Apparatus Comprising of Propulsion system...UFO Part 1of 2

#Qanon Why is deep state Censoring/Deleting Mathematics from History?
See p.1 pg. 1,3,4...etc. Don't worry, though we have original documents with uncensored mathematics; but for some reason Google is labeling this TABULAR OR GRAPHIC MATERIAL SET FORTH AT THIS POINT IS NOT DISPLAYABLE. I wonder if this is reasonably related to the recent Youtube and Media Censorships and FAKE NEWS?
FOR EDUCATIONAL USE ONLY

US PAT APP 20060027709
United States Patent Application
Title: APPARATUS COMPRISING OF PROPULSION SYSTEM
Document Type: Published Application
Published Application Number: US 20060027709 A1
Published Application Date: 2006-02-09
Application Number: 11/197787
Application Date: 2005-08-04
Inventor(s):
Pinto, Fabrizio, Monrovia, CA
Assignee(s):
InterStellar Technologies Corporation, Monrovia, CA
Attorney(s) or Agent(s):
DEMONT & BREYER, LLC, SUITE 250, 100 COMMONS WAY, HOLMDEL, NJ 07733
Priority Information
Related Information: Provisional application No. 60/598,658, filed on 2004/08/04.
Classification Information
International Classes (IPC 8): B64G-1/40
U.S. Classes (Original): 244/171.100
Drawing Pages: 7
Language: English
Abstract

A propulsion system that does not consume fuel. The system operates to modify the dispersion force (i.e., van der Waals) that arises between particles, such as neutral atoms. A lifting force is generated as a result of this modification of the dispersion force. In the illustrative embodiment, the propulsion system includes particles, a particle trap, a source of electromagnetic energy, and a piston.
Claims

Number of Claims: 20
I claim:

1. An apparatus comprising: at least one trap for confining particles; a device for delivering electromagnetic radiation to the confined particles, wherein said device delivers an amount of electromagnetic radiation that is sufficient to: (i) induce long-range interactions between said particles; and (ii) cause said particles to either accelerate or hover.
2. The apparatus of claim 1 wherein said trap is an atomic trap.
3. The apparatus of claim 1 wherein said device is a laser.
4. The apparatus of claim 1 wherein said particles are characterized by a polarizability, wherein said particles have a polarizability that is greater than a static polarizability.
5. The apparatus of claim 1 wherein said electromagnetic radiation has a wavelength, and wherein said wavelength is a near-resonance wavelength.
6. The apparatus of claim 1 wherein said particles are Rydberg atoms.
7. The apparatus of claim 1 wherein said particles are neutral atoms.
8. The apparatus of claim 1 wherein said apparatus includes at least 1x106 traps.
9. The apparatus of claim 1 further comprising a current controller, wherein said current controller causes said device for delivering electromagnetic radiation to deliver, at a minimum, an amount, W, of electromagnetic radiation, given by the expression:TABULAR OR GRAPHIC MATERIAL SET FORTH AT THIS POINT IS NOT DISPLAYABLEwherein: W: is power, in Megawatts; {overscore (R)}/a0: is the average interatomic distance, in Bohr radii; λL: is the wavelength of the electromagnetic radiation, in micrometers; αnr: is a factor (dimensionless) by which the static polarizability of a particle is increased at near resonance; and kL: is the laser light wave number (dimensionless).
10. The apparatus of claim 1 wherein said apparatus comprises a propulsion system, and wherein said propulsion system is coupled to a vehicle.
11. The apparatus of claim 10 further comprising an arrangement whereby said vehicle receives momentum from said particles.
12. The apparatus of claim 10 further comprising a surface upon which said particles are impacted, wherein said surface is coupled to said vehicle.
13. The apparatus of claim 12 wherein said surface is a part of a piston.
14. The apparatus of claim 10 wherein said vehicle is selected from the group consisting of helicopter, prop-driven aircraft, jet-aircraft, and space vehicle.
15. The apparatus of claim 10 further comprising a conventional propulsion system, wherein said conventional propulsion system is selected from the group consisting of a turboprop engine, a turbojet engine, a turbo-fan engine, ramjet engine, and chemical (rocket) engine.
16. The apparatus of claim 1 wherein said apparatus is coupled to freight.
17. An apparatus comprising: a hull; a first propulsion system, wherein said first propulsion system is disposed within said hull and comprises: (a) at least one trap for confining particles; and (b) a device for delivering electromagnetic radiation to the confined particles, wherein said device delivers an amount of electromagnetic radiation that is sufficient to: (i) induce long-range interactions between said particles; and (ii) cause said particles to accelerate; (c) a surface against which said particles are impacted, wherein said surface couples to said hull; and a source of energy for powering said device.
18. The apparatus of claim 17 comprising a second propulsion system, wherein said second propulsion system is selected from the group consisting of a turboprop engine, a turbojet engine, a turbo-fan engine, a ramjet engine, and chemical (rocket) engine.
19. An apparatus comprising: at least one atomic trap for confining particles; at least one laser for delivering electromagnetic radiation to the confined particles; a power source for powering said laser; and a controller for controlling said laser, wherein said controller, in conjunction with said power source, provide an amount of current to said laser that is sufficient to cause said laser to deliver an amount of electromagnetic radiation that is sufficient to: (i) induce long-range interactions between said particles; and (ii) cause said particles to accelerate or hover.
20. The apparatus of claim 19 further comprising a conventional propulsion system.
    Specification
    STATEMENT OF RELATED APPLICATION
    [0001] This application claims priority of U.S. Provisional Patent Application 60/598,658, which was filed on Aug. 4, 2004 and is incorporated by reference herein.
    FIELD OF THE INVENTION
    [0002] The present invention relates generally to propulsion systems.
    BACKGROUND OF THE INVENTION
    [0003] Sixty-six years after the Wright brothers made their first, sustained powered flight, Neil Armstrong walked on the Moon. Incredible progress to be sure, but can this pace of innovation be sustained? Will we soon visit neighboring planets or the nearest stars? Can we reach these destinations with the technology that got us to the Moon? If we can't, what propulsion technologies might be able to take us to these unthinkably remote places?
    [0004] Current propulsion technology is based on an action-reaction principle, whereby a gas is expelled at high-speed to propel a payload in the opposite direction. This technology is typically embodied as a chemical rocket engine. While a payload can be rapidly accelerated using a chemical rocket, fuel is quickly consumed to develop the required thrust. To illustrate the problem, consider that if a spacecraft could be powered to achieve a constant acceleration of only 1 g, the trip from Earth to Mars would require about 2-4 days. In fact, modern chemical rocket engines can achieve accelerations much greater than 1 g. But even at 1 g, the fuel would be exhausted within minutes. As a consequence, the trip to Mars from Earth via chemical rocket takes about six months.
    [0005] With current chemical-rocket technology, most of the weight at launch is fuel. For example, a typical choice for a mission to Mars would involve the Boeing Delta II 7925 or 7925H rocket stages. In its common configuration, the RS-27A engine of the Delta II first stage, along with an additional nine strap-on sold rocket motors, will have a mass of about 285,000 kilograms at launch. But of this mass, only slightly more than 1000 kilograms will reach Mars.
    [0006] As noted above, the delivery of several tons of a payload to Mars via chemical rockets is contemplated to take about six months, with total mission duration of about two to three years. For the majority of transit time, astronauts will be weightless, which is known to adversely affect the human body. Furthermore, the astronauts will be subject to exposure from harmful radiation. Additionally, the prospects of mounting a rescue or recovering from a serious malfunction are slim due to the transit times involved.
    [0007] And the distances involved in interstellar travel are so large that with this technology, a trip to even the nearest star systems would take hundreds to thousands of years.
    [0008] It seems clear that current technology does not provide a means to manned exploration of the Solar System or beyond. That being the case, can technological approaches be conceived that will send spacecraft from the Earth to destinations within the Solar System in a matter of days or weeks, as opposed to years or decades? Any such approach will face a daunting technological requirement. Namely, in order to drastically reduce travel time to “neighboring” planets and “nearby” stars, exceedingly large velocities must be achieved—velocities that are on the order of a significant fraction of the speed of light.
    [0009] Proposals that meet this mission time requirement will, therefore, typically require what can only be described as “fantastic” technologies. From a feasibility perspective, perhaps the most “promising” of those technologies that have been proposed is the matter-antimatter drive. When combined, matter and antimatter will completely annihilate, releasing unfathomable quantities of energy. But even if we were able to develop a matter-antimatter drive, its use should be proscribed. The reason is that if antimatter were to leak from its containment chamber while in the vicinity of Earth, there is a distinct possibility that the resulting energy release would destroy Earth or at least cause the extinction of all life thereon.
    [0010] Another exotic propulsion technology is the “ solar sail.” Although solar sails can produce momentum by reflecting a portion of the light that they receive from the sun, this approach, on its own, does not offer a solution to the problem of achieving interstellar or even interplanetary travel. More specifically, in order to deliver a space probe to a nearby star in less than a century, the sail must be driven by laser light aimed at it throughout the trip. The power requirement for the laser, which would be located on Earth, is on the order of hundreds of thousands of terawatts. For the sake of comparison, the current planetwide consumption of electricity is on the order of about 1 terawatt. And this approach has a further complication. Namely, the craft must be slowed from a non-trivial fraction of the speed of light to orbital velocity at its final destination using light that is coming from earth. This would require the coordination of very complex maneuvers that, if not carried out correctly, might result in the destruction of the ecosystem of the destination planet.
    [0011] In the late 1990s, NASA established and funded a program, now defunct, called the “Breakthrough Propulsion Program.” The program's charter was to evaluate entirely new propulsive principles that would enable interstellar or at least interplanetary travel. Technologies under consideration included the Schlicher thruster, Deep Dirac Energy, Podkletnov gravity shielding, Podkletnov force-beam, transient inertia, coupling between electromagnetism and spacetime, gravity modification schemes, anomalous heat effect, Biefeld-Brown effect, warp drives, wormholes, high-frequency gravitational waves, superluminal tunneling, the Slepian Drive, and the quantum vacuum (e.g., dynamical Casimir effect, etc.).
    [0012] Unfortunately, none of these approaches were deemed to be promising. For example, one study pertaining to the quantum vacuum concluded that the acceleration of a spacecraft propelled by the dynamical Casimir effect would, after ten years under acceleration, be traveling at 0.1 meters per second!
    [0013] In light of the foregoing, it seems likely that an as yet unidentified propulsion technology will be required to make routine, manned interplanetary and interstellar travel a reality.
    SUMMARY
    [0014] The illustrative embodiment of the present invention is a system and method for propulsion that avoids some of the drawbacks of the prior art. Unlike conventional propulsion technology, the propulsion system described in this specification does not consume fuel (although there is an energy requirement). In fact, the system does not even use fuel, as the term is commonly used.
    [0015] The illustrative embodiment is grounded in accepted physics principles, albeit leading-edge theoretical, experimental and applied physics. The propulsion method does not violate basic physical “laws,” such as the conversation of momentum. The equations on which the propulsion system is based are clearly established in the art, although extended to a domain of applicability and mode of use that has not been previously contemplated.
    [0016] The propulsion system operates by modifying the dispersion force (i.e., van der Waals) that arises between particles, such as neutral atoms. The following two discoveries by the inventor enable the propulsion system:
    (1) The dispersion force interaction between any two neutral atoms is affected by the presence of an external gravitational field in a way that results in a repulsive force upon the atomic pair.
    (2) The distortion of the dispersion interaction described above, which is usually quite small and is proportional to the number of atoms present, can be magnified by many orders of magnitude. This requires that (a) the atom-atom interaction to be transformed from a relatively shorter-range interaction to a relatively longer-range interaction; and (b) a very large number of atoms are present for mutual interaction.
    [0019] A method in accordance with the illustrative embodiment of the present invention comprises:
    generating a lifting force by subjecting a plurality of confined particles to a trigger acceleration; and
    exposing the particles to an amount of electromagnetic radiation that is sufficient to induce the lifting force to: (i) exhibit relatively long-range interactions; and (ii) increase the momentum of the particles; and
    transferring at least a portion of the increase in momentum to a vehicle.
    [0025] To begin the propulsion cycle, the particles must be subjected to acceleration, that is, the “trigger” acceleration. This can be accomplished, for example, by supporting a craft that contains the propulsion system in a gravitational field (i.e., the craft cannot be in free fall). Or, acceleration can be kinematic, such as by rotating the craft, or using a conventional propulsion system to accelerate the craft. The force is referred to as a “lifting” force because it's direction is opposite to the weight of the particles.
    [0026] In the illustrative embodiment, the particles are neutral atoms, the atoms are confined in an atomic trap, and a laser provides the electromagnetic energy that is required to transform the atom-atom interaction from a short-range to a long-range interaction and to increase the momentum of the atoms.
