1
00:00:00,060 --> 00:00:01,780
The following
content is provided

2
00:00:01,780 --> 00:00:04,030
under a Creative
Commons license.

3
00:00:04,030 --> 00:00:06,880
Your support will help MIT
OpenCourseWare continue

4
00:00:06,880 --> 00:00:10,740
to offer high quality
educational resources for free.

5
00:00:10,740 --> 00:00:13,350
To make a donation, or
view additional materials

6
00:00:13,350 --> 00:00:17,237
from hundreds of MIT courses,
visit MIT OpenCourseWare

7
00:00:17,237 --> 00:00:17,862
at ocw.mit.edu.

8
00:00:26,650 --> 00:00:28,910
PROFESSOR: I apologize
for the delay.

9
00:00:28,910 --> 00:00:32,080
But we are ready
for multimedia show

10
00:00:32,080 --> 00:00:34,030
with three different
applications.

11
00:00:34,030 --> 00:00:35,260
One of them are the clickers.

12
00:00:35,260 --> 00:00:37,590
I hope you all took a clicker.

13
00:00:37,590 --> 00:00:40,848
It's just two questions today,
but really good questions.

14
00:00:40,848 --> 00:00:42,640
[LAUGHTER]

15
00:00:42,640 --> 00:00:46,337
So today's actually
very nice topic.

16
00:00:46,337 --> 00:00:48,420
I think almost every lecture
we have a nice topic,

17
00:00:48,420 --> 00:00:50,570
but today's especially nice.

18
00:00:50,570 --> 00:00:53,770
It's called Unraveling
Open Quantum System.

19
00:00:53,770 --> 00:00:56,360
That may not tell you so much.

20
00:00:56,360 --> 00:00:59,310
Well unravelling means we
want to sort of untangle it,

21
00:00:59,310 --> 00:01:01,850
We want to understand
what is inside.

22
00:01:01,850 --> 00:01:05,349
And so we want to develop
a better understanding what

23
00:01:05,349 --> 00:01:08,260
is inside a density matrix.

24
00:01:08,260 --> 00:01:10,710
How can we think about
the quantum system

25
00:01:10,710 --> 00:01:13,930
in an even more microscopic
way than the density

26
00:01:13,930 --> 00:01:16,390
matrix tell us?

27
00:01:16,390 --> 00:01:20,050
And again, this finds
applications in atomic physics,

28
00:01:20,050 --> 00:01:24,275
because one application of it
is spontaneous emission optical

29
00:01:24,275 --> 00:01:25,540
block equations.

30
00:01:25,540 --> 00:01:27,910
On the other hand,
actually, this class

31
00:01:27,910 --> 00:01:29,760
will also teach you
something very general

32
00:01:29,760 --> 00:01:33,720
about quantum physics.

33
00:01:33,720 --> 00:01:41,440
So today's lecture is this part.

34
00:01:41,440 --> 00:01:44,380
We'll have three topics.

35
00:01:44,380 --> 00:01:51,360
One is I want to provide a
motivation for single quantum

36
00:01:51,360 --> 00:01:53,440
systems.

37
00:01:53,440 --> 00:01:54,980
And you should appreciate this.

38
00:01:54,980 --> 00:01:59,250
A single quantum system is not
what Schrodinger and Heisenberg

39
00:01:59,250 --> 00:02:00,620
told us about.

40
00:02:00,620 --> 00:02:03,866
Quantum mechanics was formulated
as an ensemble average.

41
00:02:03,866 --> 00:02:10,240
But today we want to talk about,
is there something in addition?

42
00:02:10,240 --> 00:02:13,060
About single quantum systems.

43
00:02:13,060 --> 00:02:19,730
Part two is-- addresses the
derivation of the quantum Monte

44
00:02:19,730 --> 00:02:22,220
Carlo wave function method.

45
00:02:22,220 --> 00:02:24,650
And this is a formalism,
which is really

46
00:02:24,650 --> 00:02:28,250
simulating in quite general
terms, quantum physics

47
00:02:28,250 --> 00:02:31,630
by following-- using quantum
Monte Carlo simulations--

48
00:02:31,630 --> 00:02:34,250
trajectories of individual
quantum systems.

49
00:02:34,250 --> 00:02:36,990
So here the individual
quantum system--

50
00:02:36,990 --> 00:02:39,960
even if you have one, even
if you don't have one--

51
00:02:39,960 --> 00:02:43,120
becomes a computational tool.

52
00:02:43,120 --> 00:02:46,080
And eventually in
part two I want

53
00:02:46,080 --> 00:02:49,620
to give you another example
for unraveling open quantum

54
00:02:49,620 --> 00:02:56,540
system, which differ from
the models for dephasing.

55
00:02:56,540 --> 00:03:00,580
And if you remember that this
was on one of your homework

56
00:03:00,580 --> 00:03:02,000
assignment that's correct.

57
00:03:02,000 --> 00:03:04,210
I sort of want to here
take your homework

58
00:03:04,210 --> 00:03:08,100
into the classroom discussion.

59
00:03:08,100 --> 00:03:11,070
Well I think it's nice to start
this lecture with the quote

60
00:03:11,070 --> 00:03:13,832
by Erwin Schrodinger.

61
00:03:13,832 --> 00:03:16,550
Where he said, "We
never experiment

62
00:03:16,550 --> 00:03:19,370
with just one electron,
or atom, or molecule.

63
00:03:19,370 --> 00:03:23,600
In thought experiments, we
sometimes assume that we do;

64
00:03:23,600 --> 00:03:28,700
this inevitably entails
ridiculous consequences."

65
00:03:28,700 --> 00:03:32,670
Well today we want to talk about
those ridiculous consequences,

66
00:03:32,670 --> 00:03:37,380
because many people
in our field trap,

67
00:03:37,380 --> 00:03:41,255
address, and observe single
electrons and single atoms.

68
00:03:44,000 --> 00:03:48,630
Now this is related
to quantum jumps.

69
00:03:48,630 --> 00:03:53,770
Quantum jumps has even made
it into the popular realm.

70
00:03:53,770 --> 00:03:58,050
You can find lots and lots of
references on quantum jumping,

71
00:03:58,050 --> 00:04:00,610
but I warn you this
is about quantum

72
00:04:00,610 --> 00:04:02,810
jumping into different worlds.

73
00:04:02,810 --> 00:04:05,770
It's about broadening
your horizon.

74
00:04:05,770 --> 00:04:08,640
And those authors,
those people actually,

75
00:04:08,640 --> 00:04:12,780
sell you something by claiming
that people like Einstein

76
00:04:12,780 --> 00:04:17,480
and Stephen Hawking would
support the idea that you can

77
00:04:17,480 --> 00:04:21,640
jump into different universe,
experience another version

78
00:04:21,640 --> 00:04:24,920
of yourself-- probably a
better version of yourself--

79
00:04:24,920 --> 00:04:27,110
and this will
inspire you to change

80
00:04:27,110 --> 00:04:29,880
your life in the world you live.

81
00:04:29,880 --> 00:04:31,750
But-- So this is
quantum jumping.

82
00:04:31,750 --> 00:04:33,810
I'm not talking about
quantum jumping,

83
00:04:33,810 --> 00:04:37,780
I will only talk
about quantum chunks.

84
00:04:37,780 --> 00:04:44,150
So what you see here is
really a classic experiment.

85
00:04:44,150 --> 00:04:45,910
One of three
experiments which were

86
00:04:45,910 --> 00:04:47,820
done almost simultaneously.

87
00:04:47,820 --> 00:04:49,620
There have always been
races for, you know,

88
00:04:49,620 --> 00:04:51,170
the first observation
of something.

89
00:04:51,170 --> 00:04:53,880
And there was the [? Hamburg ?]
group by [? Toshak ?],

90
00:04:53,880 --> 00:04:56,150
the Seattle group
by Hunt [INAUDIBLE],

91
00:04:56,150 --> 00:04:59,000
and the Boulder
Group by Dave Wylen,

92
00:04:59,000 --> 00:05:02,580
who observed the phenomenon
of quantum jumps for the first

93
00:05:02,580 --> 00:05:03,310
time.

94
00:05:03,310 --> 00:05:05,190
Let me just tell you
what it is, because it's

95
00:05:05,190 --> 00:05:07,780
exactly what you're
going to describe.

96
00:05:07,780 --> 00:05:11,500
Here in this situation
you have a system.

97
00:05:11,500 --> 00:05:15,560
It's a trapped barium
ion with a p and s state.

98
00:05:15,560 --> 00:05:19,970
And there is very
fast transition

99
00:05:19,970 --> 00:05:22,370
where you can rapidly
scatter light, and get

100
00:05:22,370 --> 00:05:25,605
lots of fluorescence,
at 493 nanometer.

101
00:05:28,200 --> 00:05:33,870
But what happened is,
every once in a while,

102
00:05:33,870 --> 00:05:44,520
the particle has a branching
ratio to decay to d state,

103
00:05:44,520 --> 00:05:50,310
and, well, d to s 2 units
of angular momentum.

104
00:05:50,310 --> 00:05:55,910
The transition here down to
the ground state is very slow.

105
00:05:55,910 --> 00:05:57,930
It's slow because
it's forbidden.

106
00:05:57,930 --> 00:06:00,950
It's even slower
because it happens

107
00:06:00,950 --> 00:06:05,980
at longer wavelengths
of 1,600 nanometer.

108
00:06:05,980 --> 00:06:07,190
And this is slow.

109
00:06:07,190 --> 00:06:12,340
And it has a lifetime
of about 50 second.

110
00:06:12,340 --> 00:06:17,580
So you-- If you think
about it naively

111
00:06:17,580 --> 00:06:19,370
and that's probably
the correct approach--

112
00:06:19,370 --> 00:06:21,970
you would say the
particles in the s state,

113
00:06:21,970 --> 00:06:23,140
it's like a light bulb.

114
00:06:23,140 --> 00:06:26,210
It scatters many, many
photons, but then it

115
00:06:26,210 --> 00:06:28,030
branches to the d state.

116
00:06:28,030 --> 00:06:30,210
And for 50 seconds
the light bulb

117
00:06:30,210 --> 00:06:33,410
is switched off, until
the particle spontaneously

118
00:06:33,410 --> 00:06:35,710
goes down to the s state,
and then the light bulb

119
00:06:35,710 --> 00:06:37,420
is switched on.

120
00:06:37,420 --> 00:06:40,100
And this is exactly
what people observed.

121
00:06:40,100 --> 00:06:42,469
But I mean I was-- This was '86.

122
00:06:42,469 --> 00:06:44,260
That was the time I
was a graduate student,

123
00:06:44,260 --> 00:06:46,190
and I talked with
theorists in the field,

124
00:06:46,190 --> 00:06:48,650
and respectable
theorists told me

125
00:06:48,650 --> 00:06:50,640
that there was a
controversy-- probably

126
00:06:50,640 --> 00:06:53,250
not among experimentalists,
who use the naive

127
00:06:53,250 --> 00:06:55,890
approach-- but theorists
who believed too much

128
00:06:55,890 --> 00:06:57,990
in what they calculate.

129
00:06:57,990 --> 00:07:00,380
Because if you solve it
quantum mechanically,

130
00:07:00,380 --> 00:07:02,060
and look what you get.

131
00:07:02,060 --> 00:07:03,730
It's a steady state solution.

132
00:07:03,730 --> 00:07:04,670
Atom in steady state.

133
00:07:04,670 --> 00:07:06,020
Laser in steady state.

134
00:07:06,020 --> 00:07:09,560
And if you calculate the quantum
mechanical expectation value,

135
00:07:09,560 --> 00:07:13,030
the expectation value is
constant as a function of time.

136
00:07:13,030 --> 00:07:15,420
So a simple quantum
calculation would

137
00:07:15,420 --> 00:07:18,740
say the fluorescence
is on average.

138
00:07:18,740 --> 00:07:20,940
Well just the time
average and his constant,

139
00:07:20,940 --> 00:07:22,770
and his time independent.

140
00:07:22,770 --> 00:07:25,190
So there were some
serious doubts

141
00:07:25,190 --> 00:07:27,650
whether you would observe
this, or something

142
00:07:27,650 --> 00:07:30,840
which looks more like
the ensemble average.

143
00:07:30,840 --> 00:07:35,260
Well made all of the theorists
soon realize you cannot

144
00:07:35,260 --> 00:07:37,510
calculate the steady
state average.

145
00:07:37,510 --> 00:07:40,080
You have to calculate
the correlation function

146
00:07:40,080 --> 00:07:41,080
in steady state.

147
00:07:41,080 --> 00:07:43,190
And the steady state
correlation function

148
00:07:43,190 --> 00:07:45,940
has structure exactly
at 50 seconds.

149
00:07:45,940 --> 00:07:48,590
So in a steady state
system, if you're

150
00:07:48,590 --> 00:07:51,760
interested in fluctuations
and temporal behavior,

151
00:07:51,760 --> 00:07:54,320
you can't just
calculate quantities,

152
00:07:54,320 --> 00:07:57,530
you have to calculate
correlation functions

153
00:07:57,530 --> 00:07:59,250
of quantities.

154
00:07:59,250 --> 00:08:02,110
So anyway-- But then
of course, if you now

155
00:08:02,110 --> 00:08:05,270
average over many,
many realizations,

156
00:08:05,270 --> 00:08:09,730
you find exactly an exponential
decay of 50 seconds.

157
00:08:09,730 --> 00:08:11,660
And this is sort
of what you would

158
00:08:11,660 --> 00:08:14,970
get for an ensemble average.

159
00:08:14,970 --> 00:08:17,200
So this is what
quantum jumps are.

160
00:08:17,200 --> 00:08:19,090
This is what you
should always imagine

161
00:08:19,090 --> 00:08:23,340
when we talk about quantum jumps
in the rest of this lecture.

162
00:08:23,340 --> 00:08:25,910
But I also want to show
you more recent example

163
00:08:25,910 --> 00:08:29,520
from Serge Haroche,
in Paris, where

164
00:08:29,520 --> 00:08:32,860
they observed quantum jumps.

165
00:08:32,860 --> 00:08:36,520
Recording a single
photon in a cavity.

166
00:08:36,520 --> 00:08:39,750
So what happens is there's a
single photon in this cavity,

167
00:08:39,750 --> 00:08:41,730
and they can read
out whether there

168
00:08:41,730 --> 00:08:45,230
is a photon or not by sending
Rydberg atoms through.

169
00:08:45,230 --> 00:08:48,210
And a single photon
is never observed,

170
00:08:48,210 --> 00:08:51,640
but it causes a phase
shift, an AC-Stark shift,

171
00:08:51,640 --> 00:08:53,330
on the Rydberg atom.

172
00:08:53,330 --> 00:08:56,010
And then they can read out
with a Ramsey interferometer

173
00:08:56,010 --> 00:08:57,750
the AC-Stark shift.

174
00:08:57,750 --> 00:09:01,080
So every atom which
passes through the cavity

175
00:09:01,080 --> 00:09:04,910
will tell you whether
there is a photon or not.

