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  σ = ⅓ 8p b⁴ R⁶ (18) where wave number b = ω / c = 2p /λ. The dielectric constant Є is actually the relative dielectric constant which is a ratio of substance permittivity to the permittivity of free space Є_o. In order to appreciate the range of values that Equation (18) may assume, it is noted that At optical frequencies, only the electrons can respond significantly. The dielectric constants are in the range Є = 1.7 10, with Є = 2 3 for most solids. Water has Є = 1.77 1.80 over the visible range, essentially independent of temperature from 0 to 100C. With this information in mind, a graph of the behavior with frequency is also shown in Figure 15. A declining Є with frequency can only make Equation (18) even smaller as Є tends toward the limit of 1 (where the permittivity equals Є_o ).

As an example of the total cross section for scattering by a relatively good dielectric, Є = 3 can be chosen. Then, with f = 10¹⁴ Hz and R = 0.1 x 10^-6 m (thus keeping λ > R), Equation (18) is found to yield σ = 2.6 x 10^-17 m². Dividing σ by the actual cross sectional area of a microsphere ( pR² ) for comparison, scattering by a dielectric sphere of optical frequency electromagnetic radiation yields a loss of about 8 x 10^-6 in power.

In comparison, for λ > R, small conducting spheres have a total scattering cross section that is significantly larger, where

σ = ⅓ 10p b⁴ R⁶ . (19)

There is an advantage of using conducting spheres in place of the dielectric spheres which is more significant than designing for the doubling effect from possible beat frequencies. The cross section σ for a one-tenth micron-sized conducting sphere (R = 0.1 x 10^-6 m) with visible light incident (f = 10¹⁴ Hz) yields about 2 x 10^-16 m² for λ > R. Dividing this as before by the actual cross sectional area yields only 6 x 10^-5 loss of power or ten times better than the dielectric scattering cross section.

While both of these total cross section calculations still may seem very low, there seems to be an explanation for it. Since they were still within a power of ten from λ = R, Figure 16 shows there is an interference scattering effect for plane waves, within a few wavelengths of this region. Utilizing the spherical Bessel function expansion for a plane wave, similar to the inventor Mead, the solution with amplitudes and phases is found for the boundary condition that the wave function is zero at R = a but the radial velocity of the wave is zero at R = 0. As seen in Figure 16 (the textbook uses Q for total cross section), the surprise is that in the region of λ = 2pR and smaller (<1 on the abscissa), the total cross section becomes very small, tending to zero. The graph, however, reflects the boundary conditions used, such as the radial velocity v_r of the ratio S (of scattered intensity to incident intensity) = 0, the total cross section calculation with the proper Bessel function tends toward the limit of 2pR² for λ <<R, which is twice the actual cross sectional area. However, for very long wavelengths compared to the radius (λ >>R) the total cross section for plane

wave scattering by a sphere tends toward 4pR² which is four times the actual cross sectional area.

Also confirmed from Figure 17, based on the same text with a Bessel function treatment of a plane wave incident (from the left) on a rigid sphere, the scattered intensity grows larger with smaller wavelength, instead of exhibiting a trend toward resonance when λ ≈ R, tending to be spherically symmetrical in the limit where bR à 0, S à 1, and σ à 4pR². These figures seem to contradict the inventors assertion of a ZPE electromagnetic resonance with a microsphere and any number of microspheres in close proximity. Furthermore, the penetration of electromagnetic waves throughout dielectric spheres is another complication which is treated in detail in the above-mentioned theoretical physics text, including a complex index of refraction when required, to account for transmission, reflection, and absorption. These two issues diminish the feasibility of the inventors microsphere design significantly.

In the region of λ <R, classical ray theory applies since the wavelength is short compared to the radius of curvature. Fresnel equations can also be utilized, treating the surface as locally flat. This argument also leads to the standard description in physics of specular reflection.

To show the value of Bessel function analysis of plane wave scattering and the strong directional dependence, a graph of the differential cross section is plotted in Figure 18. Considering the previously mentioned advantage of using conducting spheres of the micron size, the analysis is also more straightforward. Under these circumstances, the tangential magnetic fields and normal electric fields of the electromagnetic wave will be approximately equal to the incident wave. The differential backward scattering of the incident radiation, for λ < R and θ < 10/bR is found to be,

= R²(bR)² (20)

where J₁ is a Bessel function of the first kind of order one. The forward differential scattering for θ >> 1/bR (higher θ angles) is simply R²/4. A plot of Equation (20), for the smaller angles, is the dashed line in Figure 18, with the exact solution as the solid line. Destructive interference is noted where it dips below unity. The peak in the graph of Figure 18 indicates a strong reflection back-scattering for a conductive sphere. This is a common phenomenon since silver, a very good conductor, is used often for coating glass to create mirrors. The conductive surface allows the electric field vector of the electromagnetic wave to oscillate freely upon contact, with very little resistance, thus creating the reflective wave.

Such electromagnetic radiation scattering is distinguished from Thomson scattering, Rayleigh scattering, Coulomb scattering, Compton and Rutherford scattering, which also use cross section formulae as well. Each of these, more common with particle scattering, will be discussed in the following sections.

It should be emphasized that the same two σ limits discussed above, 2pR² and 4pR², for small wavelengths (λ <<R) and large wavelengths (λ >>R) respectively, are also derived in quantum mechanics using the method of partial waves for scattering of wave packets by a perfectly rigid sphere and thus will also be applied in the further sections to follow.

For feasibility consideration of energy extraction, to collect and transduce the total scattered ZPE radiation from the vacuum flux, it would be necessary to place one sphere at the focus of an evacuated, reflecting 3-D ellipsoid cavity with the PV converter at the other, for example, instead of the spherical cavity the inventors refer to. However, in the interest of maximizing energy output per volume, it may be more convenient to engineer sheets of single spheres placed in alternate planes between planar PV converters, which may unfortunately limit the available ZPF frequencies.

An important calculation for each sphere of interest is to find whether significant scattered zero-point energy is available at these wavelengths. Therefore, the spectral density Equation (16) is integrated as,

. (21)

For the wavelength range of 0.4 to 0.7 microns (micrometers) in the visible light band, using Equation (8), the radial frequencies can be generated for the integrated energy density equation. Substituting these for ω₂ and ω₁ we find an energy density of only 22 J/m³ or 22 microjoules/cc which equals approximately 0.24 eV/μm³ (electron volts per cubic micron).

To create a simple standard calculation for the frequency band of each sphere of interest, ω₂ is chosen to correspond to the radius R and ω₁ is chosen to be 1/10 of that frequency. For a microsphere, with λ = R = 10^-6 m, the spectral energy density from Equation (21) is ρ(ω) = 0.62 J/m³ or 3.9 eV/μm³ for the decade range: Δf = 3 x 10¹³ to 3 x 10¹⁴ Hz. From Figure 10, this energy density is also comparable to the photon energy (2 eV) in the visible band.

Nanosphere Energy Scatterers

In the region of λ > R for the scattering by these nano-sized spheres (nanosphere) the classical electrodynamic equations still apply. However, with a radius R of the sphere considered to be 10^-9 m, the effect on the ZPF spectral energy density is quite dramatic. In Figure 22, it is noted that 1 nm is in the keV region. A resonant correspondence with the sphere diameter of 2 x 10^-9 m equals a full wavelength antenna, the resonant frequency will be in the range of 1 x 10¹⁷ Hz. The spectral energy density of ZPE at this frequency is substantially more promising. Using Equation (21), we find that the ZPE spectral energy density is 6.2 x 10¹¹ J/m³, which is a billion times more energy per cubic meter than was available from the ZPF for the micro-sized spheres. Converting to electronvolts per cubic nanometer, it is interesting that the ZPF offers about 390 eV/nm³ which is three orders of magnitude more energy than available to the micron-sized sphere. The advantage as well is that a billion of these spheres will fit into a cubic micrometer, if a collection was found to be coherently constructive with regards to scattering. Vacuum polarization is probably more pronounced at the nanometer dimensions, yielding more ZPE virtual particles which would be expected contribute more significantly to scattering off of nanospheres.

Evaluating Equation (19) at this resonant frequency and radius, it is found that for a conducting nanosphere, the scattering cross section σ = 1 x 10¹⁵ m² for the region λ > R. Comparing with cross-sectional area pR² for the nanospheres, it is found to be 318 times its cross-sectional area pR². (The ratio of σ to spherical surface area is also constant σ /4p R² = 83). This demonstrates that the scattering cross section σ is geometrically correlated to the objects actual cross-sectional area.

For the consideration of λ ≈ 2R, resonance is still not expected to affect the amplitude of scattered radiation appreciably.

For the consideration of λ < R the scattering profile seen in Figure 18 would still apply because quantum mechanical effects become important only when hf ≈ mc². This may be anticipated for the femtosphere.

The present state of the art for engineering capabilities in the microsphere and nanosphere regions is illustrated in Figure 19. Called nanoboxes, they are electrically conductive single crystals of silver, produced at the University of Washington, with slightly truncated edges and corners. Each box was bounded by two sets of facets (eight triangular facets and six square ones), and any one of these facets could lie against a solid substrate. The inset shows the SEM image of an individual box sitting on a silicon substrate against one of its triangular facets, illustrating the high symmetry of this polyhedral hollow nanoparticle. Octahedra and tetrahedra, such as the inset have also been produced, which approach the patent-proposed ideal of a nanosphere. The white scale bar at the bottom of Figure 19 is 100 nm in length for comparison. For 17-min and 14-min growth times, the nanocubes had a mean edge length of 115 9 and 95 7 nm, respectively. For the sake of the feasibility discussion, regarding the microspheres difficulty of predictable beat frequencies, it is noted that the tolerances quoted here are between 7% and 8%. Thus, the benefit of a single beat frequency production of two adjacent silver nanopolyhedrons is judged to be not feasible at either microsphere or nanosphere sizes because of the large manufacturing errors. Nanocubes with sides as small as 50 nm have also been obtained, though some of them were not able to evolve into complete truncated cubes.

Regarding the scale of 1 nanometer in diameter, such as the nanosphere that is proposed, the error control may not require a higher tolerance range than quoted above. As seen in Figure 20, if individual molecular crystals were used for 1 nm range, they do not vary widely in size. A sphere of carbon-60, for example, would be a real possibility, though it is not highly conductive. If metal nanopolygons are used, it is noted that nonspherical gold and silver nanoparticles absorb and scatter light of different wavelengths, depending on nanoparticle size and shape.

Interestingly, gold and silver nanoparticles have been used as sensors, since they have surface-enhanced Raman scattering and other optical effects peculiar to the ~10- to 100-nm range. Instead, using heavy metal atoms, such as Polonium with a diameter of 0.336 nm should be considered in this section because of superior spherical shape and reproducibility. Polonium may also be an interesting candidate because it is the only element known to crystallize in a primitive cubic unit cell under room temperature conditions. Therefore, the interatomic spacing is also very well known. ZPE virtual particles, or equivalent ZPE electromagnetic radiation, would not be expected to play a large part in scattering off polonium atoms however. Instead, they already are known to contribute to the Lamb shift of the 2p electron levels, with about 1.06 GHz worth of energy. Furthermore, virtual particle scattering contributes the same energy to every state, consisting of e²A²/2mc² in the nonrelativistic theory with the Hamiltonian, where A is the vector potential. Beat frequencies would be unlikely and very difficult to engineer with polonium atoms since the atoms would normally share the same energy, being at the same temperature, etc.

Picosphere Energy Resonators

In the picosphere range, it is more likely that some of the key elements of this patent may be more effectively applied. One of the reasons for this is that up until this point, there has not been a necessity for lower frequency scattering. To review some of the transduction methods available, Figure 21 shows some of the standard devices for transducing ionization into electricity. Note that ionizing radiation can also consist of electromagnetic X-rays or gamma rays since there is sufficient energy at these frequencies to cause ionization. The method of ionizing transduction relies upon the production of ion pairs in a gas or solid by the incidence of radiation. The applied electric field in Figure 21 is an excitation voltage (Exc.) used to separate the ionized positive and negative charges to produce an electromotive force. For the picosphere range, it is also expected that small individual atoms can be arranged to meet the specifications for the patent more effectively since nature has much better error tolerances than engineers can manufacture artificially.

The high frequency electromagnetic spectrum is reproduced in Figure 22, which picks up where Figure 10 left off, with wavelength decreasing from left to right. The picosphere with a radius of 10^-12 m (1 pm) and a wavelength equal to its diameter, corresponds to a frequency of 1.5 x 10²⁰ Hz using Equation (8). Using Einsteins equation E = hf, the photon energy at that frequency can be found to be about 650 keV which is useful to compare with the spectral energy density. Using Equation (21), we calculate a spectral energy density of 6.2 x 10²³ J/m³ or 390 keV/pm³.

In the range of λ > R the scattering cross section σ = 1 x 10²¹ m² is in the same proportion of 318 times sphere cross sectional area.

At the resonant wavelength of λ ≈ 2R, the amplitude of scattering can be expected to be higher. It is also anticipated that here is where the concept of beat frequency may be applied more conveniently, with greater precision than in either larger category. However, since all atomic radii vary between 50 pm (e.g., Helium) and 660 pm (e.g., Cesium), the picosphere with a proposed radius of 1 pm has to be declared to be impractical and therefore, not feasible.

In the range of λ < R the scattering seen in Figures 16 and 17 would still apply. The diffraction pattern is also very predictable. The intensity distribution of X-ray diffraction could be correlated to the theoretical scattering off a sphere from Equation (21). An example is seen in Figure 23 where the wavelength of the X-rays is 71 pm and the target is an aluminum atom, which has an atomic radius of 182 pm.

