1
00:00:00,050 --> 00:00:01,770
The following
content is provided
2
00:00:01,770 --> 00:00:04,010
under a Creative
Commons license.
3
00:00:04,010 --> 00:00:06,860
Your support will help MIT
OpenCourseWare continue
4
00:00:06,860 --> 00:00:10,720
to offer high quality
educational resources for free.
5
00:00:10,720 --> 00:00:13,330
To make a donation or
view additional materials
6
00:00:13,330 --> 00:00:17,202
from hundreds of MIT courses,
visit MIT OpenCourseWare
7
00:00:17,202 --> 00:00:17,827
at ocw.mit.edu.
8
00:00:20,465 --> 00:00:21,930
PROFESSOR: All right.
9
00:00:21,930 --> 00:00:26,860
Then let's go back to
our discussion of what
10
00:00:26,860 --> 00:00:34,050
happens to classical and quantum
mechanical magnetic moments
11
00:00:34,050 --> 00:00:40,200
when they are exposed
to magnetic fields.
12
00:00:40,200 --> 00:00:48,830
I just want to remind you
of what we did last class
13
00:00:48,830 --> 00:00:57,450
and what we want
to wrap up today.
14
00:00:57,450 --> 00:01:00,790
This is rapid adiabatic passage.
15
00:01:00,790 --> 00:01:05,650
I mentioned to you and
explained it to you
16
00:01:05,650 --> 00:01:09,390
that rapid adiabatic
passage is a powerful way
17
00:01:09,390 --> 00:01:12,430
to manipulate a classical
and quantum system,
18
00:01:12,430 --> 00:01:16,170
and what we discussed is that
when a spin points in the up
19
00:01:16,170 --> 00:01:19,810
direction and you
sweep the resonance
20
00:01:19,810 --> 00:01:22,610
of an oscillating magnetic
field-- the frequency
21
00:01:22,610 --> 00:01:26,400
of an oscillating magnetic
field, through the resonance,
22
00:01:26,400 --> 00:01:31,920
you create an effective magnetic
field in the moving frame which
23
00:01:31,920 --> 00:01:33,410
will rotate.
24
00:01:33,410 --> 00:01:37,080
And the atom, when the
change is done adiabatically,
25
00:01:37,080 --> 00:01:41,080
will follow the rotation and
therefore invert the spin.
26
00:01:41,080 --> 00:01:43,890
So it's a perfect,
very robust method
27
00:01:43,890 --> 00:01:48,905
to invert population
in spin systems.
28
00:01:52,920 --> 00:01:57,390
What I want to pick up
today is the question,
29
00:01:57,390 --> 00:01:59,120
how slow is adiabatic.
30
00:01:59,120 --> 00:02:03,040
We have to fulfill an
adiabatic condition
31
00:02:03,040 --> 00:02:06,470
and we have already
an idea already
32
00:02:06,470 --> 00:02:07,940
of what the adiabatic
condition is
33
00:02:07,940 --> 00:02:10,490
but now we want to derive it.
34
00:02:10,490 --> 00:02:13,090
That we had this
picture of a spin
35
00:02:13,090 --> 00:02:14,950
which is rapidly precessing.
36
00:02:14,950 --> 00:02:21,060
It always precesses
around the direction
37
00:02:21,060 --> 00:02:25,700
of the effective magnetic field,
and the condition adiabadicity
38
00:02:25,700 --> 00:02:30,190
is that the rotation of the
effective magnetic field
39
00:02:30,190 --> 00:02:33,060
has to be much slower
than the precession.
40
00:02:33,060 --> 00:02:35,430
Let me just make it clear
by a counter example.
41
00:02:35,430 --> 00:02:38,310
If the atom precesses
around the magnetic field
42
00:02:38,310 --> 00:02:41,330
and the magnetic field
would suddenly jump,
43
00:02:41,330 --> 00:02:43,320
then the atom would
now start precessing
44
00:02:43,320 --> 00:02:46,160
about the new direction
of the magnetic field
45
00:02:46,160 --> 00:02:48,870
and it would have
completely changed its angle
46
00:02:48,870 --> 00:02:50,560
relative to the magnetic field.
47
00:02:50,560 --> 00:02:53,660
It would have lost its alignment
with the magnetic field.
48
00:02:53,660 --> 00:02:56,040
So you clearly see
that the condition is
49
00:02:56,040 --> 00:02:59,570
the direction of the
magnetic field must not jump,
50
00:02:59,570 --> 00:03:02,330
and the only other time scheme
is the frequency of the Larmor
51
00:03:02,330 --> 00:03:05,510
precession, so our
condition for adiabadicity
52
00:03:05,510 --> 00:03:11,100
is the rotation of the
effective magnetic field
53
00:03:11,100 --> 00:03:13,445
has to be slow compared to
the precession frequency.
54
00:03:18,330 --> 00:03:24,597
And so we want to now
derive from that condition
55
00:03:24,597 --> 00:03:25,930
the conditions for adiabadicity.
56
00:03:28,586 --> 00:03:31,610
And just as an outlook
to make it interesting,
57
00:03:31,610 --> 00:03:35,150
what I will derive for you
in the classical picture
58
00:03:35,150 --> 00:03:37,260
is now something
which you will later
59
00:03:37,260 --> 00:03:40,280
encounter as the
Landau-Zener parameter.
60
00:03:40,280 --> 00:03:44,120
But Landau-Zener sweeps, we
talk about it later today
61
00:03:44,120 --> 00:03:47,030
is a quantum mechanical version
of rapid adiabatic passage,
62
00:03:47,030 --> 00:03:48,990
but we now get
classically a result,
63
00:03:48,990 --> 00:03:51,720
which will feature the
Landau-Zener parameter.
64
00:03:54,750 --> 00:04:04,000
So with that, let us write
down what we want to look at.
65
00:04:04,000 --> 00:04:13,180
It is adiabatic condition,
and to write it down in words
66
00:04:13,180 --> 00:04:19,350
is that the Larmor
frequency, omega L, which
67
00:04:19,350 --> 00:04:24,120
is given by the
effective magnetic field,
68
00:04:24,120 --> 00:04:30,970
has to be much larger
than theta dot.
69
00:04:30,970 --> 00:04:34,140
Now things in general
are rather complicated.
70
00:04:34,140 --> 00:04:36,240
If we are far away
from resonance,
71
00:04:36,240 --> 00:04:39,110
you change the frequency,
but the effective field
72
00:04:39,110 --> 00:04:40,550
is not changing a lot.
73
00:04:40,550 --> 00:04:45,050
The critical moment is really
when we have the real field, we
74
00:04:45,050 --> 00:04:48,760
add the fictitious field
and that cause a rotation.
75
00:04:48,760 --> 00:04:53,120
The critical moment is
when we are near resonance.
76
00:04:53,120 --> 00:04:56,540
So in other words, we have
to fulfill an inequality.
77
00:04:56,540 --> 00:04:59,360
The left side has to be
larger than the right side,
78
00:04:59,360 --> 00:05:05,910
but the left side
is actually smallest
79
00:05:05,910 --> 00:05:08,670
in the vicinity
of the resonance,
80
00:05:08,670 --> 00:05:12,340
and the angle theta
dot is actually
81
00:05:12,340 --> 00:05:14,820
largest near resonance.
82
00:05:14,820 --> 00:05:17,475
That's when it sort of
quickly goes to 90 degrees.
83
00:05:20,750 --> 00:05:25,420
So therefore, if we want to find
the condition of adiabadicity,
84
00:05:25,420 --> 00:05:28,210
we can derive it by
looking at the region
85
00:05:28,210 --> 00:05:29,105
around the resonance.
86
00:05:32,200 --> 00:05:40,490
So the effective magnetic
field is the real field
87
00:05:40,490 --> 00:05:47,200
minus the fictitious
field caused
88
00:05:47,200 --> 00:05:50,520
by the rotation by
the transformation
89
00:05:50,520 --> 00:05:51,735
into the rotating frame.
90
00:05:54,740 --> 00:06:04,410
So we have the magnetic field
at an angle theta with respect
91
00:06:04,410 --> 00:06:04,993
to the z-axis.
92
00:06:10,480 --> 00:06:14,750
I've just written down
the z component for you,
93
00:06:14,750 --> 00:06:20,580
and the transverse component
is-- so this is this component
94
00:06:20,580 --> 00:06:23,521
and the transverse component
is the amplitude of our drive
95
00:06:23,521 --> 00:06:24,020
field B1.
96
00:06:27,010 --> 00:06:29,035
So we can just read
it form the diagram.
97
00:06:34,070 --> 00:06:39,060
The resonance data
is 90 degrees,
98
00:06:39,060 --> 00:06:43,450
and the correction
angle is whatever
99
00:06:43,450 --> 00:06:49,420
we have of the effective
z field over B1,
100
00:06:49,420 --> 00:06:54,080
and that means that the
derivative, the angle theta
101
00:06:54,080 --> 00:06:58,220
dot, the angular velocity
at which the magnetic field
102
00:06:58,220 --> 00:07:04,350
rotates, there's a time
derivative because we
103
00:07:04,350 --> 00:07:06,180
sweep the frequency.
104
00:07:06,180 --> 00:07:10,260
So therefore, theta
dot is nothing else
105
00:07:10,260 --> 00:07:16,960
than omega dot, the sweep
rate of the frequency divided
106
00:07:16,960 --> 00:07:18,120
by gamma B1.
107
00:07:22,680 --> 00:07:29,975
But gamma B1 is nothing else
than the Rabi frequency.
108
00:07:34,500 --> 00:07:40,650
And on resonance, the Larmor
frequency is just a Rabi
109
00:07:40,650 --> 00:07:46,220
frequency, because on
resonance-- sorry for repeating
110
00:07:46,220 --> 00:07:51,240
myself-- the fictitious field
has canceled the bias field,
111
00:07:51,240 --> 00:07:54,450
and the only field left
is the rotating field,
112
00:07:54,450 --> 00:07:58,510
but the rotating field
is the Rabi frequency
113
00:07:58,510 --> 00:08:00,780
with a gamma factor.
114
00:08:00,780 --> 00:08:07,520
So therefore, we have
the adiabatic condition
115
00:08:07,520 --> 00:08:10,490
that omega dot
over omega Rabi has
116
00:08:10,490 --> 00:08:17,180
to be smaller than omega Rabi
or to say it inverts the change
117
00:08:17,180 --> 00:08:20,820
delta omega of your
drive frequency,
118
00:08:20,820 --> 00:08:27,330
the change delta omega
in one Ravi period
119
00:08:27,330 --> 00:08:29,370
has to be smaller than
the Rabi frequency.
120
00:08:34,409 --> 00:08:37,080
So omega has units of frequency.
121
00:08:37,080 --> 00:08:41,350
Omega dot is a derivative that's
in units of frequency squared,
122
00:08:41,350 --> 00:08:44,120
and this is to be smaller than
the Rabi frequency squared.
123
00:08:59,250 --> 00:09:01,280
You find that actually
quite often if you
124
00:09:01,280 --> 00:09:05,550
do an adiabatic change
of your trap frequency,
125
00:09:05,550 --> 00:09:09,610
things are adiabatic as long
as the change of the trap
126
00:09:09,610 --> 00:09:13,250
frequency in one period
of the trap frequency
127
00:09:13,250 --> 00:09:15,210
is smaller than
the trap frequency,
128
00:09:15,210 --> 00:09:18,240
and you find something else
that the derivative of your trap
129
00:09:18,240 --> 00:09:21,760
frequency has to be smaller
than the trap frequency squared.
