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

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where the first and second terms in the second line are the kinetic and potential
energies of the oscillator, respectively. The additional 1/2 appearing in the first line of
Eq. (1) is called the vacuum zero-point energy for the reason to be explained in the
next section . The first line of Eq. (1) is more commonly expressed in units of energy
(Joules) in quantum mechanics, which is obtained simply by multiplying the right-hand
side by the photon energy nw so that H ose =hw (ii +½) .
It is beyond the scope of this report to elaborate further on the entire subject of the
quantum optics. The reader should consult Reference [38] for more information.
Basic Notions on the Origin of the Quantum Vacuum Zero-Point Fluctuations
Here we discuss the basic notions of the quantum vacuum zero-point fluctuations
(ZPF), which is an important feature in quantum optics. The origin of the ZPF is
attributed to the Heisenberg Uncertainty Principle. According to this principle, q and
pare any two conjugate observables that we are interested in measuring, and they
obey the commutation relation already shown in the previous section. Their
corresponding uncertainty relation is !1q!1p?:. h/2, where !1q is the variance (a.k.a.
uncertainty) of observable q and !1p is that of the conjugate observable p. This
relation states that if one measures observable q with very high precision (i.e., its
uncertainty !1q is very small), then a simultaneous measurement of observable p will
be less precise (i.e., its uncertainty !1p is very large), and vice versa . In other words,
it is not possible to simultaneously measure two conjugate observable quantities with
infinite precision.
This minimum uncertainty is not due to any correctable flaws in measurement, but
rather reflects the intrinsic fuzziness in the quantum nature of energy and matter.
Substantial theoretical and experimental work has shown that in many quantum
systems the limits to measurement precision is imposed by the quantum vacuum ZPF
embodied within the uncertainty principle. Nowadays we rather see the Heisenberg
Uncertainty Principle as a necessary consequence, and therefore, a derived result of the
wave nature of quantum phenomena. The uncertainties are just a consequence of the
Fourier nature of conjugate pairs of quantities (observables). For example, the two
Fourier-wave-conjugates time and frequency become the pair of quantum-particle
conjugates time and energy and the two Fourier-wave-conjugates displacement and
wave number become the pair of quantum-particle conjugates position and momentum.
The Heisenberg Uncertainty Principle dictates that a quantized electromagnetic
oscillator (a.k.a. a photon state) can never come entirely to rest, since that would be a
state of exactly zero energy, which is forbidden by the commutation relation given in
the previous section. Instead, every mode of the field has liw/2 as its average minimum
energy in the vacuum, and this is called the zero-point energy (ZPE) .** This ZPE term
is added to the classical blackbody spectral radiation energy density p(w)dro [i.e., the
energy per unit volume of radiation in the frequency interval (ro,w + dro)] [25]:
*" hw is the energy of a single mode (or photon).
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