If a smaller target on the order of a picosphere were used, it is expected that the scattering pattern would be the same for λ < R.

In this region, the need for a heterodyned frequency might emerge if, for example, the ionization transducers of Figure 21 were not configured for high efficiency capturing of the ZPE scattered radiation. However, the production of a beat frequency that also resonates with the geometry of an array of atoms may be problematic, for two reasons. The array would preferably need to be a 2-D sheet only one atom thick, such as thin metal foil used for diffraction studies, to prevent destructive interference of the ZPF scattering. Secondly, the real barrier to creating a useful ZPE beat frequency atomic array is producing picospheres that vary reliably in one part in one thousand with a maximum error tolerance of one part in ten thousand. An avenue of speculative physics would require the engineer to estimate the diameter of a suitable metal atom in the ground state and pursue a manufacturing procedure to excite alternate adjacent atoms to a very long metastable state, which is known to expand its size, much like Ryberg atoms. Hypothetically, this ideal situation would achieve a small difference in diameter of adjacent atoms sufficient to produce beat frequencies of resonant scattered ZPE.

Utilizing the Fermi-Thomas model of the atom, most atomic radii can be approximated by

a ≈ 1.4 a_o / Z^⅓ (22)

where Z = atomic number and a_o = h² / me², the hydrogenic Bohr radius. Taking an excellent example of two atoms with similar size, platinum (Pt) and gold (Au) would be good candidates since they are next to each other on the periodic table and relatively inert, Noble metals. It is presumed that the diameter may resonate with a full wavelength, with 183 pm and 179 pm as the radii for Pt and Au respectively. In that case, 8.20 x 10¹⁷ Hz is the corresponding Pt frequency and 8.37 x 10¹⁷ Hz is the corresponding Au frequency, both in the soft X-ray band. Subtracting the two frequencies, the beat frequency would theoretically be a difference of 1.83 x 10¹⁶ Hz, moving it down into the UV band. If the conversion of UV incident electromagnetic energy is more efficient than transducing soft X-rays, then this method would offer a chance to collect ZPE, so long as the arrangement of multiple pairs of Pt and Au atoms could constructively interfere at their beat frequency. However, the wavelength of 1.83 x 10¹⁶ Hz is about 16 nm, which forces the placement of individual atomic pairs to be fairly distant from each other, compared to their size. With only a 2% difference in diameter, the beat frequency difference yields a power of ten difference for lower frequency detection, as the Mead-Cox resonant microsphere analysis of Figure 11. The improvement in amplitude from resonance would reasonably be only a power of two, unless a resonant cavity was used as well.

Another example, representing the smallest atomic pair that is available for this experiment, is hydrogen (H) and deuterium (D), an isotope of hydrogen with one proton and one neutron in the nucleus. The Bohr radius a_o for hydrogen is 53 pm and the atomic radius of deuterium is about the same. In fact, the H_α emission lines (Balmer series) for deuterium and hydrogen are 656.10 nm and 656.28 nm respectively, a difference of only 0.03%. Such a similar size will force the beating frequency to be more than a power of ten difference, which apparently is viewed as an advantage by the patent holders. For such gaseous atoms, the phenomenon of upscattering might be achieved with this gas at a finite temperature T with a Maxwell-Boltzmann velocity distribution if the incident ZPE virtual particles fell into a regime of low energy up to about 10 kT. This implies that it is possible for the incident particle to gain energy in a scattering collision. In the situation where the hydrogen or deuterium nuclei might be at rest, the scattering probability P(E_I à E_f ) inversely depends upon the incident particle energy E_i. However, for elastic scattering in a hydrogen (proton) gas, the scattering probability depends on the final particle energy E_f and is not zero even for E_f > E_i. In Figure 24, a graph is shown of the scattering probability for scattering of a proton gas with various incident particle energies. With resonance considerations seen in Figure 25 added to the design as well, such a regime might be a test, with a minimum of risk, for the Mead spherical collector concept in an atomic, picosphere region. However, with or without a successfully amplified beat frequency, the upscattering of virtual particles from a proton gas may still have inherent flaws for two reasons: 1) most such proton gas experiments have been conducted only with low
energy incident neutrons; 2) the dissipative effect of radiation reaction precludes spontaneous absorption of energy from the vacuum field which normally applies only to an atom in the ground state. Yet, the gain of one to two times the incident E_I = k_BT may be valuable to the energy equation as the gas transfers energy to the incident particles, even if the probability drops to 50%, since theoretically an abundant number of virtual particles are available. At room temperature (T 300K) for example, 1 k_BT = 0.026 eV which is about 10¹³ Hz or an infrared terahertz frequency.

In Figure 25, an example of a capture resonance is shown at temperature T₁, where the average cross section dramatically increases for a certain resonant incident energy E_o. Another aspect of temperature increase is also graphically demonstrated. This is called Doppler broadening caused by the Doppler shift in frequency as a thermally excited atom moves away from or toward the incident particle with greater temperature-dependent speed. Therefore, an increase in temperature causes the increased cross section of a resonant peak to be lost. Lower temperatures are important for preserving the advantage of resonance.

Thus examining the options for the resonant sphere, the two atomic pairs of deuterium and hydrogen are the best beating examples in the picosphere region, still demonstrating major unknowns in the beat frequency design concept of the Mead patent. At the present state of nanotechnology development however, the picospheres cannot be manufactured.

Quantum Femtosphere Amplifiers

With the examination of the femtosphere (R = 10^-15 m = 1fm) there are a number of phenomena that synchronize so well with this dimension that the patent being examined seems to be more compatible with the nuclear particle than any other size sphere. The first obvious advantage is the spectral energy density of Equation (21) which is found to be 6.2 x 10³⁵ J/m³ or 390 MeV/fm³. This is also interesting in that the quantum mechanical realm applies where hf is about the same as mc². Testing for this condition, both energies are calculated with a wavelength of 2 x 10^-15 m and a corresponding frequency (see Figure 22) of 1.5 x 10²³ Hz. The Einstein formula for photon energy of the femtosphere is

E = hf = 9.9 x 10^-11 J = 619 MeV . (23)

To determine the mass of the femtosphere, it is known that the radius of either the proton or neutron is about 8 x 10^-16 m. ^, This is remarkably close (within 20%) to the conceptual femtosphere radius R of 1 fm. Therefore, it is reasonable to use the average mass of either nuclear particle (1.7 x 10^-27 kg) in the Einstein equation for mass-energy, to find the energy equivalent of the femtospheres mass:

E = mc² = 1.5 x 10^-10 J = 938 MeV . (24)

Comparing Equations (23) and (24), they are the same order of magnitude, so it is determined that quantum mechanical rules apply in this region. The classical equations for energy and scattering cross section are still applicable. However, they may be regarded as classical approximations, in view of the correspondence principle, to quantum mechanical phenomena. The important quantum effects are (1) discreteness of the possible energy transfers, and (2) limitations due to the wave nature of the particles and the uncertainty principle.

In this femtosphere range, Rutherford scattering is applicable. The total nuclear Rutherford scattering cross section is,

σ = p R² (25) where z is the number of charges (particles) in the incident at a velocity v and Z is the number of charges (particles) in the target. For example, at high velocities, even for incident virtual photon radiation, the total cross section can be far smaller than the classical value of pR², which is its geometrical area. The parenthetical terms in Equation (25) can result in a reduction of 10^-24 times the geometrical area p R² of the target for an incident photon at the speed of light.

For this size of target, due to vacuum polarization, it is important to mention that more incident virtual particles from the vacuum, as discussed in Chapter 1 (with an artist rendering in Figure 3), will also be present for a charged femtosphere. Therefore, the de Broglie wavelength of the incident particle will also be important, treating the ZPF virtual particles on the same level as electromagnetic waves:

λ = h/p = h/mv . (26)

The de Broglie requirement of quantum mechanics, postulated in 1924, thus affects the possibilities for energy generation for the Mead patented design. For a nucleus of finite sizethe de Broglie wavelength of the incident particle does enterThe situation is quite analogous to the diffraction of waves by a spherical object.

Vacuum polarization will also enhance the natural electromagnetic radiation from the vacuum for the femtosphere. Since double slit experiments with particles like electrons and neutrons demonstrate the wave phenomenon of diffraction, femtosphere particles can also be regarded as wave packets.

This type of scattering utilizes the continuously distributed energy eigenvalues of quantum mechanics which consider the boundary conditions at great distances from the collision. The scattering is treated here only as elastic scattering, so there is no absorption by the target. This is different from photoelectric scattering or Rayleigh scattering, which are inelastic.

For the region of λ > R this can be represented by the low energy limit where 2pR << λ (the circumference is much less than the wavelength). For the femtosphere, the total cross section is approximately σ ≈ 10^-30 m² in this low frequency, long wavelength region.

For the region of λ ≈ 2R or 2pR, there is a uniquely quantum mechanical phenomenon of resonant scattering called resonance fluorescence that applies to the absorption of radiation by an atom, molecule, or nucleus in a transition from its ground state to an excited state with the subsequent re-emission of the radiation in other directions in the process of de-excitation. The reaction of the target when the frequency of the incident equals the binding energy of the target, the scattering becomes very great, exhibiting a formal resonance peak seen in Figure 26.

Note that a single femtosphere, such as a free neutron or proton would not possess the requisite binding energy for this type of oscillator resonance. In addition, this process reveals an inelastic form of scattering but the dramatically increased cross section apparent in the resonant peak

of Figure 26 more than makes up for any energy lost in absorption and re-emission. Resonance fluorescence brings a number of phenomena into play that

are seen in Figure 26, such as Thomson scattering, which is scattering of radiation by a free charge. It can be X-rays by electrons or gamma rays by a proton for example. Thomson scattering occurs along a baseline of higher frequencies in Figure 26 above ω_o,

where the cross section is

σ_T = . (27)

The Thomson cross section of Equation (27), evaluated for a target electron, yields 6.6 x 10^-25 cm² for σ_T where e²/mc² = 2.8 fm, the classical electron radius.

By comparison, the Rayleigh scattering (which causes the atmosphere to take on a blue color) cross section has a dependence on the fourth power of the incident frequency. Thus, the highest visible frequency (blue color) has a much larger scattering cross section and consequently, a stronger interaction with air molecules, including a wide scattering angle. Both phenomena illustrate the dominance of cross section scattering terms with cubic and quartic exponents.

Deuteron Femtosphere

One example of an intriguing femtosphere oscillator is the deuteron (a proton bound to a neutron in close contact), which also demonstrates resonant fluorescence. It would satisfy the requirement of binding energy for the target and is also treated as an oscillator. However, it should be mentioned that it is unlikely that the deuteron would exhibit beat phenomena, since the wave functions overlap making the deuteron behave as one nucleus (ionized deuterium), with a single resonant frequency and a radius of about 2 fm.

Recalling that the beat frequency design of the Mead patent is simply a means to downshift the radiated frequency, it is also possible to use the standard electrical engineering approach instead. Frequency converters, for example, utilize a local oscillator which mixes with the incoming signal to produce a subtracted intermediate frequency one or two orders of magnitude lower, for the same purpose. Once the resonant frequency of the deuteron is determined to be beyond the useful energy transduction range, an engineering recommendation therefore, would be to find a lower local oscillator in the same range, so that a substantially reduced intermediate frequency can be produced.

The deuteron has one bound state ( l = 0 ) of a proton and neutron with binding energy of E_b = 2.23 MeV. The more preferred state ( 75% ) is where the spin ½ of the proton and neutron are parallel (triplet state). The cross section for the deuteron is σ = . (28)

Equation (28) is about 70 x 10^-24 cm² for the singlet (antiparallel) state and about 20 x 10^-24 cm² for the triplet state, which computes to 550 times and 157 times the geometrical cross sectional area, respectively.

With resonance fluorescence, the condition is in a sense an almost bound state that isnt below zero energy to be a true bound state (see the dashed line levels in Figure 27). If the potential V(r) is an attractive well, then the effective potential is,

V_eff = V(r) + h² l (l+1) / 2mr² (29)

where the integer angular momentum quantum number, l > 1. Resonances are the bound states of the well at the positive energy, indicated by the dotted linesWhat happens in scattering at a resonant energy is that the incident particle has a large probability of becoming temporarily trapped in such a quasi-bound state of the well; this possibility increases the scattering cross section. In Figure 27, the depth of the potential well for the deuteron must be V_o = 36 MeV for the deuteron.

With that introduction to resonance with the deuteron, it should be mentioned that it is also an advantageous oscillator since the binding energy of E_b = 2.23 MeV corresponds to an X-ray frequency of f_b = 5.4 x 10²⁰ Hz, instead of the gamma ray frequency of 10²³ Hz that should resonate with the diameter of a femtosphere. Thus, the deuteron binding energy satisfies the need for a lowering of the resonant frequency for transduction purposes, voiced in the Mead patent. The cross section is complicated by the existence of a singlet and triplet state depending on the proton and neutron spin direction. There are no bound excited states of the deuteron. Neutron-proton scattering experiments indicate that the force between n and p in the singlet state (antiparallel spins) is just sufficiently less strong than in the triplet state to make the deuteron unstable if the spins are antiparallelthere is a small, measurable quadrupole moment.