130
00:09:21,760 --> 00:09:23,310
So these are
adiabatic conditions,
131
00:09:23,310 --> 00:09:27,430
when you tighten up magnetic or
optical confinement for atoms.
132
00:09:27,430 --> 00:09:29,450
So this is very, very genetic.
133
00:09:29,450 --> 00:09:32,090
The small rate of the
frequency you change
134
00:09:32,090 --> 00:09:34,910
has to be smaller than the
relevant frequency squared.
135
00:09:37,930 --> 00:09:41,020
As I said, we come
back to that when
136
00:09:41,020 --> 00:09:45,610
we do the quantized treatment
of rapid adiabatic passage
137
00:09:45,610 --> 00:09:48,130
and we encounter
that combination
138
00:09:48,130 --> 00:09:51,010
in the Landau-Zener
parameter, which
139
00:09:51,010 --> 00:09:55,530
takes us to our next topic.
140
00:09:55,530 --> 00:10:01,760
We want to now talk
about quantized spin
141
00:10:01,760 --> 00:10:02,870
in a magnetic field.
142
00:10:12,549 --> 00:10:14,590
And one of the first things
I will be telling you
143
00:10:14,590 --> 00:10:16,760
is that everything
we have learned
144
00:10:16,760 --> 00:10:20,410
about the classical magnetic
moment you don't have
145
00:10:20,410 --> 00:10:23,080
to unlearn or
re-learn, it exactly
146
00:10:23,080 --> 00:10:26,020
applies to the quantum spin.
147
00:10:29,750 --> 00:10:34,600
Before we look at
the Hamiltonian
148
00:10:34,600 --> 00:10:41,530
and the standard Hamiltonian
for two-level system,
149
00:10:41,530 --> 00:10:45,450
let us first look at
something more general, which
150
00:10:45,450 --> 00:10:47,655
is Heisenberg
equations of motion.
151
00:10:52,580 --> 00:10:57,570
So we want to write down
the differential equation,
152
00:10:57,570 --> 00:11:01,290
the equation of motion
for expectation values.
153
00:11:10,300 --> 00:11:15,950
So for an atom in
a magnetic field,
154
00:11:15,950 --> 00:11:20,470
our Hamiltonian is simply
the same Hamiltonian,
155
00:11:20,470 --> 00:11:24,370
which involves the gyromagnetic
ratio, the angular momentum
156
00:11:24,370 --> 00:11:28,120
operator, and the
magnetic field.
157
00:11:28,120 --> 00:11:34,220
And you know from
quantum mechanics
158
00:11:34,220 --> 00:11:47,760
that Heisenberg's equation
of motion for any operator
159
00:11:47,760 --> 00:11:53,980
are simply that the time
derivative of the operator
160
00:11:53,980 --> 00:11:58,410
equals the commutator
with the Hamiltonian.
161
00:12:02,190 --> 00:12:05,770
In some cases, if you have
an explicit time derivative
162
00:12:05,770 --> 00:12:09,230
of the operator,
you have to edit,
163
00:12:09,230 --> 00:12:11,390
but we are talking here
about the angular momentum
164
00:12:11,390 --> 00:12:13,510
operator which has no
explicit time dependents.
165
00:12:16,180 --> 00:12:22,410
So we are interested
in the operator which
166
00:12:22,410 --> 00:12:30,370
describes the magnetic moment,
but the magnetic moment
167
00:12:30,370 --> 00:12:32,920
is nothing else than
the gyromagnetic ratio
168
00:12:32,920 --> 00:12:38,510
times the angular momentum.
169
00:12:38,510 --> 00:12:42,880
So therefore, the
relevant commutator
170
00:12:42,880 --> 00:12:51,820
is the commutator of the
Hamiltonian with the angular
171
00:12:51,820 --> 00:12:54,100
momentum operator.
172
00:12:54,100 --> 00:12:57,350
Just to remind you,
the Hamiltonian
173
00:12:57,350 --> 00:13:01,530
was proportional to z.
174
00:13:01,530 --> 00:13:09,310
So what we are talking about is
commutators, not surprisingly,
175
00:13:09,310 --> 00:13:14,620
between the angular
momentum operators
176
00:13:14,620 --> 00:13:25,180
and those commutators are
just this cyclic commutation
177
00:13:25,180 --> 00:13:29,090
involving the epsilon tensor.
178
00:13:29,090 --> 00:13:36,100
So if you just put that
in component by component,
179
00:13:36,100 --> 00:13:43,260
you realize immediately that the
operator equation for the time
180
00:13:43,260 --> 00:13:47,930
derivative of the
magnetic moment
181
00:13:47,930 --> 00:13:55,170
is nothing else than the cross
product, the vector product,
182
00:13:55,170 --> 00:13:57,820
of the operator of
the magnetic moment
183
00:13:57,820 --> 00:13:59,920
times the magnetic field.
184
00:14:03,640 --> 00:14:11,400
To commence, this is
exact, but also it
185
00:14:11,400 --> 00:14:15,470
looks exactly like
the classical result.
186
00:14:22,060 --> 00:14:28,130
Well, you would say it's
an operator equation.
187
00:14:28,130 --> 00:14:31,610
Usually operator
equation some of them
188
00:14:31,610 --> 00:14:34,620
are pretty useless because you
can't calculate the operators,
189
00:14:34,620 --> 00:14:37,670
but in this case,
we can immediately
190
00:14:37,670 --> 00:14:40,730
take the expectation
value, so we
191
00:14:40,730 --> 00:14:44,170
can get some immediately
meaningfully equation
192
00:14:44,170 --> 00:14:53,900
namely that the
expectation value follows
193
00:14:53,900 --> 00:15:02,030
the same equation,
and this tells us
194
00:15:02,030 --> 00:15:04,310
that whenever we have
a quantum system,
195
00:15:04,310 --> 00:15:07,390
it has a magnetic moment.
196
00:15:07,390 --> 00:15:10,680
In an applied external
magnetic field,
197
00:15:10,680 --> 00:15:15,360
the result is simply rotation,
precession, and this is exact.
198
00:15:15,360 --> 00:15:17,210
It's not a classical
approximation.
199
00:15:17,210 --> 00:15:19,365
It's an exact result
for quantum mechanics.
200
00:15:22,890 --> 00:15:30,180
So the way how we derived
it makes it obvious that
201
00:15:30,180 --> 00:15:37,710
it's an exact result, which is
valid not only for spin-1/2,
202
00:15:37,710 --> 00:15:39,980
but it is valid for any spin.
203
00:15:39,980 --> 00:15:44,600
If you have a magnetic moment
corresponding to a spin of 10 H
204
00:15:44,600 --> 00:15:49,420
bar, this spin follows the same
equation of motion as spin-1/2.
205
00:15:55,150 --> 00:15:59,450
Of course, a special case
is valid for spin-1/2,
206
00:15:59,450 --> 00:16:04,200
but spin-1/2 is isomorphous
to a two-level system.
207
00:16:04,200 --> 00:16:08,140
Any two-level system can be
regarded as spin-1/2 system,
208
00:16:08,140 --> 00:16:13,110
therefore, this geometric
interpretation that
209
00:16:13,110 --> 00:16:17,250
the dynamics of the quantum
system is just a precession
210
00:16:17,250 --> 00:16:21,115
rigorously, exactly applies
to any two-level system.
211
00:16:33,560 --> 00:16:38,880
It's also valid, and this
will be relevant for atoms,
212
00:16:38,880 --> 00:16:41,830
if we have composite
angular momentum.
213
00:16:48,010 --> 00:16:51,500
For instance, we will encounter
the total angular momentum
214
00:16:51,500 --> 00:16:55,050
F of an atom which
has components
215
00:16:55,050 --> 00:16:56,970
from the orbital
motion of the electron,
216
00:16:56,970 --> 00:17:00,450
the spin of the electron,
and the spin of the nucleus.
217
00:17:00,450 --> 00:17:02,890
But if we have such
an angular momentum,
218
00:17:02,890 --> 00:17:07,450
F, the creation of motion
is it will precess around
219
00:17:07,450 --> 00:17:10,720
a magnetic field.
220
00:17:10,720 --> 00:17:26,810
Well the small print here is,
unless the B field is so strong
221
00:17:26,810 --> 00:17:29,580
that it de-couples
the components
222
00:17:29,580 --> 00:17:39,490
or that it breaks up the
coupling of the different parts
223
00:17:39,490 --> 00:17:40,540
of the angular momentum.
224
00:17:46,020 --> 00:17:48,330
In other words, we have
simply assumed here
225
00:17:48,330 --> 00:17:52,890
that magnetic moment is
gamma times angular momentum,
226
00:17:52,890 --> 00:17:55,630
and that requires that
the angular momentum are
227
00:17:55,630 --> 00:17:57,126
coupled in a certain way.
228
00:17:57,126 --> 00:17:58,500
If we don't fully
understand what
229
00:17:58,500 --> 00:18:00,850
coupling of angular
momentum is, we really
230
00:18:00,850 --> 00:18:03,480
talk about that when we
talk about atomic structure.
231
00:18:03,480 --> 00:18:07,520
So as long as the spin state
coupled to one total spin,
232
00:18:07,520 --> 00:18:09,480
this total spin
will just precess.
233
00:18:14,920 --> 00:18:19,340
This picture of
precession will also
234
00:18:19,340 --> 00:18:37,590
be valid for a system
of N two-level systems
235
00:18:37,590 --> 00:18:44,200
coupled to an external
field, and this
236
00:18:44,200 --> 00:18:52,550
will be the example of
Dicke superrradiance, which
237
00:18:52,550 --> 00:18:54,830
we will discussed towards
the end of the course.
238
00:19:05,500 --> 00:19:08,260
So very simple result,
but very powerful,
239
00:19:08,260 --> 00:19:10,540
and this is your
permission whenever
240
00:19:10,540 --> 00:19:13,530
you encounter any of
the systems to see
241
00:19:13,530 --> 00:19:15,700
a vector precessing
in your head.
242
00:19:15,700 --> 00:19:16,490
This is exact.
243
00:19:28,690 --> 00:19:31,025
So we've talked about
Heisenberg equation of motion
244
00:19:31,025 --> 00:19:33,150
for general spin, but--
you have a question, Nancy?
245
00:19:33,150 --> 00:19:36,935
AUDIENCE: What did you mean
by N two-level systems here?
246
00:19:36,935 --> 00:19:39,350
Are we talking about
coherent systems
247
00:19:39,350 --> 00:19:42,731
or non-coherent systems?
248
00:19:42,731 --> 00:19:47,350
PROFESSOR: We talk about
in two-level systems
249
00:19:47,350 --> 00:19:52,100
and to be more specific the
coupling comes because they all
250
00:19:52,100 --> 00:19:54,320
talk to the same magnetic field.
251
00:19:54,320 --> 00:19:58,270
So we have in two-level
systems connected
252
00:19:58,270 --> 00:20:01,710
to the modes of the
electromagnetic field.
253
00:20:01,710 --> 00:20:03,440
We start with the
symmetric state.
254
00:20:03,440 --> 00:20:05,250
The coupling is
symmetric and that
255
00:20:05,250 --> 00:20:09,310
preserves the symmetry
of the atomic state.