    [0027] Some of the increase in momentum of the atoms is transferred to a vehicle, such as a spacecraft. This transfer of momentum propels the vehicle. In the illustrative embodiment, this is accomplished by letting the atoms work against a piston, which in turn impacts against a part of the vehicle. Atoms that hit the piston are recycled to the atomic trap for the next propulsion cycle.
    [0028] Several points of explanation or definition will be useful in understanding the illustrative embodiment of the present invention and its underlying principles.
    [0029] The term “long range” interaction or force usually describes a force that decays with distance as 1/R n, where n is a positive number. The term “short range” interaction or force usually describes a force that decays with distance as exp[−R]. Those conventions are not followed in this disclosure. Rather, for the purpose of this disclosure and the appended claims, terms that “short range” and “long range” are comparative or relative terms. For example, the language “inducing the lifting force to exhibit relatively long range interactions” means that the lifting force is induced to exhibit a relatively longer-range interaction than is normally the case.
    [0030] Since the momentum that is donated by the particles propels the vehicle, there might be a tendency to characterize the particles as “fuel.” But the particles are not “ fuel” in any conventional sense of that term. They simply serve as a “momentum exchanging element.”
    [0031] It is important to recognize that the particles are not accelerated by the wall of the spacecraft but, rather, by their mutual dipole fields (van der Waals force) as distorted by the craft's acceleration (gravitational or otherwise). In other words, this method does not violate the action-reaction law, since the motion of the particles is not due to action of vehicle upon them.
    [0032] An example of “action of the vehicle” on the particles is if the atoms were accelerated due to an explosion in the trap. In that case, the net of all internal forces on the system would be zero and, at the end of the process, the craft would not gain any momentum. In accordance with the illustrative method, however, the atoms accelerate towards the piston, etc., independently of the vehicle and do transfer a net amount of momentum to it during impact.
    [0033] The lifting force that is “generated” by the method is not a new force. Rather, it is simply the vertical component of a known intermolecular force; in particular, the van der Waals force. This vertical component arises from an asymmetry in the van der Waals force that results from the introduction of a gravitational field (or acceleration). There would be no such asymmetry, nor vertical component of force, in flat space-time.
    [0034] This result—that the interaction potential between two neutral atoms in their ground state depends upon the position of the atoms in a gravitational field—is new. Previous studies pertained to the distortion of the field of two point charges, not two dipoles (atoms). While this “new” force is measurable with presently existing technology, harnessing it, such as to lift an object, is not feasible, since this force amounts to an exceedingly small correction to the total weight of each atom.
    [0035] It is useful to note that using many such atom pairs does not improve this situation, because that does not result in a larger force per atom. The reason for this is that the inter-atomic energy is a function, to a large power, of the reciprocal of distance. In other words, the atom-atom dispersion-force interaction is a realtively short-range force. As a consequence, the total “lifting” force on a large number of atoms is increased only minutely with respect to the lifting force acting on just one pair of atoms. That is, if the number of atoms is N, there will be about N pairs (for N>>1) to consider, but the mass of the system also goes up as N. So, there is no gain realized by adding atoms.
    [0036] Critical to the present invention is the inventor's recognition that if this newly discovered “ effective force” could be transformed from a relatively short-range interaction into a relatively long-range interaction, the lifting force that is available would be greatly increased. In particular, if particles could be made to interact over the long range, then the total energy of the system results from the interaction of every particle with all other particles present. For large groups of particles (N>>1), the interaction grows as N2, while the total mass of the system is only growing proportionally to N. As a consequence, a large gain in energy can be realized by using large groups of particles.
    [0037] A mechanism for transforming relatively short-range interactions into relatively long-range interactions was theoretically discovered several years ago and has been re-evaluated more recently as a way to introduce unusual behaviors in a cloud of trapped atoms. See, Kurizki et al., “New Regimes in Cold Gases Via Laser-Induced Long-Range Interactions,” . . . . The method involves isotropic illumination of atoms by lasers. That technique, with several modifications, is utilized in conjunction with the illustrative embodiment.
    [0038] As previously noted, the present propulsion system and method overcomes a key drawback of chemical engines; namely, the fact that at some point, the fuel is expended. In accordance with the illustrative embodiment of the present invention, it is possible to maintain acceleration without expelling high-speed gases. In other words, the propulsion system does not require fuel. Alternatively, if the “particles” are considered to be “fuel,” then there is no consumption of fuel due to the process.
    [0039] As previously mentioned, the illustrative propulsion systems and methods described herein are not energy free. In particular, to achieve the required transformations, an intense radiation field, such as can be generated by powerful lasers, must be developed throughout the region in which the particles are trapped. In the case of a craft destined for extremely long interplanetary or interstellar flights, the energy required to power the lasers is obtained, for example, from an on-board nuclear reactor, akin to the reactors powering some submarines.
    [0040] The propulsion system described herein has many applications. In particular, in addition to its use as a propulsion system for spacecraft, it can be used to deliver a payload into low earth orbit without requiring orbital speeds. Furthermore, a small version of the propulsion system could be attached to literally any item (e.g., a pallet of goods, a railroad car, etc.) so that the item could be readily moved (e.g., in a warehouse, loaded onto a cargo ship, etc.) as needed. The propulsion system can, of course, also be used in conventional aircraft.
    [0041] Additionally, the present propulsion system can be used to supplement a main, conventional propulsion system. In fact, this would facilitate phase-in to replace conventional technologies. For example, a propulsion system in accordance with the illustrative embodiment that is not sufficiently powered to bring a craft to a hover could be used to effectively reduce the mass of the craft, thereby improving the fuel consumption of the main propulsion system. Alternatively, it could be used as a supplemental system for emergencies.
    BRIEF DESCRIPTION OF THE DRAWINGS
    [0042] FIG. 1 depicts distortion in the spherical symmetry of the field of a simple charge, as caused by the presence of a gravitational field.
    [0043] FIG. 2 depicts distortion in the cylindrical symmetry of the field of a classical dipole, as caused by the presence of a gravitational field.
    [0044] FIG. 3 depicts a propulsion system in accordance with the illustrative embodiment of the present invention.
    [0045] FIG. 4 depicts the propulsion system of FIG. 3, wherein lasers are illuminating confined particles.
    [0046] FIG. 5 depicts the propulsion system of FIG. 3, wherein particles impact against an elastically-bound piston.
    [0047] FIG. 6 depicts the propulsion system of FIG. 3, wherein particles are pumped back to a reservoir for use in a subsequent propulsion cycle.
    [0048] FIG. 7 depicts a method in accordance for propulsion in accordance with the illustrative embodiment of the present invention.
