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M1L5q.txt
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M1L5q.txt
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#
# File: content-mit-8422-1x-captions/M1L5q.txt
#
# Captions for 8.422x module
#
# This file has 35 caption lines.
#
# Do not add or delete any lines. If there is text missing at the end, please add it to the last line.
#
#----------------------------------------
The second element we need to realize arbitrary single-qubit
operation is the beam splitter.
And I introduce the beam splitter by just saying,
hey, look, I think that's a good Hamiltonian.
Let's see what this Hamiltonian does to the two modes.
And then we realize, yes, it takes those two modes
and it mixes the modes with cosine theta sine theta
weighting factors.
And that's exactly what you expect the beam splitter to do.
I'm not showing it here now because it's part
of your homework assignment.
You will show that if you have a coherent state,
the coherent state is split by a ratio, which
is cosine square theta sine square theta.
So it's exactly what you expect from a beam splitter.
So from that, you realize now what
this angle theta is which I just put into the Hamiltonian.
The cosine square and the sine square of it
are the reflection and transmission.
It's the reflection and the transmission
of the beam splitter.
Let me now introduce a matrix representation
which will come in handy.
You remember that if you transform an operator, the mode
operator, the annihilation operator a and b,
we multiply with the beam splitter operator on the left
and on the right-- [INAUDIBLE] B a B [INAUDIBLE].
But there is often a simpler way how the right-hand side can
be written, and this is by using a matrix representation which
goes as follows.
We can say that the two operators are transformed
by the following matrix, and this matrix
represents the beam splitter.