For the region of even smaller sizes, beyond the femtosphere resonance, where λ < R, the cross section can be represented by the high energy limit where R >> λ. Here the scattering by a perfectly rigid femtosphere can be approximated, as mentioned with the microsphere, with a total cross section of

σ ≈ 2 p R ² (29)

which is twice the actual geometrical cross section area. The reason for the apparently anomalous result of Equation (29) is that the asymptotic form of the wave function is composed of the incident and the scattered wave, which also experiences interference between the two partial waves. However, so long as 2pR/λ is finite, diffraction around the sphere in the forward direction actually takes place, and the total measured cross sectionis approximately 2pR².

Electron Femtosphere

The electron is the best femtosphere for many reasons. Its classical radius r_o is calculated to be 2.8 x 10¹⁵ m or 2.8 fm and is suitable for a Mead patent test. As seen in Figure 3, the electron, like the proton, offers a steep electrical gradient at its boundary that creates a decay or polarization of the vacuum locally. It is expected that electron charge clusters like Cooper pairs or bigger boson charge bundles can offer a substantially enhanced vacuum activity in their vicinity. The patents of Ken Shoulders (US #5,018,180) and Hal Puthoff (US #5,208,844) on charge cluster devices discuss the potentials of such an approach but lack sufficient engineering feasibility to control their volatility. Therefore, an ion trap or force field confinement process is required.

QED vacuum effects such as the coupling of the atomic electron to the vacuum electromagnetic field show that the electron is more intimately connected to the vacuum flux than most other particles. The zero-point oscillations of the field contribute to the electron a certain amount of energyE_fl ~ e²h/4mca², with an upper bound of hf_max = 15 MeV for a free electron. The coupling term for the atomic electron in the Hamiltonian is (e²/2mc² )A² where A is the vector potential and the parenthetical modifier is familiar from Equation (27) as half of the classical electron radius. Since this term does not involve atomic operators, it contributes the same energy to every state in the atom. Besides the ground state contribution, called the Lamb shift, it appears that every other electron level is also shifted upwards from vacuum flux energy or virtual particles. For this reason alone, it should be emphasized that extraction of energy from the vacuum is already occurring in every atom throughout the universe, since every atomic electron and every free electron is positively energized. However within the atomic system, the effects of the vacuum field and radiation reaction cancel, so that the spontaneous absorption rate is

A₁₂ = R_VF - R_RR = ½ A₂₁ - ½ A₂₁ = 0 (30)

where A₁₂ is the Einstein A coefficient for the electron transition from the ground state to the first atomic energy level. The spontaneous emission rate sums the rate of energy absorption from the vacuum field R_VF and the radiation reaction rate R_RR to equal the Einstein A coefficient.

The energetic scattering of the vacuum flux on a free electron, clearly seen in Figure 3, may perhaps be optimally amplified in the gas state, such as within the confines of an ion trap. This author collaborated in the construction of such a trap, which proved that electron and ion densities can be increased with such a trap, as the electrons are retained in one place for measurements and
energy extraction. Such an apparatus may also work well for charge clusters, after applying inductive braking to their kinetic energy. For an applied voltage of 300V and vacuum pressures of at least a microTorr, the concentration of ions ranged between 10⁸ and 10¹⁰ ions per cc. As seen in Figure 29, the voltage profile or potential distribution inside the grid with the presence of negative space charge from the Thoriated filaments exhibits a large concentration of electrons. Assuming that charge clusters cannot be trapped by any other means without

destroying them, the nonresonant ion trap should provide a reliable method for study and possible energy extraction if an additional collection and amplification method for the femtosphere is optimized and implemented.

Weisskopf notes that if the electron is assumed to be a sphere of radius a, then only waves with a wavelength λ /2p > a will act upon the electron, while the wavelengths λ /2p >> a will not be that significant. The upper bound of hf_max assumes an electron radius of a = c/f_max while the number of vibration modes of the ZPF gives rise to a value of a ≈ r_o( hc/e²)^½ so that the fluctuation energy seemingly pushes the electron radius to even greater values

Casimir Force Electricity Generator

A fascinating example of utilizing mechanical forces from the Casimir effect and a change of the surface dielectric properties, to intimately control the abundance of virtual particles, is an optically-controlled vacuum energy transducer developed by a Jet Propulsion Lab scientist. A moving cantilever or membrane is proposed to cyclically change the active volume of the chamber as it generates electricity with a thermodynamic engine cycle. The invention proposes to use the Casimir force to power the microcantilever beam produced with standard micromachining technology. The silicon structure may also include a microbridge or micromembrane instead, all of which have a natural oscillation frequency on the order of a free-carrier lifetime in the same material. The discussion will refer the (micro)cantilever design but it is understood that a microbridge or flexible membrane could also be substituted. The invention is based on the cyclic manipulation of the dimensions of Casimir cavity created between the cantilever and the substrate as seen in Figure 30. The semiconducting membrane (SCM) is the cantilever which could be on the order of 50-100 microns in size with a few micron thickness in order to obtain a resonant frequency in the range of 10 kHz, for example.

Two monochromatic lasers (RS) are turned on thereby increasing the Casimir force by optically changing the dielectric properties of the cantilever. This frequency dependence of a dielectric constant, can be seen in Figure 15. It can vary with frequency by a few orders of magnitude inversely proportional to the frequency. The standard analysis of cavity modes usually identifies the resonant modes of the cavity, dependent on the boundary conditions. ^, However, Pintos pro-active approach is to excite a particular frequency mode in the cavity. In doing so, an applied electrostatic charge (V_b) increases as the cantilever is pulled toward the adjacent substrate (SCP) by the Casimir force. Bending the charged cantilever on a nanoscale, the Casimir attractive force is theoretically balanced with opposing electrostatic forces, in the same way as Forwards parking ramp of Figure 8. As the potential difference to the cantilever assembly is applied with reference to a conducting surface (CP2) nearby, the distance to this surface is also kept much larger than the distance between the cantilever and the substrate (SCP). Upon microlaser illumination, which changes the dielectric properties of the surface and increases the Casimir force, there is also predicted an increase in electrostatic energy due to an increase in capacitance and voltage potential. Therefore a finite electrical current can be extracted and the circuit battery is charged by an energy amount equal to the net work done by the Casimir force. Pinto estimates the Casimir force field energy transfer to be approximately 100 to 1000 erg/cm². Converting this to similar units used previously, this Casimir engine should produce in the range of 60 to 600 TeV/cm² (teraelectron volts per square centimeter) which is also equal to 0.01 to 0.1 mJ/cm² for every cycle in Figure 31.

Analysis of the Casimir engine cycle demonstrates its departure from hydroelectric, gaseous, or gravitational systems. For example, the Casimir pressure always acts opposite to the gas pressure of classical thermodynamics and the energy transfer which causes dielectric surface changes does not flow to the virtual photon gas. Altering physical parameters of the device therefore, can change the total work done by the Casimir force, in contrast to gravitational or hydroelectric systems. Unique to the quantum world, the type of surface and its variation with optical irradiation is a key to the transducer operation. Normally, changing the reflectivity of a surface will affect the radiation pressure on the surface but not the energy density of the real photons. However, in the Casimir force case, Pinto explains, the normalized energy density of the radiation field of virtual photons is drastically affected by the dielectric properties of all media involved via the source-free Maxwell equations.

Specifically, Pinto discovered that the absolute value of the vacuum energy can change just by causing energy to flow from a location to another inside the volume V. This finding predicts a major breakthrough in utilization of a quantum principle to create a transducer of vacuum energy. Some concerns are usually raised, as mentioned previously, with whether the vacuum energy is conserved. In quantum systems, if the parameters (boundary conditions) are held constant, the Casimir force is strictly conservative in the classical sense, according to Pinto. When they are changed, however, it is possible to identify closed paths along which the total work done by this force does not vanish.

To conclude the energy production analysis, it is noted by Pinto that 10,000 cycles per second are taken as a performance limit. Taking the lower estimate of 100 erg/cm² per cycle, power or wattage is calculated to be about 1 kW/m² which is on par with photovoltaic energy production. However, on the scale of interest, where s in Figure 31 is always less than 1 μm, the single cantilever transducer is expected to produce about 0.5 nW and establish a millivolt across a kilohm load, which is still fairly robust for such a tiny machine.

The basis of the dielectric formula starts with Pintos analysis that the Drude model of electrical conductivity is dependent on the mean electron energy <E> (less than hf) and estimated to be in the range of submillimeter wavelengths. The Drude model, though classical in nature, is often used for comparison purposes in Casimir calculations. The detailed analysis by Pinto shows that carrier concentrations and resistivity contribute to the estimate of the total dielectric permittivity function value, which is frequency dependent. The frequency dependence is of increasing concern for investigations into the Casimir effects on dielectrics.

Analyzing the invention for engineering considerations, it is clear that some of the nanotechnology necessary for fabrication of the invention have only become available very recently. The one-atom microlaser, invented in 1994, should be a key component for this invention since about ten photons are emitted per atom. However, it has been found that new phenomena, (1) the virtual-photon tunnel effect and (2) the virtual-photon quantum noise, both have an adverse effect on the preparation of a pure photon-number state inside a cavity, which may impede the performance of the microlaser if placed inside a cavity. Pinto concurs that such a low emission rate is necessary since the lasing must take place as a succession of very small changes

Another suggested improvement to the original invention could involve a femtosecond or attosecond pulse from a disk-shaped semiconductor microlaser (such as those developed by Bell Laboratories). The microlaser could be used in close proximity to the cantilever assembly. Such microlaser structures, called microdisk lasers measuring 2 microns across and 100 nm thick, have been shown to produce coherent light radially (see Figure 32). A proper choice of laser frequency would be to tune it to the impurity ionization energy of the semiconductor cantilever. In this example, the size would be approximately correct for the micron-sized Casimir cavity.

Pinto chooses to neglect any temperature effects on the dielectric permittivity. However, since then, the effect of finite temperature has been found to be intimately related to the cavity edge choices that can cause the Casimir energy to be positive or negative. Therefore, the contribution of temperature variance and optimization of the operating temperature seems to have become a parameter that should not be ignored. Also supporting this view is the evidence that the dielectric permittivity has been found to depend on the derivative of the dielectric permittivity with respect to temperature.

Cavity QED Controls Vacuum Fluctuations

It is known from the basic physics of cavity QED that just the presence of the walls of a cavity will cause any atoms within it to react differently. For example, the spontaneous emission rate at wavelength should be completely suppressed if the transition dipole moment is parallel to the mirror plates where the walls of the cavity are reflecting conductors. In other words, a confined antenna cannot broadcast at long wavelengths. An excited atom in a small cavity is precisely such an antenna, albeit a microscopic one. If the cavity is small enough, the atom will be unable to radiate because the wavelength of the oscillating field it would like to produce cannot fit within the boundaries. As long as the atom cannot emit a photon, it must remain in the same energy level; the excited state acquires an infinite lifetime[because] there are no vacuum fluctuations to stimulate its emission by oscillating in phase with it. Such effects are noticed for cavities on the order of hundreds of microns and smaller, precisely the range of Pintos cavity. Therefore, it can be expected that carefully choosing the fundamental resonant frequency of the cavity will provoke the emission of photons so that the dielectric effect on the walls may be enhanced with less input of energy.

Furthermore, the most important Casimir force research relating to Pintos invention may be the analysis of a vibrating cavity. If the membrane oscillation frequency is chosen, for example, to be close to a multiple frequency (harmonic) of the fundamental unperturbed field mode of the cavity, resonant photon generation will also provoked. Such resonant photon generation in a vibrating cavity like Pintos has been studied in the literature. ^,

Another aspect of the Pinto experiment apparently not discussed in his article is the relative concentration of gas molecules in the vacuum energy transducer of Figure 30. Though a complete evacuation of air would be preferable, especially when compression of the membrane could be impeded by increasing gas pressures, it is naturally expected that too many gas molecules will still remain airborne even with a high vacuum, such as 10^-10 to 10^-12 Torr. Therefore, using another characteristic of cavity QED may be recommended. First of all, the selection of the gas is important, so that the atomic transition frequency matches the cavity resonant frequency very closely. Once this is achieved, it would be recommended, from an engineering point of view, to optimize the design of the size of the cavity transducer so that the atomic transition has a slightly higher frequency than the resonant frequency of the cavity. This could easily occur with the resonant wavelength slightly longer than the resonant transition wavelength of the gas in question. In that way, the gas molecules will be repulsed from entering the cavity, thus creating a lower gas pressure inside. Logically, this would be accomplished with the cavity transducer in the maximum S_A position in Figure 30. It would thereby add to the compression force of the movable membrane. As the membrane reaches its lowest position in the engine cycle with minimum S_A position, cavity QED dictates that since the atomic transition frequency will then be lower than the resonant frequency of the cavity, the force will be attractive, pulling gas molecules toward the cavity and increasing the pressure. This condition may be accomplished as well, since the shorter wavelength of the smaller cavity size will now be less than the longer wavelength of the atomic transition wavelength of the gas. Such a condition, with extra gas molecules in the cavity, will assist in pushing the membrane upwards again. Such detailed planning with gases and cavity dimensions should create a situation where the ZPF is supplying a larger percentage of the energy output, with a minimum of nanolaser input energy. If so, the Pinto vacuum energy generator would offer an unparalleled miniature electricity source that could fill a wide range of nanotechnology needs and microelectronic needs.

Spatial Squeezing of the Vacuum

The analysis of Pintos invention is analogous to spatial squeezing of the initial states to decrease the energy density on one side of a surface, below its vacuum value, in order to increase the Casimir force. For an oscillating boundary like Pintos, this can also create a correlated excitation of frequency modes into squeezed states and sub-Casimir regions where the vacuum develops structure. Pressing zero-point energy out of a spatial region can be used to temporarily increase the Casimir force. This spatial squeezing technique is gaining increasing acceptance in the physics literature as a method for bending quantum rules while gaining a short-term benefit, such as modulating the quantum fluctuations of atomic displacements below the zero-point quantum noise level of coherent phonon (vibrational) states, based on phonon-phonon interactions.

The squeezing technique involves minimizing the expectation value of the energy in a prescribed region, such as a cavity. In general, a squeezed state is obtained from an eigenstate of the annihilation operatorby applying to it the unitary squeezing (or dilation) operator. Ideally, it seems promising to generate squeezed modes inside a cavity by an instant change of length of the cavity. The implied infinite speed or frequency for a movable membrane would not be achievable however. If it were approachable, the squeezing would cause a modification of the Casimir force so that it could become a time dependent oscillation from a maximum to minimum force. Pursuing resonance measurements may turn out to be the most realistic experimental approach in order to exploit the periodic variation in the Casimir force by squeezing.

In Figure 33, the effect of squeezing can be seen in the fundamental cavity mode n = 0 where the emission of photons is almost double that allowed by the Planck radiation law Equation (9), where there are quantized field modes. Hu found that the other field modes go to a mixed quantum state due to the intermode interaction caused by the classical Doppler effect from the moving mirrors. The theory also predicts that the significant features of the nonstationary Casimir effect are not sensitive to temperature.

Focusing Vacuum Fluctuations

Another development that may directly affect transduction possibilities of ZPE is the theoretical prediction of focusing vacuum fluctuations. Utilizing a parabolic mirror designed to be about 1 micron in size (labeled a in Figure 33), with a plasma frequency in the range of 0.1 micron for most metals, Ford predicts that it may be possible to deflect atoms with room temperatures of 300K, levitate them in a gravitational field, and trap them within a few microns of the focus F. A positive energy density results in an attractive force. Depending upon the parameters, it may alternatively result in a repulsive Van der Waals force at the focus with a region of negative energy density. This type of trapping would require no externally applied electromagnetic fields or photons. The enhanced vacuum fluctuations responsible for these effects are found to arise from an interference term between different reflected rays. The interesting conundrum is the suggestion that parabolic mirrors can focus something even in the absence of incoming light, but vacuum fluctuations are often treated as evanescent electromagnetic fields. The manifestation of the focusing phenomenon is the growth in the energy density and the mean squared electric field near the focus.

Focusing vacuum fluctuations in many ways resembles amplified spontaneous emission (ASE) which occurs in a gain medium, where the buildup of intensity depends upon the quantum noise associated with the vacuum field.

Stress Enhances Casimir Deflection

An interesting Casimir force effect, seen more and more frequently in nano-electromechanical system (NEMS), is illustrated in Figure 35. Shown is a membrane or cantilever of thickness h that covers a well of width l and height a, which is deflected in the y direction, by an amount of distance W(x) depending upon the position with respect to x. The equation describing the deflection of any point on the membrane is:

D ∂⁴W(x)/ ∂x⁴ = F . (31)

D = Eh³/(12-12p²) where E is the elastic modulus, h is the thickness (see Figure 35) of the membrane, and p is the Poisson ratio. The Poisson ratio is the ratio of the transverse contracting strain to the elongation strain. The Casimir force F in Equation (31), due to the proximity to the bottom plate, is an inhomogeneous force in this situation, varying from point to point along x as

F =

where h, W(x) and a are defined above.

Equations (31) and (32) are then equated to produce a quartic equation dependent on W(x) where the residual applied stress/strain σ can be added as a modifier. Solving for W(x) under conditions of strain (stretching) yields a tendency toward a stationary wave pattern characteristic of buckling, without any appreciable change in the center deflection. Solving for W(x) under conditions of stress (compression) reveals that compressive residual stress enhances the deflection of the bridge and reduces its [buckling] behaviour. The amount of enhancement at the center is W(0) = 0.0074a or almost 1%. However, since the Casimir force in Equation (32) increases by the fourth power of the distance (a W(x)), it is also regarded as a positive feedback system, with a tendency of increasing any deflection in a direction toward structural failure.

Casimir Force Geometry Design

Since the Casimir force is such an integral part of the experimental energy manifestations of the ZPF as well as the Chapter 4 analysis, it is worthwhile to review some of its important characteristics. First of all, the attractive Casimir force between two uniform, flat metal plates which are perfectly conducting (and therefore, a reflective surface) is

F = p² hc / 240 d⁴ (33)

where d is the spacing between the plates. Milonni points out that besides the usual vacuum fluctuations approach, one can also treat the virtual photons of the vacuum as carriers of linear momentum. This perspective yields a mathematical proof that the Casimir force can also be classically analyzed as a physical difference of radiation pressure on the two sides of each plate.

In comparison, the Coulomb force for charged plates, such as with Forwards charge foliated conductors of Figure 8, is found to be

F_Coul = V² / 8pd² . (34)

Thus, with a potential difference of only V = 17 mV at d = 1 μm, the Casimir force equals the Coulomb force. This is also the operating principle behind Pintos cavity transducer of Figure 30, as the Casimir force is varied cyclically.

In Figure 36, a comprehensive approach by Maclay is made for an arbitrarily-sized box made of perfectly conducting surfaces. As the dimensions of the box deviate from a cubic design (1 x 1 x 1) the Casimir forces change as well.

A maximum positive energy density (dark area near the origin) signifies a positive Casimir force or outward pressure. Effectively, positive energy density produces a repulsive Casimir force. The dimensions of 1 x 1 x 1.7 signify the transition zone known as zero energy density. Any further increase in size results in a negative or attractive Casimir force. It is readily apparent from these calculations that a similar system, with a movable membrane like Pintos Figure 30, offers a restoring force for either deviation from zero, as if the cavity held a compressible fluid.

In Figure 37, the Casimir forces for a perfectly conducting 1 x 1 x C rectangular box, expanding from C = 1 to an elongated size, can be traced quite closely. Again, as in Figure 36, it can be seen that as C = 1.7 the Casimir pressure P₁ crosses the zero energy line. To help distinguish the Casimir energy density E / V and Casimir energy E lines from the rest, it is noted that these two lines cross zero at the same point C = 3.5, while the Casimir pressure P₃ line stays constant past C = 1. The E / V, E, and P₃ lines are all negative when C < 1 showing the dominance of d⁴ in the denominator of Equation (33), when two surfaces approach 1 micron or less.

The discovery by Maclay of a particular box dimension (1 x 1 x 1.7), that sits in the middle of attractive and repulsive Casimir forces, presents a possible scenario for vacuum energy extraction. This interesting motion suggests that we may be organizing the random fluctuation of the EM field in such a way that changes in pressure directly result, which could lead to work being done. One interesting question is can we design a cavity that will just oscillate by itself in a vacuum. One approach to this would require a set of cavity dimensions such that the force on a particular side is zero, but if the side is moved inward, a restoring force would be created that would tend to push it outward, and vice versa. Hence a condition for oscillation would be obtained. Ideally, one would try to choose a mechanical resonance condition that would match the vacuum force resonance frequency. More complex patterns of oscillation might be possible. The cavity resonator might be used to convert vacuum fluctuation energy into kinetic energy or thermal energy. More calculations of forces within cavities are needed to determine if this is possible, what would be a suitable geometry and how the energy balance would be obtained. Maclay concedes however, that upon analyzing Forwards charged parking ramp of Figure 8, with like charges supplying the restorative force to the Casimir attractive force, that no net work would be done for any given oscillation cycle.

When dielectrics are considered, the analysis becomes more involved. Calculations of Casimir forces for situations more complicated than two parallel plates are notoriously difficult, and one has little intuition even as to whether the force should be attractive or repulsive for any given geometry. With a dielectric set of parallel plates, the characteristics of dispersive (phase velocity is a function of frequency) or non-dispersive (all frequencies equally transmitted or reflected) dielectrics enters into the equation. For example, a classic example is two dispersive dielectric parallel plates that have a Casimir energy which depends only on the distance between the plates and the dispersion of the dielectrics.

Various geometries of rectangular cavities can also be studied using the principle of virtual work where E = - ∫ F dx. With the Casimir vacuum energy E for a dielectric ball of radius a, for example, the Casimir force per unit area is,

F =

For a dilute, dispersive dielectric ball for example, the Casimir surface force is found to be attractive with inward pressure. A system of two dielectric spheres with general permittivities and some chosen values of the refractive index n has also been evaluated for Casimir forces.

One application for this type of Casimir force calculation lies with biological cells which are spheres with a high dielectric constant. Figure 38 shows a B-lymphocyte which is 1 micron across which therefore must experience and compensate for the inward Casimir pressure. Biological structures may also interact with the vacuum field. It seems possible that cells, and components of cells, for example, the endoplasmic reticulum may interact with the vacuum field in specific ways. A cell membrane, with a controllable ionic permeability, might change shape in such a way that vacuum energy is transferred. Microtubules, in cell cytoskeletons, may have certain specific properties with regard to the vacuum field. Diatoms, with their ornate geometrical structures, must create interesting vacuum field densities; one wonders if there is a function for such fields. Many of these structures that are less than one micron in size have much higher Casimir pressures to contend with, such as ribosomes which are about 0.02 micron across.

Other geometrical objects have also been analyzed for the resultant Casimir forces such as hemispheres, pistons, and flat, circular disks. Instead of solid objects, configurations such as spherically symmetric cavities have also been presented in the literature. Rectangular cavities, for example, have also been found to have a temperature dependence and edge design variations which can lead to the Casimir energy being positive or negative.

Another interesting area of possible energy extraction from the Casimir effect is in astronomical bodies such as stars. The Casimir effect has been proposed as a source of cosmic energy. In such cosmological objects as white dwarfs, neutron stars, and quasars, the volume effect of the Casimir force is theoretically sufficient to explain the huge output of quasars for example. A calculation of the shift in energy density of the ZPF due to the presence of an ideal conductor in a volume V of space relative to the case with an absence of the volume is the mean value of the stress-energy tensor of QED inside volume V. A conductive material will be conductive for frequencies below the plasma frequency ω < ω_p and transparent for frequencies above the plasma frequency ω > ω_p. The plasma frequency, for nonpropagating oscillations depending only on the total number of electrons per unit volume is, in Gaussian units,

ω_p² = 4p ne² / m . (36)

The dielectric constant for high frequencies is also dependent on the plasma frequency (compare with Figure 15),

ε (ω ) = 1 ω_p² / ω² . (37)

(In dielectric media, Equation (37) applies for ω² >> ω_p².) The shift in the vacuum energy density due to the presence of a volume of ideal conducting material is, expressed in terms of the plasma frequency,

Δ E_vac = ω_p⁴ h c / 4p² . (38)

With a dramatic increase in the electron density n due to gravitational compression in collapsing stars, an energy creation is predicted that compares with 10³⁸ J expected for a nova or 10⁴² J for a supernova if the radius of the star is compressed to approximately R ≈ 10⁷ m.

Vibrating Cavity Photon Emission

Various cavities have been analyzed so far for the net Casimir effect. However, the case of photon creation from the vacuum due to a non-stationary Casimir effect in a cavity with vibrating wall(s) is unique and has interesting ramifications. Comparing with Pintos cavity of Figure 30, the cavity chosen by Dodonov to create resonance photon generation also has one moving wall while the rest of the rectangular cavity is stationary. The fundamental electromagnetic mode is ω₁ = p c / L_o where L_o is the mean distance between the walls of the cavity. The maximum value of the energy is found to be three times the minimum value, depending on the phase. The total energy also oscillates in time and the photon generation rate tends toward a constant value as long as any detuning is less than one.

While changes in the dielectric constant of cavity walls affect the Casimir vacuum force of Pintos vibrating cavity, there are also effects from a change in the refractive index of a medium. Hizhnyakov presents evidence for the emission of photons from such a distortion of the spectrum of zero point quantum fluctuations. If the medium experiences a time-dependent refractive index, it has been demonstrated that part of the energy will be emitted as real photons. An example is a dielectric medium excited by a rectangular light pulse for about a femtosecond (10^-15 seconds). The spectral density of the photon energy is shown to depend only upon the rate of change of the refractive index over time, which is unusual. While Hawking and Unruh radiation effects are mixed thermal states, this refractive index derivative effect is said by Hizhnyakov to be a pure state equally related to the ZPF as a non-linear quantum optical effect. In terms of energy flow, a picojoule (10^-9 J) laser pulse lasting for a femtosecond produces about ten megawatts (10 MW/cm²) of power input and the input pulse has about 10^-5 cm² cross sectional area which gives about a 100 W power input. The output intensity, estimated to be about a picowatt, is calculated to be the sum of two pulses created from the leading and trailing edges of the input refractive index change.

The Unruh radiation referred to above is actually called the Unruh-Davies effect which refers to a phenomenon related to uniform acceleration. In a scalar field such as the ZPE vacuum, the effect of acceleration is to promote zero-point quantum field fluctuations to the level of thermal fluctuations. Milonni points out that it took a half a century after the birth of quantum theory for the thermal effect of uniform acceleration to be discovered. The effective temperature that would be measured by an accelerated detector in a vacuum is

T_U =

which leads to the interpretation that thermal radiation is very similar to vacuum fluctuation radiation. In Equation (38), k is the wave number (ω/c) and a is the acceleration. Both vacuum probability distribution functions and thermal distributions exhibit a Gaussian probability distribution since the vacuum distribution is the T à 0 limit of the thermal distribution.

