1
00:00:00,350 --> 00:00:01,000
PROFESSOR: OK.

2
00:00:01,000 --> 00:00:05,140
So time to complicate
the model a little bit

3
00:00:05,140 --> 00:00:08,990
to get more interesting
physics from it.

4
00:00:08,990 --> 00:00:12,130
So what am I going to do?

5
00:00:12,130 --> 00:00:15,550
I'm going to add an extra term.

6
00:00:18,590 --> 00:00:21,880
So the system goes
now to an H of t

7
00:00:21,880 --> 00:00:24,940
that is going to have
the alpha t over 2.

8
00:00:24,940 --> 00:00:28,580
And now it's gonna have a
little term of diagonal.

9
00:00:40,080 --> 00:00:44,210
So there's several ways
of thinking about this.

10
00:00:44,210 --> 00:00:49,225
H12 is going to be
constant in time.

11
00:00:52,320 --> 00:00:54,930
So it's a number.

12
00:00:54,930 --> 00:00:57,800
H12 star is it's
complex conjugate.

13
00:00:57,800 --> 00:01:10,520
If you're looking at t equals
0, H becomes just H12, H12 star.

14
00:01:10,520 --> 00:01:15,680
And the energy eigenstate
or the energy eigenvalues

15
00:01:15,680 --> 00:01:22,010
of this matrix are plus
or minus the norm of H12,

16
00:01:22,010 --> 00:01:25,220
the absolute value of H12.

17
00:01:25,220 --> 00:01:31,220
Just put a couple of lambdas,
calculate the eigenvalues.

18
00:01:31,220 --> 00:01:34,850
It involves H12 times
H12 star, which is

19
00:01:34,850 --> 00:01:37,670
the square of the norm of H12.

20
00:01:37,670 --> 00:01:39,650
And the energies are those.

21
00:01:39,650 --> 00:01:46,730
So look at what's happening
in your energy diagram

22
00:01:46,730 --> 00:01:49,990
as a function of t.

23
00:01:49,990 --> 00:01:58,800
At t equals 0, there are two
energies, H12 and minus H12.

24
00:02:02,720 --> 00:02:05,060
Those are the energies
at time equals 0.

25
00:02:05,060 --> 00:02:08,690
We're trying to get the analog
of what was going on there.

26
00:02:08,690 --> 00:02:11,180
And now you could
say, OK, at time

27
00:02:11,180 --> 00:02:14,950
equals 0, that's what I get.

28
00:02:14,950 --> 00:02:17,800
What do I get at large times?

29
00:02:17,800 --> 00:02:20,560
Well, at large times,
these are dominant.

30
00:02:20,560 --> 00:02:22,040
And these are very small.

31
00:02:22,040 --> 00:02:26,570
So you must get something
similar to these arrows here.

32
00:02:26,570 --> 00:02:35,700
So what I'll draw
here is this and this.

33
00:02:35,700 --> 00:02:41,040
And I cannot trust this
here for small time.

34
00:02:41,040 --> 00:02:46,710
But presumably, this is about
right here and about right here

35
00:02:46,710 --> 00:02:50,370
and about right here
and about right here.

36
00:02:50,370 --> 00:02:56,640
And the states must be
the same one, 0, 1 here;

37
00:02:56,640 --> 00:03:03,860
0, 1; 1, 0; 1, 0.

38
00:03:03,860 --> 00:03:08,810
So that's what you know just
without doing any calculation.

39
00:03:08,810 --> 00:03:11,140
That's what your system does.

40
00:03:11,140 --> 00:03:14,600
And now, a
Hamiltonian in general

41
00:03:14,600 --> 00:03:18,230
doesn't get levels crossing.

42
00:03:18,230 --> 00:03:23,000
That requires a coincidence like
having no off-diagonal element.

43
00:03:23,000 --> 00:03:25,190
So actually, what this
will give you is this.

44
00:03:35,800 --> 00:03:39,910
So that's how the system will
look as a function of time.

