* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download The Use and Abuse of “photon” in Nanomechanics – pdf
Quantum computing wikipedia , lookup
Density matrix wikipedia , lookup
Particle in a box wikipedia , lookup
Bell's theorem wikipedia , lookup
Copenhagen interpretation wikipedia , lookup
Path integral formulation wikipedia , lookup
Wheeler's delayed choice experiment wikipedia , lookup
Many-worlds interpretation wikipedia , lookup
Quantum group wikipedia , lookup
Quantum machine learning wikipedia , lookup
Topological quantum field theory wikipedia , lookup
Orchestrated objective reduction wikipedia , lookup
Quantum teleportation wikipedia , lookup
Relativistic quantum mechanics wikipedia , lookup
Atomic theory wikipedia , lookup
Matter wave wikipedia , lookup
Symmetry in quantum mechanics wikipedia , lookup
Hydrogen atom wikipedia , lookup
Renormalization group wikipedia , lookup
Double-slit experiment wikipedia , lookup
Coherent states wikipedia , lookup
Quantum state wikipedia , lookup
Casimir effect wikipedia , lookup
Bohr–Einstein debates wikipedia , lookup
Quantum field theory wikipedia , lookup
EPR paradox wikipedia , lookup
X-ray fluorescence wikipedia , lookup
Interpretations of quantum mechanics wikipedia , lookup
Delayed choice quantum eraser wikipedia , lookup
Aharonov–Bohm effect wikipedia , lookup
Quantum key distribution wikipedia , lookup
Scalar field theory wikipedia , lookup
Renormalization wikipedia , lookup
Hidden variable theory wikipedia , lookup
Wave–particle duality wikipedia , lookup
Quantum electrodynamics wikipedia , lookup
History of quantum field theory wikipedia , lookup
Canonical quantization wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
The Use and Abuse of the “photon” in Nanomechanics David P. Norwood Southeastern Louisiana University, Hammond LA 70402, USA [email protected] Abstract In this paper, I decry misuse of the word “photon”, primarily because of the misleading notions it creates in students and physicists alike. Like Dr. Lamb [Lamb, 1995] I blame the “particle-like” nature of the word “photon” for planting in the minds of impressionable students a host of “particle-like” ideas about EM radiation: ideas that are misleading or wrong, and persist into the young physicist’s old age. I begin by reviewing examples of particularly egregious misuse of the photon. I review the “semi-classical” and “neo-classical” approximations, and the conventional quantum theory of radiation, upon which the word “photon” is based. I discuss applications of classical and quantum theories of radiation to experiments involving the interaction of light with matter. I call attention to cases where commonly held opinions regarding the QTR are misleading or wrong. I finally discuss the relevance of this to the field of nanomechanics. Keywords: QED, semiclassical, radiation-reaction, 1. Introduction – What’s wrong with ‘photon’? Misuse of word photon falls into two classes: pretentious and wrong. An egregious example of the former would be the phrase “stream of visible photons”• instead of “light”. This is not factually wrong, but it is unfortunate that physics, a field that seeks to describe the world in the simplest terms, should produce such a tortured reference. Such examples abound, and no more need be said, except perhaps to paraphrase Albert Einstein and say that the language used to describe a physical effect should be as simple as possible, but no simpler. I move on to discuss various statements about photons that the author feels are simply wrong (or at best misleading). 1.1 Photons are small and localized This statement can be traced at least as far back as Einstein’s discussion of the production and transformation of light [Einstein, 1905], in which he describes light as consisting of “independently moving points” and “energy quanta localized at points of space”. More recent authors have gone further and characterized photons as “point particles”. In fact, neither the classical or quantum theories of radiation provide a means to even discuss the size or “position” or an electromagnetic wave. It is almost unfortunate that the symbol typically used as an index or label for the electromagnetic field, the symbol r, is the same as that used to refer to the position of an actual material particle. Unfortunate since they play such completely different roles in reference to the two different entities. But in fact, this notion (of small corpuscles of light) might not have survived had the Nobel committee not chosen to mention particularly Einstein’s description of the photoelectric effect when awarding him the Nobel Prize in 1921. (Consider that his description of the Stokes shift, mentioned immediately before the photoelectric effect in the same paper, and explained on the same basis, is seldom mentioned). 