176
00:09:04,910 --> 00:09:08,740
And-- And this is shown here.

177
00:09:08,740 --> 00:09:12,830
Blue means no,
there is no photon.

178
00:09:12,830 --> 00:09:14,720
Red means there is a photon.

179
00:09:14,720 --> 00:09:18,850
So here you repeatedly figure
out again there is no photon.

180
00:09:18,850 --> 00:09:21,060
And suddenly there
is a quantum jump

181
00:09:21,060 --> 00:09:23,455
when the cavity has
absorbed a photon,

182
00:09:23,455 --> 00:09:26,750
or when a thermal
photon was released.

183
00:09:26,750 --> 00:09:29,060
You may say, what happened here?

184
00:09:29,060 --> 00:09:30,640
Well no measurement is perfect.

185
00:09:30,640 --> 00:09:31,680
There is some noise.

186
00:09:31,680 --> 00:09:33,910
But by seeing that
these are outliers,

187
00:09:33,910 --> 00:09:37,250
you clearly see that here
is a photon in the cavity,

188
00:09:37,250 --> 00:09:42,030
and here where the red
read out of the atom

189
00:09:42,030 --> 00:09:45,610
overwhelms the blue readout
here you have no photon.

190
00:09:45,610 --> 00:09:50,690
And then as time goes by, you
can observe the death and birth

191
00:09:50,690 --> 00:09:54,450
of micro-cavity photons.

192
00:09:54,450 --> 00:09:57,320
So these are now quantum
jumps in the-- not

193
00:09:57,320 --> 00:10:01,400
in atomic population, but
quantum jumps in the photon

194
00:10:01,400 --> 00:10:04,814
number in a cavity.

195
00:10:04,814 --> 00:10:07,244
AUDIENCE:What's the
time scale on that?

196
00:10:07,244 --> 00:10:09,200
Seconds?

197
00:10:09,200 --> 00:10:10,710
PROFESSOR: Yeah, time is second.

198
00:10:10,710 --> 00:10:15,940
So what is shown here is-- I
think what is shown is here

199
00:10:15,940 --> 00:10:22,955
are 300 milliseconds, and
this is 2.5 second element.

200
00:10:22,955 --> 00:10:26,210
AUDIENCE: What was the
density of each line mean?

201
00:10:26,210 --> 00:10:29,350
PROFESSOR: Each line
is a Rydberg atom,

202
00:10:29,350 --> 00:10:32,290
which has been readout with
a Ramsey interferometer.

203
00:10:32,290 --> 00:10:32,791
And--

204
00:10:32,791 --> 00:10:34,290
AUDIENCE: So that
means that they're

205
00:10:34,290 --> 00:10:35,520
leaving no, like between--

206
00:10:35,520 --> 00:10:36,978
PROFESSOR: I think
that means there

207
00:10:36,978 --> 00:10:40,380
was a Rydberg atom in state e,
which means there is a photon.

208
00:10:40,380 --> 00:10:42,840
And blue means, the
blue bar means, there

209
00:10:42,840 --> 00:10:45,790
was a Rydberg atom in state g.

210
00:10:45,790 --> 00:10:50,340
These are two highlighting
states of the rubidium atom.

211
00:10:50,340 --> 00:10:52,300
Sort of each spike
here is one readout

212
00:10:52,300 --> 00:10:55,750
of the microchannel plate.

213
00:10:55,750 --> 00:10:57,530
So these are detector clicks.

214
00:10:57,530 --> 00:10:59,180
You can say, yes.

215
00:10:59,180 --> 00:11:02,322
You can sort of say, blue
is yes, and red is no.

216
00:11:02,322 --> 00:11:03,530
And these are the two clicks.

217
00:11:03,530 --> 00:11:06,302
AUDIENCE: Yeah, but, like the
separation between two clicks

218
00:11:06,302 --> 00:11:07,882
is not uniform.

219
00:11:07,882 --> 00:11:09,840
PROFESSOR: No, because
if you're an atomic beam

220
00:11:09,840 --> 00:11:12,600
the atoms come not
equally spaced out.

221
00:11:12,600 --> 00:11:14,700
There's a randomness
when the atom comes.

222
00:11:14,700 --> 00:11:18,700
I think it's posed only in
statistics in this experiment.

223
00:11:18,700 --> 00:11:19,790
It's a real experiment.

224
00:11:19,790 --> 00:11:21,184
It's amazing.

225
00:11:21,184 --> 00:11:21,850
Other questions?

226
00:11:21,850 --> 00:11:22,380
Team One.

227
00:11:22,380 --> 00:11:23,671
AUDIENCE: So one more question.

228
00:11:23,671 --> 00:11:25,630
So the, for example,
the first bit of blue

229
00:11:25,630 --> 00:11:29,502
is all measurements for the
same state of the cavity.

230
00:11:29,502 --> 00:11:31,217
I know blue means
photon or no photon,

231
00:11:31,217 --> 00:11:32,800
but if it does mean
photon, that means

232
00:11:32,800 --> 00:11:35,880
it's the same photon
interacting with--

233
00:11:35,880 --> 00:11:37,272
PROFESSOR: Yes.

234
00:11:37,272 --> 00:11:38,730
AUDIENCE: OK.

235
00:11:38,730 --> 00:11:41,170
PROFESSOR: Yes, because, I
mean, this is something people

236
00:11:41,170 --> 00:11:43,370
know that the photon
number in this cavity

237
00:11:43,370 --> 00:11:45,930
only changes over
very long time scales.

238
00:11:45,930 --> 00:11:49,310
And you sort of see,
kind of, exactly

239
00:11:49,310 --> 00:11:53,890
hear how-- You see
here two quantum jumps.

240
00:11:53,890 --> 00:11:55,730
One from blue to
red, and one from red

241
00:11:55,730 --> 00:11:58,858
to blue in this cavity.

242
00:11:58,858 --> 00:11:59,830
Yes?

243
00:11:59,830 --> 00:12:01,531
AUDIENCE: I don't if
this is maybe just

244
00:12:01,531 --> 00:12:04,190
my accent's not very good,
but in the bottom left graph--

245
00:12:04,190 --> 00:12:04,690
Right?

246
00:12:04,690 --> 00:12:07,650
Or on the bottom graph, in
sort of the left corner,

247
00:12:07,650 --> 00:12:09,630
it looks like
there's some overlap

248
00:12:09,630 --> 00:12:11,610
between the dark blue
and the dark red.

249
00:12:11,610 --> 00:12:14,085
Yeah.

250
00:12:14,085 --> 00:12:15,185
PROFESSOR: You mean here?

251
00:12:15,185 --> 00:12:16,560
AUDIENCE: Yeah,
like right there.

252
00:12:16,560 --> 00:12:20,885
Is that like sort of there is
a photon in that [INAUDIBLE]

253
00:12:20,885 --> 00:12:21,510
simultaneously?

254
00:12:25,745 --> 00:12:27,745
PROFESSOR: I think, to
the best of my knowledge,

255
00:12:27,745 --> 00:12:29,431
I haven't really
looked-- I've heard

256
00:12:29,431 --> 00:12:31,930
wonderful talks about it.You
can go back and read the paper,

257
00:12:31,930 --> 00:12:35,290
but this probably means in
the atom number in the cavity,

258
00:12:35,290 --> 00:12:38,070
and the algorithm they have
used to analyze the data

259
00:12:38,070 --> 00:12:41,660
says that here the photon
number was 0, jumped to 1,

260
00:12:41,660 --> 00:12:43,500
and jumped back again.

261
00:12:43,500 --> 00:12:46,320
So you-- If you see
here, for instance,

262
00:12:46,320 --> 00:12:48,310
there is some overlap,
but you would still

263
00:12:48,310 --> 00:12:51,320
say, since either there is
a photon or not a photon,

264
00:12:51,320 --> 00:12:54,010
as long as you have a--
as you have a higher

265
00:12:54,010 --> 00:12:57,040
number of red clicks
than blue clicks,

266
00:12:57,040 --> 00:12:59,710
you would say there is a photon.

267
00:12:59,710 --> 00:13:03,100
And the little overlap, there
is maybe a little bit of time

268
00:13:03,100 --> 00:13:05,970
where you don't know whether
there is a photon or not.

269
00:13:05,970 --> 00:13:10,410
But this really-- This is
real, because every measurement

270
00:13:10,410 --> 00:13:12,560
takes a certain amount of time.

271
00:13:12,560 --> 00:13:14,450
Since you have a
few random clicks,

272
00:13:14,450 --> 00:13:16,125
if they are a few
random clicks, you

273
00:13:16,125 --> 00:13:19,070
do not know yet if
the system has jumped,

274
00:13:19,070 --> 00:13:20,960
you have to wait a little bit.

275
00:13:20,960 --> 00:13:23,075
But this is really
also showing how

276
00:13:23,075 --> 00:13:28,480
our knowledge about the
quantum system comes with time.

277
00:13:28,480 --> 00:13:29,880
And therefore,
the wave function,

278
00:13:29,880 --> 00:13:33,130
or the statistical
operator changes with time,

279
00:13:33,130 --> 00:13:35,710
because of wave function
and statistical operator

280
00:13:35,710 --> 00:13:38,950
are simply a description of
our knowledge about the quantum

281
00:13:38,950 --> 00:13:39,450
system.

282
00:13:42,372 --> 00:13:47,242
AUDIENCE: [INAUDIBLE]
is too dense

283
00:13:55,224 --> 00:13:57,390
PROFESSOR: You could have
a situation, for instance,

284
00:13:57,390 --> 00:13:59,270
you could prepare
the cavity, that it

285
00:13:59,270 --> 00:14:01,970
has an equal probability
foreseeable in one.

286
00:14:01,970 --> 00:14:05,510
And then what you would read out
may just be completely random.

287
00:14:05,510 --> 00:14:08,200
So that may be possible,
but in their experiment,

288
00:14:08,200 --> 00:14:12,570
I think, the nature
here is that you

289
00:14:12,570 --> 00:14:14,410
have most of the
time stable photons,

290
00:14:14,410 --> 00:14:17,440
because the cavity is cooled
to very low temperature,

291
00:14:17,440 --> 00:14:21,410
and only occasionally does
the thermal distribution,

292
00:14:21,410 --> 00:14:23,870
the Bose-Einstein distribution
of photons in the cavity

293
00:14:23,870 --> 00:14:24,780
have a one.

294
00:14:24,780 --> 00:14:27,540
And what we observe are
the thermal fluctuations

295
00:14:27,540 --> 00:14:30,610
between mainly 0 and
occasionally a 1.

296
00:14:35,250 --> 00:14:38,260
If you would make the cavity
and little bit hotter,

297
00:14:38,260 --> 00:14:41,320
it would have maybe
half of the time a 1

298
00:14:41,320 --> 00:14:43,640
and half of the time 0.

299
00:14:43,640 --> 00:14:46,540
But that wouldn't mean that
you have completely randomness

300
00:14:46,540 --> 00:14:50,280
in your results, because
there is a thermal relaxation

301
00:14:50,280 --> 00:14:52,740
time over which a
photon is created,

302
00:14:52,740 --> 00:14:54,360
and a photon goes back.

303
00:14:54,360 --> 00:14:57,430
The photon would not jump
back and forth from the cavity

304
00:14:57,430 --> 00:15:00,290
to the walls of the
cavity in microseconds.

305
00:15:00,290 --> 00:15:03,600
It would probably
have an exchange time,

306
00:15:03,600 --> 00:15:06,274
and you could sort of measure
the thermal correlation

307
00:15:06,274 --> 00:15:07,440
time of photons in a cavity.

308
00:15:10,150 --> 00:15:11,400
It's a really good experiment.

309
00:15:15,040 --> 00:15:15,590
OK.

310
00:15:15,590 --> 00:15:18,550
So in that sense I
think I showed you

311
00:15:18,550 --> 00:15:23,300
that what Schrodinger said
is ridiculous is real,

312
00:15:23,300 --> 00:15:26,450
is the subject of
current research,

313
00:15:26,450 --> 00:15:29,180
and since for
quantum information

314
00:15:29,180 --> 00:15:30,720
processing in
quantum computers, we

315
00:15:30,720 --> 00:15:33,830
need single quantum system
is at the heart of quantum

316
00:15:33,830 --> 00:15:36,220
information processing.

317
00:15:36,220 --> 00:15:46,410
So therefore, we
want to now describe

318
00:15:46,410 --> 00:15:47,740
single quantum systems.

319
00:16:01,090 --> 00:16:04,710
But when I do that,
I-- come very --

320
00:16:04,710 --> 00:16:06,640
It will become clear at
the end of this unit,

321
00:16:06,640 --> 00:16:10,790
but I want to give you
the warning right away.

322
00:16:10,790 --> 00:16:14,630
The way how I describe now
a single quantum system

323
00:16:14,630 --> 00:16:15,820
may not be unique.

324
00:16:19,110 --> 00:16:21,660
There will be,
there is actually,

325
00:16:21,660 --> 00:16:23,420
an infinite number of ways.

326
00:16:27,330 --> 00:16:29,040
And I've already
mentioned that to you,

327
00:16:29,040 --> 00:16:31,160
that there is an
infinite number of ways

328
00:16:31,160 --> 00:16:34,720
to unravel a density matrix.

329
00:16:34,720 --> 00:16:37,420
I mentioned that you can
write every density matrix

330
00:16:37,420 --> 00:16:40,380
as a probability of pure states.

331
00:16:40,380 --> 00:16:42,810
Weighted probability of
being in pure states,

332
00:16:42,810 --> 00:16:44,417
but there's always
different choices.

333
00:16:44,417 --> 00:16:46,750
You can get an infinite number
of different combinations

334
00:16:46,750 --> 00:16:50,320
of pure states, and construct
your density matrix.

335
00:16:50,320 --> 00:16:53,250
So, so this was sort of in
at the level of the density

336
00:16:53,250 --> 00:16:56,730
matrix, but similarly--
and this will come out

337
00:16:56,730 --> 00:17:04,452
of this discussion today--
there is also an infinite number

338
00:17:04,452 --> 00:17:11,089
of ways to unravel the
dynamics-- the dynamics

339
00:17:11,089 --> 00:17:14,010
involved in the evolution
of an open quantum system.

340
00:17:20,630 --> 00:17:28,590
So-- so with that I want to
describe the quantum Monte

341
00:17:28,590 --> 00:17:30,340
Carlo wave function method.

342
00:17:33,120 --> 00:17:37,040
So in the quantum Monte
Carlo wave function method,

343
00:17:37,040 --> 00:17:40,380
we perform-- We don't
need an experiment.

344
00:17:40,380 --> 00:17:42,490
We don't need
sophisticated experiments

345
00:17:42,490 --> 00:17:44,280
to observe a single
quantum system.

346
00:17:44,280 --> 00:17:46,420
It's a theoretical
method, but we

347
00:17:46,420 --> 00:17:49,115
perform thought, or
gedanken experiments.

