AAWSAP DIRD, Quantum Tomography of Negative Energy States in the Vacuum, January 2011

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UNCLASSIFIED//FOR 055ICIAl 1481!! er•t I
a'=rt\1 +(l - ri)1' 2 a2 from Section IIIB-2 to define the annihilation operators of the
.
, ,,
1/ 2 • 1
1/ 2 -
- ·
1/2 • 1
1/ 2 -
­
detected light modes a 1 = ri a1 +(1- ri) b1 and a 2 =ri a2 +(1 - ri) b2 , where b, and
b2 are the annihilation operators of the vacua entering the second unused ports of the
fictitious beam splitters. The annihilation operators a; and a~ describe the light modes
(or fields) emerging from the 50:50 beam splitter where the signal is optically mixed
with the LO. Again, the LO is an intense field compared with the signal so it can be
treated classically. Therefore, we do some algebra to compute the difference photon
number ii21 = n; - n7= a;ta; - a7ta7 , but retain only the leading terms with respect to
aw , and obtain the final result [38]:
(12)
The symbol H C in Eq. (12) denotes the Hermitian conjugate of the other part of an
expression and b=i- 12(b2-b1) . The fluctuation mode operator b corresponds to the
optical mixing of the fictitious vacuum-noise modes b1 and b2, and it obeys the bosonic
commutation relation [ b,b']=1 (e.g., see Sect. 111B-2). Because the interference of
vacuum with vacuum yields vacuum, the fluctuation mode b can be regarded as a
bosonic mode, being in the vacuum state as well.
UNCLASSIFIED//FOR OFFI@IAl YSfI or•t'&'
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