1.2 Photons are required to understand the photoelectric effect In this introduction, I include quotes without references in order to protect the guilty. This is simply one example of a great many physical effects that are taken as proof that the electromagnetic field must be quantized. To be sure, there are physical effects that must be described by a quantum theory of radiation. But as is discussed in the section on applications, the photoelectric effect and many others can be described at least qualitatively, and in many cases, quantitatively by a classical theory of electromagnetism coupled with a quantum theory of matter. This is the so-called semiclassical approximation and is described later. Particularly entertaining is a statement in the introduction of a quantum mechanics textbook that “the frequency dependence of the [photoelectric] effect is not comprehensible within that framework [i.e., in a classical theory of electromagnetic waves]”, a textbook that goes on to present a full description of the photoelectric effect using a classical theory of electromagnetism. 1.3 There is no delay in photoelectron emission with decrease in intensity of radiation This statement is simply untrue. While for large intensities, the onset of photoelectron flux begins immediately and the flux scales with the intensity, for sufficiently low intensities, the time between the emission of photoelectrons becomes longer and longer. These are the “photon-counting” experiments. And this shows that there IS a delay in order for an electron to accumulate enough energy to be liberated from the bound state. 1.4 A photon wave function collapses to transfer energy This point is sometimes raised when discussing the absorption of electromagnetic energy by an electron. As the electron is so much smaller than the “photon” (its size taken to be it’s wavelength), there must be a “collapse” in order for the energy to be transferred. Again, this simply fails to understand classical electrodynamics. When subjected to an incident EM wave, an electron will generate a dipole field that, when added to the incident field will “channel” the electromagnetic energy to the dipole. An electron absorbing energy from visible light is no more mysterious than a miniature radio with a 10 cm antenna absorbing energy from a radio wave with a 1000 m wavelength. 1.5 Photons are required if light is to carry and transfer momentum This very common assertion simply reflects a poor understanding of classical electrodynamics. It is well understood that a classical electromagnetic wave can carry and transfer energy, momentum and angular momentum. And the energy and momentum of classical fields (in a unit volume) even satisfies the relation often touted as quantum mechanical: E = pc. 1.6 Photons are required to explain light scattering anisotropically This claim is typically mentioned in the context of the Compton effect, in which light incident upon a solid or molecule is scattered in a well-defined direction, rather than isotropically. The claim is made that a classical theory cannot explain the Compton effect because a classical theory predicts that light will be scattered isotropically. This again reflects a poor understanding of classical electrodynamics. While it is true that in the limit (v/c)~0, classical electrodynamics predicts isotropic scattering, for a relativistic electron (which condition obtains in the Compton effect), radiation is highly directional. These examples are typical of the sorts of claims found in introductory textbooks in quantum mechanics and modern physics. And they likely reflect less upon the writer’s expertise as a physicist than upon the “cookie cutter” approach to textbook writing. If fact, the very best textbooks begin by simply talking about how to do QM, not what it means. (Landau and Lifshitz 1958, Griffiths, 1994) In fact, the physical effects that prove that the electromagnetic field is quantized are subtle and generally hard to find (as opposed to being forced upon us in the way that quantum effects in atoms and molecules are). This is particularly relevant to general problems in the interaction of light with matter, including in the field of nanomechanics. Here, the primary interest lies NOT in proving that the electromagnetic field is quantized but rather in solving problems in the interaction of electromagnetic fields with matter (including some cases in which the field is not deliberately applied – e.g., the Casimir effect, to be discussed below). 