02,580
an operator which
acts on the modes,

1043
01:13:02,580 --> 01:13:05,770
on the reservoir on the
vacuum, and an operator which

1044
01:13:05,770 --> 01:13:06,628
acts on the vacuum.

1045
01:13:10,090 --> 01:13:13,990
Now, we want to make
the explicit assumption

1046
01:13:13,990 --> 01:13:18,291
that the initial state of the
reservoir is the vacuum state.

1047
01:13:18,291 --> 01:13:18,790
It's empty.

1048
01:13:21,800 --> 01:13:25,755
And I want to show you
what is the structure

1049
01:13:25,755 --> 01:13:29,540
of the operators we obtain.

1050
01:13:29,540 --> 01:13:32,600
And so if you put
v into here, you

1051
01:13:32,600 --> 01:13:37,100
have first the commutator with
rho, which I write down here,

1052
01:13:37,100 --> 01:13:40,790
and then we have to take
another commutator with v.

1053
01:13:40,790 --> 01:13:47,680
And the result of
that is the following,

1054
01:13:47,680 --> 01:13:52,090
that when it comes
to relaxation pauses,

1055
01:13:52,090 --> 01:13:55,940
based on the general
structure of the time

1056
01:13:55,940 --> 01:14:01,450
evolution of quantum mechanics,
we have this double commutator.

1057
01:14:01,450 --> 01:14:16,180
And the operator which, couples
our system to the environment

1058
01:14:16,180 --> 01:14:18,740
is erasing and
lower an operator.

1059
01:14:18,740 --> 01:14:21,560
I mean, the atom because it
interacts with the environment

1060
01:14:21,560 --> 01:14:24,820
either absolves the photon
or emits the photon.

1061
01:14:24,820 --> 01:14:29,420
But those operators, sigma
plus and sigma minus,

1062
01:14:29,420 --> 01:14:34,100
appear now always as
products because we

1063
01:14:34,100 --> 01:14:35,860
have two occurrences
of the interaction,

1064
01:14:35,860 --> 01:14:41,410
v. But if you look at the
double commutator structure,

1065
01:14:41,410 --> 01:14:44,790
the operator sigma plus
sigma minus appears.

1066
01:14:44,790 --> 01:14:46,370
This is just the
general structure

1067
01:14:46,370 --> 01:14:48,900
of this double
commutator appears

1068
01:14:48,900 --> 01:14:51,930
to the left side of the
atomic density matrix

1069
01:14:51,930 --> 01:14:54,390
to the right side of the
atomic density matrix

1070
01:14:54,390 --> 01:14:57,430
and then there is the coarse
term where the atomic density

1071
01:14:57,430 --> 01:14:59,210
matrix is in the
middle of the two.

1072
01:15:09,230 --> 01:15:11,770
So this is actually
something which

1073
01:15:11,770 --> 01:15:15,540
is very general
and very important

1074
01:15:15,540 --> 01:15:19,550
in the theory of
open quantum system.

1075
01:15:19,550 --> 01:15:26,050
What I'm discussing with you now
is this famous Lindblad form.

1076
01:15:26,050 --> 01:15:30,030
And the story goes like that.

1077
01:15:30,030 --> 01:15:34,960
You want to know what are
possible environments, not just

1078
01:15:34,960 --> 01:15:35,587
empty vacuum.

1079
01:15:35,587 --> 01:15:36,920
You can have fluctuating fields.

1080
01:15:36,920 --> 01:15:39,320
You can have, you name it.

1081
01:15:39,320 --> 01:15:44,470
But if you are saying that
your environment interacts

1082
01:15:44,470 --> 01:15:47,330
with your system
through an operator,

1083
01:15:47,330 --> 01:15:53,130
and our operator is now the
operator sigma minus, which

1084
01:15:53,130 --> 01:15:58,800
is a spontaneous emission,
you need heavy t.