348
00:17:51,700 --> 00:17:57,850
So our gedanken experiments is
that we perform measurements.

349
00:17:57,850 --> 00:18:05,540
So we assume-- we
have-- When we want

350
00:18:05,540 --> 00:18:07,180
to describe an
ensemble of atoms,

351
00:18:07,180 --> 00:18:09,360
we assume we have a single atom.

352
00:18:09,360 --> 00:18:11,730
We assume it emits
photon, and then we

353
00:18:11,730 --> 00:18:14,210
assume that the
photon is detected.

354
00:18:14,210 --> 00:18:17,080
And the mom-- So the
moment the photon

355
00:18:17,080 --> 00:18:19,380
is detected we
know, for instance,

356
00:18:19,380 --> 00:18:21,230
that the atom is in
the ground state.

357
00:18:21,230 --> 00:18:24,830
And so our quantum Monte
Carlo simulation follows now

358
00:18:24,830 --> 00:18:28,870
the trajectory of the single
quantum system by saying,

359
00:18:28,870 --> 00:18:31,110
I toss a coin, there's
a certain probability

360
00:18:31,110 --> 00:18:32,370
for detecting the photon.

361
00:18:32,370 --> 00:18:34,420
If I detect the photon,
the system I know

362
00:18:34,420 --> 00:18:35,650
is in the ground state.

363
00:18:35,650 --> 00:18:37,510
If I don't detect
the photon, it's

364
00:18:37,510 --> 00:18:39,150
not in the ground
state, and such.

365
00:18:39,150 --> 00:18:43,080
And these probabilistic
approach is just simulated

366
00:18:43,080 --> 00:18:46,340
in the quantum
Monte Carlo sense.

367
00:18:46,340 --> 00:18:49,950
So-- So what is
in essence-- what

368
00:18:49,950 --> 00:18:51,850
is the essence of the
quantum Monte Carlo

369
00:18:51,850 --> 00:18:56,900
method is that we allow a
small time step delta t.

370
00:18:56,900 --> 00:18:59,530
It's a real time simulation.

371
00:18:59,530 --> 00:19:02,260
And there is a small
probability that the system

372
00:19:02,260 --> 00:19:05,320
has decayed in that time.

373
00:19:05,320 --> 00:19:10,840
So we may now toss a coin-- use
a random function generator--

374
00:19:10,840 --> 00:19:14,695
and ask, has spontaneous
emission taken place?

375
00:19:17,210 --> 00:19:20,840
And then the two
possibilities are yes,

376
00:19:20,840 --> 00:19:23,450
and we continue
whatever the outcome,

377
00:19:23,450 --> 00:19:25,130
whatever the quantum state is.

378
00:19:25,130 --> 00:19:32,880
Or we say no, and then with
the probability-- with a-- so

379
00:19:32,880 --> 00:19:35,060
with a probability,
p, we say yes.

380
00:19:35,060 --> 00:19:37,380
With a probability
1 minus p we say no,

381
00:19:37,380 --> 00:19:39,710
and then we continue the
simulation accordingly.

382
00:19:42,500 --> 00:19:44,420
So this will become
much, much clearer

383
00:19:44,420 --> 00:19:47,200
when I write down the
formalism for you,

384
00:19:47,200 --> 00:19:51,660
but there is one important
aspect, which I really

385
00:19:51,660 --> 00:19:52,990
want to discuss with you.

386
00:19:52,990 --> 00:19:55,420
And this is the
essence of the method.

387
00:19:55,420 --> 00:19:59,370
Namely, how does
the wave function

388
00:19:59,370 --> 00:20:00,840
change by your measurement?

389
00:20:04,450 --> 00:20:09,210
So-- so this is the essential,
the only conceptional part

390
00:20:09,210 --> 00:20:10,960
for the quantum Monte
Carlo wave function.

391
00:20:10,960 --> 00:20:12,315
Afterwards I just show you.

392
00:20:12,315 --> 00:20:14,320
Once you have this,
once we have cleared up

393
00:20:14,320 --> 00:20:17,340
this conceptional part, the
rest is just a few equations.

394
00:20:17,340 --> 00:20:20,680
And we immediately show that
the quantum Monte Carlo wave

395
00:20:20,680 --> 00:20:23,700
function method is equivalent
to the optical block equation.

396
00:20:23,700 --> 00:20:26,170
But let's really feature
and take some time

397
00:20:26,170 --> 00:20:30,170
for discussing the concept,
namely, the change over wave

398
00:20:30,170 --> 00:20:33,060
function by a
quantum measurement.

399
00:20:36,550 --> 00:20:38,670
And maybe what this
carries home to you

400
00:20:38,670 --> 00:20:42,730
is that the nature of an open
system-- we have a quantum

401
00:20:42,730 --> 00:20:45,220
system, which interacts
with the environment--

402
00:20:45,220 --> 00:20:50,100
is, actually, that the
environment permanently

403
00:20:50,100 --> 00:20:52,170
performs measurements.

404
00:20:52,170 --> 00:20:54,890
And you will actually see
that the idea of performing

405
00:20:54,890 --> 00:20:59,620
measurements, formalized in
the quantum Monte Carlo sense,

406
00:20:59,620 --> 00:21:02,540
gives us an equation for
the density matrix, which

407
00:21:02,540 --> 00:21:05,400
is identical to the
optical block equation.

408
00:21:05,400 --> 00:21:07,550
So in other words, you
can say I am re-deriving

409
00:21:07,550 --> 00:21:10,780
for you master equation now,
but from a very, very different

410
00:21:10,780 --> 00:21:11,680
perspective.

411
00:21:11,680 --> 00:21:14,370
From the perspective
of an environment

412
00:21:14,370 --> 00:21:16,205
which constantly
performs measurement.

413
00:21:18,770 --> 00:21:19,270
OK.

414
00:21:19,270 --> 00:21:21,140
So let's go through that.

415
00:21:21,140 --> 00:21:25,280
How-- What happens
in a measurement?

416
00:21:25,280 --> 00:21:33,150
Well if our initial state is
the ground state-- starting

417
00:21:33,150 --> 00:21:36,400
very in trivial ways-- well
there is no measurement,

418
00:21:36,400 --> 00:21:39,865
because the particle will stay
in the ground state forever.

419
00:21:44,660 --> 00:21:45,160
OK.

420
00:21:45,160 --> 00:21:48,940
Now let's go to
the next situation

421
00:21:48,940 --> 00:21:52,830
that our atom starts out
in the excited state.

422
00:21:52,830 --> 00:22:01,140
Now in a time step delta t,
we have two possibilities.

423
00:22:01,140 --> 00:22:05,460
A photon can be observed,
or nor photon is observed.

424
00:22:09,720 --> 00:22:15,210
When the photon is
observed, we know for sure

425
00:22:15,210 --> 00:22:20,130
that the system is now
projected into the ground state.

426
00:22:20,130 --> 00:22:23,870
If no photon is
observed, well the system

427
00:22:23,870 --> 00:22:25,320
is still in the excited state.

428
00:22:27,881 --> 00:22:28,380
OK.

429
00:22:28,380 --> 00:22:31,092
So we've discussed what happens
when the system is initially

430
00:22:31,092 --> 00:22:33,050
in the ground, and when
the system is initially

431
00:22:33,050 --> 00:22:35,280
in the excited state.

432
00:22:35,280 --> 00:22:39,390
But now comes the
following question.

433
00:22:39,390 --> 00:22:42,600
We prepare the system
in a superposition

434
00:22:42,600 --> 00:22:46,820
of ground and excited state.

435
00:22:46,820 --> 00:22:53,010
And we want to
discuss what happens

436
00:22:53,010 --> 00:22:54,250
when the photon is observed?

437
00:22:57,740 --> 00:23:00,970
What happens when no
photon is observed?

438
00:23:00,970 --> 00:23:04,130
I think if you observe
the photon it's

439
00:23:04,130 --> 00:23:07,150
pretty clear we have to
assume after photon emission

440
00:23:07,150 --> 00:23:10,050
the system is in
the ground state.

441
00:23:10,050 --> 00:23:16,060
But if no photon is observed, I
want to give you three choices,

442
00:23:16,060 --> 00:23:20,480
and you should address
them with the clicker.

443
00:23:20,480 --> 00:23:30,010
One choice is that the particle
stays in the same state.

444
00:23:30,010 --> 00:23:36,440
The second choice is
that the particle stays

445
00:23:36,440 --> 00:23:40,780
in a superposition state,
but the probability

446
00:23:40,780 --> 00:23:45,540
to be in the excited
state decays.

447
00:23:45,540 --> 00:23:47,380
Also no photon
has being emitted,

448
00:23:47,380 --> 00:23:50,460
or no photon has been observed.

449
00:23:50,460 --> 00:23:55,020
And the third possibility
is something else.

450
00:24:01,180 --> 00:24:05,410
So in other words, assume
you're an observer,

451
00:24:05,410 --> 00:24:08,440
and you know with 100%
detection probability--

452
00:24:08,440 --> 00:24:11,330
let's not discuss technical
issues-- you have completely

453
00:24:11,330 --> 00:24:14,410
surrounded your system
with perfect detectors.

454
00:24:14,410 --> 00:24:18,880
And you know in the time
step delta t, in the time t,

455
00:24:18,880 --> 00:24:21,200
no photon was observed.

456
00:24:21,200 --> 00:24:23,240
And now you should
make a prediction

457
00:24:23,240 --> 00:24:25,240
what happens next to
the quantum system.

458
00:24:25,240 --> 00:24:27,950
And you make the
prediction by assuming

459
00:24:27,950 --> 00:24:31,010
that the system is described
by a wave function or density

460
00:24:31,010 --> 00:24:32,150
matrix.

461
00:24:32,150 --> 00:24:37,220
And I'm asking you, which
of the three choices

462
00:24:37,220 --> 00:24:41,760
correctly describes what the
system will do afterwards?

463
00:24:41,760 --> 00:24:44,900
What the system will do next?

464
00:24:44,900 --> 00:24:46,910
Is the question very clear?

465
00:24:46,910 --> 00:24:48,395
Good.

466
00:24:48,395 --> 00:24:52,355
[TAPPING]

467
00:24:59,160 --> 00:24:59,660
OK.

468
00:25:10,013 --> 00:25:11,196
All right.

469
00:25:11,196 --> 00:25:13,880
Pretty good.

470
00:25:13,880 --> 00:25:19,520
A lot of people when I taught
the class before said it's A,

471
00:25:19,520 --> 00:25:21,570
the wave function
is not changing,

472
00:25:21,570 --> 00:25:24,310
because no photon
has been emitted.

473
00:25:24,310 --> 00:25:37,150
Now in-- For those people, but
there only few among those,

474
00:25:37,150 --> 00:25:39,405
I would have now asked
the next question.

475
00:25:43,380 --> 00:25:55,990
If you have-- If you have
a 50-50 superposition,

476
00:25:55,990 --> 00:25:59,980
of ground and excited
state-- And let's

477
00:25:59,980 --> 00:26:03,630
say the spot time for
spontaneous emission

478
00:26:03,630 --> 00:26:17,150
is 10 nanoseconds-- But
now you wait one second,

479
00:26:17,150 --> 00:26:23,010
and after 1 second, no
photon has been observed.

480
00:26:27,120 --> 00:26:28,980
What is your prediction?

481
00:26:28,980 --> 00:26:33,620
In what state is the
system at this point?

482
00:26:33,620 --> 00:26:36,030
Is the system in
the ground state?

483
00:26:36,030 --> 00:26:40,570
Or is the system in,
still in, a superposition

484
00:26:40,570 --> 00:26:42,090
of ground and excited state?

485
00:26:45,090 --> 00:26:47,250
Well since we have to
clicker, you can tell me.

486
00:26:47,250 --> 00:26:49,159
Is it-- Is it still a
superposition state,

487
00:26:49,159 --> 00:26:50,700
or is the system in
the ground state?

488
00:27:29,040 --> 00:27:31,020
OK.

489
00:27:31,020 --> 00:27:36,200
So now you have to help me out,
because the people who said B,

490
00:27:36,200 --> 00:27:42,410
if they would go back to
here-- And most people said

491
00:27:42,410 --> 00:27:48,140
it is B. If you put in gamma 10
nanoseconds, the inverse of 10

492
00:27:48,140 --> 00:27:50,940
nanosecond, and the
time is 1 second,

493
00:27:50,940 --> 00:27:54,910
the excited state amplitude,
the excited stated mixture,

494
00:27:54,910 --> 00:27:58,620
is 10 to the minus nothing.

495
00:27:58,620 --> 00:28:00,670
So-- OK.

496
00:28:00,670 --> 00:28:03,545
If you want to be a
mathematician and say--

497
00:28:03,545 --> 00:28:04,760
[LAUGHING]

498
00:28:04,760 --> 00:28:07,860
--there is epsilon 10
to the minus 10, yes.

499
00:28:07,860 --> 00:28:10,150
Then you can say there is
still a small admixture

500
00:28:10,150 --> 00:28:11,680
to the excited state.

501
00:28:11,680 --> 00:28:14,590
But by making it so
extreme-- 10 nanoseconds

502
00:28:14,590 --> 00:28:19,050
versus a second-- at least
in all practical terms,

503
00:28:19,050 --> 00:28:22,010
the system is in
the ground state.

504
00:28:22,010 --> 00:28:23,810
And I think this
should teach you

505
00:28:23,810 --> 00:28:26,260
what it means if you
have a system which

506
00:28:26,260 --> 00:28:29,970
is in the superposition of
ground and excited state,

507
00:28:29,970 --> 00:28:33,060
and after 1 second it
has not emitted a photon.

508
00:28:33,060 --> 00:28:35,180
I mean--

509
00:28:35,180 --> 00:28:37,230
Then you would say,
it had its chance.

510
00:28:37,230 --> 00:28:39,340
It had plenty of
chances to emit,

511
00:28:39,340 --> 00:28:42,180
but it decided it
doesn't want to emit.

512
00:28:42,180 --> 00:28:45,130
And that would mean
your 50/50 superposition

513
00:28:45,130 --> 00:28:47,020
of ground and
excited state means

514
00:28:47,020 --> 00:28:50,760
the atom decided that
it's in the ground state.

515
00:28:50,760 --> 00:28:53,200
You should also realize
a 50% superposition

516
00:28:53,200 --> 00:28:55,630
of ground and
excited state means

517
00:28:55,630 --> 00:28:59,150
in half, in 50% of the
cases there will never

518
00:28:59,150 --> 00:29:02,470
be a photon emitted.

519
00:29:02,470 --> 00:29:04,920
So in other words,
what you should realize

520
00:29:04,920 --> 00:29:09,580
is if your 50-50% superposition
of ground and excited state,

521
00:29:09,580 --> 00:29:13,070
that the quantum
evolution of this system

522
00:29:13,070 --> 00:29:23,470
is-- after a long
time the system

523
00:29:23,470 --> 00:29:29,680
is definitely positively,
absolutely in the ground state.