2. Modeling the Interaction of Light with Matter 2.1 Semi-classical approximation While it has different meanings in different contexts, here semiclassical approximation refers to a treatment of the interaction of light and matter in which the material particles are described quantum mechanically and the electromagnetic fields are described classically, using Maxwell’s equations. The first mention the author has found of this sort of calculation is by Erwin Schroedinger, presented in four lectures given in March of 1921 [Schroedinger, 1928]. In Shroedinger’s presentation, he uses this semiclassical approximation to show how electromagnetic radiation induces transitions between electronic states of energy Ek and El, so long as the frequency, f, of the field (i.e., the perturbation) satisfies f = ∆E/h (termed the “frequency-rule” of Bohr), where ∆E is the energy difference between the two electronic states and h is Planck’s constant. Here we see energy being transferred in discrete amounts, not because a photon is a “point particle that cannot be divided” but rather because the electron can only accept energy in discrete amounts as it transitions between discrete states. And the presence of Planck’s constant, h, is not due to the inherently quantum properties of the photon, but rather due to the inherently quantum properties of the electron. In these lectures, Schroedinger also described some other aspects of the interaction of light with matter: the absence of certain transitions that might be naively expected (selection rules), dispersion (including anomalous dispersion near a resonance), and the polarization and intensity relations for the Stark and Zeeman effects. In summary, the typical implementation of the semiclassical approximation follows Schroedinger’s approach and treats electromagnetic radiation as a perturbation to a system described by Schroedinger’s equation. Transitions to and/or from a continuum are usually described using Fermi’s Golden Rule. This type of calculation is well described by recent introductory quantum mechanics texts, and will not be described further here [Cohen-Tannoudji 1977, Griffiths 1994, Merzbacher 1998]. In the section on Applications, I describe some successful uses of this approach (as well as some for which it is inadequate). 2.2 Neo-classical approximation While he did not develop it, Schroedinger provides a hint of what is called the “neoclassical” approximation in those same lectures of March, 1921. In it, he interprets the wave function as representing an electron charge density by writing: ρ = -e Ψ∗Ψ (i.e, as the elementary charge times the magnitude squared of the electron wavefunction). But the neoclassical approximation was extensively developed by Jaynes and Cummings and others [see Jaynes and Cummings, 1963 and references therein]. In this approximation, the material system is again described by quantum mechanics and the electromagnetic fields classically (using Maxwell’s equations), the difference being that the matter wavefunctions are used as source terms in the EM field equations. That is, the charge density is taken to be that shown above (by Schroedinger) and the current density by the standard probability flux equation from quantum mechanics (multiplied by the elementary charge): J ( r, t ) = − e[ψ * ∇ ψ − ψ ∇ ψ *] (1) (This expression for the probability flux is found in every introductory quantum mechanics book with which the author is familiar and will not be justified here). This “neoclassical” approximation is often in the form of a self-consistent solution of the quantum mechanical equations describing matter and Maxwell’s equation describing the electromagnetic fields. This method extends the range of applicability beyond the set of problems successfully attacked by the previously described semiclassical approximation, but the important conclusions from above remain – in most cases described by this formalism, the “quantum” effects observed come from the inherent quantum properties of matter rather than those of the electromagnetic field. 