1085
01:15:58,800 --> 01:16:02,740
The mathematical structure
of a valid quantum field

1086
01:16:02,740 --> 01:16:06,490
remains that if your system
interacts with an environment

1087
01:16:06,490 --> 01:16:09,280
by emitting a
photon sigma minus,

1088
01:16:09,280 --> 01:16:14,050
this is now the structure
of the master equation.

1089
01:16:14,050 --> 01:16:16,910
This is the structure of the
time evolution of the density

1090
01:16:16,910 --> 01:16:17,730
matrix.

1091
01:16:17,730 --> 01:16:21,120
So the operator
sigma minus and its

1092
01:16:21,120 --> 01:16:23,920
and its Hermitian conjugate
sigma plus, all of this

1093
01:16:23,920 --> 01:16:28,330
have to appear in
this combination.

1094
01:16:28,330 --> 01:16:28,830
Yes, Collin.

1095
01:16:28,830 --> 01:16:31,716
AUDIENCE: This is still
in my interaction picture.

1096
01:16:31,716 --> 01:16:32,216
Right?

1097
01:16:32,216 --> 01:16:34,696
There's no dynamical
phase evolution

1098
01:16:34,696 --> 01:16:38,144
that we put back in there.

1099
01:16:38,144 --> 01:16:38,810
PROFESSOR: Yeah.

1100
01:16:38,810 --> 01:16:39,570
OK.

1101
01:16:39,570 --> 01:16:43,460
What we do in general if the
system is driven by a laser

1102
01:16:43,460 --> 01:16:46,230
beam, for instance,
Rabi oscillation,

1103
01:16:46,230 --> 01:16:49,460
we simply add up the dynamics
of the Rabi oscillation

1104
01:16:49,460 --> 01:16:54,294
of the unitary time evolution
to the time evolution

1105
01:16:54,294 --> 01:16:55,210
done by the reservoir.

1106
01:16:55,210 --> 01:16:58,654
AUDIENCE: So this form is always
in the interaction picture?

1107
01:16:58,654 --> 01:17:00,340
PROFESSOR: Well, this
is, you would say,

1108
01:17:00,340 --> 01:17:04,540
this form is what is
the relaxation provided

1109
01:17:04,540 --> 01:17:05,890
by the environment.

1110
01:17:05,890 --> 01:17:07,980
And if you drive the
system in addition

1111
01:17:07,980 --> 01:17:11,450
with the coherent field,
unitary time evolution,

1112
01:17:11,450 --> 01:17:15,250
you would add it to it.

1113
01:17:15,250 --> 01:17:17,970
So in other words,
what I'm telling

1114
01:17:17,970 --> 01:17:21,340
you here is that this is
the general structure,

1115
01:17:21,340 --> 01:17:23,400
and if you have a
system which interacts

1116
01:17:23,400 --> 01:17:26,180
with an environment in
five different ways,

1117
01:17:26,180 --> 01:17:29,090
with a dipole wand, with a
magnetic moment and such,

1118
01:17:29,090 --> 01:17:32,320
you have maybe five
interaction terms

1119
01:17:32,320 --> 01:17:35,920
and then you have to perform
the sum over five operators,

1120
01:17:35,920 --> 01:17:39,150
and here one of them is
a sigma minus operator.

1121
01:17:39,150 --> 01:17:42,410
So in other words,
if you want to know

1122
01:17:42,410 --> 01:17:46,550
what is the whole world of
possibilities for quantum

1123
01:17:46,550 --> 01:17:49,950
system to relax and dissipate
with an environment,

1124
01:17:49,950 --> 01:17:52,610
you can pretty much
take any operator which

1125
01:17:52,610 --> 01:17:55,990
acts on your system,
but then put it

1126
01:17:55,990 --> 01:17:59,060
into this so-called
Lindblad form

1127
01:17:59,060 --> 01:18:01,380
and you have a
possible environment.

1128
01:18:04,560 --> 01:18:07,410
And I mentioned
I think last week

1129
01:18:07,410 --> 01:18:11,060
that people are now in
our field actively working

1130
01:18:11,060 --> 01:18:13,260
on environmental engineering.

