Topics in Quantum Optics and Quantum Information by Man Hong, Yung
University of Illinois at Urbana-Champaign
last updated 12 March 2007
Page 1 of 2
Problem Set #7a
Question 1
The operation of the beam splitter is as follows:
(
)
( )
1 1' 2'
2 2' 1'
/ 2
/ 2
a a ia
a a ia
+ + +
+ + +
→ +
→ +
and the polarization is preserved.
First consider
( )
( )( ) ( )( )
( )
,1 ,2 ,1 ,2
,1' ,2' ,2' ,1' ,1' ,2' ,2' ,1'
,1' ,2' ,1' ,2' ,1' ,1' ,2' ,2 '
,1' ,2 ' ,1' ,2' ,1' ,1' ,2'
10
2
1
0
2 2
1
2 2
H V V H
H H V V V V H H
H V V H V H V H
V H H V H V V H
a a a a
a ia a ia a ia a ia
a a a a ia a ia a
a a a a ia a ia a
ψ
± + + + +
+ + + + + + + +
+ + + + + + + +
+ + + + + + +
= ±
→ + + ± + +
− + +
=± − + +
( )
( )
( )
,2'
,1' ,1' , 2' ,2 '
,1' ,2' ,1' , 2'
0
0 for
2
10 for
2
V H V H
H V V H
ia a a a
a a a a
ψ
ψ
+
+ + + + +
+ + + + −
+
=
−
Next,
( )
( )( ) ( ) ( )
( )
,1 ,2 ,1 ,2
,1' ,2' ,2' ,1' ,1' ,2 ' ,2' ,1'
,1' ,2' ,1' ,2' ,1' ,1' ,2' ,2 '
,1' ,2' ,1' ,2' ,1' ,1' ,2'
10
2
1
0
2 2
1
2 2
H H V V
H H H H V V V V
H H H H H H H H
V V V V V V V V
a a a a
a ia a ia a ia a ia
a a a a ia a ia a
a a a a ia a ia a
φ
± + + + +
+ + + + + + + +
+ + + + + + + +
+ + + + + + +
= ±
→ + + ± + +
− + +
=± − + +
( )
( ) ( )
,2'
,1' ,1' ,2' ,2 ' ,1' ,1' ,2 ' ,2 '
0
0
2 2
H H H H V V V V
i i
a a ia a a a ia a
+
+ + + + + + + +
= + ± +
Note that only for
ψ
−
do the photons leave the beam splitter by separate ports.
Question 2
Given
( )
1
2
A B A B
AB
H H V Vφ
+
= +
, we shall need the following substitutions:
