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

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UNCLASSIFIED//FOR 8FPl€1J!tt U.!I! l>NL I
Quadrature components q0 are definedtttt with respect to a certain reference phase 8
that can be varied experimentally.
The principle scheme of a balanced homodyne detector is depicted in Figure 10. The
signal interferes with a coherent laser beam at a well -balanced 50 :50 beam splitter.
The laser light field is called the local oscillator {LO), and it provides the phase
reference 8 for the quadrature measurement. It is assumed that the signal and the LO
have a fixed phase relation, as is the case in most experiments applying homodyne
detection, because both fields are ultimately generated by a common master laser. The
LO should be intense with respect to the signal for providing a precise phase reference.
It is also assumed that the LO is powerful enough to be treated classically, i.e., we
totally neglect the quantum fluctuations of the LO. After the optical mixing of the signal
with the LO, each emerging beam is directed to a linear-response photodiode. The
photocurrents Ii and h are measured, electronically processed, and finally subtracted
from each other. The difference current h 1 = h - /1 is the quantity of interest because
it contains the interference term of the LO and the signal. It is assumed for simplicity
that the measured photocurrents Ti and h are proportional to the photon numbers ii1
and n2 of the beams striking each detector, which are given by ,'i 1 =a;ta; and
Yli =ll~ta~ in terms of the mode operators a; =T112(a- aw) and a~ = T 112(a +aw) of
the fields emerging from the beam splitter [38] . Here adenotes the annihilation
operator of the signal and aw is the complex amplitude of the LO.
The difference current hi is proportional to the difference photon number (assuming
perfect quantum efficiency) ii21 = ii2- ii1= a~0a+ a wat , where a~ is the complex
conjugate of aw. The phase of the LO is 0, and so we note from the definition of q0
that the measured quantity hi is indeed proportional to q0 because n21 =2'12 la wl q9 ,
which is a result that has been verified by more sophisticated theories of homodyne
detection [38). A balanced homodyne detector measures q0. The reference phase 0 is
provided by the LO and can be varied by adjusting the LO using a piezo-electrically
movable mirror, for example. An experimental method for finding the scaling of q0 in
the difference current hi is to keep a record of the sum current because the sum of ft
and h is proportional to la w l2 to leading order [38]. This can be experimentally
important because the intensity of the LO is usually an unknown quantity.
tm We note that phase shifting rotates the quadratures, q0 = (;t (0) qU(0) = qcos0+ psin0 and
Pe =rJt(0) pU(0) = -qsin0 + pcos0 , via the quadrature decomposition defined in Sect. IIB-1 and the phase
shifting property of the annihilation operator defined in Sect. IIIB-1.
UNCLASSIFIED// FOR OFPICIAL 091! er•t t
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