**6. References**

20 Will-be-set-by-IN-TECH

We can represent the position of two bells in polar coordinate as (*α*0,±*ϕ*) (Schleich et al., 1991). For *ϕ* = 0 and *ϕ* = *π*, the two bells overlap: There are no interferences in this case, but are quantum transitions between two different quantum states overlapped (Dragoman, 2001). On the other hand, for *ϕ* = *π*/2, they are separated from each other with maximum distance. The microscopical pattern of the structure of interference is directly related to the value of *φ*. Of course, these rules mentioned in this paragraph are equally applied to the case of the

The WDF plays a crucial role in analyzing nonclassical characteristics of quantum states. You can confirm the nonclassical features for the superposed squeezed states from negative values in interference structure displayed in Fig. 8, which are very similar to that of previous section. The methods for interpreting quantum superpositions are different from that for simple addition of probability distributions, because, in quantum mechanics, we deal with superpositions of probability amplitudes instead of those of probabilities themselves. This is closely related to the appearance of interference terms in the distribution functions of probability. The novel effects of nonclassical states that admits no analogue in classical mechanics have drawn special attention both in theoretical and experimental physics thanks to their applicability in modern technology employing optical and/or other dynamical

The development of modern technology in experimental photon engineering have made it possible to produce Schrödinger cat states and/or kitten states (small Schrödinger cat states) on the basis of effective nonlinear operations that can be realized via projective measurements and post-selection. Projective measurements based on the Hilbert space formulation of quantum theory produce complete determinations of the post-measurement states through the projection-valued measures of a Hermitian operator (von Neumann, 1932). Kitten states can be produced by squeezing a single-photon. An interesting and useful way to obtain a squeezed single-photon is subtracting one photon from a single-mode squeezed vacuum beam generated by an optical nonlinear process, so-called degenerate optical parametric down-conversion (Ourjoumtsev et al., 2006). A sufficiently large Schrödinger cat states with a smaller overlap between two constituent states can be created by subtracting multiphoton from a squeezed vacuum beam (Neergaard-Nielsen et al., 2011). Other methods for preparation of superposition states include a squeezed Schrödinger cat state prepared by conditional homodyning of a two-photon Fock state (Ourjoumtsev et al., 2007), high-fidelity superposition states prepared using cavity QED technology (de Queirós et al., 2007), and

preparation of entangled non-local superposition states (Ourjoumtsev et al., 2009).

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**0**

**3**

Peeter Saari

*Estonia*

**Photon Localization Revisited**

Whilst quantum electrodynamics (QED) underwent an impressive development and reached its maturity in the middle of the last century, one of its basic concepts—the photon wave function in free space—was deprived of such fortune. Although the photon wave function in coordinate representation was introduced already in 1930 by Landau and Peierls, the concept was found to suffer from inherent difficulties that were not overcome during the century —see review (Bialynicki-Birula, 1996). The common explanation presented in textbooks, e.g., (Akhiezer & Berestetskii, 1965; Mandel & Wolf, 1995), may be summed up as follows: (i) no position operator exists for the photon, (ii) while the position wave function may be localized near a space-time point, the measurable quantities like the electromagnetic field vectors, energy, and the photodetection probability remain spread out due to their non-local

However, just before the turn of the century both of these widely-espoused notions were disproved (Bialynicki-Birula, 1998; Hawton, 1999) and in the new century a fresh interest in the photon localization problem seems to have been awakened, see, e.g., (Bialynicki-Birula & Bialynicka-Birula, 2009; Chan et al., 2002; Hawton, 2007; Keller, 2000; 2005), meeting the needs of developments in near-field optics, cavity QED, and quantum computing. Recently, into the study of quantum phenomena in general and photon localization in particular, the so-called localized waves were involved (Belgiorno et al., 2010; Besieris et al., 1994; Ciattoni & Conti, 2007; Jáuregui & Hacyan, 2005; Saari et al., 2005). These belong to the propagation-invariant non-diffracting localized solutions to the linear wave equation—a research subject emerged in the 1980-ies, see the 1st collective monograph on the field (Hernández-Figueroa et al., 2008). Experimental feasibility of some of the localized waves has been demonstrated already (Alexeev et al., 2002; Bowlan et al., 2009; Grunwald et al., 2003; Reivelt & Saari, 2002; Saari

(Bialynicki-Birula, 1998) writes that the statement "even when the position wave function is strongly concentrated near the origin, the energy wave function is spread out over space asymptotically like *r*−7/2 " —citation from (Mandel & Wolf, 1995), p. 638—is incorrect and that both wave functions may be strongly concentrated near the origin. He demonstrates, on one hand, that photons can be essentially better localized in space—with an exponential falloff of the photon energy density and the photodetection rates. On the other hand, he establishes—and it is even somewhat startling that nobody has done it earlier—that certain localization restrictions arise out of a mathematical property of the positive frequency solutions which therefore are of a universal character and apply not only to photon states but hold for all particles. More specifically, it has been proven in the Letter (Bialynicki-Birula,

**1. Introduction**

relation with the position wave function.

et al., 2010; Saari & Reivelt, 1997; Sõnajalg et al., 1997).

*Institute of Physics, University of Tartu*

