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In what follows, only finite-size systems are studied. The Fock states are based upon the
Fock vacuum. The Fock vacuum is a direct product of states 0 of the i’th Hilbert space H
i i
| i
characterizing mode i; i.e.,
0 = 0 = 0 0 0
i 1 2 3
| i | i | i ⊗| i ⊗| i ⊗···
i I
Y∈
= 0 = 0 ,0 ,0 ,... , (76)
i 1 2 3
| { }i |{ }i
i I
[∈
where again I is an index set characterizing all different field modes labeled by i. “0 ” stands for
i
0 (no) quantum (particle) in the state characterized by the quantum numbers i. Likewise, more
generally,“N ”standsforN quanta(particles)inthestatecharacterizedbythequantumnumbers
i
i.
The annihilation operators a are designed to destroy one quantum (particle) in state i:
i
a 0 =0 , (77)
j
| i
a 0 ,0 ,0 ,...,0 ,N ,0 ,... =
j 1 2 3 j 1 j j+1
|{ − }i
= N 0 ,0 ,0 ,...,0 ,(N 1),0 ,... . (78)
j 1 2 3 j 1 j j+1
|{ − − }i
p
The creation operators a†i are designed to create one quantum (particle) in state i:
a†j |0 i= |{0 1,0 2,0 3,...,0 −1,1 j,0 j+1,... . (79)
j
}i
More generally, N operators (a†j)Nj create an N j-quanta (particles) state
j
(a†j)Nj |0 i∝|{0 1,0 2,0 3,...,0 −1,N j,0 j+1,... . (80)
j
}i
For fermions, N 0,1 because of the Pauli exclusion principle. For bosons, N N . With
j j 0
∈ { } ∈
propernormalization[whichcanmotivatedbythe(anti-)commutatorrelationsandby (X,X)2 =
| |
1],astate containingN quanta(particles)inmode1,N quanta(particles)inmode2,N quanta
1 2 3
(particles) in mode 3, etc., can be generated from the Fock vacuum by
(a†i)Ni
N N ,N ,N ,... = 0 . (81)
i 1 2 3
| { }i≡|{ }i √N ! | i
i I i I i
[∈ Y∈
As has been stated by Glauber ([43], p. 64),
... in quantum theory, there is an infinite set of complex numbers which specifies the
state of a single mode. This is in contrastto classical theory where each mode may be
described by a single complex number. This shows that there is vastly more freedom
in quantum theory to invent states of the world than there is in the classical theory.
We cannot think of quantum theory and classical theory in one-to-one terms at all. In
quantum theory, there exist whole spaces which have no classicalanalogues,whatever.
C Quantum interference
Inwhatfollows,weshallmakeuseofasimple“toolbox”-schemeofcombininglosslesselementsofan
experimentalsetupforthe theoreticalcalculation[48]. Theelementsofthis“toolbox”arelistedin
Table1. These“toolbox”rulescanberigorouslymotivatedbythefullquantumopticalcalculations
(e.g.,[98,91])but aremucheasierto use. Inwhatfollows,the factoriresultingfromaphase shift
of π/2 associated with the reflection at a mirror M is omitted. However, at a half-silvered mirror
beamsplitter,therelativefactoriresultingfromaphaseshiftofπ/2iskept. (Adetailedcalculation
[21]showsthatthisphaseshiftofπ/2isanapproximationwhichisexactlyvalidonlyforparticular
system parameters). T and R=√1 T2 are transmission and reflection coefficients. Notice that
−
the “generic” beam splitter can be realized by a half-silvered mirror and a successive phase shift
of ϕ = π/2 in the reflected channel; i.e., a (b+ic)/√2 (b+ie iπ/2c)/√2 (b+c)/√2.
−
− → → →
Note also that, in the “second quantization” notation, for i<j,
|i i|j i≡a†ia†j |0 i= |i i⊗|j i= |0 1,0 2,0 3,...,0 −1,1 i,0 i+1,...,0 −1,1 j,0 j+1,... . (82)
i j
i
19
physical process symbol state transformation
reflection at mirror a b=ia
a M →
b
“generic” beam splitter a (b+c)/√2
✏b✏ →
a
✏
P
P
P
c
transmission/reflection a (b+ic)/√2
→
by a beam splitter a Tb+iRc,
→
(half-silvered mirror) T2+R2 =1, T,R [0,1]
S 1 ∈