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

Text

This text came with the file. It may have mistakes. When the text and the image disagree, the image is right.

UNCLASSIFIED/ /POI': orr1c11tt U.!I! Oflt I
bosonic commutation relations: [a;' a:,;]
a, 'a,;,]
and [a;' a:,,]
al 'a,,.] =0 /
=
[
=
=
[
81 Ill
where 6c111 (= l if e= m and 0 if t ":/: m) is the Kronecker delta and the indices (t , m) are
integers [38].
A beam splitter is a four-port device not only in the case of two incoming light modes
interfering to produce two emerging light modes; a beam splitter is always a four-port
device. Even if only one beam is split into two beams, if literally nothing behind the
semitransparent mirror is interfering with the incident beam, quantum mechanically this
nothing means a vacuum state. The very possibility that the second light mode behind
the mirror might be excited makes a difference. The vacuum fluctuations carried by the
empty mode (and entering the apparatus via the so-called unused input port of the
beam splitter) do cause physical effects. Therefore, the vacuum fluctuations entering
the second (unused) input port of the beam splitter must always be assigned a formal
mode operator, a2, in order for the system to
conserve energy, 2) obey the beam
1 )
splitter's aforementioned bosonic commutation relations and 3) guarantee that the two
outgoing beams are independent bosonic light modes.
first input
a1
second output
A/
a2
second input
a2
first output
A /
al
Figure 8. Schematic of an Ideal Lossless Beam Splitter. Two
incident spatial-temporal light modes (with the annihilation operators a 1
and a2) interfere optically to produce two emerging light modes (with
the annihilation operators a; and a; ) (courtesy of Ulf Leonhardt).
In Figure 9 we illustrate the effect of vacuum fluctuations for the case of a fictitious
beam splitter, which is a model for describing linear absorption or, equivalently,
UNCLASSIFIED)) FOR orrtClltt U.!I! OHL¥
25