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

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Morris and Thorne [8] and Caves [45] point out that if one squeezes the vacuum, i.e., if
one puts vacuum rather than laser light into the input port of a squeezing device, then
one gets at the output an electromagnetic field with weaker fluctuations and thus less
energy density than the vacuum at locations where cos2[co(t- z/c)] ~ l and
sin2 [ co(t- z/c)] < < 1; but with greater fluctuations and thus greater energy density
than the vacuum at locations where cos2[co(t-z/c)] << 1 and sin2 [ co(t - z/c)]~1.
Since the vacuum is defined to have vanishing energy density, any region with less
energy density than the vacuum actually has a negative (renormalized) expectation
value for the energy density. Therefore, a squeezed vacuum state consists of a
traveling electromagnetic wave that oscillates back and forth between negative energy
density and positive energy density, but has positive time-averaged energy density.
In quantum optics the squeezed state is generated by the unitary squeezing operator:
(5)
where sis a real number that parameterizes the deviation of the variances !),,q and !),,p
from their vacuum values and is called the squeezing parameter. From Eq. (5) we
obtain the squeezed vacuum state lcp) = S(~)IO). The squeezing operator S(~) is simply
an evolution operator that describes the result of the nonlinear squeezing interaction
Hamiltonian H im =x( b*a2 - bat2 ). The squeezing parameter~ contains the product of
the amplitude b, the coupling constant X, and the interaction time.
But this is not the entire story. Since we will be dealing with high-quality lasers in what
follows, we also need to know about another important quantum optics operator that
acts on coherent states. We introduce the unitary displacement operator
D(a)=exp(aa: -a·a). D(a) displaces the amplitude aby the complex number a
according to b\a)aD(a)=a + a. To show why D(a) has anything to do with coherent
states, we apply a negative displacement to la). From the basic property of D(a), we
see that
abc-a) la) =D(-a)Dt(-a)GD(-a) la)
= be- a) (a- a)Ia)
(6)
= 0.
Equation (6) equals zero because of the definition Eq. (3) of coherent states. This
result implies that D(-a)la) = I0), which is the vacuum state. Therefore, coherent
states Ja) are displaced vacua la) = D(a)J0). This does not mean that coherent states
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