524
00:29:29,680 --> 00:29:33,300
It's always in the ground
state after a long time.

525
00:29:33,300 --> 00:29:41,970
In 50% of the cases with
emission of a photon,

526
00:29:41,970 --> 00:29:47,750
in 50% of the cases without
emission of a photon.

527
00:29:54,360 --> 00:29:56,150
If you have a
radioactive-- ensemble

528
00:29:56,150 --> 00:30:01,840
of radioactive nuclei-- and
let's say half of the atoms

529
00:30:01,840 --> 00:30:05,832
are in the excited state
and [INAUDIBLE] and half are

530
00:30:05,832 --> 00:30:10,430
in the ground state-- and you
send the sample to your friend,

531
00:30:10,430 --> 00:30:14,820
and your friend waits many,
many, many half times,

532
00:30:14,820 --> 00:30:17,680
common sense tells you that
all the particles are now

533
00:30:17,680 --> 00:30:19,870
in the ground state.

534
00:30:19,870 --> 00:30:22,615
Also only half of them have
decayed, because the other half

535
00:30:22,615 --> 00:30:24,610
were in the ground
state to begin with.

536
00:30:24,610 --> 00:30:26,930
I mean this is what
quantum physics tells us.

537
00:30:26,930 --> 00:30:29,320
This is what a 50/50
superposition is.

538
00:30:32,640 --> 00:30:35,520
But quantitatively
this is, of course,

539
00:30:35,520 --> 00:30:41,995
included in this
result-- which magically

540
00:30:41,995 --> 00:30:45,120
that almost all of you
got right-- namely there

541
00:30:45,120 --> 00:30:47,430
is a superposition of
ground and excited state,

542
00:30:47,430 --> 00:30:49,940
but the excited state
amplitude decays.

543
00:30:49,940 --> 00:30:53,870
And to the limit, in the limit
that the time is much longer

544
00:30:53,870 --> 00:30:57,110
than the decay time, no matter
what the coefficient initially

545
00:30:57,110 --> 00:30:59,050
was in front of
the excited state,

546
00:30:59,050 --> 00:31:01,191
this coefficient
has decayed to 0.

547
00:31:11,020 --> 00:31:12,902
Any question about that?

548
00:31:15,790 --> 00:31:20,150
Let me just
emphasize that, again

549
00:31:20,150 --> 00:31:24,410
in giving you also an example in
the reference, a 0 measurement.

550
00:31:32,910 --> 00:31:38,360
A 0 measurement modifies
the wave function.

551
00:31:38,360 --> 00:31:42,620
You can also put it like this,
what the wave function is

552
00:31:42,620 --> 00:31:44,920
is the best knowledge you
would have about this system.

553
00:31:44,920 --> 00:31:48,110
The wave function is a way
to predict what happens next,

554
00:31:48,110 --> 00:31:50,820
and it's the most precise
way to predict it.

555
00:31:50,820 --> 00:31:53,930
More accurate predictions than
the wave function always cannot

556
00:31:53,930 --> 00:31:56,910
be made, because of Heisenberg's
uncertainty relation.

557
00:31:56,910 --> 00:32:00,010
But what happens is,
if you have a 50/50%

558
00:32:00,010 --> 00:32:02,170
superposition of ground
and excited state,

559
00:32:02,170 --> 00:32:05,910
and the system has not emitted,
then you would say initially

560
00:32:05,910 --> 00:32:08,860
my guess was half of the atoms
were excited, half of the atoms

561
00:32:08,860 --> 00:32:10,460
are in the ground state.

562
00:32:10,460 --> 00:32:12,770
But if none of them
has decayed for awhile,

563
00:32:12,770 --> 00:32:15,670
you would say I revise
my estimate now.

564
00:32:15,670 --> 00:32:18,820
Now I have to assume that
more are in the ground state.

565
00:32:18,820 --> 00:32:22,660
Because having not decayed means
there is a higher probability

566
00:32:22,660 --> 00:32:26,260
that the system has actually--
is in the ground state

567
00:32:26,260 --> 00:32:27,460
to begin with.

568
00:32:27,460 --> 00:32:32,210
So this is how this way of
dealing with a wave function

569
00:32:32,210 --> 00:32:35,150
automatically adjusts
for the knowledge

570
00:32:35,150 --> 00:32:38,090
you have gained
about the system.

571
00:32:38,090 --> 00:32:40,520
Let me give you another
example which carries home

572
00:32:40,520 --> 00:32:44,350
the same message, and I take
it from a pedagogical paper

573
00:32:44,350 --> 00:32:54,888
from Dickey, from American
Journal of Physics, 49, 926,

574
00:32:54,888 --> 00:32:57,850
1981.

575
00:32:57,850 --> 00:33:02,750
And what he discussed
is the following.

576
00:33:02,750 --> 00:33:09,310
You have a box and originally
your atomic wave function

577
00:33:09,310 --> 00:33:11,400
is completely localized.

578
00:33:11,400 --> 00:33:14,310
Your Bose-Einstein condensate
if you want in this box,

579
00:33:14,310 --> 00:33:16,980
or let's just assume one
atom to be more precise,

580
00:33:16,980 --> 00:33:19,860
and it's completely delocalized.

581
00:33:19,860 --> 00:33:25,130
But now you focus a laser
beam into the system.

582
00:33:25,130 --> 00:33:28,180
And the laser beam
would ionize your atom,

583
00:33:28,180 --> 00:33:31,630
and you could count the
ion with 100% probability.

584
00:33:31,630 --> 00:33:35,150
And for a short moment,
you flash on the laser

585
00:33:35,150 --> 00:33:42,210
and the result is
yes, you count.

586
00:33:42,210 --> 00:33:44,490
You get a count.

587
00:33:44,490 --> 00:33:47,340
In that moment you
would say my wave

588
00:33:47,340 --> 00:33:50,240
function is more localized.

589
00:33:50,240 --> 00:33:51,450
I revise my estimate.

590
00:33:55,140 --> 00:33:58,340
Well maybe I
shouldn't say ionized.

591
00:33:58,340 --> 00:34:00,710
If it's ionized you
destroy the atom.

592
00:34:00,710 --> 00:34:03,410
Let's just say you observe
fluorescence so the atom is

593
00:34:03,410 --> 00:34:04,170
still alive.

594
00:34:04,170 --> 00:34:06,440
But now you have to
say that the atom is

595
00:34:06,440 --> 00:34:09,030
localized in that region.

596
00:34:09,030 --> 00:34:14,710
However, if you don't
observe anything,

597
00:34:14,710 --> 00:34:17,820
you would say, well
now it's actually

598
00:34:17,820 --> 00:34:20,670
more probable that
the atom is outside.

599
00:34:20,670 --> 00:34:23,600
Because if it had been
in the laser beam,

600
00:34:23,600 --> 00:34:26,670
with a certain probability, I
would have detected a photon.

601
00:34:26,670 --> 00:34:28,730
But the non-detection
of a photon

602
00:34:28,730 --> 00:34:31,090
means that I revise my estimate.

603
00:34:31,090 --> 00:34:33,000
And I say it's
much more probable

604
00:34:33,000 --> 00:34:34,625
that the atom is
outside the laser beam

605
00:34:34,625 --> 00:34:36,300
than inside the laser beam.

606
00:34:36,300 --> 00:34:42,130
So this here is exactly how a
non-observation of anything,

607
00:34:42,130 --> 00:34:46,530
a 0 measurement, modifies
your wave function.

608
00:34:46,530 --> 00:34:49,340
And the formalism
to incorporate that

609
00:34:49,340 --> 00:34:53,250
is what we have
discussed before.

610
00:34:57,350 --> 00:34:57,920
Yes?

611
00:34:57,920 --> 00:34:58,419
Cody?

612
00:34:58,419 --> 00:35:01,680
AUDIENCE: So your arguments
here so far work the same way

613
00:35:01,680 --> 00:35:03,730
as if it's in a
wave function, you

614
00:35:03,730 --> 00:35:06,042
have just an entire
Pascal probability,

615
00:35:06,042 --> 00:35:09,539
and we're updating our
understanding of probability.

616
00:35:09,539 --> 00:35:11,471
Like you haven't
included anything

617
00:35:11,471 --> 00:35:13,536
about the relative phase
between these two--

618
00:35:13,536 --> 00:35:14,910
PROFESSOR: I want
to do that now.

619
00:35:14,910 --> 00:35:19,402
I mean, I-- This is sort of just
to address the basic concepts

620
00:35:19,402 --> 00:35:20,860
about what does a
measurement mean,

621
00:35:20,860 --> 00:35:22,320
what does a 0 measurement mean.

622
00:35:22,320 --> 00:35:24,700
But now I want to write
down everything for you

623
00:35:24,700 --> 00:35:25,990
in amplitudes.

624
00:35:25,990 --> 00:35:30,280
We want to do exactly the
time evolution of the system.

625
00:35:30,280 --> 00:35:31,910
Nancy?

626
00:35:31,910 --> 00:35:33,856
AUDIENCE: Is this,
like, inconsistent

627
00:35:33,856 --> 00:35:35,824
with [INAUDIBLE]
quantum equation?

628
00:35:35,824 --> 00:35:38,776
Like especially the [INAUDIBLE],
because when you are saying

629
00:35:38,776 --> 00:35:41,728
that it is admixture of the
ground and excited state,

630
00:35:41,728 --> 00:35:43,204
and we did not observe--

631
00:35:53,435 --> 00:35:54,810
PROFESSOR: Then
the superposition

632
00:35:54,810 --> 00:35:56,430
say that ground
and excited state.

633
00:35:56,430 --> 00:35:59,720
We are not starting out in
an energy heightened state,

634
00:35:59,720 --> 00:36:04,040
so therefore, we--
what we have is not

635
00:36:04,040 --> 00:36:07,890
a sharp value of the energy,
but an expectation value.

636
00:36:07,890 --> 00:36:10,890
And also if you repeat
the measurement many,

637
00:36:10,890 --> 00:36:14,695
many times, the average
energy is conserved,

638
00:36:14,695 --> 00:36:17,320
and this is exactly what energy
conservation in quantum physics

639
00:36:17,320 --> 00:36:17,955
says.

640
00:36:17,955 --> 00:36:18,830
AUDIENCE: [INAUDIBLE]

641
00:36:26,990 --> 00:36:29,480
PROFESSOR: Well if your
particle, if your system,

642
00:36:29,480 --> 00:36:32,360
is not in an eigenstate
of the Hamiltonian.

643
00:36:32,360 --> 00:36:34,380
It doesn't have a sharp energy.

644
00:36:34,380 --> 00:36:38,170
And that would mean that when
you measure the energy now,

645
00:36:38,170 --> 00:36:39,720
you have fluctuations.

646
00:36:39,720 --> 00:36:42,830
Sometimes you measure higher,
sometimes you measure lower.

647
00:36:42,830 --> 00:36:44,600
It's only when
you're in eigenstate

648
00:36:44,600 --> 00:36:46,470
that you measure a sharp value.

649
00:36:46,470 --> 00:36:48,830
And therefore you know
from the beginning

650
00:36:48,830 --> 00:36:51,080
that you will measure
an energy distribution.

651
00:36:51,080 --> 00:36:53,310
And sometimes you
measure higher,

652
00:36:53,310 --> 00:36:56,460
and sometimes you measure
lower than the average value.

653
00:36:56,460 --> 00:36:58,510
There's nothing wrong
about it, but your question

654
00:36:58,510 --> 00:36:59,320
is a very good one.

655
00:36:59,320 --> 00:37:02,450
You should-- I
mean those examples

656
00:37:02,450 --> 00:37:06,220
are really-- On the one
hand they are trivial,

657
00:37:06,220 --> 00:37:08,940
but on the other hand, it's
very profound what they tell us

658
00:37:08,940 --> 00:37:11,860
about quantum physics
and how to apply

659
00:37:11,860 --> 00:37:13,190
conservation laws and such.

660
00:37:17,270 --> 00:37:17,770
OK.

661
00:37:17,770 --> 00:37:26,540
I think-- Let's-- Let me now
formalize exactly how this is

662
00:37:26,540 --> 00:37:31,120
done with all the
bells and whistles.

663
00:37:31,120 --> 00:37:37,580
We assume we have an initial
wave function, which is now

664
00:37:37,580 --> 00:37:43,720
an arbitrary superposition state
of ground and excited state.

665
00:37:46,670 --> 00:37:51,240
And we have the environment,
which, in this case,

666
00:37:51,240 --> 00:37:53,490
we assume is the 0 photon state.

667
00:37:53,490 --> 00:37:56,400
It's a vacuum.

668
00:37:56,400 --> 00:38:01,350
And we want to exactly
solve the Schrodinger

669
00:38:01,350 --> 00:38:04,000
equation for that system.

670
00:38:04,000 --> 00:38:06,940
But we restrict
ourselves too much,

671
00:38:06,940 --> 00:38:08,820
to a very small time step.

672
00:38:08,820 --> 00:38:12,840
The small time step is smaller
than, you know, anything else.

673
00:38:12,840 --> 00:38:15,050
Than the natural decay time.

674
00:38:15,050 --> 00:38:17,410
Maybe the inverse
[INAUDIBLE] frequency

675
00:38:17,410 --> 00:38:19,540
if you drive the system.

676
00:38:19,540 --> 00:38:22,020
You're not doing it
here, but I will later

677
00:38:22,020 --> 00:38:25,850
show how you can add a laser
beam and drive the system.

678
00:38:25,850 --> 00:38:30,600
Or it should also be smaller
than the inverse tuning.

679
00:38:30,600 --> 00:38:35,920
So what is very important here
is that we want to deal only

680
00:38:35,920 --> 00:38:38,630
with simple cases,
namely one photon

681
00:38:38,630 --> 00:38:40,680
has been emitted or not photon.

682
00:38:40,680 --> 00:38:42,780
So you want to make
sure that there

683
00:38:42,780 --> 00:38:50,390
is, at most, one
spontaneous emission event

684
00:38:50,390 --> 00:38:51,865
during the time delta t.

685
00:38:54,910 --> 00:38:58,840
So what I'm writing down now
is the total wave function

686
00:38:58,840 --> 00:39:02,030
of the system plus
the environment.

687
00:39:02,030 --> 00:39:04,450
And now we can evolve it.

688
00:39:04,450 --> 00:39:09,370
We can ask what happens?

689
00:39:09,370 --> 00:39:13,890
What is the wave
function time later?

690
00:39:13,890 --> 00:39:16,530
This can be exactly done by
time-dependent perturbation

691
00:39:16,530 --> 00:39:17,870
theory.

692
00:39:17,870 --> 00:39:20,250
And the result is
that the system

693
00:39:20,250 --> 00:39:27,290
will be in a superposition
of ground and excited state.

694
00:39:27,290 --> 00:39:29,110
And we want to calculate
the coefficients

695
00:39:29,110 --> 00:39:30,540
alpha prime and beta prime.

696
00:39:35,980 --> 00:39:39,040
And we still have
the vacuum state.

