-
Notifications
You must be signed in to change notification settings - Fork 2
/
M1L7s.txt
155 lines (152 loc) · 6.29 KB
/
M1L7s.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
#
# File: content-mit-8422-1x-captions/M1L7s.txt
#
# Captions for 8.422x module
#
# This file has 146 caption lines.
#
# Do not add or delete any lines. If there is text missing at the end, please add it to the last line.
#
#----------------------------------------
OK.
The last example is the squeezed light interferometer.
I just want to mention it briefly
because we've talked so much about squeezed light.
So now I want to show you that squeezed light can also
be used to realize Heisenberg limited interferometry.
So the idea is when we plot the electric field versus time,
and if we do squeezing in one quadrature,
then for certain times, the electric field
has lower noise and higher noise.
We discussed that.
And the idea is that if there is no noise at the zero crossing,
that this means we can determine the zero crossing of the light,
and therefore, a phase shift with higher accuracy.
You may also argue if you have these quasi-probabilities,
and with squeezing, we have squeezed the coherent light
into an ellipse and things propagate with e
to the i omega t, then you can determine a phase shift, which
is sort of an angular sector in this diagram,
with higher precision if you have squeezed the circle
into an ellipse like that.
So that's the idea.
And well, it's fairly clear that squeezing, if done correctly,
can provide a better phase measurement.
And what I want to show you here in a few minutes
is how you can think about it.
So we have discussed at lengths the optical interferometer,
where we have just a coherent state at the input.
This is your standard laser interferometer.
But of course, very importantly, the second input part
has the vacuum state.
And we discussed the importance of that.
So the one difference we want to do now
is that we replace the vacuum at the second input
by the squeezed vacuum, where r is the parameter
of the squeezed vacuum.
OK.
So that's pretty much what we do.
We take the state, we plug it into our equation,
we use exactly the same formulas we
have used for coherent states, and the question,
what is the result?
Well, the result will be that the squeezing factor appears.
Just as a reminder, for our interferometer,
we derived the sensitivity of the interferometer.
We had the quantities x and y.
And the noise is delta in this y operator squared divided by x.
This was the result when we operate the interferometer
at a phase shift of 90 degrees.
Just a reminder.
That's what we have done.
That's how we analyze the situation
with the coherent state.
The signal x is the number measurement, a dagger a.
And we have now the input of the coherent state and b dagger b.
We have an input mode a and a mode b.
They get split and then we measure at the output.
And we can now, at the output, have photons a dagger a which
come from the coherent state, and b dagger b which
come from the squeezed vacuum.
So this is now using the beamsplitter formalism
applied to the interferometer.
So this is now the result we obtain.
And in the strong local oscillator approximation,
it is only the first part which contributes.
And this is simply the number of photons in the coherent beam.
The expectation value of y is 0 because it involves
the b and b dagger operator.
And the squeezed vacuum has only--
if you write it down in the n basis, in the Fock state basis,
has only even n.
So if you change n by 1, you lose
overlap with the squeezed vacuum.
So therefore, this expectation value is 0.
For the operator y is squared, you take this and square it.
And you get many, many terms, which I don't want to discuss.
I use the strong local oscillator limit
that a dagger and a can be replaced
by the eigenvalue alpha of the coherent state.
So therefore, I factor out alpha squared
in the strong local oscillator limit.
And then what is left is b plus b dagger squared.
And since we have squeezed the vacuum,
this gives us a factor e to the minus r.
So if we put all those results together,
we find that the phase uncertainty is now
what we obtained when we had a coherent state
with the ordinary vacuum.
And in the strong local oscillator limit,
the only difference to the ordinary vacuum
is that in this term, we've got the exponential factor e
to the minus r.
And since we have taken a square root, it's e to the minus r/2.
So that result would actually suggest
that the more we squeeze, that delta phi should go to 0.
So it seems even better than the Heisenberg limit.
However, well, this is to good to be true.
What I've neglected here is the following.
When you squeeze more and more, the more you
squeeze the vacuum, the more photons
are in the squeezed vacuum, because this ellipse stretches
further and further out, and does overlap with Fock
states at higher and higher photon number.
So therefore, when you go to the limit of infinite squeezing,
you squeeze out of the limit where you can regard
the local oscillator as strong.
Because the squeezed vacuum has more photons
than your local oscillator.
And then you have to consider additional terms.
So let me just write that down.
However, the squeezed vacuum has non-zero average photon number.
And the photon number of the squeezed vacuum
is, of course, apply b dagger b to the squeezed vacuum.
This gives us a sinht function.
And we can call this the number of photons
in the squeezed vacuum.
So we have to consider now this contribution to y square.
So we have to consider the quadrature
of the ellipse, the long part of the ellipse,
the non-squeezed quadrature component.
And we have to consider that when
we calculate the expectation value of y square.
And then we find additional terms, which
I don't want to derive here.
And the question is then, if you squeeze too much, we lose.
So there's an optimal amount of squeezing.
And for this optimal amount of squeezing,
the phase uncertainty becomes approximately 1
over the number of photons in the coherent state
plus the number of photons in your squeezed vacuum.
So this is, again, very close to the Heisenberg limit.
So the situation with squeezed light
is less elegant, because if you squeeze too much,
you have to consider additional terms.
This is why I gave you the example of the squeezed light
and the squeezed vacuum as the last.
But again, the Heisenberg limit is really fundamental,
as we discussed.
And for an optimum arrangement of the squeezing,
you can also use a squeezed vacuum input
to the interferometer to realize the Heisenberg limit.
Any questions?