45
00:03:39,910 --> 00:03:45,510
Those are the energy
levels as a function

46
00:03:45,510 --> 00:03:48,750
of time, the instantaneous
energy levels.

47
00:03:48,750 --> 00:03:52,360
At every instant of time,
you now have the energies.

48
00:03:52,360 --> 00:03:57,570
These are the energies
of this matrix,

49
00:03:57,570 --> 00:04:00,480
the energies, the eigenvalues.

50
00:04:03,570 --> 00:04:09,360
One line computation, our E
plus minus equal plus or minus

51
00:04:09,360 --> 00:04:15,430
square root of H12
squared plus alpha

52
00:04:15,430 --> 00:04:25,640
squared t squared over
4, so plus or minus.

53
00:04:25,640 --> 00:04:30,370
So this is E plus
and E minus of t.

54
00:04:34,730 --> 00:04:35,350
All right.

55
00:04:35,350 --> 00:04:39,172
So here we go.

56
00:04:39,172 --> 00:04:41,950
We have a real system.

57
00:04:41,950 --> 00:04:45,145
And we have something
quite interesting actually.

58
00:04:47,800 --> 00:04:54,340
When you let H12 go to 0,
you're back to that place.

59
00:04:54,340 --> 00:05:00,060
And you just zoom through
like here you did.

60
00:05:00,060 --> 00:05:03,350
You just go through
and go through.

61
00:05:03,350 --> 00:05:09,700
So when the levels
do that, you just

62
00:05:09,700 --> 00:05:14,260
continue through any
particular state you were in.

63
00:05:14,260 --> 00:05:17,290
You don't start here
and then go here.

64
00:05:17,290 --> 00:05:18,430
You just go through.

65
00:05:18,430 --> 00:05:20,960
That's what you prove here.

66
00:05:20,960 --> 00:05:28,120
And therefore, when you take
the limit of H12 going to 0,

67
00:05:28,120 --> 00:05:32,400
if H12 goes to 0,
these things collapse.

68
00:05:32,400 --> 00:05:33,340
And you're back there.

69
00:05:33,340 --> 00:05:35,660
And you zoom by through.

70
00:05:35,660 --> 00:05:42,170
So when H12 will be
very, very small,

71
00:05:42,170 --> 00:05:45,820
you will be likely,
in fact, more likely

72
00:05:45,820 --> 00:05:49,750
to make the transition
than not to make it.

73
00:05:49,750 --> 00:05:52,210
So if you have a system for--

74
00:05:52,210 --> 00:06:01,360
this is just a micron separated
here, a milli-electron volt--

75
00:06:01,360 --> 00:06:04,100
let's be more precise here--

76
00:06:04,100 --> 00:06:07,660
you're more likely to
zoom through and make

77
00:06:07,660 --> 00:06:11,680
the nonadiabatic transition
because this will not

78
00:06:11,680 --> 00:06:13,390
be an adiabatic process.

79
00:06:13,390 --> 00:06:16,670
They're getting to
close to each other.

80
00:06:16,670 --> 00:06:25,010
On the other hand,
if you are far away,

81
00:06:25,010 --> 00:06:27,770
you're going to be
very unlikely to make

82
00:06:27,770 --> 00:06:33,050
the nonadiabatic transition
because you are very separated.

83
00:06:33,050 --> 00:06:36,470
The nice thing
about this problem

84
00:06:36,470 --> 00:06:40,220
is that it can be
solved analytically.

85
00:06:40,220 --> 00:06:43,850
Not terribly easy.

86
00:06:43,850 --> 00:06:47,330
It involves a little bit
of hypergeometric functions

87
00:06:47,330 --> 00:06:48,840
and some differential equations.

88
00:06:48,840 --> 00:06:51,030
But it can be solved.

89
00:06:51,030 --> 00:06:54,740
You will solve it
numerically in the homework.

90
00:06:54,740 --> 00:06:57,320
I will say a few words about it.

91
00:06:57,320 --> 00:07:01,020
And we'll discuss the
transition in this case.

92
00:07:01,020 --> 00:07:03,710
And the answer is
known analytically.

93
00:07:03,710 --> 00:07:07,430
It's a very famous result.
In fact, many people

94
00:07:07,430 --> 00:07:10,970
have written papers trying
to give simple derivations

95
00:07:10,970 --> 00:07:14,120
of this answer.

96
00:07:14,120 --> 00:07:16,750
So in order to just
write the answer

97
00:07:16,750 --> 00:07:21,130
and to see how it looks--

98
00:07:21,130 --> 00:07:24,880
so this is the answer
for a transition--

99
00:07:24,880 --> 00:07:31,840
we try to discuss the notion
of adiabatic process here.

100
00:07:31,840 --> 00:07:35,470
When is this
transition adiabatic?

101
00:07:35,470 --> 00:07:38,320
And what we do is this.

102
00:07:38,320 --> 00:07:50,150
You can imagine taking a tangent
here and another tangent here.

103
00:07:50,150 --> 00:07:55,370
And when it hits the lines,
the linear lines that we

104
00:07:55,370 --> 00:07:58,380
plotted here, bring them down.

105
00:07:58,380 --> 00:08:00,290
So you get a rectangle here.

106
00:08:03,570 --> 00:08:07,780
So you hit the
alpha t over 2 line.

107
00:08:07,780 --> 00:08:17,740
And you call this 2 tau
d or 2 tau, no, tau d.

108
00:08:17,740 --> 00:08:18,620
OK.

109
00:08:18,620 --> 00:08:24,140
So let's figure out
what that is because I

110
00:08:24,140 --> 00:08:30,080
say that this time is
the time that going up

111
00:08:30,080 --> 00:08:36,320
on the line, alpha t over
2, this time 2 tau d,

112
00:08:36,320 --> 00:08:39,650
you get the height, H12.

113
00:08:39,650 --> 00:08:50,360
That means that H12 is alpha
times 2 tau d divided by 2.

114
00:08:50,360 --> 00:09:00,630
So tau d is H12 over alpha.

115
00:09:00,630 --> 00:09:08,350
So tau d is usually thought
as the timescale associated

116
00:09:08,350 --> 00:09:13,950
to the change in
the Hamiltonian.

117
00:09:13,950 --> 00:09:16,680
That is you have an
original Hamiltonian, that's

118
00:09:16,680 --> 00:09:19,660
where this linear [INAUDIBLE].

119
00:09:19,660 --> 00:09:23,460
And over a time 2d,
the shape is changed

120
00:09:23,460 --> 00:09:26,520
into the final Hamiltonian.

121
00:09:26,520 --> 00:09:32,220
This is the process in
which the original system

122
00:09:32,220 --> 00:09:36,810
is changed into the new
system within this timescale.

123
00:09:36,810 --> 00:09:39,780
The timescale tau d.

124
00:09:39,780 --> 00:09:44,550
But we have another time
that is interesting here.

125
00:09:44,550 --> 00:09:47,310
You see, while this
change is happening,

126
00:09:47,310 --> 00:09:55,230
the separation of the energy
levels is by this distance H12.

127
00:09:55,230 --> 00:09:59,540
So the Hamiltonian
at t equals 0,

128
00:09:59,540 --> 00:10:08,640
the Hamiltonian looks
like 0, H12, H12, here.

129
00:10:08,640 --> 00:10:14,280
And if you have two states
governed by such Hamiltonian

130
00:10:14,280 --> 00:10:17,380
that is valid near
time equals 0,

131
00:10:17,380 --> 00:10:20,910
there's going to be oscillations
between states here.

132
00:10:20,910 --> 00:10:24,720
You can go from the first
state to the second state

133
00:10:24,720 --> 00:10:28,800
with some frequency
governed by this number.

134
00:10:28,800 --> 00:10:37,420
That is called the Rabi
frequency, Rabi oscillation.