256
00:20:09,310 --> 00:20:14,920
In other words, we will have
a situation where the angular
257
00:20:14,920 --> 00:20:19,110
momentum is the maximum
angular momentum we
258
00:20:19,110 --> 00:20:22,420
can get in two-level
system, and the dynamics
259
00:20:22,420 --> 00:20:25,330
of this two-level system,
the description of Dicke
260
00:20:25,330 --> 00:20:29,410
superradiance has the
geometric visualization
261
00:20:29,410 --> 00:20:30,720
of this precessing motion.
262
00:20:33,510 --> 00:20:35,880
I know I'm not
explaining it exactly.
263
00:20:35,880 --> 00:20:39,230
I want to sort of whet your
appetite for what comes later
264
00:20:39,230 --> 00:20:42,839
and also sort of prep you that
some of the simple pictures
265
00:20:42,839 --> 00:20:44,380
will really carry
through the course.
266
00:20:48,180 --> 00:20:51,670
OK, so this is for
very general spin.
267
00:20:51,670 --> 00:20:56,810
Let's now talk about
features of, yes,
268
00:20:56,810 --> 00:21:01,370
the most important spin for us,
namely the two-level system,
269
00:21:01,370 --> 00:21:02,680
which is spin-1/2.
270
00:21:16,440 --> 00:21:27,530
Well, the most generic
system is an electron
271
00:21:27,530 --> 00:21:38,170
in a magnetic field B times ez.
272
00:21:41,060 --> 00:21:46,465
And well, why don't we start
with a clicker question.
273
00:21:51,940 --> 00:21:54,540
So the question is, what
is the level structure
274
00:21:54,540 --> 00:21:57,610
of the electron in
the magnetic field?
275
00:21:57,610 --> 00:22:01,970
It's a two-level system,
and you have two options
276
00:22:01,970 --> 00:22:06,940
A and B. One option is the
upper state is spin up.
277
00:22:06,940 --> 00:22:17,740
The ground state is spin
down or the opposite.
278
00:22:17,740 --> 00:22:24,800
So in other words, and
tell me whether an electron
279
00:22:24,800 --> 00:22:27,620
is in the lower state
when the speed is up
280
00:22:27,620 --> 00:22:29,250
or when the spin is down.
281
00:22:38,560 --> 00:22:40,440
You would say it's
a stupid definition,
282
00:22:40,440 --> 00:22:42,730
but we talk all the
time about an electron
283
00:22:42,730 --> 00:22:44,960
is in the spin down
state or the spin
284
00:22:44,960 --> 00:22:47,900
up state, which is the lower
energy state of the electron.
285
00:22:52,390 --> 00:22:54,720
So I think by exchanging
the better recent number
286
00:22:54,720 --> 00:22:57,730
of responses has
considerably gone up.
287
00:23:04,291 --> 00:23:04,790
OK.
288
00:23:10,030 --> 00:23:14,770
The answer is the electron
is in the ground state.
289
00:23:14,770 --> 00:23:24,190
Spin down is the lowest
state for the electron.
290
00:23:28,340 --> 00:23:31,890
To say it in words, of
course, a compass needle
291
00:23:31,890 --> 00:23:34,430
wants to be aligned
with the magnetic field.
292
00:23:34,430 --> 00:23:41,450
So you want the magnetic
moment to be aligned
293
00:23:41,450 --> 00:23:43,330
with the magnetic
field, and that
294
00:23:43,330 --> 00:23:45,340
means the magnetic
moment has to point
295
00:23:45,340 --> 00:23:49,300
in the plus e direction,
which up but the electron has
296
00:23:49,300 --> 00:23:52,500
negative charge, the
gamma factor is negative,
297
00:23:52,500 --> 00:23:56,170
and that's why for the
electron, the vector of the spin
298
00:23:56,170 --> 00:24:00,360
and the vector of the
magnetic moment are opposite.
299
00:24:00,360 --> 00:24:02,700
So just try to find some
main-mode technical thing.
300
00:24:02,700 --> 00:24:04,380
The electron lives
in the basement.
301
00:24:04,380 --> 00:24:06,630
It wants to be down, spin down.
302
00:24:06,630 --> 00:24:10,170
This is the lowest
state for the electron.
303
00:24:10,170 --> 00:24:13,270
However, if you
have a system which
304
00:24:13,270 --> 00:24:16,500
has a positive gyromagnetic
ratio, which would correspond
305
00:24:16,500 --> 00:24:26,820
to, well, positive charge,
nucleus if it has somewhat
306
00:24:26,820 --> 00:24:31,590
normal magnetic moment, then
in that case, the spin up state
307
00:24:31,590 --> 00:24:33,870
is less energetic than
the spin down state.
308
00:24:38,440 --> 00:24:39,920
So let me just write that down.
309
00:24:39,920 --> 00:24:45,200
So the correct
answer is this one,
310
00:24:45,200 --> 00:24:50,470
and it involves that
gamma is negative.
311
00:24:50,470 --> 00:24:52,580
The gamma is a
gyromagnetic ratio,
312
00:24:52,580 --> 00:24:55,270
the ratio between magnetic
moment and angular momentum,
313
00:24:55,270 --> 00:24:57,065
and for negative
charges, it's negative.
314
00:25:01,280 --> 00:25:04,400
For positive gamma, the
situation is inverted.
315
00:25:07,770 --> 00:25:10,850
So let's just use for
a moment the result
316
00:25:10,850 --> 00:25:13,860
we got from Heisenberg's
equation of motion.
317
00:25:13,860 --> 00:25:16,150
We know the classical result.
318
00:25:18,960 --> 00:25:21,710
We have already derived
for the classic spin,
319
00:25:21,710 --> 00:25:24,270
the classical result
for the expectation
320
00:25:24,270 --> 00:25:25,620
value of the magnetic moment.
321
00:25:28,540 --> 00:25:31,420
But now I want to sort of
relate it to something quantum
322
00:25:31,420 --> 00:25:34,920
mechanical because we know that
the classical solution equals
323
00:25:34,920 --> 00:25:37,660
the quantum mechanical solution.
324
00:25:37,660 --> 00:25:45,330
So if you have a
two-level system,
325
00:25:45,330 --> 00:25:48,900
the z component of
the magnetic moment
326
00:25:48,900 --> 00:25:57,450
is the difference between
spin down and spin up.
327
00:26:01,750 --> 00:26:04,860
And because of conservation
probability, p up and p down
328
00:26:04,860 --> 00:26:11,180
is unity, we can also
write that as 2 times--
329
00:26:11,180 --> 00:26:13,880
let me now introduce
e for excited state,
330
00:26:13,880 --> 00:26:16,750
just I know it's hard to keep
track of spin up, spin down.
331
00:26:16,750 --> 00:26:19,450
I want to make sure that I
mean now the excited state,
332
00:26:19,450 --> 00:26:23,850
so the excited state for
the electron is spin up.
333
00:26:23,850 --> 00:26:41,550
So we have this condition, so
therefore, the excited state
334
00:26:41,550 --> 00:26:48,070
fraction of a two-level system
is related to the expectation
335
00:26:48,070 --> 00:26:53,150
value of the magnetic
moment in that wave,
336
00:26:53,150 --> 00:26:56,320
and now we want to use the
classical result we derived.
337
00:27:01,540 --> 00:27:05,550
We derived the
classical result when
338
00:27:05,550 --> 00:27:21,170
for t equals 0 all the spins
were in the ground state,
339
00:27:21,170 --> 00:27:26,150
and by using the result
we derived previously,
340
00:27:26,150 --> 00:27:28,930
we have the 1/2 from
the previous line
341
00:27:28,930 --> 00:27:33,400
and then the magnetic
moment, mu z.
342
00:27:33,400 --> 00:27:41,510
We found an expression which
involved the Rabi frequency,
343
00:27:41,510 --> 00:27:55,108
and the off resonant
Rabi frequency
344
00:27:55,108 --> 00:28:02,290
are sine square generalized Rabi
frequency times time over 2.
345
00:28:05,380 --> 00:28:10,200
So the two factors
of 1/2 and 1/2
346
00:28:10,200 --> 00:28:17,230
cancel, and what we find now for
the quantum mechanical system
347
00:28:17,230 --> 00:28:19,930
using Heisenberg's
equation of motion
348
00:28:19,930 --> 00:28:23,560
is that if you prepare
a system initially
349
00:28:23,560 --> 00:28:27,290
in the ground state,
the excited state
350
00:28:27,290 --> 00:28:33,460
probability, the fraction,
the excited state oscillates
351
00:28:33,460 --> 00:28:37,300
with a Rabi frequency,
and this is,
352
00:28:37,300 --> 00:28:41,660
I think, the second time in this
course and not the last time
353
00:28:41,660 --> 00:28:44,150
that we see the
Rabi, that we obtain,
354
00:28:44,150 --> 00:28:45,620
the Rabi transition probability.
355
00:28:55,350 --> 00:28:56,300
But let's go further.
356
00:28:56,300 --> 00:28:58,400
We have now discussed
the classical spin.
357
00:28:58,400 --> 00:29:01,540
We have sort of done classical
quantum correspondence
358
00:29:01,540 --> 00:29:03,730
with Heisenberg's
equation of motion.
359
00:29:03,730 --> 00:29:09,490
We know that this in general,
it implies Rabi oscillations,
360
00:29:09,490 --> 00:29:15,200
but now we want to go deeper
into the quantum domain
361
00:29:15,200 --> 00:29:19,440
by talking about the
spin-1/2 Hamiltonian.
362
00:29:25,760 --> 00:29:29,880
So in other words, we want to
go beyond expectation values.
363
00:29:34,480 --> 00:29:37,490
We want to talk about
the wave function itself.
364
00:29:41,350 --> 00:29:49,200
So the Hamiltonian,
which we will
365
00:29:49,200 --> 00:29:53,250
use for major parts
in this course,
366
00:29:53,250 --> 00:29:56,519
it's one of the fundamental
Hamiltonians in physics.
367
00:29:56,519 --> 00:29:58,310
Of course, there's the
harmonic oscillator.
368
00:29:58,310 --> 00:30:00,630
There is a hydrogen
atom, but then there's
369
00:30:00,630 --> 00:30:06,000
this Hamiltonian, which is a
two-level system with splitting
370
00:30:06,000 --> 00:30:07,790
omega naught.
371
00:30:07,790 --> 00:30:17,170
And then we make, which
is often a simplification,
372
00:30:17,170 --> 00:30:26,270
where we use a pure exponential,
so a single frequency
373
00:30:26,270 --> 00:30:30,890
in complex notation where
the complex exponential e
374
00:30:30,890 --> 00:30:34,170
to the i omega t
is the drive term.
375
00:30:37,610 --> 00:30:40,100
So this is one of the
simplest Hamiltonian
376
00:30:40,100 --> 00:30:41,230
for this kind of system.
377
00:30:41,230 --> 00:30:46,130
It's a two-level system with a
splitting, and now it is driven
378
00:30:46,130 --> 00:30:49,780
and the simplest drive term
is not cosine omega t or sine
379
00:30:49,780 --> 00:30:51,810
omega t as we will see.
380
00:30:51,810 --> 00:30:55,430
The simplest drive term
is e to the i omega t.
381
00:30:55,430 --> 00:30:58,870
So this is now our Hamiltonian.