2.3 Radiation Reaction This is an effect well known from classical electrodynamics [Jackson, 1975], frequently scorned (by those who are even familiar with it) because of its counterintuitive and “unrealistic” predictions. (One wonders how quantum mechanics in general, and quantum electrodynamics in particular, might have fared had the same sort of skepticism prevailed during the development of those theories). In fact, this appears in both classical and quantum theories of electrodynamics, and plays a significant role in both. To review briefly, this is the effect of the field created by charge upon that very charge. One of two effects that produce comparatively modest discomfort is the modification of the mass of the electron, essentially modifying the “bare” electron mass by adding the Coulomb energy of the electron divided by c2 (and suitably “cut off” to avoid the divergence of the Coulomb energy). The other effect that seems less disturbing to physicists is the damping of the motion of an oscillating charge due to the interaction of the charge with the field it radiates. That the motion of the charged particle must be damped as energy is radiated away seems clear, and so this effect also leads to comparatively little discomfort (and in fact, this is the effect for which radiation reaction is named). On the other hand, two other effects that lead physicists to such scorn for radiation reaction are preacceleration and “runaway” solutions. In the simplest case, at least, one can choose which of these effects obtains. “Runaway” solutions are solutions in which the motion of the charge tends to infinity with time (an obviously unsatisfactory state of affairs). Preacceleration is an acausal effect in which a charge begins to accelerate under an influence that has not yet reached the location of the charge. While the author shares the concern over exponentially divergent speeds and accelerations, the nonlocal nature of preacceleration seems less distasteful, particularly in view of experiments in quantum mechanics for which nonlocal effects remain a conceivable explanation. (In short, while it seems clear that nonlocal effects are forbidden macroscopically by the unequivocal success of relativity – a classical theory – it does NOT seem resolved that such effects are forbidden to quantum objects). In any event, some authors claim to have resolved these questions in both a classical [Valentini, 1988] and quantum mechanical [Moniz, 1977] context, and term preacceleration and “runaway” solutions as artifacts of an improper development. 3. Outline of Quantum Electrodynamics The generally accepted (and thus most commonly used) quantum theory of radiation is based upon the Hamiltonian formulation of electrodynamics. Quantization of the electromagnetic field proceeds much as the quantization of matter. That is, one begins with a Hamiltonian, recasts the dynamical variables from real numbers to operators and imposes upon those operators the uncertainty relation (in the form of a nonzero commutation relation). But as many practicing physicists are not well versed in the Hamiltonian formulation of even classical electrodynamics•, it may be no accident that they are likewise uncertain of the actual predictions of quantum electrodynamics (which then leads them to make unsupportable comments regarding the position and size of a photon). I will briefly and superficially sketch out the In which category, the author, alas, qualified for most of his career. Which perhaps explains his missionary zeal on this topic. typical development of the quantum theory of radiation.[ Cohen-Tannoudji, 1989; Gerry and Knight, 2005] I begin by noting that in the Lagrangian development of electrodynamics (which leads directly to the Hamiltonian formulation), one does not construct a Lagrangian, but rather a Lagrangian density (related to the energy density of the electromagnetic field). One begins with the classical energy density of an electromagnetic field, which can be transformed to be formally the same as a harmonic oscillator: H= ( ) ( 1 1 2 ε 0 E2 + c 2B = pˆ + ω 2 qˆ 2 2 2 ) (2) where the carets denote that the classical fields are now considered operators. (Note that I have included no term which allows the field to interact with matter – while this certainly makes this Hamiltonian useless from a practical point of view, it suffices to make the points necessary here). The solution of this problem is well known and results in a quantized energy in a unit