697
00:39:39,040 --> 00:39:43,150
But then we have the possibility
that a photon has been emitted.

698
00:39:43,150 --> 00:39:46,490
In that case we are
in the ground state.

699
00:39:46,490 --> 00:39:49,040
And the direct product
with the environment

700
00:39:49,040 --> 00:39:53,750
involves now the photon emitted
into a certain direction

701
00:39:53,750 --> 00:39:58,400
with wave vector k, with
the polarization epsilon.

702
00:39:58,400 --> 00:40:03,630
And we have coefficients
beta, k epsilon,

703
00:40:03,630 --> 00:40:07,090
and we have to sum over all
possibilities for the photon

704
00:40:07,090 --> 00:40:07,680
to be emitted.

705
00:40:11,220 --> 00:40:15,430
So we call this-- So the wave
function of the total system

706
00:40:15,430 --> 00:40:17,200
has now two parts.

707
00:40:17,200 --> 00:40:20,190
It's a wave function,
which I call psi 0.

708
00:40:20,190 --> 00:40:22,840
This is a wave
function which involves

709
00:40:22,840 --> 00:40:25,830
no photon in the environment.

710
00:40:25,830 --> 00:40:28,296
And the wave function
psi 1 involves

711
00:40:28,296 --> 00:40:29,545
one photon in the environment.

712
00:40:32,280 --> 00:40:35,190
So this is the [INAUDIBLE],
which seems very natural,

713
00:40:35,190 --> 00:40:37,330
but it's also-- you
could immediately

714
00:40:37,330 --> 00:40:39,810
prove that these are the
only possibilities, how

715
00:40:39,810 --> 00:40:41,700
the system can evolve.

716
00:40:41,700 --> 00:40:46,221
And we can verify this by using
time-dependent perturbation

717
00:40:46,221 --> 00:40:46,720
theory.

718
00:40:57,140 --> 00:40:59,730
It's actually almost
everyone of you

719
00:40:59,730 --> 00:41:04,310
has seen it in either 8.21 or in
a more basic course on quantum

720
00:41:04,310 --> 00:41:05,030
physics.

721
00:41:05,030 --> 00:41:06,690
It's time-dependent
perturbation theory

722
00:41:06,690 --> 00:41:08,680
for the emission of a photon.

723
00:41:08,680 --> 00:41:11,920
But usually when you see
those treatments there

724
00:41:11,920 --> 00:41:18,100
reservoir, the environment,
is not treated as explicitly

725
00:41:18,100 --> 00:41:21,520
as we do it here.

726
00:41:21,520 --> 00:41:24,140
The theory where this
is treated exactly

727
00:41:24,140 --> 00:41:28,210
in this way, how we need it,
is the Viegener Biscoff theory.

728
00:41:33,080 --> 00:41:35,640
Which is nothing else in
the perturbative approach,

729
00:41:35,640 --> 00:41:37,890
but it's really a
perturbation theory.

730
00:41:37,890 --> 00:41:40,920
Not just for the atomic system--
how we sometimes present it

731
00:41:40,920 --> 00:41:43,720
in a simplified version--
it's perturbation theory

732
00:41:43,720 --> 00:41:47,060
for the total wave function
of the complete system.

733
00:41:47,060 --> 00:41:48,044
Philmore?

734
00:41:48,044 --> 00:41:50,504
AUDIENCE: So you mentioned
the time interval [INAUDIBLE]

735
00:41:50,504 --> 00:41:52,480
is much smaller than
a lot of things.

736
00:41:52,480 --> 00:41:57,279
But you were ignoring, I take
it, the counter rotating terms?

737
00:41:57,279 --> 00:41:59,279
In which case we're still
at a large enough time

738
00:41:59,279 --> 00:42:02,977
step that's average
out, so to speak?

739
00:42:07,907 --> 00:42:09,879
I simply say this,
because I expected

740
00:42:09,879 --> 00:42:13,330
to be an e with a photon term.

741
00:42:13,330 --> 00:42:16,781
From the g going
up, so to speak.

742
00:42:28,970 --> 00:42:29,800
PROFESSOR: Yes.

743
00:42:29,800 --> 00:42:45,180
Well-- The counter
rotating term--

744
00:42:45,180 --> 00:42:46,960
The first answer I
wanted to give you,

745
00:42:46,960 --> 00:42:49,030
no this is exact
perturbation theory.

746
00:42:49,030 --> 00:42:55,300
But I think I get
myself in trouble

747
00:42:55,300 --> 00:43:01,970
if I would allow-- I would get
myself in trouble if I would

748
00:43:01,970 --> 00:43:06,500
allow the time step to be
extremely short, because then

749
00:43:06,500 --> 00:43:09,490
I'm in shorter
than 1 over omega.

750
00:43:09,490 --> 00:43:12,110
Because during the
time 1 over omega,

751
00:43:12,110 --> 00:43:18,400
we have-- you can say we
have counter rotating terms.

752
00:43:18,400 --> 00:43:20,990
Or in other words, this
particle in the ground

753
00:43:20,990 --> 00:43:23,240
state during a
time 1 over omega,

754
00:43:23,240 --> 00:43:26,330
particle in the ground state can
emit a photon and reabsorb it.

755
00:43:26,330 --> 00:43:28,050
These are where
those weird diagrams

756
00:43:28,050 --> 00:43:30,440
with virtual states,
which we discussed earlier

757
00:43:30,440 --> 00:43:32,370
in this course.

758
00:43:32,370 --> 00:43:39,400
So I think I want to assume
here, which I haven't done,

759
00:43:39,400 --> 00:43:43,730
that the time step is larger
than omega to the minus 1.

760
00:43:43,730 --> 00:43:46,290
And as you know the
counter rotating term

761
00:43:46,290 --> 00:43:50,020
has a detuning delta,
which is 2 omega,

762
00:43:50,020 --> 00:43:52,060
so therefore it is excluded.

763
00:43:52,060 --> 00:43:54,580
Or in other words, when
we talk about photons

764
00:43:54,580 --> 00:43:56,650
sent into the
environment, we want

765
00:43:56,650 --> 00:44:01,550
to talk about real photons,
and not virtual photons.

766
00:44:01,550 --> 00:44:02,380
Yes, good point.

767
00:44:07,900 --> 00:44:10,340
OK.

768
00:44:10,340 --> 00:44:15,340
So if you do simple lowest order
time dependent perturbation

769
00:44:15,340 --> 00:44:20,710
theory, in the Viegener
Biscoff approach,

770
00:44:20,710 --> 00:44:25,040
you get an exact
result for beta prime.

771
00:44:25,040 --> 00:44:29,640
You find that beta prime
in this superposition

772
00:44:29,640 --> 00:44:34,750
is the original amplitude
beta, but it has decayed with e

773
00:44:34,750 --> 00:44:38,780
to the minus gamma over 2dt.

774
00:44:38,780 --> 00:44:42,470
And since we are only
interested in small time steps,

775
00:44:42,470 --> 00:44:48,050
we can do a linear
expansion of that.

776
00:44:48,050 --> 00:44:53,160
The probability, dp, that
a photon has been emitted

777
00:44:53,160 --> 00:45:02,410
is the norm of this
wave function psi 1.

778
00:45:02,410 --> 00:45:06,500
And this, using Viegener
Biscoff perturbation theory,

779
00:45:06,500 --> 00:45:12,240
is gamma dt times the
amplitude squared,

780
00:45:12,240 --> 00:45:17,010
that the system was
excited to begin with.

781
00:45:17,010 --> 00:45:26,790
And because of the conservation
of the norm, the wave function,

782
00:45:26,790 --> 00:45:29,350
the norm of the wave
function psi 0-- which

783
00:45:29,350 --> 00:45:32,640
is a wave function without
a photon being emitted--

784
00:45:32,640 --> 00:45:34,040
is 1 minus dp.

785
00:45:39,180 --> 00:45:43,490
So what I'm telling
you here is that this

786
00:45:43,490 --> 00:45:46,010
is an exact result of
time-dependent perturbation

787
00:45:46,010 --> 00:45:47,330
theory.

788
00:45:47,330 --> 00:45:51,000
Often when this-- when textbooks
treat spontaneous emission,

789
00:45:51,000 --> 00:45:52,980
they're more interested
in getting Fermi's gold

790
00:45:52,980 --> 00:45:54,340
rule at this rate.

791
00:45:54,340 --> 00:45:57,990
But the same approach-- if
you just write it down--

792
00:45:57,990 --> 00:46:02,090
tells you how the amplitude
in the excited state evolves.

793
00:46:06,030 --> 00:46:09,680
So therefore, what we learn
from perturbation theory,

794
00:46:09,680 --> 00:46:14,970
that alpha prime and beta prime,
the wave, the coefficients

795
00:46:14,970 --> 00:46:22,090
of the wave function,
without emission of a photon,

796
00:46:22,090 --> 00:46:26,240
this evolution of
the wave function

797
00:46:26,240 --> 00:46:35,140
occurs with a
non-emission Hamiltonian.

798
00:46:47,000 --> 00:46:49,330
And this non-emission
Hamiltonian

799
00:46:49,330 --> 00:46:54,560
is our Hamiltonian
for the atomic system.

800
00:46:54,560 --> 00:46:59,460
But then we have to account
for spontaneous decay,

801
00:46:59,460 --> 00:47:02,695
and this is done by
this non-emission part.

802
00:47:06,090 --> 00:47:10,750
So occasionally
you may have heard

803
00:47:10,750 --> 00:47:13,150
that people wave
their hands and say,

804
00:47:13,150 --> 00:47:15,820
your system is described
by a Hamiltonian, which

805
00:47:15,820 --> 00:47:18,550
has an imaginary part for decay.

806
00:47:18,550 --> 00:47:20,990
And this is sort of
phenomenological.

807
00:47:20,990 --> 00:47:24,430
This is definitely not
the case, because, even

808
00:47:24,430 --> 00:47:26,260
if you have a
Hamiltonian like this,

809
00:47:26,260 --> 00:47:29,740
a pure state would simply
decay and remain a pure state.

810
00:47:29,740 --> 00:47:32,490
What we are doing
here explicitly

811
00:47:32,490 --> 00:47:36,410
is we deal with probability
in the correct way.

812
00:47:36,410 --> 00:47:39,120
Every time the
system can branch out

813
00:47:39,120 --> 00:47:43,650
into different possibilities,
two different dimensions

814
00:47:43,650 --> 00:47:47,750
of the density matrix-- one
has probability delta p,

815
00:47:47,750 --> 00:47:50,610
one has probability
1 minus delta p--

816
00:47:50,610 --> 00:47:53,330
and it is only the wave
function associated

817
00:47:53,330 --> 00:47:56,260
with probability 1
minus delta p, which

818
00:47:56,260 --> 00:47:59,120
evolves with this Hamiltonian.

819
00:47:59,120 --> 00:48:02,770
So what I'm telling you
is the exact solution

820
00:48:02,770 --> 00:48:06,970
for the evolution
of the total system,

821
00:48:06,970 --> 00:48:10,000
in terms of a density matrix
for the atomic system.

822
00:48:13,260 --> 00:48:14,510
And this is exact.

823
00:48:18,430 --> 00:48:34,060
So that means we
can now-- write down

824
00:48:34,060 --> 00:48:38,620
the-- Just get some extra space.

825
00:48:41,816 --> 00:48:42,315
Oops.

826
00:48:42,315 --> 00:48:43,520
How does it do it?

827
00:48:51,392 --> 00:49:06,690
Nope So we have to
insert new page.

828
00:49:06,690 --> 00:49:09,070
Yep.

829
00:49:09,070 --> 00:49:09,570
OK.

830
00:49:15,540 --> 00:49:16,040
OK.

831
00:49:16,040 --> 00:49:24,200
So the procedure, how we
implement this exact solution

832
00:49:24,200 --> 00:49:27,590
of perturbation theory,
is the following.

833
00:49:27,590 --> 00:49:31,740
We have a time step delta t.

834
00:49:31,740 --> 00:49:36,290
We compute what
is the probability

835
00:49:36,290 --> 00:49:43,220
that a photon will be emitted.

836
00:49:46,090 --> 00:49:53,650
Then we need a ran-- Then we
need a random number generator.

837
00:49:53,650 --> 00:49:56,630
So we need a number
epsilon, which

838
00:49:56,630 --> 00:50:03,290
is a random number chosen
in the interval 0 and 1.

839
00:50:03,290 --> 00:50:11,780
If this random number turns
out to be smaller than delta p,

840
00:50:11,780 --> 00:50:17,660
then we continue our time
evolution on the computer.

841
00:50:17,660 --> 00:50:22,170
That psi is now in
the ground state,

842
00:50:22,170 --> 00:50:24,660
and maybe there is a laser
beam which excites it again.

843
00:50:24,660 --> 00:50:26,880
And such I will add a few
bells and whistles later.

844
00:50:29,540 --> 00:50:36,380
Otherwise psi,
our wave function,

845
00:50:36,380 --> 00:50:42,490
is now the wave function
which is the time evolution

846
00:50:42,490 --> 00:50:46,310
with a non-emission operator of
the original wave function psi.

847
00:50:46,310 --> 00:50:48,200
And since we have a
real wave function now

848
00:50:48,200 --> 00:50:51,837
with probability 1, we have to
re-normalize the wave function

849
00:50:51,837 --> 00:50:52,670
by this denominator.

850
00:51:02,490 --> 00:51:05,510
And then we execute
the next time step.

851
00:51:05,510 --> 00:51:09,720
This means go to 2 and
do the next time step.

852
00:51:09,720 --> 00:51:12,850
And then you have, so to speak,
if you do it many, many times,

853
00:51:12,850 --> 00:51:15,980
you get a time direct
trajectory of, so to speak,

854
00:51:15,980 --> 00:51:17,590
one experiment.

855
00:51:17,590 --> 00:51:20,510
And then you start again with
your system in a wave function

856
00:51:20,510 --> 00:51:23,530
psi, and you accumulate
a second experiment.

857
00:51:23,530 --> 00:51:26,750
And maybe you do it 10,000
times to get enough statistics.

858
00:51:26,750 --> 00:51:28,820
And then you sum up,
you know, everything

859
00:51:28,820 --> 00:51:31,130
you want and calculate
all the expectation

860
00:51:31,130 --> 00:51:34,240
values you want to know
about your quantum system.

861
00:51:39,260 --> 00:51:50,810
So the claim is that this
method, called quantum Monte

862
00:51:50,810 --> 00:51:53,760
Carlo wave function
method, is fully

863
00:51:53,760 --> 00:51:58,260
equivalent to the
optical block equations.

864
00:51:58,260 --> 00:52:04,080
And I want to prove
it to you by showing

865
00:52:04,080 --> 00:52:11,850
that, if I take a density
matrix, which is an ensemble

866
00:52:11,850 --> 00:52:22,080
average, over many
realizations of those quantum

867
00:52:22,080 --> 00:52:25,510
Monte Carlo trajectories.

