Is it possible to create a huge amount of negative energy with quantum effects already known to us (such as Casimir Effect or squeezed vacuum)? Can we create a large amounts of negative energy for traversable wormholes, FTL, time machines? 
I read that Casimir Effect can create negative energy (very small amounts of it) between plates. Maybe it is possible to amplify this effect?
 A: The Casimir effect occurs because the set of frequencies of virtual photons directed normal to the plates seperated by $d$ are in a discrete form $\omega~=~n\pi c/a$ while the photons that pass through the plate have a continuous set of frequencies $\omega~=~ck$ If we orient the plates in the $x,~y$ plane the total frequency of a wave is
$$
\omega~=~c\sqrt{k_x^2~+~k_y^2~+~\frac{n^2\pi^2}{d^2}}.
$$
The Hamiltonian for a given frequency is $H~=~\frac{1}{2}\hbar\omega$, where the total Hamiltonian requires a summation or integration over all frequencies. We can evaluate the energy by integrating over all the frequencies
$$
\langle E\rangle~=~\hbar \frac{A}{4\pi^2}\int dk_xdk_y\sum_{n=1}^{\infty}\omega_n
$$
The Wikipedia page uses zeta function regularization. This involves a replacement $\omega_n~\rightarrow~\omega_n|\omega_n|^{-s}$ so that the summation defines a zeta function. After evaluation $s~\rightarrow~0$. The integrations over $k_{x,y}$ are evaluated in a standard way. The result important for physics is that
$$
\langle E\rangle~=~-\frac{\hbar c\pi^2 A}{720 d^3},
$$
which is a negative energy. Negative energy is involved with the squeezed states of bosons and also these exotic solutions to Einstein's field equations. The force is found with the work-energy $W~=~\int Fdx$ or the magintude of this force is $F~=~\partial\langle E\rangle/\partial d$ and so
$$
F~=~-\frac{\hbar c\pi^2 A}{240 d^4},
$$
which is not terrible large.
Let us run some numbers. We have for a $1m^2$ plate (which is much larger than most experimental setups) $\frac{\hbar c\pi^2 A}{240}~=~1.3\times 10^{-27}jm^3$. This gives the energy density of the vacuum, when divided by $3$. The energy of the vacuum between these plates drops by this factor. Dividing by $c^2$ and multiplying by the area of the plates $1m^2$ and using $d~=~1m$ means the energy drop between the plates is equivalent to $10^{-19}$ times the mass equivalent of a proton! This is very tiny. 
So how could one amplify this effect? The only variable you have is the distance. For ordinary matter it is probably only possible to get the distance to the diameter of an atom. This gives a force $F~\simeq~-10^{13}N$ and the vacuum energy between the plates is reduced by $-10^{3}j$. However, at this point and really before other effects take over, such as electrostatic interactions between the surface atoms of the plates etc, so this is not realistic. 
It also must be mentioned there is no free energy here. If you configure the plates so they are attracted you are right in thinking that work has been performed. However, to reconfigure the plates you must exert a force to separate them again, and the work involved then goes into raising the energy of the vacuum between the plates. So this is not a way of getting energy out of the vacuum. What energy you get out of it is just exactly what you put into it, and with thermodynamic losses with the mechanics of moving plates less so. 
A: To be frank with you, I won't even begin to pretend I know the math behind any of the Wikipedia material on this page. Casimir Effect
But I do know what the words: "it has been speculated", "X theorised", "controversial" , "arbitrary"  and "if"       all mean:  unfortunately just hope and guessing.
But as soon as a frog is levitated Frog Levitation  In Magnetic Field by utilising the Casimir effect, I will believe :).

There are few instances wherein the Casimir effect can give rise to repulsive forces between uncharged objects. Evgeny Lifshitz showed (theoretically) that in certain circumstances (most commonly involving liquids), repulsive forces can arise. This has sparked interest in applications of the Casimir effect toward the development of levitating devices. An experimental demonstration of the Casimir-based repulsion predicted by Lifshitz was recently carried out by Munday et al.[39] Other scientists have also suggested the use of gain media to achieve a similar levitation effect,though this is controversial because these materials seem to violate fundamental causality constraints and the requirement of thermodynamic equilibrium (Kramers-Kronig relations). Casimir and Casimir-Polder repulsion can in fact occur for sufficiently anisotropic electrical bodies; 
It has been suggested that the Casimir forces have application in nanotechnology,[42] in particular silicon integrated circuit technology based micro- and nanoelectromechanical systems, silicon array propulsion for space drives, and so-called Casimir oscillators.
The Casimir effect shows that quantum field theory allows the energy density in certain regions of space to be negative relative to the ordinary vacuum energy, and it has been shown theoretically that quantum field theory allows states where the energy can be arbitrarily negative at a given point,[44] Many physicists such as Stephen Hawking,[45] Kip Thorne,[46] and others[47][48][49] therefore argue that such effects might make it possible to stabilize a traversable wormhole. Miguel Alcubierre has suggested[50] using the effect to obtain the negative energy density required for his Alcubierre Drive.