volume of (n+1/2) hf, where n is the number of photons of mode frequency, f, and h is Planck’s constant. Note that even when there are zero photons in this mode, there is a nonzero energy. And while the mean field in such a state is zero (reflecting the fact that the operators for the number of photons and electric field do not commute), the field has a nonzero second moment. These are the so-called vacuum fluctuations, which are invoked as the sole explanation for the effects of spontaneous emission, the Lamb shift and the Casimir effect (this latter effect of particular interest to those in the field of nanomechanics). But what is not often acknowledged are the cosmological effects of this non-zero vacuum energy density. In fact, the energy density is divergent unless some high frequency cutoff is invoked to prevent the divergence. I discuss this in the section on Applications. 4. Applications In this section I discuss a more or less random selection of physical phenomena for which it commonly held that there is no classical or semiclassical explanation. That is, phenomena that are taken to establish the need for a quantum theory of radiation, with particular attention to those taken to establish the reality of the vacuum electric field. It is my desire NOT to deny the need for a quantum theory of radiation (that need I feel is well established) but rather to establish some guidelines as to where and when such a quantum theory is truly necessary and point out the effectiveness of some alternatives. And if, in the process, I can clean up some sloppy and unsupported language in reference to the photon, then my time has not been ill spent. 4.1 Spontaneous emission Frequently taken as definitive proof that the vacuum electric field has observable physical effects, in fact there are other approaches that can predict spontaneous emission. The basic issue is that an electron in an excited, but stationary (i.e., time independent) state should remain there. In order to decay, there must be a perturbation that disturbs the electron in its state, causing a transition to a lower state. The conventional description has it that the omnipresent vacuum electric field provides just this perturbation. But semiclassical descriptions that include radiation reaction [Crisp and Jaynes, 1969; Stroud, Jr. and Jaynes, 1970] produce spontaneous emission with the correct Einstein coefficient, although other authors [Nesbet, 1971] claim that this SC description cannot give the correct thermal (i.e., Planck) distribution of radiation, while other authors compare a QED and SC theory (including radiation reaction) of spontaneous emission from harmonic oscillator states and find no difference between the theories [Doslic and Bosanac, 1995]. More recent papers [Camparo, 2001] compare a SC theory of spontaneous decay of atoms in a microcavity to measurements and find good agreement. Perhaps more significant, it has been shown that radiation reaction (in QED) and vacuum fluctuations are complementary explanations for spontaneous emission [Ackerhalt, et al., 1973, Milonni and Smith, 1975], and in fact, whether spontaneous emission, and the widths and shifts of the lines, is attributable to vacuum fluctuations or to a theory of QED including radiation reaction is shown to depend upon the ordering of commuting atomic and field operators [Milonni, et al., 1973]. 4.2 Lamb shift The Lamb shift has much in common with spontaneous emission. Not in the essential effect (spontaneous emission is a downward electronic transition accompanied by the emission of radiation, while the Lamb shift is a measured difference between two electronic states), but in their physical descriptions. The Lamb shift is another effect taken to prove the existence of the vacuum electric field, and in fact, most of the references above address both spontaneous emission and the Lamb shift and much of that discussion would be replicated here. Additionally, there is a SC description of a “two photon” Lamb dip, which agrees with experiment, but only qualitatively [Shimizu, 1974]. 