    [0049] FIG. 8 depicts a schematic of a vehicle that incorporates the propulsion system of FIG. 3.
    [0050] FIG. 9 depicts a schematic of the nuclear power subsystem of the vehicle of FIG. 8.
    DETAILED DESCRIPTION
    [0051] This Detailed Description proceeds with Section 1.1, which provides a description of propulsion system 100 and a method for propulsion in accordance with the illustrative embodiment of the present invention. Section 1.2 discloses a vehicle that incorporates propulsion system 100. The remaining sections, which include Sections 2.1-2.4 and 4 provide a theoretical development for propulsion system 100 and performance estimates. 1.1 Propulsion System 100
    [0052] FIG. 3 depicts propulsion system 100 in accordance with the illustrative embodiment of the invention. Propulsion system 100 includes particles 102, chamber 104, piston 108, source(s) of electromagnetic radiation 110, return line(s) 112, and pump(s) 114, interrelated as shown.
    [0053] FIG. 7 depicts method 700 for propulsion, which can be used in conjunction with propulsion system 100. In accordance with operation 702 of method 700, a plurality of particles are confined.
    [0054] FIG. 3 depicts propulsion system 100 at the beginning phase of the propulsion cycle. Particles 102, which in some embodiments are ground-state atoms, are confined in particle trap 106 of chamber 104 in known fashion and in accordance with operation 702.
    [0055] In operation 704, particles 102 are subjected to a trigger acceleration. This can be accomplished, for example, by supporting propulsion system 100 in a gravitational field. Assuming propulsion system 100 is in a vehicle, such support is provided, for example, if the vehicle is at rest on the surface of the Earth or in flight, as long the vehicle is not in free fall. In some alternative embodiments, the trigger acceleration can be kinematic, such as by rotating the craft, or by using a conventional propulsion system to accelerate the craft.
    [0056] At operation 706, the particles are exposed to an amount of electromagnetic radiation that is sufficient to:
21. induce an effective inter-particle force that arises between said particles to exhibit long-range interactions; and
22. increase the momentum of the particles. This operation is depicted in FIG. 4, wherein sources of electromagnetic radiation 110 (“EM sources 110 ”) are activated and directed toward particles 102. In some embodiments, the EM sources are high-power lasers. In the FIG. 4, two EM sources 110 are shown. Depending upon the power required for a given embodiment, far more EM sources might be required. Power requirements for driving the propulsion system are described in
    [0059] The EM radiation causes an upward acceleration of particles 102 with respect to a vehicle, etc., that houses propulsion system 100. In the case of an ideal propulsion system, particles 102 remain trapped in place (in particle trap 106) as they mutually interact and the craft is accelerated upward by the reaction of the atoms themselves against whatever forces are used to keep them in trap 106.
    [0060] It is possible, if not likely, that once particles 102 are accelerated by conducting operations 704 and 706, they will escape from particle trap 106. This is a non-ideal situation, which yields less than the ideal momentum. But, if particles are allowed to escape, this relaxes the constraints on particle traps 106. That is, suitable traps can be readily constructed with existing technologies. See, e.g., H. J. Metcalf and P. van der Straten, Laser Cooling and Trapping (Springer, N.Y., 1999); http://ww.rle.mit.edu/cua/research/project 02/project 02.vandp.htm. These references describe techniques for trapping atoms at extremely low temperatures.
    [0061] Although it is necessary for the generation of the lifting force itself, it is desirable to trap particles at very low temperatures because the thermal speed of, for example, atoms, even at room temperature, is comparable to the maximum speeds that can be obtained by this method. It is, therefore, “easier” to illuminate the atoms for an appropriate length of time if they are not moving at very high thermal speeds.
    [0062] Continuing with the description of method 700, operation 708 recites impacting the accelerated particles against a surface, thereby transferring some of the momentum of the particles to the surface. This operation is depicted in FIG. 5, wherein particles 102 impact piston 108 at high speed.
    [0063] Piston 108 functions as a shock absorber. That is, the piston provides an area against which particles 102 can impact and which can transfer momentum to the vehicle non-destructively at every forward stroke. In propulsion system 300, non-destructive momentum transfer is indicated by spring 116, which elastically couples piston 108 to a vehicle. Shock absorber technology for aerospace applications is well developed within the context of pyrotechnic release technology. See, e.g., N. Butterfield, Pyrotechnic Release Devices, in Space Vehicle Mechanisms, P. Conley, Ed. (Wiley, N.Y., 1998).
    [0064] As illustrated in FIG. 5, in propulsion system 100, as piston 108 moves upward, return lines 112 are accessed. The return lines provide a route back to particle trap 106. As particles move away from piston 108, the piston drops back to a seated position against chamber 104.
    [0065] FIG. 6 depicts particles 102 in return lines 112, being pumped via pumps 114 toward a gas reservoir (not depicted) for reuse in a subsequent propulsion cycle. The propulsion cycles occur at a rapid and substantially continuous pace.
    [0066] It is very important to recognize that in propulsion system 100, and in accordance with method 700, particles 102 are NOT being accelerated by the walls of the vehicle or by the chamber in which they reside. Rather, they are accelerated by their mutual dipole fields, as distorted by vehicle acceleration (i.e., gravitational or otherwise). There is, therefore, no concern that this scheme violates the action-reaction law, since the motion of particles 102 is NOT due to an action upon them by the vehicle.
    [0067] An example of a situation in which the walls of the vehicle are acting on particles 102 is if the particles were accelerated by an explosion in chamber 104. In such a case, the net sum of all internal forces on the system would be zero and, at the end of the process, the vehicle would not gain any net momentum. But using the methods and apparatus described herein, particles 102 are accelerated toward piston 108 INDEPENDENTLY of the vehicle and, on impact, transfer a net amount of momentum to it. 1.2 Vehicle Incorporating Propulsion System 100
    [0068] FIG. 8 depicts vehicle 800, which incorporates propulsion system 100 in accordance with the illustrative embodiment of the present invention. As depicted in FIG. 8, vehicle 800 includes propulsion subsystem 100, nuclear power subsystem 810, crew quarters 830, and shielding 840, arranged as shown.
    [0069] The presence of nuclear power subsystem 810 requires the use of shielding 840 to protect crew quarters 830 and propulsion subsystem 100. Those skilled in the art will be capable of designing and building shielding suitable for this purpose.
    [0070] The purpose for nuclear power subsystem 810 is to generate electricity to power EM source(s) 110. Nuclear power is used as a power source due to the ability of a nuclear reactor to provide continuous power for extended periods of time (e.g., several years, etc.). Operation of nuclear power subsystem 810 is described in more detail below in conjunction with FIG. 9.