382
00:30:58,870 --> 00:31:02,740
and since it is
so important, let
383
00:31:02,740 --> 00:31:14,580
me ask you a clicker question
whether this Hamiltonian can
384
00:31:14,580 --> 00:31:32,775
be exactly realized in nature
or it is an approximation.
385
00:31:44,710 --> 00:31:51,550
For instance, that you always
have cosine omega t as a drive,
386
00:31:51,550 --> 00:31:53,880
and cosine omega t is
e to the plus i omega t
387
00:31:53,880 --> 00:31:55,410
and e to the minus i omega t.
388
00:31:55,410 --> 00:31:57,700
And then maybe with the
rotating wave approximation,
389
00:31:57,700 --> 00:31:59,430
you throw away a term.
390
00:31:59,430 --> 00:32:01,500
So I sort of wanted
to ask you, is this
391
00:32:01,500 --> 00:32:05,620
an idealization that we
have a coupling which
392
00:32:05,620 --> 00:32:10,240
is a simple, complex
exponential, this nature always
393
00:32:10,240 --> 00:32:15,600
more complicated or
is there a simple way
394
00:32:15,600 --> 00:32:17,470
to realize this
Hamiltonian in nature?
395
00:32:20,420 --> 00:32:21,850
So what do you think?
396
00:32:36,510 --> 00:32:37,980
Stop display.
397
00:32:44,080 --> 00:32:45,470
What's funny about that?
398
00:32:45,470 --> 00:32:46,853
AUDIENCE: Equal to.
399
00:32:46,853 --> 00:32:48,700
AUDIENCE: There are
two [INAUDIBLE].
400
00:32:48,700 --> 00:32:50,700
PROFESSOR: Oh, they
couldn't make up their mind.
401
00:32:54,140 --> 00:32:57,660
The system should
reject those votes.
402
00:32:57,660 --> 00:33:02,290
Anyway, it means most of
you anticipate what I want
403
00:33:02,290 --> 00:33:05,950
to derive to you that we can
actually exactly get this
404
00:33:05,950 --> 00:33:11,210
Hamiltonian and indeed, this is
the Hamiltonian I will derive
405
00:33:11,210 --> 00:33:14,510
for you in the next few minutes,
which is the Hamiltonian
406
00:33:14,510 --> 00:33:21,960
of spin-1/2 in a magnetic field
coupled to a rotating magnetic
407
00:33:21,960 --> 00:33:23,720
field.
408
00:33:23,720 --> 00:33:26,210
So we start out
with fields which
409
00:33:26,210 --> 00:33:29,700
are real, real field,
no imaginary numbers,
410
00:33:29,700 --> 00:33:30,940
no complex numbers.
411
00:33:30,940 --> 00:33:34,410
These are real fields, but
when we write down how the real
412
00:33:34,410 --> 00:33:39,706
fields couple to spin-1/2, we
get e to the i omega t and e
413
00:33:39,706 --> 00:33:42,025
to the minus omega t
without any approximation.
414
00:33:46,330 --> 00:33:52,430
So since at least 80%
of you know the result,
415
00:33:52,430 --> 00:33:58,770
just regard it as an exercise to
introduce how we spin matrices
416
00:33:58,770 --> 00:34:02,850
in a nice way, also
this will help you
417
00:34:02,850 --> 00:34:06,770
how to do pre-set number 1.
418
00:34:06,770 --> 00:34:12,750
So the Hamiltonian is the spin
coupled to a magnetic field,
419
00:34:12,750 --> 00:34:18,130
and if we express that by
angular momentum operators,
420
00:34:18,130 --> 00:34:20,949
the gyromagnetic ratio
it involves the operator
421
00:34:20,949 --> 00:34:22,603
for the spin in the z direction.
422
00:34:25,170 --> 00:34:30,920
So the two-level system has
a splitting of H bar omega,
423
00:34:30,920 --> 00:34:35,260
so it's plus 1/2
minus 1/2 H bar omega,
424
00:34:35,260 --> 00:34:37,540
and that means
the diagonal part,
425
00:34:37,540 --> 00:34:47,969
the non-driven part is simply
given by the Pauli spin matrix
426
00:34:47,969 --> 00:34:55,719
sigma z and omega naught
is the energy splitting,
427
00:34:55,719 --> 00:35:00,220
which is proportional to
the applied magnetic field.
428
00:35:00,220 --> 00:35:04,580
And up-down, and
excited and ground
429
00:35:04,580 --> 00:35:15,280
are the eigenstates of this
Hamiltonian with energies
430
00:35:15,280 --> 00:35:19,290
plus minus H bar
omega naught over 2.
431
00:35:22,150 --> 00:35:31,450
So this is the same Hamiltonian,
but now add a real rotating B
432
00:35:31,450 --> 00:35:35,520
field B1.
433
00:35:40,510 --> 00:35:46,950
So the drive Hamiltonian H1
is the same magnetic moment
434
00:35:46,950 --> 00:35:49,935
but now coupled to a
time-dependent rotating field.
435
00:35:52,770 --> 00:35:54,440
The amplitude of
the rotating field
436
00:35:54,440 --> 00:35:58,670
is the Rabi frequency
divided by gamma,
437
00:35:58,670 --> 00:36:09,970
and we assume that the field
is rotating in the xy plane.
438
00:36:09,970 --> 00:36:18,960
So it's ex ey cosine
omega t sine omega t.
439
00:36:18,960 --> 00:36:22,066
And now I put in two minus
signs here for convenience.
440
00:36:26,860 --> 00:36:31,180
If you want, I've just shifted--
it's just a definition.
441
00:36:31,180 --> 00:36:34,250
I've changed the definition of
the amplitude by a minus sign.
442
00:36:41,130 --> 00:36:53,900
So this is the Rabi frequency,
and the magnetic moment
443
00:36:53,900 --> 00:36:59,040
divided by gamma is
nothing than the spin.
444
00:36:59,040 --> 00:37:03,440
Magnetic moment is
gamma times the spin,
445
00:37:03,440 --> 00:37:19,220
so therefore, we are back
to the spin operators
446
00:37:19,220 --> 00:37:27,210
and the spin operators,
if I factor out 1/2 H bar,
447
00:37:27,210 --> 00:37:33,325
are now the Pauli spin
matrices sigma x and sigma y.
448
00:37:40,720 --> 00:37:48,970
And if you look at
those spin matrices,
449
00:37:48,970 --> 00:37:54,500
then you'll realize that we
go complex in our Hamiltonian,
450
00:37:54,500 --> 00:37:58,930
not because we have approximated
a real field cosine omega
451
00:37:58,930 --> 00:38:03,290
t by some e to the i
omega t, but because when
452
00:38:03,290 --> 00:38:06,950
we have a rotating field and
we write down it in Pauli spin
453
00:38:06,950 --> 00:38:12,035
matrices, we get imaginary units
form the sigma y spin matrix.
454
00:38:17,800 --> 00:38:24,060
So that means we have
now for this system
455
00:38:24,060 --> 00:38:28,940
rewritten the coupling term took
you to the rotating field H1
456
00:38:28,940 --> 00:38:38,720
as 0 0 e to the plus i omega
t e to the minus i omega t.
457
00:38:38,720 --> 00:38:48,720
And therefore,
the Hamiltonian is
458
00:38:48,720 --> 00:38:54,800
the famous two-level
Hamiltonian with omega naught
459
00:38:54,800 --> 00:39:02,190
and the Rabi frequency,
which I wrote down
460
00:39:02,190 --> 00:39:07,660
at the beginning
of this chapter.
461
00:39:12,270 --> 00:39:21,030
So we'll leave that here, but we
will use it even more in 8.421.
462
00:39:21,030 --> 00:39:25,810
This is the famous
dressed atom Hamiltonian.
463
00:39:29,016 --> 00:39:32,590
It is the starting point
to calculate eigenstates
464
00:39:32,590 --> 00:39:36,960
and eigenvalues, not just
in perturbation theory.
465
00:39:36,960 --> 00:39:43,550
You can go to oscillating
fields at arbitrary strengths,
466
00:39:43,550 --> 00:39:47,510
so you can solve exactly in
the dressed atom picture using
467
00:39:47,510 --> 00:39:51,480
this Hamiltonian, the
problem of a two-level system
468
00:39:51,480 --> 00:39:54,530
plus one mode of the
electromagnetic field
469
00:39:54,530 --> 00:39:56,430
no matter what the
drive term, what
470
00:39:56,430 --> 00:39:58,620
the strengths of the
electromagnetic field
471
00:39:58,620 --> 00:39:59,610
in the spin mode is.
472
00:40:04,060 --> 00:40:11,460
So this describes
the two-level system
473
00:40:11,460 --> 00:40:23,300
plus one mode of the
electromagnetic field
474
00:40:23,300 --> 00:40:24,870
with arbitrary strengths.
475
00:40:34,210 --> 00:40:39,350
And as I said, we talk
about some things here
476
00:40:39,350 --> 00:40:50,500
but others I explored in 8.422.
477
00:40:50,500 --> 00:40:53,220
Questions?
478
00:40:53,220 --> 00:40:53,855
Yes, Will.
479
00:40:53,855 --> 00:40:57,990
AUDIENCE: We refer to the
eigenstates and eigenenergies
480
00:40:57,990 --> 00:41:01,410
of this Hamiltonian as dressed
in states in the same way
481
00:41:01,410 --> 00:41:04,924
as we refer to address states
in a fully quantized fiction?
482
00:41:04,924 --> 00:41:10,492
Do we refer to both
cases as dressed states?
483
00:41:10,492 --> 00:41:12,360
PROFESSOR: Yes.
484
00:41:12,360 --> 00:41:13,460
That's a good question.
485
00:41:13,460 --> 00:41:16,260
The question is now, what are
the dressed states, and Will,
486
00:41:16,260 --> 00:41:18,066
I think you are
referring that there
487
00:41:18,066 --> 00:41:23,480
are two ways to talk about the
coupling of a two-level system
488
00:41:23,480 --> 00:41:25,570
to one mode of the
electromagnetic field.
489
00:41:25,570 --> 00:41:28,100
It is this same
classical picture
490
00:41:28,100 --> 00:41:33,220
where we introduce and let
me say an analog amplitude
491
00:41:33,220 --> 00:41:35,730
of the electromagnetic
field which drives it.
492
00:41:35,730 --> 00:41:38,190
And then there is a
fully quantized picture
493
00:41:38,190 --> 00:41:40,930
where you first quantize
electromagnetic field
494
00:41:40,930 --> 00:41:44,920
and you couple to
photon number states.
495
00:41:44,920 --> 00:41:48,110
Actually the beauty of
it that the two solutions
496
00:41:48,110 --> 00:41:51,060
are exactly the same.
497
00:41:51,060 --> 00:42:01,490
So in other words, how to
say, if you couple an atom
498
00:42:01,490 --> 00:42:04,490
to one mode of the
electromagnetic field.
499
00:42:04,490 --> 00:42:06,780
We have two ways
how we can solve it.
500
00:42:06,780 --> 00:42:13,860
One is, we introduce a
coherent electromagnetic field,
501
00:42:13,860 --> 00:42:17,940
and there is an exact unitary
transformation which tells us
502
00:42:17,940 --> 00:42:21,130
if we have the quantized
field in a coherent state,
503
00:42:21,130 --> 00:42:23,540
we can do unitary
transformation,
504
00:42:23,540 --> 00:42:26,490
and what we get is
exactly this Hamiltonian.
