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Physical Review A

Space-time description of photon emission from an atom

Ole Keller

Published Jul 18, 2000• 44 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.62.022111

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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\ensuremathω0,E=\ensuremath{\Elzxh}{\ensuremath{\omega}}_{0},E=\ensuremath\Elzxh\ensuremathω0​, 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.
Recent attempts to establish a space-time description of single-photon emission from single atoms (or molecules) are reviewed. In the wake of historical remarks on the so-called Lorenz zone of molecular influence, characteristics of the power-law confined birth domain of the photon are discussed within the framework of a field-unquantized formalism. In contrast to the common believe it is shown that the birth domain can be (and is) exponentially localized if the photon emerges from a pure spin transition. As a paradigm of exponential localization we discuss the emergence of a radio-frequency photon from the hydrogen hyperfine structure. Starting from the first-quantized description of the energy wave function of a polychromatic photon in vacuo, we introduce a propagator description of the photon embryo, and argue that the single-photon tunnelling process is a fingerprint of our inability to localize photons in space-time. With this understanding optical tunnelling appears as an indispensable part of all single photon-atom scattering processes. In the final part of the review we briefly describe how the formalism can be upgraded from single-photon wave mechanics to a field-quantized description based on wave packet (wave train) electrodynamics. In this new formalism, in which the polychromatic photon embryo is included as an integral part of the polychromatic field quanta, the electrons of the atom (or molecule) move not only in the self-consistent Coulomb potential but also in the transverse potential created by the particles themselves. A rigorous relativistic quantum electrodynamic theory of wave train photon emission from single atoms emerges in the horizon.
Abstract A local dynamical evolution equation for a two-spinor photon wave function is established. The vectorial spinor components are those of the two helicity components A T , ± ( + ) of the analytical part of the gauge invariant transverse vector potential. The theory’s Hamilton operator is spatially local and without singularities. Thereby previously established field-based formulations leading to various nonlocal integro-differential equations for the dynamical evolution of the photon wave function are simplified. The dynamical evolution equations for the spinor components can be written in the compact form F ± ( + ) ( r , t ) = 0 , where F ± ( + ) ( r , t ) are the analytical Riemann–Silberstein–Oppenheimer–Bialynicki vectors. The theory after extension to the QED level, allows one to describe the dynamical time evolution of single-photon wave packets. The bridge between the wave mechanical and the QED formulations is an operationally suitable mean position state | R ⟩ ( r , t ) for a transverse photon in Hilbert space. The inner product ⟨ R ( r , t ) ⟩ R ( r , t ) relates to our inability to localize a photon in space. The normalization condition for the two-spinor wave packet is established. A dynamical evolution equation describing the photon-source entanglement is established starting from the Heisenberg equation of motion for the annihilation operators associated to the two helicity species. We show how each released photon is labelled by the transverse current density operator dynamics of the charged source particles. The basic theory is applied to a study of single-photon emission from a rectlinear magnetic dipole tube (string) carrying a rotationally symmetric surface current. Numerical results are obtained for photon wave train emission. The transition from the photon wave mechanical formulation to the magnetostatic Aharonov–Bohm vector potential field is established. The generality of the A T , ± ( + ) -thoery is underlined linking it to the Minimal Coupling Principle, magnetic monopole electrodynamics, spatial photon localization, the perfe
A review is given of the space-time wave mechanics of single photons, a subject with an almost century long history. The Landau-Peierls photon wave function, which is related nonlocally to the electromagnetic field is first described, and thereafter the so-called energy wave function, based on the positive-frequency Riemann-Silberstein vectors, is discussed. Recent attempts to understand the birth process of a photon emerging from a single atom are summarized. The polychromatic photon concept is introduced, and it is indicated how the wave mechanics of polychromatic photons can be upgraded to wave train quantum electrodynamics. A brief description of particle (photon) position operators is given, and it is shown that photons usually are only algebraically confined in an emission process. Finally, it is demonstrated that the profile of the birth domain of a radio-frequency photon emitted in a hyperfine spin transition shows an exponential falloff with the distance from the atomic nucleus. The length parameter in this falloff is the Bohr radius.
A field-quantized electromagnetic propagator description emphasizing the spatial localization of light emitted from a single-electron atom is presented. With the aim of obtaining a better insight into the quantum electrodynamics in the near-field zone of the atom, two approaches, formally different but equivalent in respect to their physical predictions, are studied. From the Maxwell equations, among the transverse operators, a transverse propagator picture in which the source region for the field is identified with the spatial domain occupied by the transverse current density of the atom emerges. In this picture the field propagation is isotropic and always retarded. By identifying the source region with that of the total atomic current, the transverse propagator becomes anisotropic, and attains a nonretarded self-field part and a near-field part different from zero only for spacelike events. By changing the Coulomb Lagrangian, a nonrelativistic Hamiltonian formalism adequate for reconciling the photon concept with the anisotropic propagator picture is introduced. In this formalism the transverse self-field energy is transferred to the particle Hamiltonian, and the interaction Hamiltonian now includes an interaction between the retarded field and the transverse self-field dynamics. The link between the standard (nonpropagator) theory, often used to argue that the total electromagnetic field is always retarded, and the transverse propagator theory is established. The equivalence between the two approaches is proved by demonstrating that the standard theory in fact includes a nonretarded response in the spatial domain occupied by the longitudinal part of the induced atomic current density.
A first-quantized free photon is a complex massless vector field A=(Aμ) whose field strength satisfies Maxwell’s equations in vacuum. We construct the Hilbert space H of the photon by endowing the vector space of the fields A in the temporal-Coulomb gauge with a positive-definite and relativistically invariant inner product. We give an explicit expression for this inner product, identify the Hamiltonian for the photon with the generator of time translations in H, determine the operators representing the momentum and the helicity of the photon, and introduce a chirality operator whose eigenfunctions correspond to fields having a definite sign of energy. We also construct a position operator for the photon whose components commute with each other and with the chirality and helicity operators. This allows for the construction of the localized states of the photon with a definite sign of energy and helicity. We derive an explicit formula for the latter and compute the corresponding electric and magnetic fields. These turn out to diverge not just at the point where the photon is localized but on a plane containing this point. We identify the axis normal to this plane with an associated symmetry axis and show that each choice of this axis specifies a particular position operator, a corresponding position basis, and a position representation of the quantum mechanics of a photon. In particular, we examine the position wave functions determined by such a position basis, elucidate their relationship with the Riemann-Silberstein and Landau-Peierls wave functions, and give an explicit formula for the probability density of the spatial localization of the photon.
We extend classical Maxwell field theory to a first quantized theory of the photon by deriving a conserved Lorentz four-current whose zero component is a positive-definite number density. Fields are real and their positive (negative) frequency parts are interpreted as absorption (emission) of a positive-energy photon. With invariant plane-wave normalization, the photon position operator is Hermitian with instantaneously localized eigenvectors that transform as Lorentz four-vectors. Reality of the fields and wave function ensure causal propagation and zero net absorption of energy in the absence of charged matter. The photon probability amplitude is the real part of the projection of the photon's state vector onto a basis of position eigenvectors and its square implements the Born rule. Manifest covariance and consistency with quantum field theory is maintained through use of the electromagnetic four-potential and the Lorenz gauge.