Looking at Hawking radiation, which is emitted from a black hole, it is based on the premise that pair production from the vacuum can occur anywhere, even at the event horizon of a black hole. The treatment is related to a mathematical manipulation called Wick rotation, where the metric is rotated into the complex plane with time t à - i t, so that the temperature is inversely equal to the period. Solving for the region just outside the event horizon r > 2GM, where G is the gravitational constant, the Hawking temperature is found to be

T_H =

where M is the mass of the black hole. Since Plancks constant is included in Equation (39), Zee notes that Hawking radiation is indeed a quantum effect. The similarities between Equations (38) and (39) are referred to by Hizhnyakov.

Fluid Dynamics of the Quantum Vacuum

In the analysis of Figure 37, it was mentioned that the Casimir force within cavities of Pinto and Maclay, possessing one movable wall, behave like a compressible fluid since a restoring force is present for any deviation from the zero-force position. It turns out that more exact analogies to fluids are possible for the quantum vacuum. A hydrodynamic model of a fluid with irregular fluctuations has been proposed by Bohm and Vigier for the vacuum, which also satisfies Einsteins desire for a causal interpretation of quantum mechanics. Their work also includes a proof that the wave function probability density P = |ψ|² used in quantum theory approaches the standard formula for fluid density with random fluctuations. There is also a suggestion of further work regarding how a fluid vortex provides a very natural model of the non-relativistic wave equation of a particle with spin.

A computational fluid dynamics approach to the ZPF, with the ambitious aim of reducing flight resistance at superluminal speeds has been proposed by Froning and Roach. The negative energy density region seen in Figure 37 between Casimir plates is also implicated in spacetime warping concepts and a theoretical increase in the speed of light. Resistance to flight in air and space have interesting parallels in this theory. In Figure 39, the aerodynamic viscous drag resistance to increased speed is compared to the electromagnetic zero-point vacuum resistance to increased speed, which is perceived as inertia. Drawing upon the separate works by Puthoff and Haisch (cited in Chapter 2), this

approach takes their ZPE-related gravity and inertia theory to the engineering level of experimental simulation. In Figure 40, the analogy is drawn between the well-known equation for the speed of light c = ( μ_oε_o )^-½ and the aerodynamic gas equation for the speed of sound c = (gRγT)^-½ with compressible fluid graphics for each. The aerodynamic resistance of viscous drag exerted on the substructure of a vehicle is compared to the Lorentz force exerted on the substructure of the vehicle by the ZPF, which is also proposed to be a Casimir-like force exerted on the exterior by unbalanced ZPE radiation pressures. The conclusion drawn from this first-order analysis is that μ_o and ε_o can be perturbed by propagation speed and possibly vehicle inertia, accompanied by a distortion of the zero-point vacuum.

A fundamental part of the Fronig and Roach approach to the fluid dynamic simulation of superluminal speeds is the proposal that μ_o and ε_o can be reduced significantly by nonabelian electromagnetic fields of SU(2) symmetry. It is proposed that EM fields of nonabelian form have the same symmetry that underlies gravity and inertia. Their approach is particularly to use alternating current toroids with resonant frequencies. That nonabelian gauge symmetry offers a higher order of symmetry has been seen elsewhere in the literature. Zee, for example, notes that the square of the vector potential A² would normally be equal to zero in the abelian gauge, which all standard (trivial) electromagnetic theory texts use. Instead, he notes that a field strength such as F = dA + A² can be formulated easily in the nonabelian gauge and shown to be nonzero and gauge covariant (though not invariant). Furthermore, the nonabelian analog of the Maxwell Langrangian, called the Yang-Mills Langrangian, includes cubic and quartic terms that describe self-interaction of nonabelian bosons (photons), as well as a nonabelian Berrys phase that is intimately related to the Aharonov-Bohm phase. (The Aharonov-Bohm phase depends exclusively on the vector potential.) Even the strong nuclear interaction is accurately described by a nonabelian gauge theory. Pure Maxwell theory is free and so essentially trivial. It contains a noninteracting photon. In contrast, pure Yang-Mills theory contains self-interaction and is highly nontrivialFields listen to the Yang-Mills gauge bosons according to the representation R that they belong to, and those that belong to the trivial identity representation do not hear the call of the gauge boson.

According to Froning and Roach, the representation R can be changed by surrounding a saucer-shaped spaceship with a toroidal EM field that distorts and perturbs the vacuum sufficiently to affect its permeability and permittivity. The vacuum field perturbations are simulated by fluid field perturbations that resulted in the same percentage change in disturbance propagation speed within the region of perturbation. The computational effort was simplified by solving only the Euler equations of fluid dynamics for wave drag. The resulting μ_o and ε_o perturbation solutions are shown in Figure 41.

In his discussion of the 1910 Einstein-Hopf model, Milonni describes their derivation of a retarding force or drag on a moving dipole as a result of its interaction with the vacuum zero-point field, which acts to decrease its kinetic energy. Assuming v << c, the retarding force due to motion through the ZPF thermal field is described with a fluid dynamics equation, F = R v, where v is the velocity of the dipole and R is a formula depending upon dipole mass and ZPF spectral energy density. Milonni also notes that due to recoil associated with photon emission and absorption, which are both in the same direction, the ZPF also acts to increase the kinetic energy of a dipole. Equilibrium is established when the increase in kinetic energy due to recoil balances the decrease in kinetic energy due to the drag.

It has been proposed by Rueda and Haisch, with a contract from NASA, that the ZPF can lose its Einstein-Hopf drag as the absolute temperature approaches zero, which would leave only the accelerating recoil force left. Furthermore, they propose that the ZPF can provide a directional acceleration to monopolar particles more effectively that to polarizable particles. They also suggest that if valid, the mechanism should eventually provide a means to transfer energy, back and forth, but most importantly forth, from the vacuum electromagnetic ZPF into a suitable experimental apparatus.

Quantum Coherence Accesses Single Heat Bath

One of the main criticisms of energy extraction from the ZPF is that it represents a single low-temperature bath and the second law of thermodynamics prohibits such an energy conversion. It is well-known that Carnot showed that every heat engine has the same maximum efficiency, determined only by the high-temperature energy source and the low-temperature entropy sink. Specifically, it follows that no work can be extracted from a single heat bath when the high and low temperature baths are the same.

However, a new kind of quantum heat engine (QHE) powered by a special quantum heat bath has been proposed by Scully et al. which allows the extraction of work from a single thermal reservoir. In this heat engine, radiation pressure drives the piston and is also called a Photo-Carnot engine. Thus, the radiation is the working fluid, which is heated by a beam of hot atoms. The atoms in the quantum heat bath are given a small bit of quantum coherence (phase adjustment) which becomes vanishingly small in the high-temperature limit that is essentially thermal. However, the phase associated with the atomic coherence, provides a new control parameter that can be varied to increase the temperature of the radiation field and to extract work from a single heat bath. The second law of thermodynamics is not violated, according to Scully et al., because the quantum Carnot engine takes more energy, with microwave input, to create the quantum coherence than is generated.

Figure 42 Photo-Carnot heat cycle diagram. Q_in is provided by hot atoms from bath.

The Photo-Carnot engine, shown in Figure 42, creates radiation pressure from a thermally excited single-mode field that can drive a piston. Atoms flow through the engine from the T_h heat bath and keep the field at a constant temperature for the isothermal 1à 2 portion of the Carnot cycle (Figure 43). Upon exiting the engine, the bath atoms are cooler than when they entered and are reheated by interactions with the blackbody at T_h and 'stored' in preparation for the next cycle.

The stimulus for the work came from two innovations in quantum optics: the micromaser and microlaser (Figure 32) and lasing without inversion (LWI). In micromasers and microlasers, the radiation cavity lifetime is so long that a modest beam of excited atoms can sustain laser oscillation. In LWI, the atoms have a nearly degenerate pair of levels making up the ground state. When the lower level pair is coherently prepared, a small excited state population can yield lasing (without inversion). In the QHE, the 'engine' is a microlaser cavity in which one mirror is a piston driven by the radiation pressure given by

P V = n h p c / L (40)

where P is the radiation pressure, V is the cavity volume, n is the average number of thermal photons in the right mode (about 10³), and L is the length of the cavity. In Figure 43, an engine cycle diagram is shown which is a temperature versus entropy graph for the Photo-Carnot engine, where |ρ_bc| ≈ 3 x 10^-6 is an off-diagonal density matrix element in the extension of the quantum theory of a laser without inversion. Figure 43 contains a closed cycle of two isothermal and two adiabatic processes (compare with Pintos Figure 31). Q_in is the energy absorbed during the isothermal expansion and Q_out is the energy given to the heat sink during the isothermal compression. However, instead of two states which would render this a classical engine, the QHE has three states, which can result in quantum coherence. If there is a non-vanishing phase difference between the two lowest atomic states, then the atoms are said to have quantum coherence. This can be induced by a microwave field with a frequency that corresponds to the transition between the two lowest atomic states. Quantum coherence changes the way the atoms interact with the cavity radiation by changing the relative strengths of emission and absorption. In Figure 42, as the atoms leave the blackbody at temperature T_h they pass through a microwave cavity that causes them to become coherent with phase Φ before they enter the optical cavity. The temperature that characterizes the radiation is T_Φ which is

T_Φ = T_h (1 n ε cos Φ) (41)

where ε is the magnitude of quantum coherence, ε = 3|ρ_bc|. The second term in Equation (41) is also used in the QHE efficiency equation,

η_Φ = η ( T_c/T_h ) n ε cos Φ (42)

Thus, depending on the value of Φ, the efficiency of the quantum Carnot engine can exceed that of the classical engine even when T_c = T_h . It can therefore extract work from a single heat bath.

Inexplicably, Scully et al. fail to cite a previous work by Allahverdyan and Nieuwenhuizen that utilizes more rigorous physics for same purpose of extraction of work from a single thermal bath in the quantum regime with quantum coherence. The authors, perhaps, have more controversial statements in the article regarding free energy extraction. Using the quantum Langevin equation for quantum Brownian motion, they note that it has a Gibbs distribution only in the limit of weak damping, thus preventing the applicability of equilibrium thermodynamics. The reason is related to quantum entanglement and the necessary mixed state. Our main results are rather dramatic, apparently contradicting the second law: We show that the Clausius inequality dQ < TdS can be violated, and that it is even possible to extract work from the bath by cyclic variations of a parameter (perpetuum mobile). The physical cause for this appalling behavior will be traced back to quantum coherence in the presence of the near-equilibrium bath. It is also emphasized that the quantum coherence is reflected in the quantum noise correlation time which exceeds the damping time 1/ Γ.

Regarding the ZPF, it is interesting that the quantum Langevin equation is a consequence of the fluctuation-dissipation theorem of Equation (11). The authors note that part of the equation includes the fluctuating quantum noise, which has a maximum correlation time and therefore has a long memory (quantum coherence) at low temperature. The Brownian particle of interest also has semi-classical behavior due to its interaction with the bath, where there is a transfer of heat, even for T = 0.

The possibility of extracting energy from the bath is due to the nonequilibrium state, which is ensured by the switching energy ½ γ Γ<x²>_o that is also a purely classical effect. The switching energy depends upon γ which is the damping constant and Γ which is the cutoff frequency. Both harmonic and anharmonic oscillation potentials are considered. The Brownian quantum particle strongly interacts with the quantum thermal bath, described by the Fokker-Planck equations.

Two formulations of the second law, namely, the Clausius inequality and the impossibility to extract work during cyclical variations, can be apparently violated at low temperatures. One could thus speak of a perpetuum mobile of the second kind. We should mention, however, that the number of cycles can be large, but not arbitrarily large. As a result, the total amount of extractable work is modest. In any case, the system energy can never be less than its ground state energyWe call them apparent violations, since, the standard requirements for a thermal bath not being fulfilled, thermodynamics just does not apply. Let us stress that also in the classical regime the harmonic oscillator bath is not in full equilibrium, but there noise and damping have the same time scale 1/ Γ, allowing the Gibbs distribution to save the day and thermodynamics to apply. Our results make it clear that the characterization of the heat bath should be given with care. If it thermalizes on the observation time, standard thermodynamics always applies. Otherwise, thermodynamics need not have a sayThe finding that work can be extracted from quantum baths may have a wide scope of applications such as cooling.'

In a physics commentary, Linke defends the second law by insisting that the work done by the Scully et al. piston (in Figure 42) is less than the work required to establish quantum coherence. Linke clarifies the Photo-Carnot process by stating, When the phase difference is adjusted to the value p, destructive interference reduces the likelihood of photon absorption, whereas emission from the upper level is not affected. This deviation from detailed balance between photon absorption and emission increases the photon temperature. The resulting temperature difference between photon gas and heat bath allows the photon Carnot engine to produce work in the absence of a hot bath.

Thermodynamic Brownian Motors

There is still another aspect of the ZPF that presents the possibility of energy extraction, which are nonequilibrium fluctuations a different representation of the single heat bath. Biasing the Brownian motion of a particle in an anisotropic medium without thermal gradients, the force of gravity, or a macroscopic electric field is a way that usable work is theoretically generated from nonequilibrium fluctuations, such as those generated externally or by a chemical reaction far from equilibrium. Fluctuation-driven transport is one mechanism by which chemical energy can directly drive the motion of particles and macromolecules and may find application in a wide variety of fields, including the design of molecular motors called Brownian motors and pumps.