    [0071] Propulsion subsystem 100 couples to shielding 840, which receives momentum transferred from piston 108 (see FIGS. 3-7 and the accompanying description). This substantially continuous transfer of momentum from particles 102 to piston 108 to vehicle 800 (e.g., shielding 840) drives the vehicle.
    [0072] In the embodiment that is depicted in FIG. 8, propulsion subsystem 100 is disposed an end of vehicle 800. This location draws maximum advantage from a rotational trigger acceleration while providing the crew, in crew quarters 830, with appropriate gravity-like conditions.
    [0073] FIG. 9 depicts an embodiment of nuclear subsystem 810 suitable for use to provide electricity to drive EM sources 110 (e.g., lasers, etc.) in propulsion subsystem 100. The embodiment that is depicted in FIG. 9 is a direct Rankine cycle continuous power system. (See, e.g., M. W. Edenburn, “Models for Multimegawatt Space Power Systems,” Sandia Report SAND86-2742 (June 1990). This type of system is suitable for use with vehicle 800 due to its ability to provide continuous power for several years.
    [0074] Nuclear subsystem 810 includes nuclear reactor 812, separator 814, turbine 816, radiator 818, pump 820, generator 822, and power conditioning unit 824.
    [0075] Reactor 812, which is liquid metal cooled, boils potassium and sends the saturate vapor to turbine 816 for power generation. Since the fluid leaving the “hot” end of reactor 812 is unlikely to be pure vapor, separator 814 is used to separate the saturated vapor from its accompanying liquid. The liquid is recirculated to the “cold” end of reactor 812.
    [0076] Waste heat is rejected by space radiator 818 . Since the system rejects heat from a condensing working fluid, the radiator operates nearly isothermally and radiates a relatively large amount of heat per unit area. Condensed liquid is returned to the “cold” end of reactor 812 via pump 820.
    [0077] Electricity that is produced by generator 822 is appropriately conditioned in power conditioning unit 822 to provide EM sources 110 with a suitable supply of electrical power. 2.1 Distorted Dipole-Dipole Potential
    [0078] It is already a well-known fact that a gravitational field can introduce novel forces acting on a single charge or on a dipole. An example is the self-interaction of a point charge in a Schwarzschild geometry [8], ultimately due to the term Linet [9] discovered has to be added to the Copson potential [10] in order to satisfy the appropriate asymptotic boundary conditions for this problem. Commenting about the very extreme conditions nearby a miniblack-hole, Smith and Will wrote that “[I]t is amusing to note that . . . the test particle's electrostatic self-force would suffice to support it against the hole's gravity, without the help of any external force.”
    [0079] Unfortunately, however, such fascinating conclusion is undermined by the fact that, as pointed out by these authors, “it is meaningless to talk of an electron being held fixed at, say, 10−13 cm from a miniblack hole, when the Compton wavelength of an electron is two orders of magnitude larger than this.” Following this approach, the self-interaction of a static dipole has also been calculated [11], but Parker has shown that this force has no effect on the Hamiltonian of a neutral atom in free-fall [12].
    [0080] Another example is the “electrostatic levitation of a dipole,” predicted on the basis of the distortion caused by a uniform gravitational field [13]. This author found that “one is unlikely to witness such levitation,” which could only be observed in a fixed classical dipole whose electron charge separation is 1.4x10−15 m. The outlook for detection of these field distortion phenomena was effectively summarized by Boyer, who stated that “Clearly our example may be instructive from a theoretical point of view, but it does not lend itself to easy experimental measurement.” [14]
    [0081] In this section, we consider the effect of a weak gravitational field upon intermolecular forces. In particular, the effect of gravitation on the van der Waals hydrogenic interatomic potential in the unretarded regime is discussed within non-relativistic first-order perturbation theory. The quantitative conclusion of these computations is that the system proposed herein shows extreme promise for direct experimental verification although the effect is certainly too small to be of any practical engineering use in the field of propulsion.
    [0082] The first step to obtain the distorted dipole-dipole potential is the calculation of the electrostatic potential, and thus of the electric field, of a point dipole in the presence of gravitation. For this purpose, let us start by considering the potential field of a single point charge q located at a position r0=x0i in the quasi-homogeneous gravitational field caused by a relatively distant spherically symmetrical mass distribution M located at a radial distance R from the dipole. This has been the subject of several investigations, starting with the pioneering work of Whittaker [15].
    [0083] Since we are considering a charge in a gravitational field g antiparallel to the z-axis and located at a position other than the origin, we transform the unprimed Rindler coordinates defined by the usual metricTABULAR OR GRAPHIC MATERIAL SET FORTH AT THIS POINT IS NOT DISPLAYABLEby introducing new primed coordinates given by t=(1+ gz0/c2)−1t′, xi =x0i+x′i. By substituting these definitions into Eq. (1), it is simple to show that the metric in the new coordinates is:TABULAR OR GRAPHIC MATERIAL SET FORTH AT THIS POINT IS NOT DISPLAYABLEwhere GR≡g/(1+gz0/c2). With this result, we can transform Whittaker's expression for the electrostatic potential in Kottler-Whittaker coordinates into our transformed Rindler frame (for simplicity of notation, we neglect to write the primes in what follows):TABULAR OR GRAPHIC MATERIAL SET FORTH AT THIS POINT IS NOT DISPLAYABLEwhere |r−r0|2=(x− x0)2+(y−y0)2+ (z−z0)2, |ρ−ρ0 |2=(x−x0)2+ (y−y0)2. This expression of course approaches the Coulomb potential as gxi/c2, gx0 i/c2→0. In this paper, we shall neglect the Linet term [9] responsible for the self-interaction discussed above since this contribution will be shown to be negligible with respect to the effects treated herein.
    [0084] In order to write down the electrostatic dipole field in the presence of a gravitational field, we use the formulation of Léauté and Linet, originally designed to calculate the self-interaction of an electric dipole [11]. For a point dipole A of moment dA=dA i, with k=1 . . . 3, located at rA, the result found by these authors, neglecting the self-interaction term [ 12], can be written as:TABULAR OR GRAPHIC MATERIAL SET FORTH AT THIS POINT IS NOT DISPLAYABLEwhere VC is the Copson potential, which, in the quasi-homogeneous field limit, coincides with our solution above.
    [0085] Since, to the best of this author's knowledge, this potential has never been graphically represented, it is shown, along with the corresponding electric field lines, in FIG. 3 for a point dipole dAi=dAk.