868
00:52:25,510 --> 00:52:30,882
That then this density matrix
follows the differential

869
00:52:30,882 --> 00:52:32,840
equation, which is your
optical block equation.

870
00:52:37,670 --> 00:52:40,710
And the proof is shown here.

871
00:52:40,710 --> 00:52:45,490
So what I told you is the
density matrix after time delta

872
00:52:45,490 --> 00:52:49,000
t, has now two matrix elements.

873
00:52:49,000 --> 00:52:51,660
One with probability
delta p, and this

874
00:52:51,660 --> 00:52:53,190
is-- photon has been emitted.

875
00:52:53,190 --> 00:52:57,190
System is in the ground state,
with probability 1 minus delta

876
00:52:57,190 --> 00:52:59,154
p.

877
00:52:59,154 --> 00:53:04,210
We evolve the quantum state
with a non-emission Hamiltonian.

878
00:53:07,570 --> 00:53:11,990
And then all what is done
in the next few steps we

879
00:53:11,990 --> 00:53:14,320
assume that delta t is small.

880
00:53:14,320 --> 00:53:16,870
We do a Taylor expansion
of the exponent.

881
00:53:16,870 --> 00:53:19,390
We neglect quadratic
terms in delta t,

882
00:53:19,390 --> 00:53:20,950
and then we come to this line.

883
00:53:24,410 --> 00:53:28,640
And if we write this as
rho of t plus delta t,

884
00:53:28,640 --> 00:53:32,840
minus the original
rho of t, we find

885
00:53:32,840 --> 00:53:38,130
that this follows differential
equation, which is exactly

886
00:53:38,130 --> 00:53:40,730
the optical block equations.

887
00:53:40,730 --> 00:53:43,320
So therefore what we
sort of implemented

888
00:53:43,320 --> 00:53:45,930
as a form of doing many,
many quantum measurements

889
00:53:45,930 --> 00:53:50,130
in the environment
is a procedure,

890
00:53:50,130 --> 00:53:54,155
which is rigorously the
same as the optical block

891
00:53:54,155 --> 00:53:58,100
equations, which we derived
using a master equation

892
00:53:58,100 --> 00:53:58,600
approach.

893
00:54:06,030 --> 00:54:07,310
OK.

894
00:54:07,310 --> 00:54:11,960
This method is very powerful
for the following reasons.

895
00:54:11,960 --> 00:54:16,746
If you simulate a density
matrix with, you know,

896
00:54:16,746 --> 00:54:21,600
an internal states, or external,
internal states-- I should've

897
00:54:21,600 --> 00:54:23,700
said with in quantum
states, you need

898
00:54:23,700 --> 00:54:27,320
n times n matrix elements,
which can become quite a memory

899
00:54:27,320 --> 00:54:29,230
hog for your computer.

900
00:54:29,230 --> 00:54:33,820
The wave function, at any given
moment, as only n components.

901
00:54:33,820 --> 00:54:38,150
So therefore there is a
computational advantage

902
00:54:38,150 --> 00:54:42,360
in using a stochastic
wave function

903
00:54:42,360 --> 00:54:46,120
approach over simulation
of the density matrix.

904
00:54:46,120 --> 00:54:48,990
The second thing is that a
lot of people, especially

905
00:54:48,990 --> 00:54:51,930
experimentalists, like
sort of this approach,

906
00:54:51,930 --> 00:54:56,150
because it reflects directly
what they do in the experiment.

907
00:54:56,150 --> 00:55:00,050
And I think it's pretty
clear that with this approach

908
00:55:00,050 --> 00:55:06,890
you can deal with a great
variety of situations.

909
00:55:06,890 --> 00:55:10,050
Let me just mention
two obvious extensions.

910
00:55:10,050 --> 00:55:14,360
One is polarization.

911
00:55:21,510 --> 00:55:22,500
A photon is detected.

912
00:55:25,040 --> 00:55:28,570
That means your random number
produced a number epsilon,

913
00:55:28,570 --> 00:55:31,270
which was smaller than delta p.

914
00:55:31,270 --> 00:55:41,260
At that point you can create
a second random number,

915
00:55:41,260 --> 00:55:48,680
which determines if your pull--
if the photon has been detected

916
00:55:48,680 --> 00:55:51,510
with sigma plus, sigma
minus, or pi polarization.

917
00:55:54,090 --> 00:55:57,770
Or you can discuss recoil.

918
00:56:02,270 --> 00:56:07,646
If the photon is-- If
the photon is detected,

919
00:56:07,646 --> 00:56:11,620
you use-- you throw on
another random function

920
00:56:11,620 --> 00:56:24,250
generator, which determines what
the k vector of the photon is.

921
00:56:24,250 --> 00:56:30,730
So what direction each part
k the photon has taken,

922
00:56:30,730 --> 00:56:34,560
and this determines
now what is the recoil,

923
00:56:34,560 --> 00:56:37,540
the change in
momentum of your atom.

924
00:56:37,540 --> 00:56:39,050
So you see, kind
of, you can start

925
00:56:39,050 --> 00:56:40,840
with an atom at 0 momentum.

926
00:56:40,840 --> 00:56:43,250
You can see it emits
a photon, and then

927
00:56:43,250 --> 00:56:45,110
your computer always
makes a choice

928
00:56:45,110 --> 00:56:46,550
based on the random number.

929
00:56:46,550 --> 00:56:48,190
And then you say,
OK my photon has now

930
00:56:48,190 --> 00:56:51,170
received a recoil
kick at 45 degrees.

931
00:56:51,170 --> 00:56:54,270
Now your wave function of
the photon is such and such,

932
00:56:54,270 --> 00:56:56,600
and if you add a laser
to the Hamiltonian,

933
00:56:56,600 --> 00:56:59,430
the recoil may have kicked--
may have Doppler shifted

934
00:56:59,430 --> 00:57:03,340
the resonance, but everything
is easily taken into account.

935
00:57:03,340 --> 00:57:05,980
So you see this quantum
Monte Carlo wave function

936
00:57:05,980 --> 00:57:08,960
method can easily be
extended to describe

937
00:57:08,960 --> 00:57:11,930
external degrees of freedom,
multiple laser fields, and all

938
00:57:11,930 --> 00:57:14,750
that.

939
00:57:14,750 --> 00:57:15,835
Any questions?

940
00:57:21,740 --> 00:57:22,390
OK.

941
00:57:22,390 --> 00:57:29,910
Let me now just
generalize the thought,

942
00:57:29,910 --> 00:57:34,500
but I think you know
already everything about it.

943
00:57:40,040 --> 00:57:42,760
We talked about the
master equation.

944
00:57:42,760 --> 00:57:45,820
We talked about the most
general master equation

945
00:57:45,820 --> 00:57:49,220
in the [INAUDIBLE] platform,
where those operators L

946
00:57:49,220 --> 00:57:51,420
are the jump operators.

947
00:57:51,420 --> 00:57:54,960
And the prominent
example for jump operator

948
00:57:54,960 --> 00:57:57,740
was the signal minus
operator, which

949
00:57:57,740 --> 00:58:00,780
takes a particle from the
excited state to the ground

950
00:58:00,780 --> 00:58:01,300
state.

951
00:58:01,300 --> 00:58:04,810
And this is the jump operator
for spontaneous emission.

952
00:58:04,810 --> 00:58:06,710
But you may have
many jump operators.

953
00:58:06,710 --> 00:58:09,770
May be spontaneous emission
of different photons

954
00:58:09,770 --> 00:58:11,700
with different
polarization and such.

955
00:58:11,700 --> 00:58:14,690
Or the cavity may lose
a photon, and then we

956
00:58:14,690 --> 00:58:16,190
have a jump operator
for the cavity.

957
00:58:16,190 --> 00:58:18,370
We went through that.

958
00:58:18,370 --> 00:58:22,370
Well that means
now in an exact way

959
00:58:22,370 --> 00:58:24,200
that the non-emission
Hamiltonian is

960
00:58:24,200 --> 00:58:27,820
the original Hamiltonian
with an imaginary part, which

961
00:58:27,820 --> 00:58:29,630
comes because of
those jump operators.

962
00:58:32,450 --> 00:58:40,680
And the general quantum Monte
Carlo wave function procedure

963
00:58:40,680 --> 00:58:44,630
is that first you ask,
has something happened?

964
00:58:44,630 --> 00:58:46,510
Has any jump happened?

965
00:58:46,510 --> 00:58:50,320
And this gives you the
probability delta p.

966
00:58:50,320 --> 00:58:55,610
And if-- If no
jump has happened,

967
00:58:55,610 --> 00:58:57,110
you just do a time
evolution with

968
00:58:57,110 --> 00:58:59,510
a non-emission Hamiltonian.

969
00:58:59,510 --> 00:59:10,290
But if a jump has
happened, then the jump

970
00:59:10,290 --> 00:59:14,690
is now-- the wave function is
projected by the jump operator.

971
00:59:14,690 --> 00:59:16,320
If you have a sigma
minus operator,

972
00:59:16,320 --> 00:59:18,160
it takes a particle
to the ground state.

973
00:59:18,160 --> 00:59:21,230
If you have several operators,
a different jump operator

974
00:59:21,230 --> 00:59:24,250
may take your particle
to another state.

975
00:59:24,250 --> 00:59:29,330
And you-- And you then
have a branching ratio

976
00:59:29,330 --> 00:59:31,500
that you know first
a jump has happened.

977
00:59:31,500 --> 00:59:34,800
And then the question is,
which jump has happened?

978
00:59:34,800 --> 00:59:37,110
And you just play the
probabilities game.

979
00:59:37,110 --> 00:59:39,690
So that you see that the quantum
Monte Carlo wave function

980
00:59:39,690 --> 00:59:42,400
method can be immediately
be generalized.

981
00:59:46,430 --> 00:59:53,510
So let me come back
to the special case

982
00:59:53,510 --> 00:59:56,290
of spontaneous emission.

983
00:59:56,290 --> 00:59:58,840
In that case, we have
only one jump operator,

984
00:59:58,840 --> 01:00:00,840
which is sigma minus.

985
01:00:00,840 --> 01:00:05,140
And the normalization is
the square root of gamma.

986
01:00:05,140 --> 01:00:12,460
And that actually means that
our quantum trajectory ground

987
01:00:12,460 --> 01:00:16,160
state, excited state,
is very simple.

988
01:00:16,160 --> 01:00:21,310
If you start with the system
100% in the excited state,

989
01:00:21,310 --> 01:00:24,110
it's very trivial what happens.

990
01:00:24,110 --> 01:00:25,370
Nothing happens.

991
01:00:25,370 --> 01:00:27,870
But if then the jump
occurs, the particle

992
01:00:27,870 --> 01:00:30,090
is in the ground state.

993
01:00:30,090 --> 01:00:34,370
Then nothing happens
then until infinite time.

994
01:00:34,370 --> 01:00:38,020
You then have to start--
The next trajectory

995
01:00:38,020 --> 01:00:40,000
you start in the excited state.

996
01:00:40,000 --> 01:00:44,240
And this time, by chance of
the random number generator,

997
01:00:44,240 --> 01:00:46,070
the jump happens later.

998
01:00:46,070 --> 01:00:49,260
Another time the
jump happens earlier.

999
01:00:49,260 --> 01:00:53,270
And what you then have to
do is, you have to average,

1000
01:00:53,270 --> 01:00:56,200
over all of those
realizations, what

1001
01:00:56,200 --> 01:00:58,830
is the probability
for the particle

1002
01:00:58,830 --> 01:01:00,750
to be in the excited state.

1003
01:01:00,750 --> 01:01:06,080
And this is now your estimator
for the excited state diagonal

1004
01:01:06,080 --> 01:01:08,648
matrix element of
the density matrix.

1005
01:01:08,648 --> 01:01:13,260
And if you-- If you've
written your code correctly,

1006
01:01:13,260 --> 01:01:17,150
you will find that you get a
wonderful exponential decay

1007
01:01:17,150 --> 01:01:18,960
exactly as you have expected.

1008
01:01:29,475 --> 01:01:29,975
Questions?

1009
01:01:35,350 --> 01:01:38,660
Now-- Yes?

1010
01:01:38,660 --> 01:01:42,140
Let-- Let me now
talk about dephasing.

1011
01:01:49,530 --> 01:01:53,410
What I've shown you so far is
how we can unravel the density

1012
01:01:53,410 --> 01:01:56,530
matrix in many
microscopic realizations.

1013
01:01:56,530 --> 01:01:59,280
And I've given you
a specific example.

1014
01:01:59,280 --> 01:02:04,660
We assume probability
for spontaneous emission,

1015
01:02:04,660 --> 01:02:07,790
and with that we propagate
our wave function.

1016
01:02:07,790 --> 01:02:11,560
But now I come to, sort of,
the nitty gritty details,

1017
01:02:11,560 --> 01:02:16,220
or the dirty truth, that what
we have assumed is by no means

1018
01:02:16,220 --> 01:02:17,330
unique.

1019
01:02:17,330 --> 01:02:20,910
And I want to explain it to
you first by reminding you

1020
01:02:20,910 --> 01:02:25,080
of a very, very nice homework
assignment you have solved.

1021
01:02:25,080 --> 01:02:28,600
And this was about we have
optical block equations

1022
01:02:28,600 --> 01:02:30,810
for the density matrix,
and the dense --

1023
01:02:30,810 --> 01:02:34,920
and the solution of the optical
block equation is that we have

1024
01:02:34,920 --> 01:02:39,050
population damping
with a damping time t1.

1025
01:02:39,050 --> 01:02:41,320
And the off diagonal
matrix elements

1026
01:02:41,320 --> 01:02:44,370
are damped with a time t2.

1027
01:02:44,370 --> 01:02:48,000
Remember if you've only
spontaneous emission t2

1028
01:02:48,000 --> 01:02:50,070
is 2 times t1.

1029
01:02:50,070 --> 01:02:53,310
But we can have a lot of
other processes, which

1030
01:02:53,310 --> 01:02:56,250
can lead to much,
much faster defacing

1031
01:02:56,250 --> 01:02:59,990
time than the population, than
the time of population changes.

1032
01:02:59,990 --> 01:03:04,850
And in your homework, you have
discussed three possibilities.

1033
01:03:04,850 --> 01:03:10,680
One is, well, spontaneous
emission is energy loss.

1034
01:03:10,680 --> 01:03:14,690
But you have discussed three
different possibilities for t2.

1035
01:03:14,690 --> 01:03:17,620
One-- Three different
possibilities

1036
01:03:17,620 --> 01:03:20,500
how dephasing, loss of
coherence, and phase damping

1037
01:03:20,500 --> 01:03:22,030
can happen.

1038
01:03:22,030 --> 01:03:25,980
And in this homework
assignment, you assumed either

1039
01:03:25,980 --> 01:03:29,810
that there is a random phase,
that the elastic collision

1040
01:03:29,810 --> 01:03:32,400
which project onto
the excited state,

1041
01:03:32,400 --> 01:03:35,560
and then you got the most
crazy and artificial model.