Brownian motion, the random collisions with solvent molecules by a particle in a liquid, has been studied historically by Einstein as well as by Langevin. Langevins equation, as noted in the previous section, suggested that the forces on the particle due to the solvent can be split into two components: (1) a fluctuating force that changes direction and magnitude frequently compared to any other time scale of the system and averages to zero over time, and (2) a viscous drag force that always slows the motions induced by the fluctuation term. Related to the fluctuation-dissipation theorem of Equation (11), the amplitude of the fluctuating force is governed by the viscosity of the solution and by temperature, so the fluctuation is often termed thermal noise. Even in an anisotropic medium, the fluctuations are symmetric as required by the second law of thermodynamics, as are the dissipative drag force. Therefore, when all components of fluctuation-driven system are treated consistently, net motion is not achieved if it is an isothermal system, despite the anisotropy of a ratchet's teeth. However, a thermal gradient in synergy with Brownian motion can cause directed motion of a ratchet and can be used to do work but these are hard to maintain in microscopic and molecular systems. Recent work has focused, instead, on the possibility of an energy source other than a thermal gradient to power a microscopic motor.

Astumian proposes a fluctuating electrical potential that causes the uphill transport of a particle. A fluctuating potential energy profile is provided with an anisotropic sawtooth function U_saw and periodically spaced wells with no net macroscopic force. When the potential is off, the energy profile is flat with a uniform force everywhere. When the potential is turned on again, the particle is trapped in one of the wells. The result is resolved into two components: the downhill drift and the diffusive spreading of the probability distribution. For intermediate times, it is more likely for a particle to be trapped in one of the uphill wells if the potential were turned back on, than between the first and second well. Thus, turning the potential on and off cyclically can cause motion to the right and uphill against gravity despite the net force to the left. The theory has been successfully tested with colloidal particles with anisotropic electrodes turned on and off, as well as with an optical trap modulated to create a sawtooth potential.

Figure 44 shows the basic mechanism by which a fluctuating or oscillating force can cause directed motion along a ratchet potential U_saw and also some statistics of an example. The presence of thermal noise allows a subthreshold fluctuating force to cause flow. The force is modulated between + F_max where the average velocity <v> is calculated as a function of F_max where the dashed lines (B) indicate the threshold forces. Another example is shown (C) where average velocity is plotted as a function of the thermal noise strength k_BT, where k_B is Boltzmanns constant and T is the absolute temperature and F_max = 0.4 pN. The (C) graph shows that, up to a point, increasing the noise can actually increase the flow induced by a fluctuating force. However, the (B) graph shows that for forces near the optimum (about 1.5 pN in this example), the velocity decreases with increasing noise.

A more general approach is suggested for analyzing fluctuation-driven transport using the diffusion equation with a probability density given by the Boltzmann distribution P(x) ≈ exp(-U/ k_BT).

Since modulating the potential certainly requires work, Astumian believes there is no question of these devices being perpetual motion machines. The surprising aspect is that flow is induced without a macroscopic force. All of the forces involved are local and act on a length scale of the order of a single period of the potential. Yet the motion persists indefinitely, for many periods. However, the direction of the flow depends upon how the modulation is applied. Such devices are indicated to be consistent with the behavior of molecules.

In a dramatic confirmation of the Astumian theory, two subsequent experiments were performed by Linke et al. which applied an electron ratchet in a tunneling regime with a rocking-induced current (tunneling through and excitation over the ratchets energy barrier) flow in opposite directions. Thus the net current direction depends on the electron energy distribution at a given temperature. The practical aspect of this experimental approach is that a square wave source-drain voltage is applied with the time-averaged electric field being zero, similar to AC electricity, and yet the output net current is DC, similar to a rectifier.

Figure 45A shows in the inset the asymmetrical darker regions which are etched trenches that laterally confine a two-dimensional sheet of electrons located parallel to the surface of a GaAs/AlGaAs heterostructure. Figure 45B inset compares the barrier height Vo with the electron energy |eU| < 1 meV. The trench's periodic variation in width induces a corresponding variation in electron confinement energy that creates asymmetric energy barriers at each constriction. When the square wave source-drain bias voltage is applied, the resulting current I is plotted for two devices R10 and R1. Because of the geometric asymmetry, the electric field along the channel produced by the voltage deforms the barriers in a way that depends on the polarity of the voltage, trapping the electrons in one of the side gates along the x direction. The quantum ratchet is thus established by confining electrons to an asymmetric conducting channel of a width comparable to the electron wavelength. The experiment demonstrates importance of resonant design and fulfills the theoretical prediction that a fluctuating voltage is sufficient to cause a unidirectional current.

The detailed theoretical model of the tunneling ratchet is shown in Figure 46. The difference ∆t between the transmission functions for the barrier potential V_o in Figure 45B inset is graphed (solid curve) for a rocking voltage U_o = 0.5 mV. The bottom of Figure 46 shows visually the approximate equality of the barrier height V_o and the Fermi energy μ_F of 12 meV. The thin curves are identified by temperature and are graphed against the energy range ∆f of the right hand ordinate axis. The Figure 46 top inset shows the dramatic difference between the solid curve quantum mechanical behavior (qm) and the classical (cl) transmission function versus electron energy ε for U_o = 0. The classical step function occurs at ε = V_o which is the Fermi energy μ_F. Note the Figure 46 lower inset which summarizes the net current I in picoamperes versus temperature, for rocking voltages U_o = 0.7 mV (dashed curve), 0.5 mV (solid curve), and 0.3 mV (dotted curve). All temperatures in this experiment were within a few degrees of absolute zero.

Transient Fluctuation Theorem

Another development in the thermodynamics of microscopic systems has recently redefined the concept of work. In a system connected to a single heat bath, uncertainties on the order of k_BT will arise from the Boltzmann distribution of energies in the initial and final states, as well as from energy exchange with the heat bath as the system goes from initial to final energy states. Because of the thermal fluctuations or energy uncertainties, it has now been proven by theory and experiment, that the work cannot be uniquely specified, even if the path is known. When the system is microscopic, the fluctuations are significant and a transient fluctuation theorem has evolved to account for the behavior which takes the form,

P(W) / P(-W) = e^W (43)

where W is the work divided by k_BT and P(W) is the probability of performing positive work over an interval of time, while P(-W) is the probability of performing negative work over the same period of time . It needs to be clarified that in these microscopic systems, work is measured as it is delivered to a vessel but half the time the system goes in reverse, apparently violating the law of entropy. In Figure 47, a Poisson distribution is shown of an optical trap interacting with an experimental vessel having micron-sized beads. Though the trap exerted a restoring force, as for the spring in Figure 2, the experimentally-determined values measured for the bead position in the integrated fluctuation theorem showed a nonzero probability for negative work, for up to two seconds.

Imagine, as is often the case, that after a certain time, the bead has a higher energy than it had initially. Then, if the work done by the trap on the vessel (bead plus bath) is negative, energy has been delivered to both the bead and the optical trap interacting with the vessel. That energy came from the water bathjust the sort of energy transfer prohibited by the second law in the thermodynamic limit of infinitely large systems: Heat has been converted to work with 100% efficiency.

The generalized transient fluctuation theorem has various forms depending upon the application. For example, the probability ratio in Equation (43) can instead relate to entropy production w for forward and reverse time, where microscopic reversibility is an essential requirement. The theorem finds its greatest utility in the application to single heat baths and systems driven by a stochastic (random), microscopically-reversible (time-symmetric), periodic process. Then, the only Gaussian distribution that satisfies the fluctuation theorem has a variance that is twice the mean 2<w> = <(w-<w>)²>, which is a version of the standard fluctuation-dissipation relation of Equation (11). It is noted that entropy production, which is irreversible dissipation, is directly related to the fluctuations.

An example of a systematic application of the fluctuation theorem, reminiscent of the quantum ratchet concept seen previously, is a Metropolis Monte Carlo simulation by Crooks, illustrated in Figure 48. A single particle occupies a finite number of positions in a one-dimensional box with periodic boundaries and coupled to a heat bath of T = 5. The energy surface E(x) is outlined by a series of dashed () lines. The equilibrium distribution to be expected from such a potential well is shown as a symmetric hill with open circles Ο since the particle is free to move with equal probability to the right, left, or to stay put. However, every eight time intervals, the energy surface moves to the right by one position.

In this way, the system is driven away from equilibrium and settles into a time-symmetric, nonequilibrium steady state distribution shown by black dots ●, skewed to the left. The master equation for this system can be solved exactly to compare with the theory.

Power Conversion of Thermal Fluctuations

While many recent scientific advances in the treatment of nonequilibrium energy fluctuations have been reviewed so far, there are also developments in the area of nonlinear equilibrium thermal fluctuations. Based on the Nyquist theorem (another name for the fluctuation-dissipation theorem) and van Kampens work on nonlinear thermal fluctuations in diodes, Yaters pioneering work involves the use of microscopic diodes in a simple circuit, whose results gave higher conversion efficiencies than the Carnot cycle for certain limiting cases as these model sizes decreased.' While the use of two heat baths suggests a thermionic energy source, his detailed analysis makes it clear that the energy fluctuation conversion is added to a heat pump thermal conversion cycle, for example, yielding a factor of 10 improvement.

Yater offers a simplified master equation for the output rectified current from an independent particle model,

I (N) ≈ exp [ (β α)(N ½ ) (βm + αn)] 1 (44)

where β = q² / k_BT_cC, N = the number of excess electrons in the total circuit capacitance C, and α = q² / k_BT_rC. The designation of n and m are related: n = C_cV/q and m = C_rV/q where C = C_c + C_r. The forward and reverse diode currents in Figure 49 also combine in textbook fashion to produce the total current of Equation (44):

I (N) ≈ I_1a I_2b I_2a I_1b (45)

A Schottky barrier diode, for example, at the liquid Helium temperature of T_c = 1K can be used for the cold bath T_r in Figure 49. For such a diode the nonlinearity factor is β = 1.16 x 10⁴ e/C where C is the capacitance of the diode (C ~ 10^-16F) and e is the charge on the electron. A Schottky diode is also known to be formed between a semiconductor and a metal, with nonlinear rectifying characteristics and fast switching speeds.

Yater notes that for the long range design goals, sub-micron circuit sizes are required if all the high power goals of megawatts per square meter are to be achievedThe results of an analysis of the independent particle model for both classical and quantum effect, show that the reversible thermoelectric converter with power conversion of energy fluctuations has the potential of achieving the maximum efficiency of the Carnot cycle. The potential applications of this device can be seen to be universal. His patent #4,004,210 clarifies that the electric energy fluctuations are transmitted from the higher temperature diode to the lower temperature diode while the heat transfer is in reverse, which is unusual.

Upon reviewing the literature, Yater summarizes his findings: The relation of the second law of thermodynamics to the power conversion of fluctuation energy has been of recurring interest and study. The results of these studies have ranged from the conclusion that conversion of fluctuation energy is prohibited by the second law to the conclusion that the conversion of fluctuation energy is not limited by the second law of thermodynamics.

Rectifying Thermal Noise

Particles that move aperiodically due to thermal or external noise, in the presence of asymmetric periodic potentials, have also been called stochastic ratchets. These systems have the intriguing ability to rectify symmetric correlated noise and thus have the ability to produce a net electrical current. As a consequence of the fluctuation-dissipation theorem, the ratchet does not drift (as in the Metropolis Monte Carlo simulation of Figure 48) if it is in interaction only with a thermal bath. An external forcing must be used to produce the drift, according to the researchers, Ibarra-Bracamontes et al.

By describing the ratchet system in the Brownian particle regime with the Langevin equation (like Allahverdyan and Nieuwenhuizen), thermal noise is considered to have finite correlation times. This treatment yields a prediction of current production due to a time-dependent external force. In this case, the external force is either sinusoidal or stochastic. The generalized Langevin equation is an embellishment of Newtons equation, F = ma, describing a particle of mass m moving in an asymmetric periodic potential V(x) that forms the ratchet,

m d²x/dt² = d(t) Γ(t t) dx(t)/dt dV(x)/dx + f(t) + F_ext(t) . (46)

Γ(t) is the dissipation kernel, which in this case has memory and correlated with friction. The term, f(t), is the stochastic fluctuating thermal force exerted by the bath which has a usual stochastic property of being Gaussian with zero mean. By a numerical solution, Equation (49) does not show a net current in the absence of external forces (if F_ext(t) = 0). However, adding a time-symmetric external force in general is a necessary and sufficient condition for the Langevin equation to yield a current flow in one direction.

Rather than argue in favor of nonequilibrium fluctuations being present with a symmetric correlated force, the emphasis is made in this case in favor of an external origin for the force. In accordance with the second law of thermodynamics, the assertion is also made here that 'one cannot extract a current from a thermal bath, whether white or colored. Colored noise is where some frequencies dominate the noise spectrum and is also referred to as pink noise. The average position of the electron <x> is shown in Figure 50 for five different external force amplitudes versus time. The solution demonstrates the Metropolis Monte Carlo simulation, with a different ratchet design than Figure 48. However, it is not clear that Ibarra-Bracamontes et al. have succeeded in proving their controversial claim that any external forcing may be used to produce the drift but the rectification of thermal noise due to an asymmetric external potential has been demonstrated.

An experimental demonstration of rectifying random thermal fluctuations involves ferrofluids, which are colloidal suspensions of nanoparticle ferromagnetic grains (~10 nm). In this case, the external potential is an anharmonic oscillating magnetic field created by Hemholtz coils H_y at a distance from the ferrofluid. It is regarded as a rotating magnetic field since there is a static magnetic field H_x at right angles to the oscillating field which sums to create rotation and angular momentum (Figure 51). The angular momentum of electromagnetic fields is a principle of classical physics. Without thermal fluctuations, relaxation dynamics tend to cause the particles to align with the field in the x-y plane and no average rotation nor torque is created. In the presence of thermal fluctuations however, stochastic transitions occur due to the magnetic field asymmetry, yielding slightly different probabilities for the magnetic orientational motion of the ferromagnetic grains, as discovered by Engel et al.