4.3 “Photon counting” As mentioned in the Introduction, “photon counting” is a phenomenon in which extremely low light intensities are seen to delay the absorption of light by an electron (the lower the light intensity, the longer the delay). Also as mentioned, this effect is taken as evidence of a “photon wave function” collapse (to enable the transfer of the energy from the extended electromagnetic wave to the compact electron), thus demonstrating the quantum nature of electromagnetic radiation. (And perversely, the delay produced as the light intensity decreases is taken as further evidence of a quantum nature, despite the fact that no delay at higher intensities is likewise “proof” of light’s quantum nature). Low light levels (i.e., small number of photons) is generally taken as a line between classical and quantum descriptions (similar to, and perhaps because of, a similar criterion in the quantum theory of matter). In fact, there are SC theories of “photon counting” [Mandel and Wolf, 1966] or other low light level experiments [Ugaz, 1994; Koshino and Ishihara, 2004]. 4.4 Nonlinear optics Under this general heading, I consider the interaction of matter with intense electromagnetic radiation, giving rise to effects such as two- and multi-photon absorption, nonlinear refraction, multiphoton ionization of atoms and intensity dependent index effects. These effects, again, have a long history of SC and QED explanations, again with varying levels of success. It was found early on that the SC and QED calculations of two-photon absorption give the same results [Carasutto, et al., 1967]. Other authors [Chang and Stehle, 1971; Gordov and Tvorgov, 1973] claimed that a SC explanation is inadequate to describe the behavior atoms interacting with high intensity fields (those giving rise to nonlinear effects). Note what is being claimed: that the SC approximation is untenable at high intensities – i.e., in the limit of large numbers of photons. This is refuted by other authors who continue to present SC theories of twophoton [Burrows and Salzman, 1977] and multi-photon [Salzman, 1977] absorption, and multiphoton ionization of atoms in strong fields [Akberg, et al., 1991]. Other nonlinear effects successfully treated by the SC approximation are bleaching by excitons in polymers [Li, et al., 1992] and two photon Doppler free spectroscopy [Cesar and Kleppner, 1999]. Comparisons of QED and SC theories of high order harmonic generation conclude that the two descriptions agree at low harmonic orders but disagree at high orders (and that the QED calculations better describe the experiments) [Gao, et al., 2000]. Relevant to nanomechanics is a SC description of frequency conversion in a microcavity [Martinez, 2005] and of a two-photon nonlinearity of an exciton in a quantum dot [Koshino and Ishihara, 2005]. Also, QED and SC descriptions of nonlinear optics in one-dimensional photonic crystals [Centini, et al., 2005] are found to agree with experiment and one another. 4.5 Compton effect Again, this was described in the introduction, and there exist SC and QED descriptions of the effect. A SC description is said to be equivalent to the QED description (and exactly equivelant if the coherent state is used in the quantum description [Frantz, 1965]). A later paper corrects previous SC and QED descriptions, but comes to the same conclusions [Kormendi and Farkas, 1999]. 4.6 Mixed states and entanglement Under this heading, I consider experiments designed to be sensitive to the entanglement of the wavefunction of two photons that were created in such an entangled state. These experiments generally are inspired by Bell’s consideration [Bell, 1964] of the Einstein-Podolsky-Rosen paradox [Einstein, et al., 1935] or are otherwise related to it. Generally, an inequality is calculated that distinguishes between a SC and QED description, such as that by Franson [Franson, 1991A; Franson, 1991B]. This does in fact, seem to be an intrinsically quantum effect that has no SC description. But it gave rise to the unfortunate statement that these experiments provide "graphic illustration of the lack of objective realism of the electric field". Other scientists who obtained the same experimental results made less dramatic statements [Pryde, et al., 2005]. There are also cavity measurements that seem to require a QED description due to interference effects [Jyotsna and Agarwal, 1994] although other cavity measurements can be described by a SC approximation for “large” fields (number of photons greater than 2) [Thompson, et al., 1998]. A superficially related effect is that of photon “antibunching” in which a photon is “forced” to “decide” whether to (for example) reflect or transit through a beam splitter. Experimental results from these sorts of experiments [Beveratos, et al., 2001] seem to require a quantum theory of radiation. 