-
Notifications
You must be signed in to change notification settings - Fork 0
/
M5L25k.txt
219 lines (215 loc) · 8.59 KB
/
M5L25k.txt
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
#
# File: content-mit-8-421-5x-subtitles/M5L25k.txt
#
# Captions for 8.421x module
#
# This file has 209 caption lines.
#
# Do not add or delete any lines.
#
#----------------------------------------
So that's the basic phenomenon of superradiance.
Let's assume we have the same system
and we just convinced ourselves, yes it's superradiant.
Photons are emitted N times faster.
Now, what would you think will happen when we are not
looking for spontaneous emission,
but we shine a laser light on it and we are
asking for induced emission?
Or the other way around, we ask--
and you know that from Einstein's treatment,
it's completely reciprocal-- we'll
be asking the question what happens to the absorption
process?
So is a stimulated emission process or an absorption
process, are they also enhanced N times?
I don't know.
Do you have any opinions about that?
It should be, because it's [INAUDIBLE] new system.
So we forget that they were in atoms.
We just treat it as one new system, and those [INAUDIBLE],
so a stimulated emission [INAUDIBLE].
Yes.
It's a subtle way of counting.
I've shown you with that certain matrix element, especially
the matrix element when the spin is sort of in the middle,
is at 90 degrees, that we have matrix elements which
are N times enhanced.
And of course, if we ask for absorption or stimulated
emission, we're talking to a system, which has an N times
enhanced matrix element, and you would
say things go N times faster.
Why don't you hold this thought for a moment.
Now let me just wear another hat and say
we have assumed that we have n independent spins
that are closely next to each other
but they're not interacting.
And now I take these N spins, and for stimulated emission
and absorption we can just use the picture
of Rabi oscillation.
On the first Rabi cycle, we emit.
On the next Rabi cycle, we absorb.
So if I take now-- and why don't you think about,
not about these pseudospins, electronically,
atoms with electronic excitation,
just think of real spins which have a magnetic moment
and you drive them with a magnetic field.
So now you have your N little spins.
You apply a magnetic field to them--
time-dependent magnetic field-- and the time-dependent magnetic
field is now driving the spin in Rabi oscillations.
And the external drive field talks to one spin,
talks to the next, talks to all of them, but each of the spins
does exactly the same Rabi oscillation
it would do if all the other atoms were not present.
So the picture is you have an external field, all the atoms
coupled to the external field, but the coupling
of each atom to this external field
is exactly the same as of a single atom
coupling to the external field, and the Rabi frequency
for each atom is exactly the Rabi frequency you
would get for a single atom.
So therefore, based on this picture,
I would expect that I have my N spins,
and these can now be real spins with a magnetic moment
or can be atoms in the electronically excited state.
I coherently drive them with a drive field
and they will do Rabi oscillations,
but the frequency of Rabi oscillations
will be the same as for a single atom.
OK.
We've just held the thought that there
are matrix elements in the Dicke states
which are N times-- square root N times larger--
and it seems-- the Dicke states seem to suggest something
to us, which would say there should
be an enhancement, whereas the analysis in independent atoms
which are driven by an external field also seems compelling.
So now we have to reconcile the two approaches.
Is the question clear?
So we want to figure out, we have matrix elements
in the Dicke state which suggest enhancement,
but the simple picture of N independent atoms
driven by an external field would
say there is no enhancement.
OK.
So let me just write down more formally what I explained.
When we have an initial state, which
is all the atoms are in the ground state to the power N,
and the state N would evolve when it is
driven in the state phi of t.
So now the-- let me-- how do I say--
the wave function for-- the exact wave function for our N
particles is nothing else than the time-dependent solution
of Schrodinger's equation for a single particle--
so this is single particle-- to the power N.
So therefore-- I mean, this is pretty much
a mathematical proof.
So it takes exactly 1/2 a single atom Rabi period
to completely invert the population
exactly as for a single atom.
So that's the result. However, if you describe the system
by Dicke states, you have matrix elements
which are proportional to N. However,
if you want to-- I mean, I've sort of described it just
as a two level system.
Each atom does Rabi oscillation.
Then I've said, OK, the system of N atoms
is just N individual systems.
But if you insist to describe it as a collective spin,
then we have the Dicke states, then
we have the N times enhancement of the matrix element, but then
we also have to go through N states.
So we have N steps in the Dicke ladder, and one can say now--
and this is sort of the exact argument-- you have N steps.
You take each step N times faster.
But the total time is the same.
Sort of N times 1/N is 1.
OK.
But now, when we talk about spontaneous emission,
we are not driving the system with an external field.
It's really driven by the system itself, which emits photons
into the empty mode.
Spontaneous emission, each step is
proportional to the matrix element squared
because we're talking about Fermi's golden rule
in spontaneous emission.
So this is proportional to N squared.
And if you say that we have N steps, well,
then we have N squared over N. Then we have a speedup.
Each step is N squared faster.
We divide by N and we get the superradiant speedup,
which is N.
So superradiance is something which you observe
in spontaneous emission.
But you cannot absorb it in a driven system,
because in a driven system you can say you have a classical
external field and this external field talks to one atom,
to N atoms exactly in the same way.
It is really the interference of spontaneously emitted photons
which is at the heart of superradiance.
As a side remark, we're talking here
about the coherent effect, which is N times enhanced,
and you can actually regard that as a kind
of bosonic enhancement in the emission of photons
because the photons are bosons.
When Bose-Einstein condensation was discovered
and people were thinking about basic experiments, of course
one thing which was on our mind is
we wanted to show that there are processes
in the Bose-Einstein condensate which are N times enhanced.
For fermions they would be suppressed.
This is sort of the flip side.
Weak enhancement for bosons, complete suppression
for fermions.
And we found it.
For instance, formation of the condensate had an N times
enhancement.
There was a stimulation factor.
But we also thought, you should actually--
there may be ways where you can observe suppression
of light scattering or enhancement
of light scattering.
But we thought about it with two laser beams backscattering,
and the ideas seemed compelling.
And then we said no, wait a moment.
If we use laser beams, everything is stimulated.
You can observe bosonic enhancement
and fermionic suppression only when
you have spontaneous events.
If you drive it in a unitary time evolution,
you will not be able to see quantum statistical suppression
or enhancement.
And the same thing as we have seen here.
When you have a stimulated system,
everything is sort of undergoing a unitary time evolution,
and the unitary time evolution for N atoms
is the same as for a single atom.
You need the element of spontaneous emission.
So it --- I'm not proving it to you.
But I'm just making it as a remark.
We have seen here that the superradiance only
shows up in spontaneous emission and not when
we drive the system.
A driven system is a unitary evolution,
and the same conclusion which we just got for superradiance
also applies if you want to observe fermionic suppression
or bosonic enhancement in quantum gasses.
It needs an element of spontaneous scattering
or spontaneous emission.
[Question] If we can look at it in terms of interference of photons,
how does that tie in here?
Because if the stimulated emitted photons are still
interfering, then you could get addition
or destructive interference, constructive interference
of those photons, right?
The quick answer is you have a classical field which
you use for-- you have a laser field for stimulating emission
or for absorption.
There are so many photons in the laser field
that the few photons which your system emits do not matter.
We're really talking to a classical field,
and it doesn't matter whether the other N minus 1 atoms
have emitted a photon, because you have zillions of photons
in your laser field and they determine
the dynamics of the system.