1042
01:03:35,560 --> 01:03:40,430
That every time, randomly,
the phase of the excited state

1043
01:03:40,430 --> 01:03:44,040
flips form plus to minus.

1044
01:03:44,040 --> 01:03:48,990
So if I would implement that
now, with a quantum Monte Carlo

1045
01:03:48,990 --> 01:03:52,470
method, it would have
the following affect.

1046
01:03:52,470 --> 01:03:56,080
And I hope you enjoy, sort of,
the graphical representation.

1047
01:03:56,080 --> 01:03:58,620
You can really think about
it, that this is what happens.

1048
01:03:58,620 --> 01:04:01,300
That this is what inside--
what is inside the density

1049
01:04:01,300 --> 01:04:03,820
matrices described
by this process.

1050
01:04:03,820 --> 01:04:08,420
So let's assume we
have a system, which

1051
01:04:08,420 --> 01:04:11,410
would be in a superposition
of ground and excited state.

1052
01:04:11,410 --> 01:04:13,430
And that would mean
that in the left frame

1053
01:04:13,430 --> 01:04:16,770
the dipole moment would
just rotate at the resonance

1054
01:04:16,770 --> 01:04:17,870
frequency.

1055
01:04:17,870 --> 01:04:20,810
It rotates, and it is a
rotating dipole moment,

1056
01:04:20,810 --> 01:04:23,810
which emits coherent light.

1057
01:04:23,810 --> 01:04:26,630
But now you assume that
you have a random phase.

1058
01:04:26,630 --> 01:04:28,880
So if you add
random phases to it,

1059
01:04:28,880 --> 01:04:33,140
maybe because you're
fluctuating magnetic fields,

1060
01:04:33,140 --> 01:04:36,520
then the line becomes sort
of-- That doesn't look random,

1061
01:04:36,520 --> 01:04:37,620
but you know what I mean.

1062
01:04:37,620 --> 01:04:39,120
It becomes sort of jagged.

1063
01:04:39,120 --> 01:04:41,710
The phase distribution
is still sort of there,

1064
01:04:41,710 --> 01:04:43,630
but there is a
jitter on top of it.

1065
01:04:43,630 --> 01:04:46,630
And as a result, the light
emitted by this dipole

1066
01:04:46,630 --> 01:04:51,320
is spectrally broadened

1067
01:04:51,320 --> 01:04:54,680
Well in your second
model you assumed

1068
01:04:54,680 --> 01:04:58,400
that-- with a
certain randomness,

1069
01:04:58,400 --> 01:05:00,140
with a certain time
constant-- there's

1070
01:05:00,140 --> 01:05:02,720
an elastic collision in
the excited state, which

1071
01:05:02,720 --> 01:05:05,210
projects the system
into the excited state.

1072
01:05:05,210 --> 01:05:08,360
At this moment there is no
superposition state anymore,

1073
01:05:08,360 --> 01:05:11,350
and the dipole moment is 0.

1074
01:05:11,350 --> 01:05:12,790
So that's what you assume.

1075
01:05:12,790 --> 01:05:15,100
Or when you have
random phase flips,

1076
01:05:15,100 --> 01:05:18,730
you assumed that
suddenly this kind

1077
01:05:18,730 --> 01:05:21,350
of data to data
ministic sine function

1078
01:05:21,350 --> 01:05:27,600
suddenly jumps corresponding
to a minus sign in the excited

1079
01:05:27,600 --> 01:05:28,160
state.

1080
01:05:28,160 --> 01:05:30,410
And depending where
you are in the cycle,

1081
01:05:30,410 --> 01:05:32,430
it creates jumps
at random places.

1082
01:05:39,660 --> 01:05:45,295
And the question of course is,
if all those three processes--

1083
01:05:45,295 --> 01:05:48,400
and that's what you showed
in your homework-- lead

1084
01:05:48,400 --> 01:05:52,810
to the same density matrix,
which one is correct?

1085
01:05:52,810 --> 01:05:55,550
Which one is real--
is really realized

1086
01:05:55,550 --> 01:06:00,090
in an experiment on a
system which is described

1087
01:06:00,090 --> 01:06:02,640
by this kind of damping, or
by the optical block equation?

1088
01:06:06,860 --> 01:06:11,380
So you can say all or none.

1089
01:06:11,380 --> 01:06:13,670
Well you can assume
what you want,

1090
01:06:13,670 --> 01:06:15,690
it doesn't make a difference.

1091
01:06:15,690 --> 01:06:17,210
And the reason is
actually profound.

1092
01:06:20,210 --> 01:06:22,970
The reason-- The reason
is profound in that sense

1093
01:06:22,970 --> 01:06:28,250
that the way-- our goal
was to have an open quantum

1094
01:06:28,250 --> 01:06:32,380
system, which is the
atomic system, interacting

1095
01:06:32,380 --> 01:06:34,620
with a reservoir.

1096
01:06:34,620 --> 01:06:36,720
But the way, how we
phrase the question,

1097
01:06:36,720 --> 01:06:39,800
we are only interested in
what the atomic system does.

1098
01:06:39,800 --> 01:06:42,640
We didn't do extra measurements
on the environment.

1099
01:06:42,640 --> 01:06:46,760
The environment was just a dump
for photons; a dump for energy;

1100
01:06:46,760 --> 01:06:51,040
a dump for whatever we assumed
in our dephasing mechanism.

1101
01:06:51,040 --> 01:06:54,590
So let me just take the
example of putting photons

1102
01:06:54,590 --> 01:06:56,450
into the environment.

1103
01:06:56,450 --> 01:06:59,490
And this is described by
a certain damping term,

1104
01:06:59,490 --> 01:07:01,940
but there is an ambiguity.

1105
01:07:01,940 --> 01:07:05,580
And this is shown here.

1106
01:07:05,580 --> 01:07:14,140
If you evolve the system-- our,
our atomic system, the density

1107
01:07:14,140 --> 01:07:18,570
matrix evolves, but the
environment sort of also

1108
01:07:18,570 --> 01:07:19,090
evolves.

1109
01:07:19,090 --> 01:07:22,200
And for instance here,
the unitary time evolution

1110
01:07:22,200 --> 01:07:24,720
has taken the excitation
from the atom,

1111
01:07:24,720 --> 01:07:27,470
and we have emitted a
photon into the environment.

1112
01:07:27,470 --> 01:07:30,370
And you remember we got
the optical block equation

1113
01:07:30,370 --> 01:07:33,190
by doing the partial trace
here, and just focusing

1114
01:07:33,190 --> 01:07:35,650
on the atomic part.

1115
01:07:35,650 --> 01:07:37,930
But now wait a moment.

1116
01:07:37,930 --> 01:07:41,560
If we trace out
the environment, we

1117
01:07:41,560 --> 01:07:45,350
can trace it out in a
different basis set.

1118
01:07:45,350 --> 01:07:47,740
Remember, I showed you the
quantum Monte Carlo wave

1119
01:07:47,740 --> 01:07:49,900
function, we emit a photon.

1120
01:07:49,900 --> 01:07:53,510
But we can, for instance,
detect the photon

1121
01:07:53,510 --> 01:07:55,400
in our gedanken experiment.

1122
01:07:55,400 --> 01:07:57,475
We can detect it with
linear polarization

1123
01:07:57,475 --> 01:07:59,830
or with circular polarization.

1124
01:07:59,830 --> 01:08:03,560
If you have a situation where
you have a-- an s state, which

1125
01:08:03,560 --> 01:08:07,990
decays to a p state, sigma plus
takes you to m equals plus 1,

1126
01:08:07,990 --> 01:08:10,820
sigma minus takes you
to m equals minus 1.

1127
01:08:10,820 --> 01:08:14,330
So therefore, if photon is
emitted in you quantum Monte

1128
01:08:14,330 --> 01:08:17,390
Carlo procedure, you would
say, now the atom mean

1129
01:08:17,390 --> 01:08:19,470
is in the plus 1
state, or now the atom

1130
01:08:19,470 --> 01:08:21,189
is in the minus 1 state.

1131
01:08:21,189 --> 01:08:24,340
But if you detect
linearly polarized light,

1132
01:08:24,340 --> 01:08:26,229
by just putting a
polarizer here--

1133
01:08:26,229 --> 01:08:28,850
and this is a unitary
transformation in front

1134
01:08:28,850 --> 01:08:31,750
of your detector-- you
would no longer project

1135
01:08:31,750 --> 01:08:34,990
the atomic system on
plus 1 at minus 1,

1136
01:08:34,990 --> 01:08:36,760
you would project
it on m equals 0

1137
01:08:36,760 --> 01:08:39,900
or whatever is connected
to the linear polarization

1138
01:08:39,900 --> 01:08:41,790
of the measurement.

1139
01:08:41,790 --> 01:08:44,510
So therefore, if
you simply assume

1140
01:08:44,510 --> 01:08:46,720
that something has been
dumped in the environment,

1141
01:08:46,720 --> 01:08:49,359
and can be used for
measurement, there

1142
01:08:49,359 --> 01:08:52,680
are many ways, many
unitary transformations,

1143
01:08:52,680 --> 01:08:56,580
what you can do to the
information, to the energy,

1144
01:08:56,580 --> 01:08:59,609
to the photons, which have been
dumped into the environment.

1145
01:08:59,609 --> 01:09:04,640
And each of them will lead to
a very different trajectory

1146
01:09:04,640 --> 01:09:07,590
in your quantum
Monte Carlo system.

1147
01:09:07,590 --> 01:09:09,870
So therefore if you
just dump the photons

1148
01:09:09,870 --> 01:09:12,390
and not measure them,
you have equal rights

1149
01:09:12,390 --> 01:09:15,990
to assume that your quantum
Monte Carlo wave function jumps

1150
01:09:15,990 --> 01:09:19,479
to states which correspond
to a circular basis,

1151
01:09:19,479 --> 01:09:23,540
or jump to states which
correspond to linear basis.

1152
01:09:23,540 --> 01:09:26,649
And there's many,
many possibilities.

1153
01:09:26,649 --> 01:09:33,649
But each of those possibilities
is 100% correct unravelling

1154
01:09:33,649 --> 01:09:35,970
of the density matrix.

1155
01:09:35,970 --> 01:09:40,330
And if you just do it right,
assume that the measurement is

1156
01:09:40,330 --> 01:09:43,640
done in a certain basis, and
you're consistent with it,

1157
01:09:43,640 --> 01:09:47,490
you will 100% correctly
describe the time evolution

1158
01:09:47,490 --> 01:09:48,670
of your atomic system.

1159
01:09:54,540 --> 01:09:56,243
Any questions?

1160
01:09:56,243 --> 01:09:57,229
Philmore?

1161
01:09:57,229 --> 01:09:59,201
AUDIENCE: Again with the dt.

1162
01:09:59,201 --> 01:10:02,652
I'm just curious that we take
a [INAUDIBLE] very short,

1163
01:10:02,652 --> 01:10:05,610
but why don't you
run into something

1164
01:10:05,610 --> 01:10:07,582
like the quantum Zeno effect?

1165
01:10:07,582 --> 01:10:10,211
Where if every evolution
is initially quadratic,

1166
01:10:10,211 --> 01:10:13,730
like the [INAUDIBLE] frequency,
you take very short dt--

1167
01:10:13,730 --> 01:10:17,420
short measuring system--
Shouldn't for a choice

1168
01:10:17,420 --> 01:10:22,865
of sufficiently short dt you,
you know the simulation would

1169
01:10:22,865 --> 01:10:25,580
give you strange
results, because of--

1170
01:10:25,580 --> 01:10:26,330
PROFESSOR: [SIGHS]

1171
01:10:26,330 --> 01:10:29,300
AUDIENCE: --some sort of
quantum Zeno effect there?

1172
01:10:31,929 --> 01:10:33,970
PROFESSOR: That's a very
deep question, Philmore.

1173
01:10:33,970 --> 01:10:36,600
And I, I want to
think about it more,

1174
01:10:36,600 --> 01:10:40,310
but I think I've
excluded that by saying

1175
01:10:40,310 --> 01:10:43,475
that the time step, dt, is
larger than the correlation

1176
01:10:43,475 --> 01:10:45,470
time of the environment.

1177
01:10:45,470 --> 01:10:48,560
So I'm doing some kind
of Markov approximation

1178
01:10:48,560 --> 01:10:53,390
with the environment, where
s-- the quadratic part

1179
01:10:53,390 --> 01:10:55,640
of the behavior
is a coherent time

1180
01:10:55,640 --> 01:10:59,060
evolution, for very short time
steps a wave function evolves

1181
01:10:59,060 --> 01:11:02,640
quadratically, but this part
is a coherent evolution.

1182
01:11:02,640 --> 01:11:06,290
And that's related to the fact
that an atom can emit a photon,

1183
01:11:06,290 --> 01:11:09,985
but in the very first
moment, before the vacuum

1184
01:11:09,985 --> 01:11:13,330
has transported it away, it
can take the photon back.

1185
01:11:13,330 --> 01:11:16,250
And the result of that is
that at very short times

1186
01:11:16,250 --> 01:11:18,710
the exponential decay doesn't
start out exponentially,

1187
01:11:18,710 --> 01:11:21,440
it starts out a
little bit flatter.

1188
01:11:21,440 --> 01:11:24,480
We get the exponential decay,
we get Fermi's golden rule,

1189
01:11:24,480 --> 01:11:28,242
and we get optical blocks
equations only if your time

1190
01:11:28,242 --> 01:11:29,700
step is larger than
the correlation

1191
01:11:29,700 --> 01:11:32,100
time of the reservoir.

1192
01:11:32,100 --> 01:11:34,740
And the fact, you remember
when we derived the master

1193
01:11:34,740 --> 01:11:39,230
equation we had to say we do
a step, which is sufficiently

1194
01:11:39,230 --> 01:11:42,450
small for the dynamics
of the atomic system,

1195
01:11:42,450 --> 01:11:46,980
but sufficiently large, that we
are not getting into any memory

1196
01:11:46,980 --> 01:11:49,044
affects of the reservoir.

1197
01:11:49,044 --> 01:11:50,710
And we've done the
same assumption here.

1198
01:11:55,370 --> 01:11:57,580
Other questions?

1199
01:11:57,580 --> 01:11:59,146
Nicky?

1200
01:11:59,146 --> 01:12:00,146
AUDIENCE: Just confused.

1201
01:12:00,146 --> 01:12:02,664
We are not always say
the environment measures

1202
01:12:02,664 --> 01:12:07,083
the system in range at
time intervals, delta t,

1203
01:12:07,083 --> 01:12:11,993
which are large compared
to the correlation time?

1204
01:12:11,993 --> 01:12:18,376
What's more [INAUDIBLE]
the evolution of the atom.

1205
01:12:18,376 --> 01:12:21,813
But now everyone isn't
that defected the Markov

1206
01:12:21,813 --> 01:12:28,210
approximation we've made when
we derived the master equation?