505
00:42:26,490 --> 00:42:29,690
So therefore, this is also--
you may not recognize it--
506
00:42:29,690 --> 00:42:32,740
this is actually the
quantum description
507
00:42:32,740 --> 00:42:37,780
of the electromagnetic field
when it is in a coherent state.
508
00:42:37,780 --> 00:42:41,830
The other option is, we use
the dressed atom picture maybe
509
00:42:41,830 --> 00:42:45,800
following some work of
[INAUDIBLE] and others
510
00:42:45,800 --> 00:42:49,320
where we assume this single
mode of electromagnetic field
511
00:42:49,320 --> 00:42:53,665
has in photons, and
then we solve it
512
00:42:53,665 --> 00:42:57,450
for this photon number state.
513
00:42:57,450 --> 00:42:59,860
So in other words,
these are the two ways
514
00:42:59,860 --> 00:43:02,450
how we can relatively,
easily treat the problem.
515
00:43:02,450 --> 00:43:05,970
Either we assume the quantum
field is in a coherence state
516
00:43:05,970 --> 00:43:08,460
or it's in a flux state.
517
00:43:08,460 --> 00:43:12,090
But since the dressed atom
picture in the standard way
518
00:43:12,090 --> 00:43:16,270
assumes that the photon
number, N, is large,
519
00:43:16,270 --> 00:43:19,330
there is a correspondence
that in the limit of N of N
520
00:43:19,330 --> 00:43:21,840
being large, the flux
state description
521
00:43:21,840 --> 00:43:24,280
and the coherent state
description fully agree.
522
00:43:28,320 --> 00:43:30,830
And you pick what you want.
523
00:43:30,830 --> 00:43:33,240
If you introduce the
electromagnetic field
524
00:43:33,240 --> 00:43:35,220
explicitly with
it's quantum state,
525
00:43:35,220 --> 00:43:38,460
you get the dressed atom
picture as a solution of a time
526
00:43:38,460 --> 00:43:41,550
independent problem, whereas
here with a coherent state
527
00:43:41,550 --> 00:43:45,380
description, the coherent state
oscillates, cosine omega t,
528
00:43:45,380 --> 00:43:47,890
with a time-dependent problem.
529
00:43:47,890 --> 00:43:49,570
And actually I
should say whenever
530
00:43:49,570 --> 00:43:52,520
I get confused in one picture,
I look in the other picture
531
00:43:52,520 --> 00:43:55,410
and it becomes clear.
532
00:43:55,410 --> 00:43:59,510
I generally prefer
where we have N photons,
533
00:43:59,510 --> 00:44:01,830
it's because we can
discuss everything
534
00:44:01,830 --> 00:44:06,280
in a time-independent way, but
for certain intuitive aspects,
535
00:44:06,280 --> 00:44:10,010
this is also variable, so in
the end, you have to learn both.
536
00:44:10,010 --> 00:44:12,334
And in your homework,
you will actually
537
00:44:12,334 --> 00:44:14,500
write down the general
solution for this Hamiltonian
538
00:44:14,500 --> 00:44:17,210
as an exercise.
539
00:44:17,210 --> 00:44:17,710
Nancy.
540
00:44:17,710 --> 00:44:21,370
AUDIENCE: I think I'm
confused a little bit.
541
00:44:21,370 --> 00:44:24,885
So in the flux state
picture, the dressed states
542
00:44:24,885 --> 00:44:30,115
can be exactly part of an
independent matter of coupling
543
00:44:30,115 --> 00:44:33,490
between a lesser photon
on any excited state.
544
00:44:33,490 --> 00:44:38,627
So like we can write eN as 1
and g or something like that.
545
00:44:38,627 --> 00:44:39,670
PROFESSOR: Yeah.
546
00:44:39,670 --> 00:44:42,950
You couple a photon field
with N photons and energy
547
00:44:42,950 --> 00:44:46,390
in H bar omega to N
minus 1 H bar omega.
548
00:44:46,390 --> 00:44:51,890
AUDIENCE: But in this one,
is there like a direct photon
549
00:44:51,890 --> 00:44:54,390
number thing, because we
haven't quantized the field yet?
550
00:44:54,390 --> 00:44:59,890
So what do the dressed
states mean at this point?
551
00:44:59,890 --> 00:45:06,320
PROFESSOR: Well, the fact
is that if you start out
552
00:45:06,320 --> 00:45:10,470
with a coherent state, your
photon field is not in photons,
553
00:45:10,470 --> 00:45:12,600
it's a laser beam.
554
00:45:12,600 --> 00:45:15,380
The laser beam or
the coherent state
555
00:45:15,380 --> 00:45:18,970
is in a quantized description,
a superposition of many flux
556
00:45:18,970 --> 00:45:20,040
states.
557
00:45:20,040 --> 00:45:23,850
So therefore, the number of
photons in a coherent state
558
00:45:23,850 --> 00:45:27,530
fluctuates or has a large
Plutonian statistics,
559
00:45:27,530 --> 00:45:30,770
and if you take one
photon out or not,
560
00:45:30,770 --> 00:45:32,260
it doesn't make
a big difference.
561
00:45:32,260 --> 00:45:34,150
For instance, for
those of you who
562
00:45:34,150 --> 00:45:38,450
know how the coherent
state is state defined,
563
00:45:38,450 --> 00:45:42,100
the coherent state
is defined as when
564
00:45:42,100 --> 00:45:45,630
you act on the coherent state
with an annihilation operator,
565
00:45:45,630 --> 00:45:48,730
you get the eigenvalue
times the coherent state.
566
00:45:48,730 --> 00:45:51,350
So that tells you you have a
fully-quantized description
567
00:45:51,350 --> 00:45:54,100
of your laser in terms
of a coherent state.
568
00:45:54,100 --> 00:45:56,530
You take one photon
out and what you get?
569
00:45:56,530 --> 00:45:58,100
the same state back.
570
00:45:58,100 --> 00:46:01,910
And this may immediately justify
that what we write down here
571
00:46:01,910 --> 00:46:05,295
is simply the coherent
state with its amplitude
572
00:46:05,295 --> 00:46:06,920
and the amplitude of
the coherent state
573
00:46:06,920 --> 00:46:10,420
would be B1, the amplitude
of the drive field.
574
00:46:10,420 --> 00:46:12,450
And we don't really
need other states
575
00:46:12,450 --> 00:46:14,360
because a coherent
state has the property.
576
00:46:14,360 --> 00:46:17,462
You to take a photon out and
you still have the same state.
577
00:46:17,462 --> 00:46:18,920
So therefore, we
don't have to keep
578
00:46:18,920 --> 00:46:20,410
track of the coherent state.
579
00:46:20,410 --> 00:46:23,500
It's there all the time.
580
00:46:23,500 --> 00:46:25,580
But what I'm saying
can be formulated
581
00:46:25,580 --> 00:46:28,949
more exactly when we use
the appropriate formulas.
582
00:46:28,949 --> 00:46:30,240
But this is sort of the bridge.
583
00:46:30,240 --> 00:46:33,570
That's why we do not have to
keep track of the photon state.
584
00:46:33,570 --> 00:46:38,754
It's because the coherent state
has those wonderful properties.
585
00:46:38,754 --> 00:46:39,420
Other questions?
586
00:46:44,710 --> 00:46:47,149
OK.
587
00:46:47,149 --> 00:46:48,565
So this is the
famous Hamiltonian.
588
00:46:52,420 --> 00:46:55,450
And of course, if it's
the famous Hamiltonian,
589
00:46:55,450 --> 00:46:58,070
we want to solve it.
590
00:47:02,640 --> 00:47:13,520
As I said, the general solution
is left to the homework,
591
00:47:13,520 --> 00:47:16,710
but I want to sort
of show you parts
592
00:47:16,710 --> 00:47:19,190
of the solution to tell a story.
593
00:47:19,190 --> 00:47:22,550
And the question is, well, how
do we solve this Hamiltonian?
594
00:47:25,100 --> 00:47:27,690
The answer is, we
do exactly what we
595
00:47:27,690 --> 00:47:29,670
did in the classical problem.
596
00:47:29,670 --> 00:47:33,000
We transform to
the rotating frame.
597
00:47:38,090 --> 00:47:41,960
In other words, this
Hamiltonian is best
598
00:47:41,960 --> 00:47:48,150
solved by doing-- you can
actually solve it directly.
599
00:47:48,150 --> 00:47:51,040
You can just put in a tri
wave function and solve it.
600
00:47:51,040 --> 00:47:55,360
But I want to sort of
bring out the big idea here
601
00:47:55,360 --> 00:47:57,920
which is analogous
to what we have done
602
00:47:57,920 --> 00:47:59,950
in the last few
classes, namely we
603
00:47:59,950 --> 00:48:03,180
have involved rotating frames.
604
00:48:03,180 --> 00:48:09,570
So what solves this Hamiltonian
is a unitary transformation,
605
00:48:09,570 --> 00:48:17,040
and the unitary
transformation is this one.
606
00:48:27,120 --> 00:48:32,402
And so this unitary
transformation,
607
00:48:32,402 --> 00:48:37,710
let me first write it down,
it transforms the Hamiltonian
608
00:48:37,710 --> 00:48:44,250
to the time independent one.
609
00:48:44,250 --> 00:48:48,010
We have now time independent
of diagonal matrix elements.
610
00:48:52,260 --> 00:48:56,100
Our diagonal matrix
element have changed.
611
00:48:56,100 --> 00:49:02,030
Delta is now the detuning of the
tri frequency from the energy
612
00:49:02,030 --> 00:49:04,440
splitting of the
two-level system.
613
00:49:04,440 --> 00:49:07,340
In particular, when
we are on resonance,
614
00:49:07,340 --> 00:49:09,880
the diagonal matrix
elements have disappeared.
615
00:49:09,880 --> 00:49:21,460
This is the result of the
unitary transformation,
616
00:49:21,460 --> 00:49:30,240
and let me just show you this
transformation over here.
617
00:49:30,240 --> 00:49:37,730
Can be actually written
as an operator involving
618
00:49:37,730 --> 00:49:42,620
the z component of
the magnetic field.
619
00:49:42,620 --> 00:49:45,770
And what I just wrote
down for you is actually
620
00:49:45,770 --> 00:49:48,930
the quantum mechanical
operator, the rotation
621
00:49:48,930 --> 00:49:57,295
operator for performing at
rotation around the z-axis.
622
00:50:00,300 --> 00:50:06,070
So by selecting the rotation
angle to be omega t,
623
00:50:06,070 --> 00:50:10,290
that's how I can generate
the unitary transformation,
624
00:50:10,290 --> 00:50:16,201
and this unitary transformation
makes the Hamiltonian time
625
00:50:16,201 --> 00:50:16,700
independent.
626
00:50:23,180 --> 00:50:26,510
So in other words, everything
is in the classical system.
627
00:50:26,510 --> 00:50:29,262
We just go to a frame which
rotates with a [INAUDIBLE],
628
00:50:29,262 --> 00:50:30,970
and we find the
time-independent problem.
629
00:50:38,840 --> 00:50:46,470
So now this Hamiltonian
can be easily solved.
