Download Laycock, Thomas Henry (2013) Creation and manipulation of quantum states of light and cold ... PDF

TitleLaycock, Thomas Henry (2013) Creation and manipulation of quantum states of light and cold ...
LanguageEnglish
File Size14.6 MB
Total Pages204
Document Text Contents
Page 102

2.6 Single Photon Sources 88

introduction of the matrix

Bmj (Ωq̂) = χ−1mj
N∑

γ=1

e−i|kL|q̂·rγχγm,

yielding

ρ (Ωq̂) =
3Γag




j,j′

eikL·(rj−rj′)ψjψ

j′



ν

(

d̂ga · eqν
)2∑

m,n

Bmj (Ωq̂)B∗nj′ (Ωq̂)
κ∗n + κm

.

Converting this into matrix form simplifies the notation somewhat

ρ (Ωq̂) =
3Γag




ν

(

d̂ga · eqν
)2∑

j,j′

ψ̃

j′

[
B† (Ωq̂)AB (Ωq̂)

]

j′j
ψ̃j

=
3Γag




ν

(

d̂ga · eqν
)2

ψ̃†B† (Ωq̂)AB (Ωq̂) ψ̃, (2.45)

where ψ̃j = e
ikL·rjψj and the elements of the matrix A are defined as

Anm =
1

κ∗n + κm
.

Finally, using the fact that the unit vector in the direction of the transition dipole

moment can be composed into three directions defined by the polarisation and

propagation vectors of the emitted photon (eq1, eq2 and q̂) as

(

d̂ga · eq1
)2

+
(

d̂ga · eq2
)2

+
(

d̂ga · q̂
)2

= 1,

the sum over the polarisation may be found as



ν

(

d̂ga · eqν
)2


[

1−
(

d̂ga · q̂
)2
]

.

Substitution of this representation of the sum yields the final expression for the

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