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

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UNCLASSIFIED//FOA QFFIEJIAL l::ISI! 9Ht'I"
where g(ffi) is the Fourier transform of g(t) and is sharply peaked around ffi =0. In
general, almost all results of quantum field theory in a vacuum state or under the
influence of external conditions (i.e., in vacuum states "deformed" by boundary
conditions or external fields) are derivable from the spectral density. This quantity is
usually known analytically, and it is of great interest to measure it for interesting
quantum field states.
Because the ground state is stationary, the quantum noise in the Casimir cavity is time­
independent, i.e., it is independent of the phase of the LO, and thus the spectral
density is also time-independent. Marecki [S, 6] derived the diagonal part of the
spectral density for the y-components of the quantum electric field between two
parallel, perfectly conducting plates (positioned at x = 0 and x = a) in a Casimir cavity
(see Figure 2):
r r
m3
+<X)
cryy(m,x,x) =- L [Q(mnL)-Q(ml 2x-nLI)]
(17)
2
4rc
n=-CX)
for y = 0, where L = 2a is twice the distance between the plates and the function Q(x) is
defined as
.
.
Q(x) =sznx + cosx _ sznx.
x2
x
x3
Note that the diagonal terms of the spectral density are the important quantities to be
measured because they will be dominant if the photodiodes are separated by a
sufficiently large distance [5, 6]. Spectral densities reveal much finer details of the
quantum ground state than already-measured Casimir forces do. By exploring the
freedom of choosing the locations of the photodiodes inside the Casimir cavity as well
as the polarizations, phases and frequencies of the LO, one can obtain a detailed
characterization of one- and two-point functions of any state S of the quantum electric
field. Therefore, an application of this particular type of BHD measurement, via
Equations (16) and (17), amounts to a tomography of the ground state of the Casimir
cavity.
For the experimental detection of the Casimir spectral density with a BHD-type device,
the Casimir cavity plates are separated by a = 1 micrometers while the photodiodes
inside it are of submicrometer width in the x-direction and submillimeter length in the
y-direction (see Figure 16). Photodiodes of several nanometers in size have already
been constructed and their high quantum efficiency versions are under development,
see Reference [60] and the references cited therein for more technical information. As
shown in Figure 16, a coherent state in the TEl mode of the Casimir cavity with a very
small wavenumber in the y-direction provides an appropriate LO. A BHD with such a
LO and the photodiodes located as shown in Figure 17 would be sensitive only to the y­
component of the quantum electric field. Figure 18 shows a schematic of the BHD
apparatus with a LO. In the figure, the linearly polarized signal field S (if present) is
optically mixed with a coherent state (LO), which is polarized orthogonally to S, on the
polarizing beam splitter (PBSl). The half wave plate (HWP) reflects the planes of
polarization with respect to its optical axis, thereby inducing a n/4 shift of the plane of
UNCLASSIFIED//fOtt OfflEJIAL l::l§E 8,.LY
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