630
00:50:55,470 --> 00:51:00,020
And you will find
as a special case
631
00:51:00,020 --> 00:51:02,750
when you start
with an amplitude,
632
00:51:02,750 --> 00:51:05,510
initially you start
in the ground state,
633
00:51:05,510 --> 00:51:10,580
then the excited
state amplitude square
634
00:51:10,580 --> 00:51:17,750
is the Rabi
oscillation, something
635
00:51:17,750 --> 00:51:32,140
we discussed 40 minutes
ago, but before, we
636
00:51:32,140 --> 00:51:33,810
got it from the
classical quantum
637
00:51:33,810 --> 00:51:36,430
mechanical correspondence
using the Heisenberg equation
638
00:51:36,430 --> 00:51:41,600
of motion and here it comes
out by explicitly solving
639
00:51:41,600 --> 00:51:44,518
for the wave function for
the dressed Hamiltonian.
640
00:51:56,021 --> 00:51:56,520
Questions?
641
00:52:13,550 --> 00:52:26,370
I want to say a few words now
about rapid adiabatic passage,
642
00:52:26,370 --> 00:52:31,700
but this time by emphasizing
the quantum mechanical aspects.
643
00:52:36,910 --> 00:52:42,079
In other words, we have
a clear understanding
644
00:52:42,079 --> 00:52:43,120
what happens classically.
645
00:52:43,120 --> 00:52:45,070
We have a clear
understanding what
646
00:52:45,070 --> 00:52:48,070
happens in the adiabatic
limit, but I just
647
00:52:48,070 --> 00:52:50,670
want to sort of in
the next 10 minutes
648
00:52:50,670 --> 00:52:52,870
use what we have
already learned,
649
00:52:52,870 --> 00:52:56,210
combine it with the quantum
mechanical Hamiltonian
650
00:52:56,210 --> 00:53:00,950
and tell you that, well, when
you are not fully adiabatic,
651
00:53:00,950 --> 00:53:04,130
you actually have transition
probabilities between the two
652
00:53:04,130 --> 00:53:05,160
states.
653
00:53:05,160 --> 00:53:11,530
So I want to sort of bring
in the concept of transition
654
00:53:11,530 --> 00:53:19,040
probabilities to
the case of-- what
655
00:53:19,040 --> 00:53:21,362
I want to say is rapid
adiabatic passage when it's
656
00:53:21,362 --> 00:53:23,820
no longer adiabatic, but what
this just means when we sweep
657
00:53:23,820 --> 00:53:28,080
the frequency and we're
not in the adiabatic limit.
658
00:53:28,080 --> 00:53:35,400
So how do we describe
it quantum mechanically?
659
00:53:35,400 --> 00:53:46,076
We start out with a Hamiltonian,
which has-- we use our rotating
660
00:53:46,076 --> 00:53:49,510
framework for convenience
that allows to write down
661
00:53:49,510 --> 00:53:56,330
exactly the same Hamiltonian
in time-independent picture.
662
00:53:56,330 --> 00:54:03,680
So the Hamiltonian has
two parts, a diagonal part
663
00:54:03,680 --> 00:54:09,330
and an off-diagonal part.
664
00:54:14,350 --> 00:54:24,930
So if I show the energy as a
function of detuning delta--
665
00:54:24,930 --> 00:54:29,095
well, maybe I should
times 2 over H bar,
666
00:54:29,095 --> 00:54:35,190
just normalize it so then it
becomes just the straight line
667
00:54:35,190 --> 00:54:38,150
at 45 degree y equals x.
668
00:54:38,150 --> 00:54:48,530
So the unperturbed-- the
Hamiltonian without drive
669
00:54:48,530 --> 00:54:52,455
has a level crossing
at detuning 0.
670
00:54:55,770 --> 00:54:58,808
Then we add to it
the drive term.
671
00:55:05,640 --> 00:55:09,140
Well, let me just
write down, not
672
00:55:09,140 --> 00:55:15,950
the drive term but
the full Hamiltonian.
673
00:55:15,950 --> 00:55:20,280
So the full Hamiltonian with
the addition of the drive term
674
00:55:20,280 --> 00:55:23,530
has delta minus
delta and now it has
675
00:55:23,530 --> 00:55:27,370
the coupling with
the Rabi frequency.
676
00:55:27,370 --> 00:55:31,660
That means that on resonance,
the degeneracy between the two
677
00:55:31,660 --> 00:55:35,840
levels is split by
the Rabi frequency,
678
00:55:35,840 --> 00:55:38,200
and if I now show
you the energy eigen
679
00:55:38,200 --> 00:55:42,580
levels of this
two-by-two Hamiltonian,
680
00:55:42,580 --> 00:55:47,650
it will asymptotically
coincide with a dashed line
681
00:55:47,650 --> 00:55:51,130
through this and through that.
682
00:55:54,090 --> 00:55:57,440
So in other words,
I'm just reminding you
683
00:55:57,440 --> 00:56:02,250
that a non-diagonal matrix
element turns a crossing
684
00:56:02,250 --> 00:56:03,520
into an avoided crossing.
685
00:56:12,640 --> 00:56:18,800
So when we take
the frequency omega
686
00:56:18,800 --> 00:56:27,120
and we sweep the detuning,
so we change delta
687
00:56:27,120 --> 00:56:36,790
and do a sweep of the frequency
omega at a rate omega dot,
688
00:56:36,790 --> 00:56:40,960
then we sweep through the
resonance and in one limit,
689
00:56:40,960 --> 00:56:46,470
we have rapid adiabatic
passage or in general, we
690
00:56:46,470 --> 00:56:49,850
realize the Landau-Zener
problem of a sweep
691
00:56:49,850 --> 00:56:52,970
through an avoided crossing.
692
00:56:52,970 --> 00:56:56,010
So what I'm formulating
here is it's
693
00:56:56,010 --> 00:56:58,670
the so-called
Landau-Zener crossing
694
00:56:58,670 --> 00:57:01,480
or the Landau-Zener
problem, which
695
00:57:01,480 --> 00:57:07,430
is the quantum mechanical
description of you take
696
00:57:07,430 --> 00:57:12,370
a system by changing
an external parameter.
697
00:57:12,370 --> 00:57:17,300
Here, we sweep the frequency
of the rotating field,
698
00:57:17,300 --> 00:57:20,310
but by changing the
external parameter,
699
00:57:20,310 --> 00:57:24,418
we sweep the system through
the avoided crossing.
700
00:57:33,880 --> 00:57:36,600
And it has the
two limiting cases
701
00:57:36,600 --> 00:57:41,620
that when we go through this
crossing very, very slowly,
702
00:57:41,620 --> 00:57:44,640
the adiabatic field
then tells us we
703
00:57:44,640 --> 00:57:48,590
stay on one of these
adiabatic solid curves,
704
00:57:48,590 --> 00:57:51,850
and this is the case of rapid
adiabatic passage, which
705
00:57:51,850 --> 00:57:54,350
we discuss in the
classical limit.
706
00:57:54,350 --> 00:57:57,090
But it is also
the other solution
707
00:57:57,090 --> 00:58:00,510
if you would sweep through
it very, very fast,
708
00:58:00,510 --> 00:58:03,830
you're in the diabatic limit,
you follow the dashed line
709
00:58:03,830 --> 00:58:06,654
and you start up here
and you wind up there.
710
00:58:15,810 --> 00:58:17,310
The Landau-Zener
problem is actually
711
00:58:17,310 --> 00:58:20,790
a problem which you find
it in all it text books,
712
00:58:20,790 --> 00:58:22,590
but to the best of
my knowledge, there
713
00:58:22,590 --> 00:58:25,580
is no simple,
elementary derivation
714
00:58:25,580 --> 00:58:27,480
which I could give
you in a few minutes.
715
00:58:27,480 --> 00:58:29,750
And if the
mathematical problem is
716
00:58:29,750 --> 00:58:34,720
a nice, mathematical
demonstration
717
00:58:34,720 --> 00:58:41,430
of an exact, solvable model,
but to my knowledge explicitly
718
00:58:41,430 --> 00:58:43,940
deriving it is not providing
additional insight.
719
00:58:43,940 --> 00:58:47,260
It's one of the cases
where the result is
720
00:58:47,260 --> 00:58:51,210
more insightful and much
simpler than the derivation.
721
00:58:51,210 --> 00:58:53,180
So what I want to give
you is, therefore,
722
00:58:53,180 --> 00:58:54,540
simply the textbook result.
723
00:59:00,650 --> 00:59:05,590
So in the adiabatic limit,
you stay on the solid line.
724
00:59:05,590 --> 00:59:12,390
If you do not cross the avoided
crossing very, very slowly,
725
00:59:12,390 --> 00:59:16,610
you'll have a
non-adiabatic probability
726
00:59:16,610 --> 00:59:21,080
to jump from one level
to the other one.
727
00:59:21,080 --> 00:59:26,660
And this non-adiabatic
probability
728
00:59:26,660 --> 00:59:32,360
is expressed as an
exponential function which
729
00:59:32,360 --> 00:59:35,680
involves the
Landau-Zener parameter.
730
00:59:35,680 --> 00:59:46,290
And the Landau-Zener parameter
in this exact solution
731
00:59:46,290 --> 00:59:56,070
is omega Rabi squared times
this new rate, d omega
732
00:59:56,070 --> 01:00:00,048
dt or d delta dt minus 1.
733
01:00:04,780 --> 01:00:08,070
This square should go
outside the brackets,
734
01:00:08,070 --> 01:00:11,200
so therefore, what we find
is from the exact solution
735
01:00:11,200 --> 01:00:16,190
that the Landau-Zener
parameter is a quarter times--
736
01:00:16,190 --> 01:00:20,175
and this should
now look familiar,
737
01:00:20,175 --> 01:00:22,890
the omega Ravi frequencies
squared over omega dot.
738
01:00:26,160 --> 01:00:30,610
And when we discussed the limit
of adiabaticity classically,
739
01:00:30,610 --> 01:00:33,910
I hope you remember I gave
you the argument by looking
740
01:00:33,910 --> 01:00:39,520
at the adiabatic condition that
the adiabatic case requires
741
01:00:39,520 --> 01:00:43,090
omega dot to be much smaller
than omega Rabi squared.
742
01:00:43,090 --> 01:00:46,050
So here very naturally
what appears in the quantum
743
01:00:46,050 --> 01:00:49,500
mechanical problem is just the
ratio of the two quantities
744
01:00:49,500 --> 01:00:54,930
we compared when we looked
for the limit of adiabaticity.
745
01:00:54,930 --> 01:01:03,980
So therefore, the probability
for a non-adiabatic transition
746
01:01:03,980 --> 01:01:08,525
is simply involving this
ratio omega Rabi squared
747
01:01:08,525 --> 01:01:09,830
over omega dot.
748
01:01:15,240 --> 01:01:17,780
So in other words,
we know already
749
01:01:17,780 --> 01:01:19,530
from the classical
argument, but here we
750
01:01:19,530 --> 01:01:29,657
confirm it, adiabaticity require
that this inequality is met.
751
01:01:45,260 --> 01:01:52,300
OK, I could stop
here, but since we
752
01:01:52,300 --> 01:01:59,220
are using sort of diabatic
sweeps in the laboratory,
753
01:01:59,220 --> 01:02:03,950
as long as I've been involved
in doing quote "atom science,"
754
01:02:03,950 --> 01:02:09,330
I want to sort of go one
step further and teach you
755
01:02:09,330 --> 01:02:11,440
a little bit more
about this formula
756
01:02:11,440 --> 01:02:14,970
and try to provide
insight, and often
757
01:02:14,970 --> 01:02:19,120
insight is also provided when
you apply perturbation theory.
