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
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
La fonction d'onde du photon en principe et en pratique Pendant ces trois ans, nous nous sommes intéressés à quelques sujets choisis en optique et en électrodynamique quantiques. Le fil rouge de nos interrogations est la fonction d’onde du photon. Les expériences d’optique et d’électrodynamique quantique peuvent-elles être décrites de manière simple, dans l’espace des positions, à l’aide d’une fonction d’onde décrivant le ou les photon(s) impliqués dans l’expérience ? Ce n’est pas entièrement évident :la description usuelle des photons se fait dans l’espace réciproque des vecteurs d’onde. Mais ces expériences gagnent à être décrites par la mécanique ondulatoire en représentation position, comme cela est fait dans les manuels de mécanique quantique pour des situations impliquant des particules massives. De surcroît, une expérience récente[1] a conduit à l’observation de trajectoires de photons uniques à travers un interféromètre à deux fentes d’Young.Pour essayer de décrire formellement ces trajectoires, il est naturel de formuler une mécanique ondulatoire pour les photons. Nous avons donc examiné en détail la construction formelle de la fonction d’onde du photon, un objet qui est resté peu étudié jusqu’aux années 1990. Nous avons également étudié les propriétés de la fonction d’onde du photon en présence de sources, et considéré pour ce faire divers systèmes quantiques ouverts (en interaction). Nous avons vu qu’il existe, en principe, une infinité de possibilités pour le choix de la fonction d’onde du photon.Nous avons mis en évidence un certain nombre de critères sur la base desquels il apparaît que seuls trois choix parmi tous ceux possibles sont intéressants, l’un d’entre eux ramenant à un objet introduit par Glauber [2] pour étudier la détection de la lumière et les corrélations du champ électromagnétique. Nous avons également vu qu’en l’absence de sources l’équation quantique de propagation des photons est formellement identique aux équations de Maxwell.À bas nombre de photons, le formalisme de la fonction d’onde peut se révéler très pratique. Nous avons adapté l’approche aux systèmes en interaction, en nous intéressant dans un premier temps à l’électrodynamique quantique1en cavité [3], en particulier aux expériences réalisées par le groupe de Serge Haroche [4]. Nous avons proposé un modèle simple pour la description des photons dans les cavités d’électrodynamique. À l’aide de ce modèle, et de la fonction d’onde du photon, nous avons étudié la propagation des photons s’échappant de la cavité. Nous avons également construit l’équation maîtresse de Lindblad sans introduire de sauts quantiques non unitaires (voir également [5]). Nous nous sommes enfin intéressés à la question de l’évolution spatiotemporelle d’un photon émis lors d’une désexcitation d’un électron atomique. Après avoir étudié soigneusement la dynamique de la désexcitation de l’électron, notamment aux temps très courts [6, 7], nous nous sommes attachés à décrire, aussi rigoureusement que possible, le champ électromagnétique émis. Celui-ci, de manière surprenante, n’évolue pas causalement. Si cela n’est pas entièrement inattendu au vu du théorème de Hegerfeldt, qui stipule [8] que la causalité est exclue pour les systèmes décrits par un Hamiltonien dont le spectre est borné inférieurement, nous avons identifié [9] deux autres sources de non-causalité, l’une, prédite qualitativement par Shirokov [10], et l’autre, entièrement nouvelle à notre connaissance, et dont la compréhension reste à affiner.
How can we construct a picture of the photon from the wealth of observation material available to us? The photon appears to have a split personality: it is neither a wave nor a particle but something else which, depending on the experimental situation, exhibits a wave- or a particle-like behavior. In other words, in the photon (as in material particles such as the electron) the particle–wave dualism becomes manifest. Whereas classically the wave and the particle pictures are separate, quantum mechanics accomplishes a formal synthesis through a unified mathematical treatment. Let us look first at the wave aspect familiar from classical electrodynamics, which seems to be the most natural description. It makes all the different interference phenomena understandable, such as the “interference of the photon with itself” on the one hand and the appearance of spatial and temporal intensity correlations in a thermal radiation field on the other (which are obviously brought about by superposition of elementary waves emitted independently from different atoms). It might come as a surprise (at least for those having quantum mechanical preconceptions) that the classical theory is valid down to arbitrarily small intensities: the visibility of the interference pattern does not deteriorate even for very small intensities – the zero point fluctuations of the electromagnetic field advocated by quantum mechanics do not have a disturbing effect – and is valid not only for conventional interference experiments but also for interference between independently generated light beams (in the form of laser light).