1207
01:12:28,210 --> 01:12:33,220
And I was just wondering
for just-- [INAUDIBLE]

1208
01:12:33,220 --> 01:12:35,560
did you derive it by
making the master equation?

1209
01:12:35,560 --> 01:12:37,854
Maybe we don't
actually need-- Maybe

1210
01:12:37,854 --> 01:12:40,722
we don't need the idea
of constantly measuring.

1211
01:12:40,722 --> 01:12:45,084
Maybe a physical interpretation
of the Markov approximation.

1212
01:12:45,084 --> 01:12:46,500
PROFESSOR: Oh yes,
exactly, Nicky.

1213
01:12:46,500 --> 01:12:50,010
I still think you said it
really very, very nicely.

1214
01:12:50,010 --> 01:12:52,300
We derived a master
equation just

1215
01:12:52,300 --> 01:12:57,540
by assuming that the system,
you know, takes a photon.

1216
01:12:57,540 --> 01:13:00,050
And past the correlation
time, the photon

1217
01:13:00,050 --> 01:13:02,700
is taken for good
by the environment.

1218
01:13:02,700 --> 01:13:05,870
And with a Markov
approximation, we formalized

1219
01:13:05,870 --> 01:13:09,540
that there is no memory
affect, the photon

1220
01:13:09,540 --> 01:13:11,340
is not stored like in a cavity.

1221
01:13:11,340 --> 01:13:13,290
The photon has disappeared.

1222
01:13:13,290 --> 01:13:15,940
And by saying that this
time is very short,

1223
01:13:15,940 --> 01:13:18,945
this was a Markov approximation
in the derivation of the master

1224
01:13:18,945 --> 01:13:20,240
equation.

1225
01:13:20,240 --> 01:13:24,780
But that also means, if
the photon has disappeared,

1226
01:13:24,780 --> 01:13:27,050
has separated from
the atomic system,

1227
01:13:27,050 --> 01:13:30,500
we are now free to
make a measurement.

1228
01:13:30,500 --> 01:13:31,910
And of course it
shouldn't matter

1229
01:13:31,910 --> 01:13:34,840
whether we make the
measurement or not.

1230
01:13:34,840 --> 01:13:37,280
But what I was able
to do today is,

1231
01:13:37,280 --> 01:13:39,870
I was making now the
assumption, let's assume

1232
01:13:39,870 --> 01:13:41,180
we make a measurement.

1233
01:13:41,180 --> 01:13:43,610
We measure all the photons
which have been emitted,

1234
01:13:43,610 --> 01:13:46,320
and the measurement
is probabilistic.

1235
01:13:46,320 --> 01:13:50,080
And we fold this
probabilistic evolution

1236
01:13:50,080 --> 01:13:51,890
into our quantum system.

1237
01:13:51,890 --> 01:13:56,330
And what we obtained was
exactly the same time evolution

1238
01:13:56,330 --> 01:14:00,890
for the density matrix as we
got from the master equation.

1239
01:14:00,890 --> 01:14:02,610
So in other words,
this should you

1240
01:14:02,610 --> 01:14:05,190
that interactions
with an open quantum

1241
01:14:05,190 --> 01:14:10,090
system, where irreversibly
energy, photons, angular

1242
01:14:10,090 --> 01:14:13,200
momentum, or whatever--
flows into the environment.

1243
01:14:13,200 --> 01:14:16,200
Once it has flown
into the environment,

1244
01:14:16,200 --> 01:14:17,590
you can measure it.

1245
01:14:17,590 --> 01:14:20,160
And you get all the stochastics
from the measurement.

1246
01:14:20,160 --> 01:14:21,900
But even if you
don't measure it,

1247
01:14:21,900 --> 01:14:23,880
you get exactly the
same stochastics as

1248
01:14:23,880 --> 01:14:24,910
if you had measured it.

1249
01:14:24,910 --> 01:14:27,122
It's all the same.

1250
01:14:27,122 --> 01:14:27,622
Jenny?

1251
01:14:27,622 --> 01:14:28,110
AUDIENCE: Now that
I think about it,

1252
01:14:28,110 --> 01:14:31,038
it seems sort of weird to me
that we've-- that we're making

1253
01:14:31,038 --> 01:14:32,990
these measurements, or
lack of measurements,

1254
01:14:32,990 --> 01:14:36,403
these interactions,
at regular intervals.

1255
01:14:36,403 --> 01:14:37,140
And not--

1256
01:14:37,140 --> 01:14:38,640
PROFESSOR: It doesn't
really matter.

1257
01:14:38,640 --> 01:14:41,190
All we-- All we
have to make is--

1258
01:14:41,190 --> 01:14:43,940
You could actually make
delta t a random variable.

1259
01:14:43,940 --> 01:14:45,672
It wouldn't change anything.

1260
01:14:45,672 --> 01:14:47,130
The only thing we
have to make sure

1261
01:14:47,130 --> 01:14:50,300
is we have to make the
time interval short enough

1262
01:14:50,300 --> 01:14:54,150
that we don't have, maybe, two
photons emitted in that time.

1263
01:14:54,150 --> 01:14:55,810
We just want to
make sure that it's

1264
01:14:55,810 --> 01:15:00,639
a simple probabilistic
branching, yes or no.

1265
01:15:00,639 --> 01:15:02,180
And that's the only
requirement here.

1266
01:15:08,470 --> 01:15:10,595
AUDIENCE: Is this a formula
or is it based on time?

1267
01:15:10,595 --> 01:15:12,219
You say that would
be the wave function

1268
01:15:12,219 --> 01:15:13,565
evolving and then collapse.

1269
01:15:13,565 --> 01:15:19,505
And that's-- This is what--
Where the measurement comes in.

1270
01:15:19,505 --> 01:15:21,485
The equivalent, so now
say, the equivalent

1271
01:15:21,485 --> 01:15:23,960
is a wave function
evolved in a collapse,

1272
01:15:23,960 --> 01:15:26,435
when you make a measurement
you say that after it evolves.

1273
01:15:26,435 --> 01:15:28,910
So is it possible
that it could also

1274
01:15:28,910 --> 01:15:32,375
have-- like evolves
some kind of formalism?

1275
01:15:32,375 --> 01:15:33,860
Like this?

1276
01:15:33,860 --> 01:15:36,335
Where there is an operator?

1277
01:15:36,335 --> 01:15:39,800
Because still like even though
the density matrix operator,

1278
01:15:39,800 --> 01:15:42,740
maybe it is kind of like--
[INAUDIBLE] describes a state.

1279
01:15:50,620 --> 01:15:52,160
PROFESSOR: I haven't
seen it, Mark,

1280
01:15:52,160 --> 01:15:55,230
but I'm absolutely certain
that you could describe

1281
01:15:55,230 --> 01:15:57,810
the same physics in
the Heisenberg picture,

1282
01:15:57,810 --> 01:16:00,020
where the time evolution
is with operators.

1283
01:16:00,020 --> 01:16:03,780
Because what we have here is,
we have a time step, delta t,

1284
01:16:03,780 --> 01:16:07,420
where the system evolves as
an isolated quantum system,

1285
01:16:07,420 --> 01:16:09,200
and then we measure again.

1286
01:16:09,200 --> 01:16:12,870
And so, I think if you
would use operator equation,

1287
01:16:12,870 --> 01:16:16,770
you would have an
evolution of the operator

1288
01:16:16,770 --> 01:16:22,250
with the same Hamiltonian, which
is non-emission Hamiltonian.

1289
01:16:22,250 --> 01:16:25,775
And then the measurement would
do some form of projection.

1290
01:16:25,775 --> 01:16:28,160
I have to think about it,
what the projection would

1291
01:16:28,160 --> 01:16:29,700
be in terms of operators.

1292
01:16:29,700 --> 01:16:37,150
But my gut feeling is, you can
always take the transformation

1293
01:16:37,150 --> 01:16:39,390
where you put the time
dependence into the operators,

1294
01:16:39,390 --> 01:16:41,170
and not in the wave function.

1295
01:16:41,170 --> 01:16:43,870
On the other hand, I have
some little bit misgivings

1296
01:16:43,870 --> 01:16:46,460
about that, because the quantum
Monte Carlo wave function

1297
01:16:46,460 --> 01:16:51,380
was really developed in order
not to deal with matrices,

1298
01:16:51,380 --> 01:16:54,810
not to deal with something
which is dimension n times in,

1299
01:16:54,810 --> 01:16:57,550
if n is the number of
components of the wave function.

1300
01:16:57,550 --> 01:17:00,760
It was specifically formulated
to have the simpler description

1301
01:17:00,760 --> 01:17:02,720
with n coefficients
for the wave function.

1302
01:17:02,720 --> 01:17:07,790
But conceptually, I
think, it is evolution

1303
01:17:07,790 --> 01:17:09,560
with a Hamiltonian measurement.

1304
01:17:09,560 --> 01:17:11,530
Evolution with a
Hamiltonian measurement.

1305
01:17:11,530 --> 01:17:16,670
And I assume this could
also be done with operators.

1306
01:17:16,670 --> 01:17:17,250
Nancy?

1307
01:17:17,250 --> 01:17:20,112
AUDIENCE: Suddenly like
[INAUDIBLE] but when

1308
01:17:20,112 --> 01:17:22,840
we've-- these simulations
are actively done,

1309
01:17:22,840 --> 01:17:25,846
how important is the randomness?

1310
01:17:25,846 --> 01:17:31,921
Like up to epsilon [INAUDIBLE]
because no computer is actually

1311
01:17:31,921 --> 01:17:32,920
emitting random numbers.

1312
01:17:36,110 --> 01:17:39,370
PROFESSOR: Well I think how
random a random number has

1313
01:17:39,370 --> 01:17:42,940
to be is really dealt
with in computer science

1314
01:17:42,940 --> 01:17:43,830
and mathematics.

1315
01:17:43,830 --> 01:17:45,318
AUDIENCE: But due
to our reserves

1316
01:17:45,318 --> 01:17:47,798
we always get this
exponential behavior?

1317
01:17:52,812 --> 01:17:54,270
PROFESSOR: I don't
know the answer.

1318
01:17:54,270 --> 01:17:55,950
I mean here,
conceptionally, it should

1319
01:17:55,950 --> 01:17:58,330
be completely random number.

1320
01:17:58,330 --> 01:18:02,560
And I think that algorithms,
which even which produce

1321
01:18:02,560 --> 01:18:07,070
pseudo random numbers, but if
the pseudo is close enough,

1322
01:18:07,070 --> 01:18:09,940
if there is a recurrence
time of your random number--

1323
01:18:09,940 --> 01:18:12,640
if your random number--
your series of random number

1324
01:18:12,640 --> 01:18:18,380
repeat itself after several
billion random numbers,

1325
01:18:18,380 --> 01:18:20,230
this gives you a
small error bar.

1326
01:18:20,230 --> 01:18:23,580
And if you don't-- If you'd
only do a limited sampling,

1327
01:18:23,580 --> 01:18:25,930
a limited number of
time trajectories,

1328
01:18:25,930 --> 01:18:27,350
I don't think it matters.

1329
01:18:27,350 --> 01:18:29,940
Of course if you want to
have ultimate precision,

1330
01:18:29,940 --> 01:18:32,245
then also the precision
of the random number

1331
01:18:32,245 --> 01:18:34,120
may enter through the
backdoor at some point.

1332
01:18:36,845 --> 01:18:48,485
AUDIENCE: [INAUDIBLE]
lack of simulation

1333
01:18:48,485 --> 01:18:52,616
of the photons emitted?

1334
01:18:52,616 --> 01:18:54,560
But if we know the
photons emitted,

1335
01:18:54,560 --> 01:19:00,880
if we have that information,
then we can decide [INAUDIBLE].

1336
01:19:00,880 --> 01:19:02,260
PROFESSOR: That's correct.

1337
01:19:02,260 --> 01:19:14,440
If we-- If we would know--
If we would know more

1338
01:19:14,440 --> 01:19:16,490
about the environment,
if we would

1339
01:19:16,490 --> 01:19:19,970
say we measured the
photons in a certain basis,

1340
01:19:19,970 --> 01:19:23,140
or when energy is dumped
in the environment

1341
01:19:23,140 --> 01:19:26,940
by a dephasing mechanism,
we look at the environment.

1342
01:19:26,940 --> 01:19:31,070
Then, of course, we would
add extra information to it,

1343
01:19:31,070 --> 01:19:34,310
and then certain
quantum trajectories

1344
01:19:34,310 --> 01:19:37,190
would reflect the extra
knowledge we have.

1345
01:19:37,190 --> 01:19:39,870
And of course, we would
then describe everything

1346
01:19:39,870 --> 01:19:41,400
in this basis.

1347
01:19:41,400 --> 01:19:43,240
On the other hand,
what you should

1348
01:19:43,240 --> 01:19:47,490
learn from this is once the
information, once the photon

1349
01:19:47,490 --> 01:19:51,140
has escaped, it doesn't
matter for the time evolution

1350
01:19:51,140 --> 01:19:55,220
of your atomic system, in which
basis you measure the photon.

1351
01:19:55,220 --> 01:19:58,870
So the evolution of the
atomic system itself--

1352
01:19:58,870 --> 01:20:05,320
at least when you averaged--
is unaffected by the basis.

1353
01:20:05,320 --> 01:20:08,040
However if you make a
coincidence measurement,

1354
01:20:08,040 --> 01:20:10,430
that you say your atoms
which has emitted,

1355
01:20:10,430 --> 01:20:12,700
is still flying through
your vacuum chamber,

1356
01:20:12,700 --> 01:20:14,540
and now you figure
out the photon

1357
01:20:14,540 --> 01:20:17,150
is emitted as circularly
polarized light.

1358
01:20:17,150 --> 01:20:20,590
Then of course you know that
this atom, which is still

1359
01:20:20,590 --> 01:20:23,592
available, has emitted
circularly polarized light.

1360
01:20:23,592 --> 01:20:25,800
And then you have actually
obtained extra information

1361
01:20:25,800 --> 01:20:27,540
about your atom.

1362
01:20:27,540 --> 01:20:29,850
In other words, the
photon in the atom

1363
01:20:29,850 --> 01:20:32,760
was entangled, like a Bell pair.

1364
01:20:32,760 --> 01:20:36,020
And if you now do a measurement
on one part of the Bell pair,

1365
01:20:36,020 --> 01:20:38,190
you know more about
the other part.

1366
01:20:38,190 --> 01:20:40,190
But this is just how
quantum physics works.

1367
01:20:40,190 --> 01:20:42,870
If you don't use this
information for anything,

1368
01:20:42,870 --> 01:20:46,840
then you could have-- you
could have as well not measured

1369
01:20:46,840 --> 01:20:47,685
the polarization.

1370
01:20:47,685 --> 01:20:51,707
And it would have no effect
on the atomic density matrix.

1371
01:20:56,580 --> 01:20:58,290
OK that's it for today.

1372
01:20:58,290 --> 01:21:01,660
We have class at the
usual time on Wednesday.