    [0086] By computing the electric field as usual, after some very lengthy algebra [13] one obtains the general expression for the interaction potential energy Wdd(r, r 0; dAi, dBi) of two point dipoles of moments dAi and dBi, placed at positions r0 and r, respectively. In order to illustrate the physical meaning of this result, let us write it to second order in gR/c2 for the case of dipole A placed at the origin and dipole B placed at r=(R, 0, 0):TABULAR OR GRAPHIC MATERIAL SET FORTH AT THIS POINT IS NOT DISPLAYABLEwhich again yields the usual Minkowski space result [17] if the gravitational field is absent. Interestingly, the asymmetry due to the presence of the gravitational field causes a net vertical force upon each dipole in this geometry where there would of course be none in flat space-time. For instance, in the case of two antiparallel dipoles dAi=−dB i=dAk, we find that this force Fdd =−∂Wdd/∂z is, again to second order:TABULAR OR GRAPHIC MATERIAL SET FORTH AT THIS POINT IS NOT DISPLAYABLEwhich is the dipole-dipole analogy of the levitation force upon a single dipole mentioned at the beginning of this section [ 15].
    [0087] The problem of a single hydrogenic atom either held fixed or in free-fall within various assigned metrics has been discussed extensively by starting from generally covariant expressions of the Dirac equation in curved space-time [18] . The present author has discussed realistic astrophysical settings for the observation of the perturbative effects of gravitational fields on freely-falling atoms both in the static case and within the framework of possible remote gravitational wave detection [ 19].
    [0088] Tourrenc and Grossiord, in their treatment of a hydrogen atom held fixed in a Schwarzschild geometry, have shown that the most significant contribution to the perturbative Hamiltonian by far derives from what can be interpreted as the classical weight of the electron in the gravitational field. However, since the corresponding energy shift for a ground-state atom is found to be ΔE˜2(GMme/R2)a0≈6x 10−21 eV, we shall neglect it here and use the unperturbed hydrogenic wavefunctions to evaluate the interatomic gravitational self-force.
    [0089] As in the undistorted van der Waals case, the general expression for the potential Wdd(r, r0; dAi, dBi) contains only bilinear forms of the type xAixBj and thus yields no first-order contribution in the case of symmetrical hydrogenic nS. The second order correction on the other hand is [20]:TABULAR OR GRAPHIC MATERIAL SET FORTH AT THIS POINT IS NOT DISPLAYABLEwhere |φn,l,mA> are the unperturbed eigenfunctions of energy En, the total energy of the unperturbed atomic pair is −2EI, and the primed summation indicates that the |φ1,0,0A; φ1,0,0B> term is excluded.
    [0090] By inspecting Eq. (5), it is evident that, in the gravitational case, the second order intermolecular potential, to second order in gxi/c2, takes on the general form:TABULAR OR GRAPHIC MATERIAL SET FORTH AT THIS POINT IS NOT DISPLAYABLEwhere C0, C1, and C2 are appropriate dimensional constants, which in principle depend on the variables r, r0, dAi, and dBi. For instance, again in the geometry used in the examples above, one can quickly recover the well-known result that C0≈− 6e2a05 as well as show that C1 ≈−2e2a05, andTABULAR OR GRAPHIC MATERIAL SET FORTH AT THIS POINT IS NOT DISPLAYABLEwhere a0 is the Bohr radius.
    [0091] In our case, however, we are not here interested in the modification of the intermolecular forces due to the presence of gravitation, but rather we want to pursue isolating the vertical component of the gravitational self-force due to the dipole-dipole interaction of two hydrogenic atoms in the |1,0,0> eigenstate located at r0=0 and r=(R, 0, 0), respectively:TABULAR OR GRAPHIC MATERIAL SET FORTH AT THIS POINT IS NOT DISPLAYABLEExplicit evaluation [13] yields the following result to first order in gxi/c2:TABULAR OR GRAPHIC MATERIAL SET FORTH AT THIS POINT IS NOT DISPLAYABLEBy again making use of the fact that the cross terms vanish in the 1s state and that <φ1,0,0A|xA 2|φ1,0,0A>=< φ1,0,0A|1/3rA2 |φ1,0,0A> and equally for all the other squared terms, we finally find:TABULAR OR GRAPHIC MATERIAL SET FORTH AT THIS POINT IS NOT DISPLAYABLEwhere the last equality was obtained by writing the ionization energy and the polarizability as EI=e2/2a0 and α0=(2a0)3, respectively. Notice that this result does not coincide with what one might expect from näively calculating one half of the negative contribution to the gravitational mass of the London binding energy of the pair, since in that case the coefficient would be ⅜ and not 1/16.
    [0092] An explicit estimate of this result in the case of two hydrogen atoms in their ground state at, for instance, 20 a 0, yields a relative acceleration alift,H˜ 4x10−13 cm/s2. The situation improves dramatically if one considers two positronium (Ps) atoms, in which case we find alift,Ps˜8x10−10 cm/s2, which is in principle detectable via atomic interferometry, at the price of dealing with the added complications of interferometry of atoms with a finite lifetime. 2.2 Relevance of the Above Results
    [0093] The above results show that the distortion of the classical dipole-dipole field caused by gravity results in a net force upon each dipole, which is anti-parallel to the weight of each atom. This phenomenon has been known for some time, although its implications at the quantum level have not been fully explored and engineering implications do not appear to have attracted their due attention. Interestingly, the warping of the Coulomb field due to gravitation renders their mutual interaction non-central, which in turn implies that Newton's action-reaction law will not be satisfied by this system. Although there is no study of this exotic problem in the literature, it is clear that, if the problem were to be considered from the standpoint of full quantum electrodynamics (QED), it would result that the virtual photon field responsible for the charge-charge interaction is perturbed by the gravitational acceleration so as to carry a net momentum flux, part of which is simply transferred to the dipole, or dipoles, thus resulting in their upward motion.
    [0094] Importantly, this phenomenon makes the interaction energy between two hydrogen atoms in their ground state dependent upon their position within the gravitational field, which results in a force upon the pair even in the quantum case. The existence of this additional force acting on an atomic pair is by itself a new result, since previous studies had concentrated on the distortion of the field of two point charges, and not of two dipoles. Despite the fact that this new and additional force is measurable with presently existing technology, its use in an actual lifting device is unlikely, since it amounts to a very small correction of the total weight of each atom.