758
01:02:19,120 --> 01:02:22,120
So I know the adiabatic
case is very simple,
759
01:02:22,120 --> 01:02:24,830
but I want to look
at the diabatic case
760
01:02:24,830 --> 01:02:27,620
and then look at
transition probabilities
761
01:02:27,620 --> 01:02:29,240
in a perturbative way.
762
01:02:29,240 --> 01:02:34,100
This is actually the way how
we often transfer population
763
01:02:34,100 --> 01:02:34,850
in the laboratory.
764
01:02:42,870 --> 01:02:48,030
So I want to understand
better the way how we transfer
765
01:02:48,030 --> 01:02:52,000
population from one
curve to the other one.
766
01:02:52,000 --> 01:02:57,040
So if we do a fast sweep,
we call it diabatic.
767
01:03:02,040 --> 01:03:10,240
So in other words, if
we have this crossing
768
01:03:10,240 --> 01:03:15,740
and we go really fast,
well, what happens
769
01:03:15,740 --> 01:03:19,100
is this is the crossing
between spin up and spin down.
770
01:03:19,100 --> 01:03:22,160
If you go much, much faster
than the Rabi frequency,
771
01:03:22,160 --> 01:03:27,050
the spin has no opportunity
to change its orientation.
772
01:03:27,050 --> 01:03:33,130
So therefore, the wave function,
the spin has to stay up or down
773
01:03:33,130 --> 01:03:36,015
and that means the
system just goes straight
774
01:03:36,015 --> 01:03:37,800
through the crossing.
775
01:03:37,800 --> 01:03:41,790
Because spin up has
positive slopes,
776
01:03:41,790 --> 01:03:43,830
spin down has negative slope.
777
01:03:43,830 --> 01:03:48,360
Being adiabatic, staying on
this lower adiabatic curve
778
01:03:48,360 --> 01:03:50,460
would actually
require the system
779
01:03:50,460 --> 01:03:54,160
to go from spin up in this
part of the adiabatic curve
780
01:03:54,160 --> 01:03:56,600
to spin down in the other part.
781
01:03:56,600 --> 01:04:00,120
And to flip a spin cannot
be done faster than the Rabi
782
01:04:00,120 --> 01:04:05,430
frequency, so if you sweep
fast, that's what's happening.
783
01:04:05,430 --> 01:04:08,830
So we have two trivial limits,
one is the adiabatic limit
784
01:04:08,830 --> 01:04:11,770
or just the adiabatic
curve and nothing happens.
785
01:04:11,770 --> 01:04:14,660
The other limit is the
infinitely-fast limit
786
01:04:14,660 --> 01:04:17,380
and nothing happens
again when we
787
01:04:17,380 --> 01:04:21,170
look at the diabatic basis,
which is spin up and spin down.
788
01:04:21,170 --> 01:04:28,270
But now let's be
almost adiabatic,
789
01:04:28,270 --> 01:04:30,080
and this is a problem
which we really
790
01:04:30,080 --> 01:04:34,330
want to understand physically
and intuitively because that
791
01:04:34,330 --> 01:04:40,530
means the system spins the
small timelier resonance,
792
01:04:40,530 --> 01:04:51,036
and there is a small probability
to make a transition.
793
01:04:56,880 --> 01:04:58,960
So if you had one of
your hyperfine states,
794
01:04:58,960 --> 01:05:01,140
pick your favorite
hyperfine state,
795
01:05:01,140 --> 01:05:03,230
you'll rapidly
sweep the frequency.
796
01:05:03,230 --> 01:05:06,650
You will find, unless you
sweep it infinitely fast
797
01:05:06,650 --> 01:05:10,220
that there's a small probability
in the other hyperfine state.
798
01:05:10,220 --> 01:05:12,280
And that's what you
want to calculate now,
799
01:05:12,280 --> 01:05:15,220
and I want you to
understand how would you
800
01:05:15,220 --> 01:05:17,390
estimate and calculate
the small probability.
801
01:05:23,390 --> 01:05:33,230
So let's now estimate
the result, namely
802
01:05:33,230 --> 01:05:35,670
for the small probability
in perturbation theory.
803
01:05:43,820 --> 01:05:47,750
And actually what I'm
calculating for you here is,
804
01:05:47,750 --> 01:05:50,110
if you've used evaporation--
I know half of the class
805
01:05:50,110 --> 01:05:53,170
is doing that-- if you
apply an i F field,
806
01:05:53,170 --> 01:05:54,630
you don't have to
make it so strong
807
01:05:54,630 --> 01:05:56,150
that you and the
adiabatic limit.
808
01:05:56,150 --> 01:05:57,920
You are exactly in this limit.
809
01:05:57,920 --> 01:06:00,900
The atom will
slosh several times
810
01:06:00,900 --> 01:06:04,160
through the resonance in
an almost diabatic way,
811
01:06:04,160 --> 01:06:06,840
but there is a finite
spin flip probability
812
01:06:06,840 --> 01:06:09,160
and that's how you
evaporate atoms.
813
01:06:09,160 --> 01:06:11,630
And I want you now
to fully understand
814
01:06:11,630 --> 01:06:15,750
the derivation, what is the
probability of ejecting atoms
815
01:06:15,750 --> 01:06:19,570
in the almost diabatic
limit with i F spin flips.
816
01:06:19,570 --> 01:06:24,520
That's a limit where 90% of
the BEC experiments operate.
817
01:06:27,880 --> 01:06:30,640
So I hope everyone realize
it's an important question
818
01:06:30,640 --> 01:06:32,290
and also I hope
everybody understands
819
01:06:32,290 --> 01:06:36,450
the question because now I
have bigger questions for you.
820
01:06:36,450 --> 01:06:40,250
The first question
which I will ask you,
821
01:06:40,250 --> 01:06:48,460
should we calculate that
transition probability
822
01:06:48,460 --> 01:06:54,020
by using perturbation theory
for an incoherent transition
823
01:06:54,020 --> 01:06:56,490
or for coherent transition?
824
01:06:56,490 --> 01:06:59,150
Let me just explain you
what I mean and then I
825
01:06:59,150 --> 01:07:01,860
ask you for your opinion.
826
01:07:01,860 --> 01:07:05,700
Coherently, we simply say
in perturbation theory,
827
01:07:05,700 --> 01:07:10,860
we start with our
population in state 1.
828
01:07:10,860 --> 01:07:14,910
We have to do the
coupling Hamiltonian time
829
01:07:14,910 --> 01:07:18,300
dependence of the
population in state 2
830
01:07:18,300 --> 01:07:26,000
and that means if we integrate
this equation for a short time,
831
01:07:26,000 --> 01:07:29,060
we find an amplitude a2.
832
01:07:29,060 --> 01:07:31,400
And the probability
to be in the state
833
01:07:31,400 --> 01:07:35,140
2, which is the
amplitude squared,
834
01:07:35,140 --> 01:07:38,860
is proportional to the
Rabi frequency squared
835
01:07:38,860 --> 01:07:41,370
times the effective
time squared,
836
01:07:41,370 --> 01:07:45,320
the effective time of
tribing the system.
837
01:07:45,320 --> 01:07:48,160
Coherent processes are
always quadratic in time.
838
01:07:51,500 --> 01:07:55,610
If we do it incoherently,
well, the way
839
01:07:55,610 --> 01:07:57,860
how we describe
incoherent processes
840
01:07:57,860 --> 01:08:03,450
are Fermi's golden rule,
which we've all seen.
841
01:08:03,450 --> 01:08:12,510
And the probability in Fermi's
golden rule is very different.
842
01:08:12,510 --> 01:08:15,600
Well, it is proportional to
the Rabi frequency squared,
843
01:08:15,600 --> 01:08:18,250
to the matrix element squared,
but Fermi's golden rule
844
01:08:18,250 --> 01:08:22,340
gives us a constant rate,
and for constant rate,
845
01:08:22,340 --> 01:08:25,180
the probability is
rate times time.
846
01:08:25,180 --> 01:08:30,660
So now it is linear
in time and then
847
01:08:30,660 --> 01:08:37,290
because of the delta function
in Fermi's golden rule--
848
01:08:37,290 --> 01:08:40,890
I'm missing a symbol
so I use gamma here.
849
01:08:40,890 --> 01:08:43,540
It has nothing to do with
the Landau-Zener probability.
850
01:08:43,540 --> 01:08:45,534
This is just the
density of states.
851
01:08:49,890 --> 01:08:51,899
So I hope you know now
what is the difference
852
01:08:51,899 --> 01:08:54,109
between coherent or incoherent.
853
01:08:54,109 --> 01:08:57,979
The most important
part is that things
854
01:08:57,979 --> 01:09:01,014
are linear in time for an
incoherent pulses rate equation
855
01:09:01,014 --> 01:09:03,897
and at least for small
times quadratic in time
856
01:09:03,897 --> 01:09:04,730
for coherent pulses.
857
01:09:08,290 --> 01:09:15,330
So now we come to
this process where
858
01:09:15,330 --> 01:09:20,470
we take atoms from
spin up to spin down.
859
01:09:20,470 --> 01:09:25,750
We evaporate with a weak,
course, almost adiabatic
860
01:09:25,750 --> 01:09:29,560
with weaker f drive, so we are
closer to the diabatic limit.
861
01:09:29,560 --> 01:09:33,670
And so if you think
about this problem,
862
01:09:33,670 --> 01:09:41,590
I want you to tell
me if this process,
863
01:09:41,590 --> 01:09:47,170
the perturbative transition
close to the diabatic case
864
01:09:47,170 --> 01:09:51,399
is that should we use
when we apply perturbation
865
01:09:51,399 --> 01:10:03,810
theory, the coherent picture
or the incoherent picture.
866
01:10:03,810 --> 01:10:07,450
In other words, is the dynamics
of the quantum system, when
867
01:10:07,450 --> 01:10:13,855
we go relatively quickly for
the Landau-Zener crossing,
868
01:10:13,855 --> 01:10:16,630
is that a coherent or
an incoherent process?
869
01:10:24,250 --> 01:10:27,217
I could see where it doesn't
matter, but it does matter.
870
01:10:36,970 --> 01:10:40,656
So I think it's
an open question.
871
01:10:40,656 --> 01:10:41,780
Let me give you the answer.
872
01:10:44,370 --> 01:10:47,940
It is coherent, and you can
see it in the following way.
873
01:10:47,940 --> 01:10:50,736
What is the source
of incoherent here?
874
01:10:50,736 --> 01:10:52,710
We have a Hamiltonian.
875
01:10:52,710 --> 01:10:55,520
The Hilbert space
is by two-by-two.
876
01:10:55,520 --> 01:10:58,470
There is no coherence
which can be lost.
877
01:10:58,470 --> 01:11:01,700
There is no spontaneous
emission to other states.
878
01:11:01,700 --> 01:11:04,000
There is no reservoir.
879
01:11:04,000 --> 01:11:06,510
We don't have a small system
which couples to a bigger
880
01:11:06,510 --> 01:11:08,810
system, and then
the small system--
881
01:11:08,810 --> 01:11:12,680
we do that on Wednesday--
has to-- tomorrow,
882
01:11:12,680 --> 01:11:15,500
Wednesday-- has to be
described by density matrix.
883
01:11:15,500 --> 01:11:17,670
We have the none of
the physics which
884
01:11:17,670 --> 01:11:20,010
would give arise to
incoherent physics.
885
01:11:20,010 --> 01:11:20,900
It is coherent.
886
01:11:25,000 --> 01:11:26,690
But maybe I'm oversimplifying.
887
01:11:26,690 --> 01:11:29,140
Is somebody who
said it's incoherent
888
01:11:29,140 --> 01:11:31,959
who wants to maybe press
me harder and tell me
889
01:11:31,959 --> 01:11:33,250
why you think it is incoherent.
890
01:11:39,760 --> 01:11:41,770
Well, one possibility
is-- and this
891
01:11:41,770 --> 01:11:44,600
is why we often use
Landau-Zener sweeps in the lab.
892
01:11:44,600 --> 01:11:47,570
We have fluctuations of
the resonance frequency.
893
01:11:47,570 --> 01:11:50,004
And when we go and
sweep through it,
894
01:11:50,004 --> 01:11:53,400
we don't know exactly
when we hit the resonance.
895
01:11:53,400 --> 01:11:56,060
And if you would take
an ensemble of systems
896
01:11:56,060 --> 01:11:59,020
and you go through the
resonance at different times,
897
01:11:59,020 --> 01:12:01,640
you will get an ensemble
of wave functions, which
898
01:12:01,640 --> 01:12:04,860
has different phase factors
in it, and in the end,
899
01:12:04,860 --> 01:12:08,530
you will actually need a
density matrix to describe it.
900
01:12:08,530 --> 01:12:10,860
But this is now an
experimental imperfection
901
01:12:10,860 --> 01:12:13,200
which I haven't assumed here.
902
01:12:13,200 --> 01:12:16,070
So in other words, what you
should do is the following.
903
01:12:16,070 --> 01:12:18,660
Whenever you sweep through
Landau-Zener crossing,
904
01:12:18,660 --> 01:12:21,120
you start with the ground
state and what you get out
905
01:12:21,120 --> 01:12:23,890
is a superposition of
ground and excited state.
906
01:12:23,890 --> 01:12:28,720
And the Hamiltonian determines
absolutely every aspect
907
01:12:28,720 --> 01:12:31,890
of the amplitude in the ground
and the excited state including
908
01:12:31,890 --> 01:12:33,480
all phase factors.
909
01:12:33,480 --> 01:12:37,965
In other words, if you do a
Landau-Zener crossing in a way
910
01:12:37,965 --> 01:12:40,720
that you prepare
50-50% of the atoms,
911
01:12:40,720 --> 01:12:45,270
they are always face coherent
and you can use this process
912
01:12:45,270 --> 01:12:48,250
as a beam splitter in
an atomic thermometer.
913
01:12:48,250 --> 01:12:49,170
Fully coherent.
914
01:12:56,230 --> 01:13:09,470
We have five minutes,
so if we assume now,
915
01:13:09,470 --> 01:13:15,770
I hope we all agree that
the amplitude is obtained
916
01:13:15,770 --> 01:13:19,640
in this coherent wave, then,
of course, the question is,
917
01:13:19,640 --> 01:13:22,450
but what is the effective time?
918
01:13:22,450 --> 01:13:25,414
When we sweep the
resonance, we are far away.
919
01:13:25,414 --> 01:13:26,080
Nothing happens.
920
01:13:26,080 --> 01:13:26,820
Nothing happens.
921
01:13:26,820 --> 01:13:28,010
Nothing happens.
922
01:13:28,010 --> 01:13:30,620
We go through the resonance,
everything happens.
923
01:13:30,620 --> 01:13:31,300
Nothing happens.
924
01:13:31,300 --> 01:13:31,966
Nothing happens.
925
01:13:31,966 --> 01:13:32,760
Nothing happens.
926
01:13:32,760 --> 01:13:36,930
So this is the effective time
when the wave function really
927
01:13:36,930 --> 01:13:39,660
changes and we create
the coherent and mixture
928
01:13:39,660 --> 01:13:42,030
in the second state.
929
01:13:42,030 --> 01:13:45,170
And it is this
effective times squared
930
01:13:45,170 --> 01:13:51,670
which determines what
happens, what our transmission
931
01:13:51,670 --> 01:13:53,830
amplitude is.
932
01:13:53,830 --> 01:13:59,480
So therefore, the
question is, what
933
01:13:59,480 --> 01:14:02,170
is the effective time in
the Landau-Zener crossing?
934
01:14:04,940 --> 01:14:08,630
I can give you three choices.
935
01:14:08,630 --> 01:14:15,900
One is, the effective time
is, well, we change omega
936
01:14:15,900 --> 01:14:20,270
and the effective time
is, how long, what
937
01:14:20,270 --> 01:14:23,490
is the time until we
have a detuning, which
938
01:14:23,490 --> 01:14:25,200
is equal to the Rabi frequency.
939
01:14:29,520 --> 01:14:32,870
Another possibility is that
the effective time where
940
01:14:32,870 --> 01:14:35,320
we coherently
drive the system is
941
01:14:35,320 --> 01:14:39,370
1 over the Rabi frequency,
just the Rabi period.
942
01:14:39,370 --> 01:14:44,365
Or another choice, how
I can construct time out
943
01:14:44,365 --> 01:14:50,240
of the two frequencies where
it's omega Rabi and omega dot.
944
01:14:50,240 --> 01:14:52,010
These are our two elements.
945
01:14:52,010 --> 01:14:55,890
So another possibility
how I can get time
946
01:14:55,890 --> 01:15:00,680
is, omega dot is
frequency squared,
947
01:15:00,680 --> 01:15:04,290
and the square root of 1
over omega dot at least
948
01:15:04,290 --> 01:15:09,040
fulfills the dimensional
criterion that this is time.
949
01:15:09,040 --> 01:15:13,640
So my question for you
is, what do you think
950
01:15:13,640 --> 01:15:15,610
is the effective
time during which
951
01:15:15,610 --> 01:15:19,230
we drive the system coherently?
952
01:15:19,230 --> 01:15:22,360
I have to tell you before
I made up the problem,
953
01:15:22,360 --> 01:15:24,550
I do not know the answer.
954
01:15:24,550 --> 01:15:26,720
But I can also
tell you that there
955
01:15:26,720 --> 01:15:28,490
is only one answer
which is correct.
956
01:15:36,920 --> 01:15:37,600
All right.
957
01:15:45,080 --> 01:15:52,110
I have to say I expected A
to be the correct answer,
958
01:15:52,110 --> 01:15:56,040
but I convinced
myself it's only B.
959
01:15:56,040 --> 01:16:02,600
And the answer is the following,
and I know I have to stop,
960
01:16:02,600 --> 01:16:04,880
but I only need
three more lines.
961
01:16:04,880 --> 01:16:09,300
The effective time is not the
time until you are detuned,
962
01:16:09,300 --> 01:16:14,770
the effective time is which
I can call the dephasing
963
01:16:14,770 --> 01:16:21,600
time, the time during which
everything is coherent.
964
01:16:21,600 --> 01:16:29,110
What happens is, we change
the frequency delta omega,
965
01:16:29,110 --> 01:16:33,810
and the delta omega
is, of course,
966
01:16:33,810 --> 01:16:37,610
omega dot times delta t.
967
01:16:37,610 --> 01:16:42,050
So if we change the frequency
by sweeping it in such a way
968
01:16:42,050 --> 01:16:46,120
that we are detuning and
now with that detuning,
969
01:16:46,120 --> 01:16:50,950
if we let the system evolve, we
would get a phase shift of pi.
970
01:16:50,950 --> 01:16:55,630
That's sort of the maximum where
everything adds up coherently.
971
01:16:55,630 --> 01:16:58,560
If we would go longer
in time, the frequency
972
01:16:58,560 --> 01:17:00,660
has changed to the
point that what
973
01:17:00,660 --> 01:17:05,460
we add to the amplitude
of the other state
974
01:17:05,460 --> 01:17:09,266
is no longer in phase to
what we have added before.
975
01:17:09,266 --> 01:17:10,960
You can sort of look
at it like this.
976
01:17:10,960 --> 01:17:12,999
You have a little bit of
phase a2, You build up.
977
01:17:12,999 --> 01:17:13,540
You build up.
978
01:17:13,540 --> 01:17:16,600
You build up by adding
amplitude with the same phase.
979
01:17:16,600 --> 01:17:18,290
But now you are
sweeping, and this
980
01:17:18,290 --> 01:17:23,720
is the criterion where
you start to pile up
981
01:17:23,720 --> 01:17:25,220
things with the
wrong phase and then
982
01:17:25,220 --> 01:17:26,904
the phase eventually
becomes randomized
983
01:17:26,904 --> 01:17:28,570
and you're not
effectively contributing.
984
01:17:32,580 --> 01:17:35,970
This equation defines
the window delta
985
01:17:35,970 --> 01:17:41,180
t during which we effectively
add substantial amplitude
986
01:17:41,180 --> 01:17:45,550
in the second state, and it
involves delta t squared,
987
01:17:45,550 --> 01:17:49,335
so neglecting factors of
unity, the result of this
988
01:17:49,335 --> 01:17:58,640
is that, indeed, delta t
is 1 over omega dot plus.
989
01:17:58,640 --> 01:18:04,630
And indeed, if we say the
probability for coherent drive
990
01:18:04,630 --> 01:18:10,150
is Rabi frequency squared times
delta t squared, what we now
991
01:18:10,150 --> 01:18:15,480
obtain is the Rabi frequency
squared over omega dot,
992
01:18:15,480 --> 01:18:19,560
and this is our
Landau-Zener parameter.
993
01:18:19,560 --> 01:18:33,240
So in other words, if we
check with the exact result,
994
01:18:33,240 --> 01:18:41,040
the Landau-Zener probability,
1 minus p non-adiabatic--
995
01:18:41,040 --> 01:18:44,240
the Landau-Zener problem
is e to the minus 2 pi
996
01:18:44,240 --> 01:18:46,900
over gamma or gamma pi over
2 over the Landau-Zener
997
01:18:46,900 --> 01:18:51,980
parameter, if I do an expansion
of the Landau-Zener parameter
998
01:18:51,980 --> 01:18:54,880
for a small value
of the exponent,
999
01:18:54,880 --> 01:18:59,950
the exact result is
2 pi times gamma,
1000
01:18:59,950 --> 01:19:03,880
and this is indeed proportional
to omega Rabi squared
1001
01:19:03,880 --> 01:19:05,870
over omega dot.
1002
01:19:05,870 --> 01:19:07,590
So in other words,
what I've shown
1003
01:19:07,590 --> 01:19:10,510
you is that coherent
time evolution
1004
01:19:10,510 --> 01:19:15,590
with this weird effective time,
what I motivated physically,
1005
01:19:15,590 --> 01:19:18,580
exactly that produce
the limit of small gamma
1006
01:19:18,580 --> 01:19:20,790
from the exact result for
the Landau-Zener crossing.
1007
01:19:23,420 --> 01:19:24,870
OK.
1008
01:19:24,870 --> 01:19:25,930
I know time is over.
1009
01:19:25,930 --> 01:19:26,810
Any questions?
1010
01:19:30,130 --> 01:19:30,630
OK.
1011
01:19:30,630 --> 01:19:33,650
So today was officially
our Monday class,
1012
01:19:33,650 --> 01:19:38,514
so we meet again tomorrow,
the same place, the same time.