4.7 Cosmological effects I mention two effects of cosmological significance that do not have a SC explanation, but do have a QED description that incorporates radiation reaction, and thus do not call for vacuum electric fields. These are the well-known Hawking radiation and the related (but less well known) Unruh effect [Barut and Dowling, 1990], in which an accelerated charge experiences a thermal bath of radiation. But by far a more profound cosmological effect is that produced by the immense energy density provided by the vacuum electric field. If the vacuum field is to be strong enough to explain, for example, the observed Lamb shift, then it produces an energy density that will not only perturb the Kepler relation between orbital periods and orbital semimajor axes, but will destabilize the orbits of the planets [Jaynes, 1978, Jaynes, 1990]. There have been attempts to resolve this problem [e.g., Hawton, 1994] but the author remains unsatisfied. 5. Relevance to Nanomechanics The field of nanomechanics is inextricably tied to the interaction of light with matter. This might be by necessity, if the materials or devices in question are intended to generate, detect or guide electromagnetic radiation. Or if the structures are to be constructed using light, as in the technique of photolithography. Many of the papers discussed above have direct relevance to the field of nanomechanics, and an attempt was made to point these out as they arose. In addition, there are specific structures for which SC and QED calculations have been made and compared [Altewischer, et al., 2005] in which the SC approximation is successful. Other cases present only a QED calculation [Savasta, et al., 2002, ; Kiraz, et al., 2004]. But even in those cases in which there is no intention of using or interacting with light, the construction and use of nanostructures will invariably result in the intrusion of effects that are ignorable for a macroscopic structure. An example is the Casimir effect [see, Lamoreaux, 2005, and references therein], a force between two neutral conductors that has its origin in the quantum theory of radiation. The Casimir force scales as d-4, where d is the distance between the conductors and it is clear that for d sufficiently small, the Casimir force can play a significant and even dominating role in the construction, manipulation and operation of nanostructured devices [Klimchitskaya and Mostepanenko, 2006]. Other effects will no doubt come into play as the practice of nanomechanics develops. Conclusion After all that, can the basic question posed be answered: is there a hard and fast rule that can guide the physicist in deciding whether to choose a quantum or classical description of the interaction of light with matter. Several answers are typically posed. There are authors who have attempted to provide such a roadmap, some unreservedly and specifically criticizing the SC approximation [Clauser, 1972; Lee and Stehle, 1976; Raimond, et al., 2001; Braun, et al., 2004; Dente, 1975]. One typical response is that a QED prescription is to be used for small fields and a SC prescription for large (i.e., intense) fields [Oliver, 1977; Lin and Chuu, 2001]. But there exist calculations and experiments (references above) that refute both claims. There are indications (again, references above) that suggest that coherence and wavefunction entanglement require a fully quantum mechanical treatment – both matter and electromagnetic field. But many of the experiments described were painstakingly contrived to produce these entanglement effects. It is not clear that they will arise in a natural way. In fact, the author’s experience in the “original nanotechnology” (i.e., the creation and characterization of polymers) has led him to the opinion that a fully quantum mechanical description of the interaction of light with matter is seldom necessary. It may be that the nanotechnologist must make do with the same guide that every other physicist has – perform the calculation and see how it compares to experiment. References Ackerhalt, Jay R.: Peter L. Knight and Joseph H. Eberly, Phys. Rev. Lett., 30 456-460 (1973) Akberg, T., D.