As with the Photo-Carnot engine, the Fokker-Planck equation is also used in this case for a quantitative solution of the induced torque effect in the ferrofluid. Solving the equation by expansion in spherical harmonics, the transitions between deterministic solutions become possible. The spatial asymmetry and temporal anharmonicity of the potential results in slightly different rates for noise-induced increments and decrements of [phase] φ, respectively. As a result, a noise-driven rotation of the particles arises. There is viscous coupling between the ferromagnetic grains and the carrier liquid so that the individual torques add to create a macroscopic torque per fluid volume, N = μ_o M x H. However, Engel et al. admit that the time-averaged N_z is much smaller than the typical values of the time-independent N_z calculated from the Fokker-Planck equation by adopting the effective field method.

With an exploration of nonlinear perturbation techniques, Engel et al. admit that only a particularly chosen Langevin function yields the correct time-averaged z-component of the torque N_z, with static and dynamic magnetic field terms included. The expression for N_z shows that both the static magnetic field and the anharmonic part of the oscillatory component are essential for the directed rotation to occur.

Interestingly, the Brownian relaxation time is another parameter which had to be heuristically determined to achieve agreement between theory and experiment. It corresponded to a particle size of approximately 35 nm which is about three times the average particle size. Engel et al. conclude that the Brownian relaxation time must therefore correspond to the largest grains in the population rather than the average.

Engel et al. succeeded in rectifying rotational Brownian motion angular momentum. It is an experimental realization of the combined action of many individual nanoscale ratchets to yield a macroscopic thermal noise transport effect.

In regards to rectifying thermal electrical noise, it is worth mentioning the U.S. Patent #3,890,161 by Charles M. Brown that utilizes an array of nanometer-sized metal-metal diodes, capable of rectifying frequencies up to a terahertz (10¹² Hz). Brown notes that thermal agitation electrical noise (Johnson noise) behaves like an external signal and can be sorted or preferentially conducted in one direction by a diode. The Johnson noise in the diode is also generated at the junction itself and therefore, requires no minimum signal to initiate the conduction in one direction. The thermal noise voltage is normally given by V² = 4k_BTRB where R is the device resistance and B is the bandwidth in Hertz. Browns diodes also require no external power to operate, in contrast to the Yater diode invention. Brown also indicates that heat is absorbed in the system, so that a cooling effect is noticed, because heat (thermal noise) energy energizes the carriers in the first place and some of it is converted into DC electricity. In contrast, the well-known Peltier effect is the closest electrothermal phenomenon similar to this but requires a significant current flow into a junction of dissimilar metals in order to create a cooling effect (or heating). Brown suggests that a million nickel-copper diodes formed in micropore membranes, with sufficient numbers in series and parallel, can generate 10 microwatts. The large scale yield is estimated to be several watts per square meter.

Quantum Brownian Nonthermal Recifiers

While many researchers believe that the asymmetrical ratchet of one form or another is essential in the conversion of stochastic fluctuations, there are others who also find that stochastic resonance (SR) in threshold systems is a sufficient substitute. The cooperation between the signal and noise introduces coherence into the systemThis coherence is conveniently quantified as the power spectral densityof the system responseThe earliest definition of SR was the maximum of the output signal strength as a function of noise An introduction to SR is shown in Figure 52 with a potential barrier U_o separating two wells at +c. If the energy is subthreshold, the system will be monotonic but adding an amount of noise on the order of U_o allows the dichotomic system to oscillate. With a supra-threshold periodic forcing, the two wells may have different net occupation levels (as arrows indicate), with the output signal-to-noise ratio (SNR) following the input SNR closely.

Anomalous transport properties, using SR, which do not exploit the ratchet mechanism have been investigated in driven periodic tight-binding (TB) lattices near zero DC bias with the combined effects of DC and AC fields, or DC field and external noise. In particular, Goychuk et al. have found that periodic TB lattices can be driven by unbiased nonthermal noise generated from the vacuum ZPF, generating an electrical current as a result of a ratchetlike mechanism, as long as there is quantum dissipation in the system.

Theoretically introducing quantum dissipation with an ensemble multi-state model of harmonic oscillators coupled to the driven system, along with an unbaised, time-dependent random force (characterized by an external, time-dependent random electric field, η(t) = eaE_η(t)/h), yields the noise-averaged stationary quantum DC electrical current J_st = e(lim tà∞)d<q(t)>/dt and quantum diffusion coefficient D. The nonthermal fluctuations are given discrete, quantum values with probabilities as QED dictates. It is assumed that neither the temperature nor the nonthermal fluctuations can cause any essential occupation of higher energy levels for the dissipation model.

Goychuk et al. succeed in demonstrating that without dissipation, for any field, the stationary DC current is always zero. An initially localized particle, as in a crystal, does not produce a current in the absence of dissipation. However, considering the effect of adding unbiased fluctuations η(t) on Bloch oscillations, Goychuk develops a master equation which includes a real-time quantum Monte Carlo calculation yielding a good approximation for a TB particle at environmental temperatures and/or strong dissipation when transport includes sequential tunneling. Bloch functions describe an electron in a periodic lattice with a sinusoidal wave function conditioned by a lattice periodicity function. Driven tunneling dynamics in Bloch oscillation system is the subject of another paper Goychuk has coauthored with efficient determination of the optimal control of quantum coherence.

In Figure 53, the production of rectified electrical current J_st is shown with an ohmic friction factor α = 1 graphed against changes in fluctuation noise strength σ for a dichotomic (two-choiced) random process with zero mean and asymmetry parameter of ξ = ½. While the solid line is adiabatic, the dashed lines depict non-adiabatic autocorrelation times of 0.1ω_c and ω_c respectively from top down, where ω_c is the cutoff frequency. The inset graph is an interesting current reversal that occurs under a case of strong friction α = 5 which is related to SR. Because the current appears as the nonlinear response to the aperiodic external signal, the existence of this maximum [in Figure 53] can be interpreted as a signature of aperiodic quantum stochastic resonance.

The stationary current is also found to depend on temperature, which is the signature of aperiodic quantum stochastic resonance (AQSR). In Figure 54, the rectified current is shown versus temperature for a fluctuation noise strength σ =0.5ω_c (solid line) and σ = 7ω_c (dashed line) while friction α = 1 and asymmetry ξ = 1.

In the presence of unbiased, asymmetric forcing, a noise-directed current always occurs in a dissipative TB lattice, because of the ratchet-like effect of the asymmetric forcing, like the stochastic ratchets that rectify thermal noise. With stochastic resonance, nonthermal fluctuations are effectively rectified, creating a measurable current. Goychuk believes that the effect should be already observable in superlattices and/or optical lattices.

For reference, it is worth mentioning that with crystal lattices, thermal fluctuations appear at environmental temperatures, with ½ mω_o²<u²> = 3(½k_BT) energy level where m and ω_o are the mass and frequency of the harmonic oscillations and u is the displacement from a fixed lattice site. The nonthermal oscillations associated with ZPE are mω_o²<u²> = 3(½hω_o) in terms of energy, adding to the lattice thermal fluctuations.

Vacuum Field Amplification

With the introduction to AQSR along with the rectification of nonthermal noise, it makes sense to investigate the amplification of quantum noise. Milonni points out that the vacuum field may be amplifiedif the spontaneously emitted radiation inside the cavity is amplified by the gain medium, then so to must the vacuum field entering the cavity. Another way to say this is that quantum noise may be amplified. Since the SR TB lattice current output depends on the noise level, as in the Goychuk simulation, the optimum level of energy extraction depends on parameter control, as in quantum optics, which utilizes quantum noise amplification. This is similar to ASE which also uses a gain medium.

The actual content of quantum noise and vacuum polarization may still remain a mystery after all of the Chapter 4 analysis. Milonni notes that even though heavier virtual particle pairs like muons, pions, etc. may take part in virtual polarization, the majority of the manifested particles will always be electron-positron pairs because of the 1/m² mass dependence of the nonexchange term between the two current densities for the electron and the negative-energy states of the ZPF. Jackson notes that the Weisacker-Williams method of virtual quanta treats every scattering impact or close encounter between charged particles as a Fourier collection of virtual particles which are equal to the electric field pulse radiated to the target. This method, in consonance with QED, gives the frequency spectrum, cutoff, and number of virtual particles per unit energy interval.

CHAPTER 5

Summary, Conclusions and Recommendations

Summary

This study was predicated on the existing volume of data already in the scientific literature regarding the nonthermal vacuum fluctuations that comprise ZPE. The assessment of the feasibility of zero-point energy extraction by humans from the quantum vacuum for the performance of useful work in the electrical, fluidic, thermodynamic, and mechanical conversion modalities is determined in this chapter.

Analyzing the specific experiments, theories, simulations, measurements and predictions in this study offers a wealth of details concerning the energetic operation of ZPE throughout the universe. As a result, it can possibly be argued from an historical perspective that, because nature already extracts ZPE for the performance of useful work, humans eventually will be able to so as well. Acknowledging the need for robust, concentrated energy sources in the world, any study of the concept of energy extraction and production should address the corresponding utility and energy quality. Therefore, this summary addresses the electrical power output and the practicality of the conversion mode and method.

This study finds that at the present time, the categories of the present major inventive ZPE conversion modes includes 1) electromagnetic conversion, 2) Casimir cavity mechanical engine, 3) fluid dynamics techniques, and 4) quantum thermodynamic rectifiers. Under these major headings are individual methods such as 1a) focusing vacuum fluctuations, 2a) cavity QED, 2b) spatial squeezing, 2c) Casimir cavity geometry design, 2d) Casimir stress enhancement, and 2e) vibrating cavity photon emission, 3a) inertial effects, 3b) hydrodynamic model, 3c) Casimir cavity, 4a) quantum coherence, 4b) Brownian motors, 4c) transient fluctuations, 4d) thermal fluctuation rectifiers, and 4e) nonthermal Brownian rectifiers, as shown in Table 1 with relevant author names.

Table 1 - ZPE Conversion Modes & Methods

Electromagnetic Conversion

In the electromagnetic energy conversion process, the proposed Mead configuration of two spheres in close proximity was analyzed for four different size categories, each of which are a thousand times smaller than the previous one. The microsphere category seemed to match the Mead design closely but was found to be lacking in sufficient scattering intensity, even when substituting conducting spheres which have the highest scattering cross section. The decrease of dielectric constant with frequency also was a problem for most materials. The ZPF energy density and resonant photon energy for the microsphere were only moderate and are summarized in Table 2, along with the other three spheres.

The nanosphere, as shown in Figures 16 and 17, demonstrates the present state of the art in nanotechnology assembly. The spectral energy density of Equation (16) for the nanosphere increases by a billion times over the microsphere even though the sphere size is reduced by a billion times. Upon integrating over a decade of frequencies with Equation (21) (see Table 2), a thousand times increase in ZPE density is calculated with each successively smaller sphere. However, no significant vacuum polarization is available for nano-sized particles, which is an energetic, physical manifestation of ZPE.

The picosphere is interesting in that Meads beat frequency concept can theoretically be realized with pairs of atoms very close in atomic weight such as platinum and gold or hydrogen and deuterium. However, the engineering challenges of such an assembly would be prohibitive, even if one could foresee a significant overunity energy production per pair. Furthermore, even with the advanced techniques such as Fords focusing of vacuum fluctuations, textbook upscattering or resonant fluorescence, etc., paired atoms of choice still have a technological barrier, lacking compatibility with any existing amplification or conversion transducer, such as those seen in Figure 21.

With the femtosphere, QED principles inherent to ZPE, start to emerge. In one sense, the femtosphere has become almost too small to manage individual particles, if they are in contact. In another sense, the size presents other opportunities such as with the ion trap, where electron femtospheres can be collected, for example. However, even as the advantage of working with electrons as ZPE receivers becomes more apparent, it is obvious much more research is needed.

Overall, the Mead patented method for utilizing ZPE collectors and resonators certainly presents a design or a collector that amplifies scattering, though Mead only analyzes a single sphere. The ZPF energy density of Equation (21) is the most relevant, showing the quartic increase of energy with frequency even though there is a cubic decrease of volume with each successive sphere. In the final assessment, given the extent of the experimentation that is required for success with this concept for extraction of useful energy, all four spheres of interest still do not receive a feasibility rating of overall confidence that would qualify it for endorsement from scientists, engineers, or investors. Using the realistic power production level or anticipated work output as a measure of energy quality, this invention receives a poor energy quality rating.

Table 2 presents more energy data for each sphere, with photon energy for the corresponding wavelength and the Einstein energy content added for comparison. The main observation with this tabular summary is that the ZPE energy for a given spherical volume finally equals (same order of magnitude) the Einstein energy content of matter as well as the photon energy of the corresponding wavelength. While the scattering cross section may seem to offer some advantages at larger sphere diameters, the equal weighting of light, matter, and vacuum for a femtosphere has to be extraordinarily inviting for the vacuum engineer and an area worthy of further research.

Table 2 - Energy and Cross Section of the Spheres

Mechanical Casimir Force Conversion

The Casimir force presents a fascinating exhibition of the power of the ZPF offering about one atmosphere of pressure when plates are less than one micron apart. As is the case with magnetism today, it has not been immediately obvious, until recently, how a directed Casimir force might be cyclically controlled to do work. The optically-controlled vacuum energy transducer however, proposed by Pinto, presents a powerful theoretical case for rapidly changing the Casimir force by a quantum surface effect, excited by photons, to complete an engine cycle and transfer a few electrons. The exciting part of Pintos invention is the QED rigor that he brings to the analysis, offering a convincing argument for free energy production. The nano-fabrication task that is presented, however, is overwhelming. Besides mounting nanolasers inside the Casimir cavity, the process suggests that a 10 Khz repetition rate is possible with a moving cantilever, without addressing the expected lifespan. The energy production rate is predicted to be robust (0.5 nW per cell or 1 kW/m²), which could motivate a dedicated research and development project in the future. However, the Casimir engine project of Pintos appears to be a long-term, multi-million dollar investment at best.