135
00:10:37,420 --> 00:10:39,370
It's something
you've done in 805.

136
00:10:39,370 --> 00:10:42,280
In two-state systems,
you oscillate.

137
00:10:42,280 --> 00:10:47,770
And the frequency of oscillation
between the states 1 and 2

138
00:10:47,770 --> 00:11:02,240
is H12 divided by h bar or 2
pi divided by the period T12.

139
00:11:02,240 --> 00:11:05,120
That's the definition
of the frequency.

140
00:11:05,120 --> 00:11:09,450
So what do we have here?

141
00:11:09,450 --> 00:11:19,740
The process is
adiabatic if the time

142
00:11:19,740 --> 00:11:29,850
tau d for the change to happen
is much larger than the period

143
00:11:29,850 --> 00:11:33,190
capital T12 of the oscillation.

144
00:11:33,190 --> 00:11:37,320
So you can think of this
system during the time

145
00:11:37,320 --> 00:11:41,280
it spends in this box like
a two-level system separated

146
00:11:41,280 --> 00:11:45,160
by H12 that oscillates
between the two states.

147
00:11:45,160 --> 00:11:50,820
And in that time, 2 tau d,
all the change is happening.

148
00:11:50,820 --> 00:11:56,640
So you should have that
tau d is much bigger

149
00:11:56,640 --> 00:12:00,100
than the period of oscillation.

150
00:12:00,100 --> 00:12:04,950
And this corresponds
too because T12

151
00:12:04,950 --> 00:12:08,840
is the inverse of
omega 12 to omega 12

152
00:12:08,840 --> 00:12:13,350
tau d much greater
than 1, so either one.

153
00:12:16,740 --> 00:12:23,420
And that would be
adiabatic condition.

154
00:12:25,930 --> 00:12:28,580
Let's put the numbers here.

155
00:12:28,580 --> 00:12:38,200
Omega 12 is H12 norm over h bar.

156
00:12:38,200 --> 00:12:42,240
And tau d-- we often
found it there--

157
00:12:42,240 --> 00:12:46,475
is H12 over alpha.

158
00:12:50,770 --> 00:12:52,970
So over alpha.

159
00:12:52,970 --> 00:12:54,920
So this is much greater than 1.

160
00:12:54,920 --> 00:12:57,580
I don't know.

161
00:12:57,580 --> 00:13:02,870
This 2 here has nothing
to do with that formula.

162
00:13:02,870 --> 00:13:05,590
So this is the
adiabatic condition.

163
00:13:09,580 --> 00:13:12,790
And this is reasonable.

164
00:13:12,790 --> 00:13:16,510
You see, what is going
to make it adiabatic?

165
00:13:16,510 --> 00:13:20,770
The more these branches are
separated, the more difficult

166
00:13:20,770 --> 00:13:25,060
the transition, the better
the adiabatic approximation.

167
00:13:25,060 --> 00:13:27,800
And it's this thing here.

168
00:13:27,800 --> 00:13:30,700
The bigger this number,
the better you are.

169
00:13:30,700 --> 00:13:35,020
The slower alpha is,
the lower the slope

170
00:13:35,020 --> 00:13:37,210
is, the more time
it's going to take,

171
00:13:37,210 --> 00:13:40,280
the better the
adiabatic approximation.

172
00:13:40,280 --> 00:13:43,900
So this is really the thing.

173
00:13:43,900 --> 00:13:46,540
And the final formula
that I will write here

174
00:13:46,540 --> 00:13:52,420
is the probability for a
nonadiabatic transition

175
00:13:52,420 --> 00:14:02,080
is exponential of minus
2 pi omega 12 tau d.

176
00:14:02,080 --> 00:14:08,200
So the probability that you
cross the thing and jump

177
00:14:08,200 --> 00:14:12,100
from this down to here is this.

178
00:14:12,100 --> 00:14:15,850
It's suppressed by
the adiabatic factor.

179
00:14:15,850 --> 00:14:18,780
And that's what you will
check in the homework.