1131
01:18:13,260 --> 01:18:16,870
They want to expose a system
to an artificial environment

1132
01:18:16,870 --> 01:18:20,070
and hope the system is
not relaxing, let's say,

1133
01:18:20,070 --> 01:18:23,690
to a broken ground state, but
to a fancy correlated state.

1134
01:18:27,500 --> 01:18:32,160
So what this Lindblad form,
if the operators appears

1135
01:18:32,160 --> 01:18:37,770
in this way, what it
insures is the following.

1136
01:18:37,770 --> 01:18:40,800
Just imagine if we
have an equation,

1137
01:18:40,800 --> 01:18:43,020
a derivative of
the density matrix,

1138
01:18:43,020 --> 01:18:45,310
which depends on
the density matrix,

1139
01:18:45,310 --> 01:18:48,080
you could write down a
differential equation and say

1140
01:18:48,080 --> 01:18:49,430
is it possible?

1141
01:18:49,430 --> 01:18:52,240
Well, it has to be consistent
with quantum physics.

1142
01:18:52,240 --> 01:18:53,980
You have certain requirements.

1143
01:18:53,980 --> 01:18:58,660
One requirement is that
rho, the density matrix,

1144
01:18:58,660 --> 01:19:01,870
always has to be
the density matrix.

1145
01:19:01,870 --> 01:19:04,270
The trace equal 1
has to be conserved.

1146
01:19:04,270 --> 01:19:09,190
A density matrix always must
have non-negative eigenvalues,

1147
01:19:09,190 --> 01:19:10,740
otherwise, what
you write down, it

1148
01:19:10,740 --> 01:19:13,200
might be a nice differential
equation, but quantum

1149
01:19:13,200 --> 01:19:15,290
mechanically, it's nonsense.

1150
01:19:15,290 --> 01:19:20,940
But now there is one more
thing, which is also necessary.

1151
01:19:20,940 --> 01:19:26,380
This time evolution of the
system's density matrix

1152
01:19:26,380 --> 01:19:31,370
must come from a unitary time
evolution of a bigger system.

1153
01:19:31,370 --> 01:19:35,190
So you must be able to extend
your system into a bigger

1154
01:19:35,190 --> 01:19:36,890
system, which is
now the environment,

1155
01:19:36,890 --> 01:19:40,710
and this whole system must
follow a Schrodinger equation

1156
01:19:40,710 --> 01:19:43,760
with a Hamilton as a
unitary time evolution.

1157
01:19:43,760 --> 01:19:46,030
And this is where
it's restrictive.

1158
01:19:46,030 --> 01:19:49,050
You cannot just write down a
differential equation and hope

1159
01:19:49,050 --> 01:19:51,140
that this will fill
some requirement,

1160
01:19:51,140 --> 01:19:56,350
and what people have shown is
under very general assumptions,

1161
01:19:56,350 --> 01:20:01,320
it is the Lindblad form
which allows for it.

1162
01:20:01,320 --> 01:20:05,360
So some operator always
has to appear in this form.

1163
01:20:08,120 --> 01:20:13,310
So often in this
Lindblad form, you

1164
01:20:13,310 --> 01:20:18,070
have an operator, which is
called a jump operator, which

1165
01:20:18,070 --> 01:20:20,390
is responsible for
the measurement, which

1166
01:20:20,390 --> 01:20:22,610
the environment
does on your system.

1167
01:20:22,610 --> 01:20:24,820
The jump operator is
here, the operator

1168
01:20:24,820 --> 01:20:28,740
which takes the atom from the
excited to the ground state.

1169
01:20:28,740 --> 01:20:32,480
With that you need a photon,
and the photon can be measured.

1170
01:20:32,480 --> 01:20:37,910
So often you can describe a
system by a jump operator,

1171
01:20:37,910 --> 01:20:41,130
and if the jump operator is
put into this Lindblad form,

1172
01:20:41,130 --> 01:20:49,760
then you have a valid master
equation for your system.

1173
01:20:52,270 --> 01:20:53,610
So let me wrap up.

1174
01:21:05,924 --> 01:21:10,320
If you take now the definition
of the raising and lowering

1175
01:21:10,320 --> 01:21:14,540
operator, and you take the
form I showed you, the Lindblad

1176
01:21:14,540 --> 01:21:21,052
form, you'll find now
this differential equation

1177
01:21:21,052 --> 01:21:22,135
for your two-level system.

1178
01:21:24,900 --> 01:21:27,930
And this is one part of
the Optical Bloch Equation.

1179
01:21:27,930 --> 01:21:30,420
Now, coming to
Collin's question,

1180
01:21:30,420 --> 01:21:32,280
if you include
the time evolution

1181
01:21:32,280 --> 01:21:35,600
of the classical field,
this is a coherent evolution

1182
01:21:35,600 --> 01:21:37,610
of the Bloch vector,
which I showed

1183
01:21:37,610 --> 01:21:39,790
at the beginning of
the class, and we

1184
01:21:39,790 --> 01:21:44,620
add this and what I wrote down
at the beginning of the class.

1185
01:21:44,620 --> 01:21:48,792
Then we find the famous
Optical Bloch Equations

1186
01:21:48,792 --> 01:21:50,000
in the Jaynes-Cummings model.

1187
01:21:55,610 --> 01:22:00,730
So these are now the
Optical Bloch Equations,

1188
01:22:00,730 --> 01:22:03,760
and I hope you enjoy now after
this complicated discussion,

1189
01:22:03,760 --> 01:22:04,910
how simple they are.

1190
01:22:04,910 --> 01:22:07,700
And it is this simple
set of equations,

1191
01:22:07,700 --> 01:22:10,650
which will be used in
the rest of the course

1192
01:22:10,650 --> 01:22:13,480
to describe the time
evolution of the system.

1193
01:22:20,430 --> 01:22:22,630
Just because I did
some generalizations

1194
01:22:22,630 --> 01:22:26,020
about the Lindblad equation,
I copied that into the lecture

1195
01:22:26,020 --> 01:22:29,230
notes from Wikipedia,
and probably now you

1196
01:22:29,230 --> 01:22:33,680
sort of understand what is the
most general Lindblad equation.

1197
01:22:33,680 --> 01:22:36,760
It has a Hamiltonian
part and then it

1198
01:22:36,760 --> 01:22:41,070
has jump operators like
you sigma minus operator,

1199
01:22:41,070 --> 01:22:45,100
but it has to come in
the form that the jump

1200
01:22:45,100 --> 01:22:49,130
operator and its complex
conjugate, emission. conjugate,

1201
01:22:49,130 --> 01:22:53,310
is on the left side, on the
right side, and left and right

1202
01:22:53,310 --> 01:22:55,130
of your density matrix.

1203
01:22:55,130 --> 01:22:57,210
So this is the generalization
I've mentioned.

1204
01:23:02,980 --> 01:23:05,720
Yeah. with that I think with
that we've derived the master

1205
01:23:05,720 --> 01:23:09,250
equation, and on
Wednesday, we will

1206
01:23:09,250 --> 01:23:13,270
look at rather simple solutions,
transient and steady-state

1207
01:23:13,270 --> 01:23:16,830
solution of the
Optical Bloch Equation.

1208
01:23:16,830 --> 01:23:20,110
Any questions?

1209
01:23:20,110 --> 01:23:22,410
One reminder about the
schedule, this week

1210
01:23:22,410 --> 01:23:25,300
we have a lecture on
Friday because I will not

1211
01:23:25,300 --> 01:23:28,120
be on town next
week on Wednesday.

1212
01:23:28,120 --> 01:23:31,700
And of course, you know today
in a week, next week on Monday.

1213
01:23:31,700 --> 01:23:32,890
It's [INAUDIBLE] day.

1214
01:23:32,890 --> 01:23:35,260
So we have three
classes this week,

1215
01:23:35,260 --> 01:23:38,890
no class the following week,
and then the normal schedule

1216
01:23:38,890 --> 01:23:41,040
for the rest of the semester.