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quantum

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Tbs(ω,α,β,ϕ)
21
0 0
✲ ′ ✲
❍ ❍P 1,α+β P 3,ϕ ✟✟✟
❍ ✟
❍ ✟
❍ ✟
❍ ✟
❍ ✟
❍ ✟
❍✟
✟❍ S(ω)
✟ ❍
P 2,β ✟✟ ❍
❍
✟ ❍
✟ ❍
✟ ❍
1 ✲✟✟ ❍ ❍❍ 1 ′ ✲
a)
TMZ(α,β,ω,ϕ)
21
M
0 0
✲ ′ ✲
❅ P 1,α+β (cid:0)(cid:0)❅ (cid:0)(cid:0)
❅ (cid:0) ❅ b P ,ϕ (cid:0)
4
❅ (cid:0) ❅ (cid:0)
P ,ω
S (1) ❅ (cid:0) 3 ❅ (cid:0)S (1)
1 2 (cid:0)❅ ❅(cid:0) 2 2
(cid:0) ❅ (cid:0) ❅
(cid:0) ❅ (cid:0) ❅
(cid:0) c❅ (cid:0) ❅
1 ✲(cid:0) P 2,β ❅❅(cid:0) ❅❅ 1 ′ ✲
M
b)
Figure 4: Elementary quantum interference device. An elementary quantum interference device
can be realized by a 4-port interferometer with two input ports 0,1 and two output ports 0,1.
′ ′
Any twodimensional unitary transformationcan be realized by the devices. a) shows a realization
by a single beam splitter S(T) with variable transmission t and three phase shifters P ,P ,P ; b)
1 2 3
showsarealizationwith50:50beamsplittersS (1)andS (1)andfourphaseshiftersP ,P ,P ,P .
1 2 2 2 1 2 3 4
22
P : 1 1eiβ , (90)
2
→
S : 0 T 1 +iR0 , (91)
′ ′
→
S : 1 T 0′+iR1′ , (92)
→
P : 0 0eiϕ . (93)
3 ′ ′
→
1 0
If 0 0 and 1 1 and R(ω) = sinω, T(ω) = cosω, then the corre-
≡ ′ ≡ 0 ≡ ′ ≡ 1
(cid:18) (cid:19) (cid:18) (cid:19)
sponding unitary evolution matrix which transforms any coherent superposition of 0 and 1 into a
superposition of 0 and 1 is given by
′ ′
iei(α+ϕ) sinω eiα cosω −1
Tbs(ω,α,β,ϕ) = eiβ
21 eiϕ cosω i sinω
(cid:20) (cid:18) (cid:19)(cid:21)
ie i(α+ϕ) sinω e iϕ cosω
= e −iβ − e− − . (94)
iα cosω i sinω
−
(cid:18) − (cid:19)
The elementary quantum interference device TMZ depicted in Fig. (4.b) is a (rotated) Mach-
21
Zehnder interferometer with two input and output ports and three phase shifters. According to
the “toolbox” rules, the process can be quantum mechanically described by
P : 0 0eiα+β , (95)
1
→
P : 1 1eiβ , (96)
2
→
S : 1 (b+ic)/√2 , (97)
1
→
S : 0 (c+ib)/√2 , (98)
1
→
P : c ceiω , (99)
3
→
S : b (1 +i0)/√2 , (100)
2 ′ ′
→
S : c (0′+i1′)/√2 , (101)
2
→