We discuss the limits of the photon wave function (PWF) formalism, which is experiencing a revival these days as a result of new practical applications in photonics and quantum optics. We build a Dirac-like equation for the PWF written in a manifestly covariant form and show that, in the presence of charged matter fields, it reproduces the standard formulation of (classical) electrodynamics. This shows that attempts to construct a full quantum theory of interacting photons (mutually or interacting with matter) based on the so-called photon wave function approach can lead only to results already described by standard quantum electrodynamics (QED). The PWF formalism can then be used only to provide an easier description of some particular situations---for example, the propagation of free photons or photons propagating in a medium as described in Bialynicki-Birula [in Progress in Optics, edited by E. Wolf (Elsevier, Amsterdam, 1996), pp. 245--294] especially when the photon number remains fixed in time but not to replace QED in toto.
We discuss the limits of the photon wave function (PWF) formalism, which is experiencing a revival these days as a result of new practical applications in photonics and quantum optics. We build a Dirac-like equation for the PWF written in a manifestly covariant form and show that, in the presence of charged matter fields, it reproduces the standard formulation of (classical) electrodynamics. This shows that attempts to construct a full quantum theory of interacting photons (mutually or interacting with matter) based on the so-called photon wave function approach can lead only to results already described by standard quantum electrodynamics (QED). The PWF formalism can then be used only to provide an easier description of some particular situations---for example, the propagation of free photons or photons propagating in a medium as described in Bialynicki-Birula [in Progress in Optics, edited by E. Wolf (Elsevier, Amsterdam, 1996), pp. 245--294] especially when the photon number remains fixed in time but not to replace QED in toto.
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 relation with the position wave function.
During these three years we focused on several topics in quantum otpics and quantum electrodynamics. A central theme in our investigations is that of the photon wave function. Can quantum optics and quantum electrodynamics experiments be described simply, in position space, with the help of a wave function describing the photon(s) featured in the experiment ? The answer to that question is not quite obvious: the usual description of photons takes place in the reciprocal space of wave vectors. But these experiments call for a wave mechanical description in the position representation, as is done in quantum mechanics textbooks in situations featuring massive particles. Moreover, in a recent experiment [1], single photon trajectories through a Young two-slit setup have been observed. In order to try and describe these trajectories formally, it is natural to build a wave mechanical formalism for photons. We therefore studied in detail the formal construction of the photon wave function, an object which was little studied until the 1990s. We also studied the properties of the photon wave function in the presence of sources.To do that, we considered several open (interacting) quantum systems. We saw that there exists in principle an infinite number of possibilities when defining the photon wave function. We emphasised several criteria on the basis of which it appears that only three choices for the wave function are interesting. One of them coincides with an object introduced and used by Glauber [2] to study light detection andthe correlations of the electromagnetic field in the quantum regime. We also saw that, in the absence of sources, the propagation equation for a single photon is formally equivalent to Maxwell’s equations. At low photon numbers, the wave function formalism can be very useful. We adapted it to interacting systems,first, to cavity quantum electrodynamics (QED) [3], in particular to the experiments carried out by Serge Haroche’s group [4]. We proposed a simple model to describe photons in QED cavities. With this model, and with the helpof the photon wave function, we studied the propagation of photons escaping a cavity. We also constructed the Lindblad master equation without introducing nonunitary quantum jumps (also see [5]). We finally investigated the spacetime evolution of a photon which is emitted during the decay of an atomic electron. After having carefully studied the dynamics of the electronic decay, especially at very short times [6, 7], we set out to describe the emitted electromagnetic field as rigorously as possible. This emitted field, surprisingly, does not evolve causally. Though this is not entirely unexpected in view of Hegerfeldt’s theorem, which states [8] that causality is impossible for quantum systems which are described by a Hamiltonian with a spectrum which is bounded by below, we identified [9] two other sources of non causality. One of them was predicted qualitatively by Shirokov [10], while the other one, which is completely new as far as we can tell, is still to be better understood
One of the key frameworks for developing the theory of light–matter interactions in modern optics and photonics is quantum electrodynamics (QED). Contrasting with semiclassical theory, which depicts electromagnetic radiation as a classical wave, QED representations of quantized light fully embrace the concept of the photon. This tutorial review is a broad guide to cutting-edge applications of QED, providing an outline of its underlying foundation and an examination of its role in photon science. Alongside the full quantum methods, it is shown how significant distinctions can be drawn when compared to semiclassical approaches. Clear advantages in outcome arise in the predictive capacity and physical insights afforded by QED methods, which favors its adoption over other formulations of radiation–matter interaction.
When rewritten in an appropriate manner, the microscopic Maxwell-Lorentz equations appear as a wavemechanical theory for photons, and their quantum physical interaction with matter. A natural extension leads from photon wave mechanics to quantum electrodynamics (QED). In its modern formulation photon wave mechanics has given us valuable new insight in subjects such as spatial photon localization, near-field photon dynamics, transverse photon mass, photon eikonal theory, photon tunneling, and rim-zone electrodynamics. The present review is based on my plenary lecture at the SPIE-Europe 2009 Optics and Optoelectronics International Symposium in Prague.
Few-photon systems are best described by their wave function rather than by the usual quantum field formalism. In this work, we develop a photon wave function (PWF) formalism suitable for analyzing a wide variety of quantum optical problems related to propagation, diffraction and imaging with quantum states of light. We establish a generalized Huygens-Fresnel (GH-F) principle that describes the propagation of any paraxial N-photon state. This tool is very helpful for predicting photo-detection correlations in space and time due to an initial N-particle entanglement, even in complex situation. The effect of lenses, beam splitters, filters ... on the photon paths can be easily taken into account. We apply the PWF formalism and the GH-F principle to three specific problems in quantum optics. First, we revisit the Hong-Ou-Mandel two-photon interference effect and analyze the effect of photon shape mismatch in space, time and polarization using the PWF formalism. Second, we show how to use the GH-F principle to analyze "ghost" imaging and diffraction experiments with entangled photon pairs such as those realized by Strekalov et al. [Phys. Rev. Lett. 74, 3600 (1995)] and Pittman et al. [Phys. Rev. A 52, R3429 (1995)] in the nineties. Finally, we use the GH-F principle to analyze the resolution enhancement in a recent quantum imaging proposal based on N incoherent single-photon sources [Phys. Rev. Lett. 99, 133603 (2007) and Phys. Rev. A 80, 013820 (2009)].