Utilizing some of the latest cavity QED techniques, such as mirrors, resonant frequencies of the cavity vs. the gas molecules, quantum coherence, vibrating cavity photon emission, rapid change of refractive index, spatial squeezing, cantilever deflection enhancement by stress, and optimized Casimir cavity geometry design, the Pinto invention may be improved substantially. The process of laser irradiation of the cavity for example, needs to be replaced with one of the above-mentioned quantum techniques for achieving the same variable Casimir force effect, with less hardware involved. At the present stage of theoretical development, the Pinto device receives only a moderate rating of feasibility. Its energy quality rating, however, is very high.

Fluid Dynamics

In the fluid dynamics analysis of the vacuum presented by Froning, it was convincingly argued that the permittivity and permeability of the vacuum can be reduced effectively by nonabelian electromagnetic fields, specifically by utilizing alternating current toroids at resonant frequencies.

While this research does not directly produce electricity, the energy extraction indirectly achieved by the use of the Froning prototype is in the form of energy conservation. By reducing the drag and inertia normally experienced by a spaceship in space, it will save a significant amount of energy, which is equivalent to generating it. The referenced information from Rueda and Haisch as well as from Maclay supports the validity of Fronings fluid dynamic approach. At its present stage of development, the feasibility rating is low, with an energy quality rating of high.

Thermodynamic Conversion

The Photo-Carnot engine is an interesting theoretcial device that relies upon quantum coherence to yield a cyclical radiation pressure for the piston-driven engine. The phase induced with the quantum coherence, provides a new control parameter that can be varied to increase the temperature of the radiation field and to extract work from a single heat bath. The claim is made that the second law of thermodynamics is not violated, according to Scully et al., because the quantum Carnot engine takes more energy, with microwave input, to create the quantum coherence than is generated. However, it is possible that as efficiency improvements are made, the output will exceed the input as is the trend with the other thermodynamic engines analyzed in this study. After all, depending on the value of the phase Φ, the efficiency of the quantum Carnot engine can already exceed that of the classical engine even when T_c = T_h. The capability of extracting heat from a single reservoir should be regarded as a requirement for a ZPE thermodynamic transducer. With the added endorsement of Allahverdyan and Nieuwenhuizen, the Photo-Carnot engine is rasied to a perpetuum mobile of the second kind. The number of cycles cannot be arbitrarily large apparently, and the total amount of extractable work is modest. However, the standard requirements for a thermal bath are not fulfilled, according to Allahverdyan and Nieuwenhuizen, so thermodynamics just does not apply. For these reasons, the Photo-Carnot invention has great potential for becoming a ZPE energy producer and receives a high feasibility rating, with moderate energy quality rating.

The Brownian motors proposed by Astumian (Figures 44) utilize Langevins equation, also mentioned in connection with the quantum coherence of Scully et al. Astumian emphasizes the fluctuating energy source (thermal noise) and the dissipation (viscous drag) that is essential to the fluctuation-dissipation theorem, fundamentally important to the ZPF. With Astumians Brownian motors, the oscillating potential also needs an asymmetrical ratchet to ensure one-way transport. The ratchet concept, while very feasible and proven by two experiments, offers only a limited production of current with underunity efficiency. Astumian notes that with the viscous drag of the solvent, all of the energy gained by the ratchet steps is dissipated with an overall thermodynamic efficiency of less than 5%.

The tunneling electron ratchet experiment performed by Linke et al., seen in Figures 45-46, is an encouraging demonstration of the Brownian motor. Linke generates a maximum of 0.2 nA with about a millivolt of source-drain rocking voltage, at the picowatt or picojoule level, which is encouraging. However, with only between 1% and 5% rectification of the total current, the efficiency is also quite as low as Astumian.

With the analysis of the transient fluctuation theorem, it becomes apparent that with microscopic systems, the performance of negative work has a high probability, apparently violating the law of entropy. The Metropolis Monte Carlo simulation of Crooks in Figure 48 is similar to the quantum ratchet concept however, and doesnt offer an advantage over the other techniques.

The Yater method for power conversion of energy fluctuations is in the same category as rectifying thermal noise. While the Yater invention has an impressive assembly of patents and journal articles, the process requires two heat sources separated by a large spread in temperature. This makes the overall analysis of the device difficult to analyze except by conventional means with underunity energy output projections. His claims for a 10 times improvement over heat pumps is intriguing and the detailed plans in his patent encourage further research, with a reasonable feasibility projected and high energy quality.

The work by Ibarra-Bracamontes et al. is another confirmation of rectifying thermal fluctuation noise. It is interesting for theoretical analysis but the type of signals that are possible for external forcing is not made clear. Rectifying random thermal fluctuations with ferrofluids adds a new twist that is unique, especially when rotational energy is not available directly from the ZPF. Engel et al. offer a fascinating experiment for consideration, consistent with the rectification of thermal fluctuations for linear motion. However, the driving potential is a complex oscillating magnetic field with a field intensity as high as the static field which is also required (several kA/m). Both are generated by a commercial electromagnet. The work output that should be calculated in time-averaged torque multiplied by rotational distance will predictably be only a few percent of the input, at best. It is a good demonstration but does not seem to represent a practical concept for motoring or rotational work. Therefore, it receives a high feasibility rating but low energy quality.

The Brown patent rectifying thermal electrical noise with nano-sized metal-metal diodes is probably the most exciting invention analyzed in Chapter 4. Though the inventor does not acknowledge a ZPE contribution to Johnson noise, it is reasonable to project that the Brown diode arrays will rectify nonthermal fluctuations as well as thermal noise. With no external input needed for conduction, nor a minimum voltage to overcome the usual diode barrier, the potential for free energy production seems quite high. The attractiveness and projected consumer interest for such a solid state, zero-maintenance device is also very high. Not only is the fabrication understandable and straightforward but the description of a cooling effect (negative kinetic energy) from the conversion of thermal noise (positive kinetic energy) is also scientifically and thermodynamically acceptable. The energy density of several watts per square meter is reasonable and robust. However, this quantity should be calculated in watts per cubic meter, since the filled Millipore sheets can easily be stacked vertically as well. Another important calculated parameter for space power is the amount of watts per kilogram, which is probably moderate to high in this case. It is possible with modern nanotechnology that this invention could compete with the battery market. Not only is the feasibility given the highest rating for this invention, but the energy quality rating is also given the highest rating as well.

Stochastic resonance is an emerging energy field that now is being used to substitute for ratcheting in the Brownian rectifiers. The work by Goychuk demonstrates the aperiodic quantum stochastic resonance (AQSR) that is essential for these rectifiers to work in a solid state environment of a tight-binding crystal lattice. With near zero DC bias, the invention is very attractive for many reasons. The ratchetlike mechanism of Goychuk is a valuable substitute for the Astumian style of Brownian motor requiring physically fabricated ratchets. It does not require a static bias, which is a distinct improvement over previously analyzed Brownian motor requirements. The stationary current is also found to be nonzero for unbiased noise, demonstrating a DC rectification, as long as there is some degree of asymmetry in the noise. As a result, the invention combines the noise and the asymmetric driving force into one signal, which is also an advantage over lesser ZPE models. The AQSR design has the ability to rectify asymmetric, unbiased, nonthermal noise, including quantum fluctuations as well, producing a measurable electrical current in a solid state crystal lattice. The only remaining variables are the amount of quantum dissipation required for the effect, the optimum operating temperature, the anticipated energy efficiency and the projected difficulty inherent in creating asymmetry with nonthermal noise that naturally tends to be symmetric in time and space. These variables may be of a sufficiently minor concern for the Goychuk invention to actually offer a gateway to the future of ZPE electricity generation. Many parts of the invention fit the ideal impedance matching of energy source behavior with energy transducer behavior. For example, quantum fluctuations are shown by Goychuk to simply require quantum dissipation and a slight asymmetry, which is less energy intensive overall than creating a quantum coherence. This invention is given the highest rating for feasibility and the highest rating for energy quality.

Conclusions

The risk analysis that is often integrated into a feasibility study that is dedicated to a single development plan is really a process to assign a degree of likelihood to stages of a project. The feasibility rating standard adopted in Chapter 5 is equivalent to such an analysis.

The results of this study finds varying feasibility ratings and energy quality ratings for the four modes of energy conversion from the ZPF. For the Electromagnetic modality, in the present stage of development, the overall method is rated unfeasible with poor energy quality, given the limitations of todays technology capabilities. The Mechanical modality fares better with a moderate feasibility rating, at the overall present stage of development, with a very high energy quality rating. The Fluid Dynamic modality drops back with a low feasibility rating but high energy quality rating. The Thermodynamic modality shines with the highest feasibility rating and the highest energy quality rating.

The overall conclusions drawn from this study support the introductory physical description of the quantum vacuum. Furthermore, the hypothesis of a ZPF vibrating with measurable mechanical pressure, electromagnetic activity, and nonthermal energy is also supported by the scientific evidence uncovered by this study. The fluid dynamics information about ZPE was a reassuring fulfillment of the fourfold modality expectation. There is also a consistency with previous research going back to the early days of QED, which adds a reliability and confidence level to the normally unsettling nature of ZPE. Further research is needed however, as outlined briefly in the next section, to fully exploit the discoveries of energy extraction from the quantum vacuum.

The implications of this study to the emerging field of discipline called vacuum engineering are enormous and far-reaching: An up-to-date assessment of the state of the art has been accomplished by this comprehensive study. Based on this engineering physics achievement, with the feasibility and energy quality ratings therein, it can be reasonably expected that at least one business plan will be generated for a ZPE invention, perhaps for the first time in history. Such a development offers the business world an opportunity to benefit from the most plentiful energy source that also now has been found to have a certain level of practicality and moderate risk assessment, compatible with competing enterprises. As a result of this study, an opportunity has emerged for the public to benefit from some of the ZPE unusual ubiquitous qualities, such as making many completed ZPE transducers completely portable and possibly installing lifetime ZPE transducers in every appliance. The implications of this study are that future generations may finally relinquish fossil fuels in favor of ZPE.

Recommendations

Based on the quality of research uncovered and the level of agreement between theory and experiment demonstrated, specifically by the thermodynamic mode of ZPE conversion, it is recommended that further attention and funding be primarily dedicated to the exploitation of zero-point energy extraction, beginning with the microscopic realm. While the other three modalities offer interesting and promising developments, the feasibility rating and energy quality rating is the highest with the thermodynamic mode. In particular, it is recommended that 1) metal-metal nanodiodes should be researched, with attention to the Johnson noise voltage and purported lack of diode barrier, along with the possible mass production of high density substrates; 2) more ratchet and ratchetlike asymmetries should be researched, by government, industry and academia, so that a TB lattice or diode assembly may one day offer a truly solid state transducer for ZPE; 3) research should continue into quantum coherence, refractive index change, and stochastic resonance with a goal of reducing the present relatively large energy investment, so that more robust avenues of product development in ZPE thermodynamics may be achieved. Brownian motors, thermal fluctuation rectifiers, and quantum Brownian nonthermal rectifiers utilizing AQSR have already achieved a level of theoretical and experimental confidence where further physics research and engineering studies can offer fruitful rewards in the production of rectified DC electricity. This mode of ZPE conversion research and development needs to be continued with earnest in order to expand mankinds woefully limited portfolio of energy choices.

A broad outline of how to undertake the recommended development work would include specific tasks and milestones associated with a) the confirmation of ZPE quantum effects described in this study on a larger scale; b) replication of results but also optimization of results; and c) engineering tasks of conductor and semiconductor design, nanowires and ohmic contacts. All of these, along with other tasks not mentioned, need to be included. The project would also include estimates of output current and energy production with any given geometry. Parallel development paths in research and development will always accelerate the completion of the optimum design. A market study should also accompany the work, so a clear focus on the existing niche to be filled is maintained. A national or international project proposal that estimates the required project scope, resources, break-even point and identifies major milestones, has to be formulated, if major progress in ZPE usage is to be achieved. Simply commissioning another study to follow up this study will lead only to institutionalizing the effort without accomplishment of set goals.

This feasibility study of ZPE extraction for useful work has presented a balanced and detailed assessment with scientific integrity, engineering utility and the likelihood of success for further development. It can be concluded that zero-point energy is deserving of more attention by engineers and entrepreneurs as a serious and practical energy source for the near future. The proposed project plan for ZPE development, yet to be written, has been reduced to a business endeavor and an exercise in return on investment.

FIGURE CREDITS

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Ibid., p. 740

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Ibid., p. 14746

Gourley, P. Nanolasers Scientific American, March, 1998, p. 57

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Maclay, J. Unusual properties of conductive rectangular cavities in the zero point electromagnetic field: resolving Forwards Casimir energy extraction cycle paradox Proceedings of Space Technology and Applications International Forum (STAIF), Albuquerque, NM, January, 1999, Figure 1, p. 3

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Sagan, Carl. Cosmos, Random House, New York, 1980, p. 37

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Goychuk, I. et al. Nonadiabatic quantum Brownian rectifiers Physical Review Letters, Vol. 81, No. 3, 1998, Figure 1, p. 651

Ibid., Figure 2, p. 651

REFERENCES

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