    [0095] The important consideration, critical to the present invention, is that adding many atomic pairs does not result in a larger force per atom. The reason for this is that the interatomic energy is a function of a large power of the reciprocal of the distance—which results in the atom-atom dispersion interaction being a short-range force. Therefore, the total lifting force on a large number of atoms is increased only very slightly with respect to that acting on just one pair. Therefore, if the number of atoms is N, there will be ˜N pairs to consider but the mass of the of the system also went up as N. Thus no gain is made.
    [0096] The critical, and non-obvious, point of the present invention is to transform the new force outlined above from a short-range force into a long-range force, as we consider in the next section. In principle, if point-like particles interact via a long-range force, the total energy of the system results form the interaction of every atom with all the others, something which is not possible in the short-range. For instance, if a system is made up of three particles, the total energy will result from the interaction of particle 1 with 2, particle 1 with 3, and particle 2 with 3. However, if the number of particle is now ten, we will have to consider the interaction of particle 1 with 2, 3 . . . , 10, and so fort, which results, in the case of a large number of particles. For large N (N>>1), in this case the number of interaction grows as N 2, while the total mass of the system is still only growing proportionally to N.
    [0097] This point would only be of philosophical importance if we did not have available a mechanism to indeed transform atom-atom interactions from short-range into long-range. Such mechanism was discovered theoretically several years ago and has been reproposed recently as a way to introduce unusual behaviors in a cloud of trapped atoms. Atomic traps have become one of the hottest subjects of scientific research in recent years. The method proposed by the researchers to cause the atom-atom interaction to become a long-range force can of course also be used in our case to leverage the presence of a large number of atoms so as to make the lifting force due to the gravitational distortion we have seen above much larger by many orders of magnitude. The technological price to pay is that, in order for this transformation to occur, powerful lasers must be pumping an intense radiation field throughout the region where the atoms are trapped. 2.3 Trapped Gases in Curved Space-Time: the Effect of Radiation Fields
    [0098] It is well-known that an intense directional radiation field, such as that produced by a laser, alters the nature of intermolecular forces [21]. For instance, in the near zone region, where kLR <<1 and kL is the laser light wavenumber, the power dependence of the force becomes ≡−1/R3. Importantly, if a molecular pair is allowed to “tumble” with equal probability in all directions with respect to the radiation field, the unretarded force averages out and the only term left is that due to the retarded part of the Hamiltonian. The resulting contribution, which can be produced by appropriate laser beams [22] and is still attractive, is [21]:TABULAR OR GRAPHIC MATERIAL SET FORTH AT THIS POINT IS NOT DISPLAYABLEwhere a αA, B(kL) are the dynamic polarizabilities of the two atoms, I is the intensity of the beams, and the domain of validity of this result is everywhere in space except where the atom-atom exchange interactions become important. It is simple to see numerically that this energy is much smaller than the usual van der Waals force at near range but, as originally pointed out by Thirunamachandran [21], the long-range nature of the force offers the potential to actually achieve remarkable effects.
    [0099] Let us now consider the distortion of Thirunamachandran's long-range interaction due to an external gravitational field, such as that present in a ground-based laboratory. One is fully justified by both our results in the unretarded case and by dimensional considerations to assume that a molecular pair interacting through the gravity-like attractive long-range force at Eq. (12) will also undergo a lifting force of the type:TABULAR OR GRAPHIC MATERIAL SET FORTH AT THIS POINT IS NOT DISPLAYABLEFor the purpose of order of magnitude estimation, let us write the total potential energy of N atoms contained in a spherical volume of diameter D interacting through a mean field determined by Eq. (12) simply as Ugas˜−N2(IkL 2/c)αA(kL)αB(kL)/D. Therefore, the total lifting force acting upon the center-of-mass of the trapped gas is Flift,zgas˜N2 Flift,zAB, assuming all atoms to be of species A, and the corresponding acceleration is:TABULAR OR GRAPHIC MATERIAL SET FORTH AT THIS POINT IS NOT DISPLAYABLEThe relevant figures of merit to judge the feasibility to bring such an atomic cluster to a hover are the number of atoms in the trap and its size, the intensity and wavelength of the laser light, and the average intermolecular distance. Consider N=102 atoms in a trap with D≈2x10−7 cm, which yields an intermolecular distance {overscore (R)}˜D/N1/3≈ 10a0. Now let us have 18 (six triads [22]) high-power lasers each outputting 2.5 kW into a 0.5 cm diameter beam at a wavelength λL≈1000 Å. Appropriately focused onto the trap size, this would yield an intensity I≈1.3x 1017 W/cm2. By moderate off-resonance detuning, it is possible to obtain dramatic increases in atomic polarizability over its static value [23] (another approach may consist of using a cold gas of highly excited Rydberg atoms, since, in this case, the polarizability is proportional to n7, where n is the principal quantum number). By adopting αA(k L)≈3x105α0, and by substituting the above numerical values into Eq. (14) in c.g.s. units, we find alift,zgas/g≈1.5, that is, the system will hover unsupported in the gravitational field of the earth or accelerate upward. 2.4 Relevance of the Above Results
    [0100] The mechanism outlined above represents the first novel and realistic proposal to achieve lift in the history of flight since the great inventions of the airplane and of the rocket in the 20th century. As we shall see in the detailed numerical estimates below, more atoms than just 102 must be present in the trap for this mechanism to be technologically convenient, although it is possible to trade off a higher laser power for a lower number of atoms in the trap. What is important to stress at this time is that this mechanism represents a non-obvious use of well understood quantum laws in the presence of gravitation for the purpose of creating fuel-free propulsion. Of course fuel-free propulsion does not imply energy-free propulsion. In other words, a source of energy is still needed to achieve the needed thrust. In the case of extremely long interplanetary and interstellar flights it is expected that on-board nuclear energy production will continue to grow in engineering importance, given the fact that no other source has been able to achieve similarly convenient power outputs.
    [0101] The above invention, however, removes the greatest problem in the way of achieving fuel-free propulsion, that is, the fact that, independently of the energy source used, at some point the fuel available on board is exhausted. For instance, if one could power a spacecraft so as to achieve a constant acceleration of 1 g, the trip from Earth to Mars would require approximately 2-4 days. Although the acceleration of 1 g is smaller than the larger accelerations achievable by present-day rocket technologies, it is absolutely impossible to maintain those accelerations for times longer than minutes at the most, simply because of quick fuel exhaustion. With the scheme outlined in this invention, on the other hand, it is possible to maintain the needed acceleration without any need to expel high speed gases, provided that the required laser illumination is constantly at work transforming the atom-atom interactions from short-range into long-range ones.