You might think that the great success of quantum electrodynamics (QED) would settle the debate on the nature of light to provide a clear view of its behavior, where the photon is regarded as the unit of excitation associated with a quantized mode of the electromagnetic (radiation) eld. However, Heisenberg’s uncertainty principle tells us that a state of denite momentum, energy, and polarization associated with a plane wave used as a basis function of quantization must be completely indenite in space and time. It suggests the diµculty of spatial localization of a photon as a particle. In fact, Newton and Wigner showed that a free photon, as a massless particle with spin 1, has no localized states on the basis of natural invariance requirements that localized states for which operators of the Lorentz group apply should be orthogonal to the undisplaced localized states, a§er a translation (Newton and Wigner 1949). According to them, one can obtain a general expression for a position operator for massive particles and for massless particles of spin 0 or 1/2, not for massless particles with nite spin, which indicates that there is no probability density for the position of the photon, and thus a position-representation wave function cannot be consistently introduced. It has also been shown that photons are not localizable, on the basis of imprimitive representations of the Euclidean group (Wightman 1962). It is now believed that photons are only weakly localizable, although single-photon states with arbitrarily fast asymptotic fallo of energy density exist, and that a lack of strict localizability is directly related to the absence of a position operator for a photon in free space and a positionrepresentation photon wave function (Hawton 1999).
The study of light through the centuries has produced a number of theories and experiments. Newton advanced the corpuscular theory, building on the early Greek view of matter composed of atoms. Then Young essentially proved the wave nature of light through the interference experiments. Maxwell provided the firm theoretical basis for the wave nature. During the late 1800s and early 1900s, further experiments showed predictions based on a continuous wave led to the well-known ultraviolet catastrophe. Plank and Einstein originated and developed the notion of the photon as an elementary quantum of propagating electromagnetic (EM) energy. The first half of the 20th century saw the development of the field quantization and the quantum electrodynamics (QED). Feynmann labeled QED as ‘‘the best theory we have’’ to describe the matter-light interaction. The second half of the 20th century witnessed the development of the coherent and squeezed optical states attributable to Glauber and Yuen. The study of light and electromagnetic fields has a long history. This chapter explores the nature of light in terms of quantum optics.
Starting from the postulate that the electromagnetic field appearing in the transverse set of microscopic Maxwell-Lorentz equations governing field-matter interactions, properly normalized, can be looked upon as describing one-photon-emission and -absorption processes in space and time, a first-quantized initiation of photon emission by a single atom is presented. The wave function for the emerging photon is introduced as a six-vector object constructed from the complex analytical signals of the Riemann-Silberstein vectors belonging to opposite photon helicities. When the atom is no longer electrodynamically active, the emitted photon is described in first quantization by the so-called energy wave function well known for photons in free space. From the momentum representation of the emerging photon wave function a condition on the analytical part of the transverse atomic current density is established which ensures that precisely one photon is emitted. A propagator description of the emerged photon dynamics in the coordinate representation is established. The photon propagator is introduced as a two-component spinor, where upper and lower tensor components are constructed, respectively, from positive and negative helicity combinations of the propagators describing the time-space evolution of the transverse electric and magnetic fields. It is shown that the emission region for the photon coincides with the region in space where the transverse atomic current density is nonvanishing. For a photon emitted in an electric dipole transition the emission region essentially is the near-field zone of the atom, and this zone therefore determines the initial (and best) spatial confinement of the photon. The photon emerging from an atom active for a finite time necessarily is of the polychromatic sort and the associated wave packet essentially is confined between spherical shells moving outwards with the vacuum speed of light. To illustrate the main principles of the fundamental theory in a heuristic fashion we apply it to a study of the emission of a one-photon sinusoidal wavetrain from a pointlike atom. It is found that the atomic current density needed to create just one photon is independent of the oscillation period in the train and thus depends only on the number of periods in the wave train. An explicit expression for the one-photon energy is derived, and it is shown that only for extremely short pulse trains pronounced deviations from the textbook result, $E=\ensuremath{\Elzxh}{\ensuremath{\omega}}_{0},$ occur. The radial energy flow in the coupled atom-photon system in the near-field zone of the atom is investigated, and the cycle-averaged outwards energy transport carried by the emerging photon in a given distance from the atom is determined.