-S. Guo, J. Ruscheinski, and Bernd Crasemann, Phys. Rev. A, 44, 3169-3178 (1991) Altewischer, E.; Y. C. Oei, M. P. van Exter, and J. P. Woerdman, Phys. Rev. A, 72, 013817 (2005) Barut, A. O., and Jonathan P. Dowling, Phys. Rev. A, 41, 2277-2283 (1990) Bell, John S., Physics, 1, 195-200 (1964) Beveratos, Alexios; Rosa Brouri, Thierry Gacoin, Jean-Philippe Poizat, and Philippe Grangier, Phys. Rev. A, 64, 061802 (2001) Braun, Lars; Walter T. Strunz and John S. Briggs, Phys. Rev. A, 70, 033814 (2004) Burrows, M. D., and W. R. Salzman, Phys. Rev. A, 15, 1636-1639 (1977) Camparo, James; Phys. Rev. A, 65, 013815 (2001) Carasutto, S., G. Fornaca and E. Polacco, Phys. Rev., 157, 1207-1213 (1967) Chang, C. S., and P. Stehle, Phys. Rev. A, 4, 641-661 (1971) Centini, M.; J. Peina, Jr., L. Sciscione, C. Sibilia, M. Scalora, M. J. Bloemer, and M. Bertolotti, Phys. Rev. A, 72, 033806 (2005) Cesar, Claudio L. and Daniel Kleppner, Phys. Rev. A, 59, 4564-4570 (1999) Clauser, John F., Phys. Rev. A, 6, 49-54 (1972) Cohen-Tannoudji, Claude, Bernard Diu and Franck Laloe, Quantum Mechanics, Wiley and Sons, NY (1977) Cohen-Tannoudji, Claude, Jacques Dupont-Roc and Gilbert Grynberg, Photons and Atoms: Introduction to Quantum Electrodynamics, Wiley and Sons, NY (1989) Crisp, M. D. and E. T. Jaynes, Phys. Rev., 179, 1253-1261 (1969) Dente, Gregory C., Phys. Rev. D, 12, 1733-1738 (1975) Doslic,N. and S. Danko Bosanac, Phys. Rev. A, 51, 3485-3494 (1995) Einstein, Albert; Ann. Physik 17, 132 (1905) as presented in Einstein’s Miraculous Year, John Stachel, ed., Einstein, A., B. Podolsky and N. Rosen, Phys. Rev., 47, 777-780 (1935) Princeton University Press, New Jersey (1998) Franson, J. D., Phys. Rev. Lett., 67, 290-293 (1991) Franson, J. D., Phys. Rev. A, 44, 4552-4555 (1991) Frantz, Lee; Phys. Rev. 139, B1326-B1336 (1965) Gao, J., F. Shen and J. G. Eden, Phys. Rev. A, 61, 043812 (2000) Gerry, Christopher C. and Peter L. Knight, Introductory Quantum Optics, Cambridge University Press, Cambridge, 2005 Gordov, E. P., and S. D. Tvorgov, Phys. Rev. D, 8, (1973) Griffiths, David J.; Introduction to Quantum Mechanics, Prentice-Hall, New Jersey (1994) Hawton, Margaret; Phys. Rev. A, 50, 1057-1061 (1994) Jackson, J. D.; Classical Electrodynamics: Second Edition, Wiley and Sons, NY (1975) Jaynes, E. T., in Coherence and Quantum Optics, ed. Mandel and Wolf, Plenum, NY. (1978) Jaynes, E.T., Probability in Quantum Theory, Workshop on Complexity, Entropy and the Physics of Information, Sante Fe, NM, May 29 - June 2, 1989; Conference Proceedings, W.H. Zurek, editor, Addison-Wesley, (1990) Jyotsna, I. V., and G. S. Agarwal, Phys. Rev. A, 50, 1770-1776 (1994) Kiraz, A.; M. Atatüre and A. Imamoglu, Phys. Rev. A, 69, 032305 (2004) G. L. Klimchitskaya and V. M. Mostepanenko, Cont. Phys., 47, 131-144 (2006) Kormendi, F. F., and Gy. Farkas, Phys. Rev. A, 59, 4172-4177 (1999) Koshino, Kazuki and Hajime Ishihara, Phys. Rev. Lett., 93, 173601 (2004) Koshino, Kazuki and Hajime Ishihara, Phys. Rev. A, 71, 063818 (2005) Lamb, Jr., W. E.; Appl. Phys. B 60, 77-84 (1995) Lamoreaux, Steven K., Rep. Prog. Phys. 68, 201–236 (2005) Landau, L. D. and E. M. Lifshitz, Volume 3 of Course of Theoretical Physics: Quantum Mechanics, Non-relativistic Theory, Addison-Wesley, Reading, MA. (1958) Lee, H. W., and P. Stehle, Phys. Rev. Lett., 36, 277-279 (1976) Li, X., and D. L. Lin, and Thomas F. George, and Xin Sun, Phys. Rev. B, 46, 13035-13041 (1992) Lin, Kao-Chin and Der-San Chuu, Phys. Rev. B, 64, 235230 (2001) Mandel, L. and E. Wolf, Phys. Rev., 149, 1033-1037 (1966) Martinez, J. C., Phys. Rev. A, 71, 015801 (2005) Merzbacher, Eugen; Quantum Mechanics: Third Edition, Wiley and Sons, NY (1998) Milonni, Peter W. and Jay R. Ackerhalt and Wallace Arden Smith, Phys. Rev. Lett., 31, 958-960 (1973) Milonni, Peter W. and Wallace Arden Smith, Phys. Rev. A, 11, 814-824 (1975) Moniz, E. J. and D. H. Sharp; Phys. Rev. D 15, 2850-2865 (1977) Nesbet, R. K., Phys. Rev. A, 4, 259-264 (1971) Oliver, Guy; Phys. Rev. A, 15, 2424-2429 (1977) Pryde, G. J.; J. L. O’Brien, A. G. White, T. C. Ralph, and H. M. Wiseman, Phys. Rev. Lett., 94, 220405 (2005) Raimond, J. M.; M. Brune, and S. Haroche, Rev. Mod. Phys., 73, 565-582 (2001) Salzman, W. R., Phys. Rev. A, 16, 1552-1558 (1977) Savasta, Salvatore; Omar Di Stefano, and Raffaello Girlanda, Phys. Rev. A, 65, 043801 (2002) Schroedinger, Erwin; Four Lectures on Wave Mechanics in Collected Papers on Wave Mechanics, Blackie and Sons, Ltd., London (1928) Shimizu, Fujio, Phys. Rev. A, 10, 950-959 (1974) Stroud, Jr., C. R., and E. T. Jaynes, Phys. Rev. A, 1, 106-121 (1970) Thompson, R. J., Q. A. Turchette, O. Carnal and H. J. Kimble, Phys. Rev. A, 57, 3084-3104 (1998) Ugaz, Eduardo, Phys. Rev. A, 50, 34-38 (1994) Valentini, Antony; Phys. Rev. Lett. 61, 1903-1905 (1988)