    [0102] It is important to stress another characteristic of the propulsive method of this invention. All the calculations carried out so far imply that the atoms are “at rest,” that is, not in “free-fall.” This is extremely important, since it is only if the atoms are somehow supported by an external force against the gravitational field that a relative acceleration due to that field can affect their mutual interactions. Such is not the case if the atoms are freely falling. In that case, in fact, the acceleration felt by a freely falling object is rigorously zero, because of the Principle of Equivalence at the foundation of the General Relativity theory. To use a well-known popular example, if two atoms are at rest with respect to the walls of a freely falling elevator, they will not feel the presence of any gravitational field—the acceleration of the elevator exactly cancels exactly the gravitational acceleration. In other words, locally, there is no distortion of the dipole-dipole field and thus no change to the van der Waals force (see below for further subtle clarifications on this point).
    [0103] The Principle of Equivalence can be stated by saying that, locally, there exists no experiment that can indicate the difference between the acceleration due to the presence of a gravitational field and that due to the kinematics of the system. For instance, once could simulate the presence of a gravitational field in the same elevator travelling through outer space by simply accelerating it “upwards” at the same rate as the free-fall acceleration it would have in the gravitational field to simulate. In fact, our Eq. (1) above was obtained exactly by making this assumption. Therefore, the atoms will undergo the lifting force at the basis of this invention whether or not there is a gravitational field against which to lift—their behavior is due to their being within an accelerated reference frame no matter what the reason for the existence of such frame.
    [0104] In principle, this represents an operational limitation of the present invention, in the sense that, if a spacecraft were to be left to freely-fall in the gravitational field of a massive body, the lifting mechanism would not be operating. For this purpose, the vehicle must be provided with an initial acceleration through other means in order for the thrust cycle described below to commence. For instance, this happens if the craft is at ground level initially, or somehow hovering under the action of an external force. Once the cycle starts, it is only necessary to coordinate the laser illumination of the (n+1)-th cycle to occur during the transfer of momentum due to the atoms that were accelerated during the n-th cycle. The dipole-dipole field during that time will behave as though under the effect of a gravitational field of that acceleration, because of the Principle of Equivalence.
    [0105] From the practical standpoint, it is appropriate to stress that every propulsive or lifting system has an appropriate envelope of performance which, is exceeded or not met, will result in insufficient or abnormal behavior. For instance, the lifting force due to the wings of an airplane will cease to be effective if the airflow detaches from the wings because of a stall condition. Therefore, pilots are trained to operate so as to remain well clear of the conditions that might lead to a stall of the airfoils, such as, for instance, excessively low speed for a fixed wing aircraft. Similarly, it is expected that the propulsive system of this invention, in its simplest embodiment, must be operated under appropriate conditions of initial acceleration, in both magnitude and direction. This is much less complicated than it may appear. For instance, in outer space, an initial acceleration can be imparted by causing the entire spacecraft to rotate around an axis by means of a reaction wheel. Once the entire vehicle is rotating like a rigid disk, from the standpoint of the atoms in the propulsive subsystem the rotational acceleration will be indistinguishable from that caused by gravity. Once the thrust cycle is successfully started, the spacecraft can then be despun while the engine provides its own acceleration. In a fixed wing application, it might be possible to make use of the lift provided by the wings themselves in order to start the process.
    [0106] Finally, it is also important to stress that the Principle of Equivalence only rigorously applies to a volume of space that is infinitely small. Therefore, there exists some distortion of the dipole-dipole field even in the case of freely-falling atoms, although this distortion is far smaller than that of supported atoms, in the sense described above. The effects of this distortion were studied by this author in several papers (see for instance [ 19]) and it is therefore possible that lift might be obtained in some embodiments even if the craft is initially in free-fall. 4. Performance
    [0107] Estimating the performance of an aerospace propulsion system based upon entirely novel physical principles naturally presents some difficulties. For instance, the typical concept of specific impulse [29] is undefined in the case in which thrust is obtained without the ejection of high speed gases. At the same time, since the approach calls for the use of high power lasers to engineer the atom-atom interactions into a long-range force, it is of interest to determine whether the thrust thus obtained is in fact larger than that which would be obtained if the laser power utilized were, for instance, projected from the spacecraft into a particular direction in space or whether a laser beam of the same power were to be aimed at a hypothetical laser sail on the spacecraft [3]. In the following subsections we obtain some important order of magnitude estimates both in equation and in graphic form of a few important quantities in order to gain a more realistic understanding of the potential capabilities of a vehicle propelled by means of the physical principle of the present invention.
    [0108] The conclusions below will clearly establish that the propulsion concept of this invention offers great potential from the standpoint of realistic engineering applications, although such parameters as the exact laser wavelength and power, trap size, atomic mass, and number of atoms of course will have to be optimized according to both accurate theoretical modeling and prototype testing.
    [0109] In what follows, in order to make a firm connection between the theoretical treatment, which was developed here in the c.g.s. system (centimeter-gram-second), and the more typical engineering M.K.S. units (meter-kilogram-second), the cgs or MKS subscripts will be appended as appropriate. If no subscript is used, the quantity should be assumed as expressed in the cgs system. No use is made if English units throughout (such as, for instance, lbf for thrust). Also, for improved legibility, all order of magnitude signs will be replaced by equal signs. 4.1 Fundamental Equations 4.1.1 Atomic Physics of Trapped Atoms in the Accelerated Propulsive System
    [0110] Let us consider a gas of NA identical atoms of mass mA, polarizability αA2 (kL), confined within an appropriate trap of such dimensions as to correspond to an average interatomic distance {overscore (R)}. In what follows, we shall assume that the number of atoms, N A, the size of the trap, D, and the average interatomic distance, {overscore (R)}, are related simply as D˜{overscore (R)}NA1/3. In addition, Thirunamachandran's theory of dispersion forces under the effect of illumination also requires the constraint that λL>>{overscore (R)} [21].
    [0111] The polarizability αA2(k L) can be made several orders of magnitude larger than its static value, α0=(2a0)3, where Bohr's radius is a0=ħ2/μ ee2 and μe is the reduced electron mass (μ≈me), by choosing an appropriate near-resonance wavelength. Without getting into the details of atomic physics calculations, in this section we shall rely upon the well-established theoretical and experimental fact that such near-resonance condition can be satisfied, that is, the polarizability can be made larger than the static value by a factor, αnr, which can be as large as αnr˜105 [23].
    [0112] Another strategy to produce values of the polarizability that are vastly larger than the static value is to use Rydberg atoms. A qualitative argument in favor of this choice is that, as we have seen above,