Zofia Bialynicka-Birula

The role of the Riemann–Silberstein vector in classical and quantum theories of electromagnetism

It is shown that the use of the Riemann–Silberstein (RS) vector greatly simplifies the description of the electromagnetic field both in the classical domain and in the quantum domain. In this review, we describe many specific examples where this vector enables one to significantly shorten the derivations and make them more transparent. We also argue why the RS vector may be considered as the best possible choice for the photon wavefunction.

Canonical separation of angular momentum of light into its orbital and spin parts

It is shown that the photon picture of the electromagnetic field enables one to determine unambiguously the splitting of the total angular momentum of the electromagnetic field into the orbital part and the spin part.

Beams of electromagnetic radiation carrying angular momentum: The Riemann–Silberstein vector and the classical–quantum correspondence

All beams of electromagnetic radiation are made of photons. Therefore, it is important to find a precise relationship between the classical properties of the beam and the quantum characteristics of the photons that make a particular beam. It is shown that this relationship is best expressed in terms of the Riemann–Silberstein vector – a complex combination of the electric and magnetic field vectors – that plays the role of the photon wave function. The Whittaker representation of this vector in terms of a single complex function satisfying the wave equation greatly simplifies the analysis. Bessel beams, exact Laguerre–Gauss beams, and other related beams of electromagnetic radiation can be described in a unified fashion. The appropriate photon quantum numbers for these beams are identified. Special emphasis is put on the angular momentum of a single photon and its connection with the angular momentum of the beam.

Motion of vortex lines in quantum mechanics

In quantum theory, vortex lines arise in the hydrodynamic interpretation of the wave equation. In this interpretation, which is originally due to Madelung, the flow of the probability density for a single particle is described in terms of the hydrodynamic variables. For the sake of simplicity, the standard time-dependent Schrodinger equation, and the related vortex lines embedded in the probability fluid of the quantum particle, are considered here. A vortex line in this case is simply the curve defined by equating the wave function to zero. The linearity of the Schrodinger equation enables us to obtain a large family of exact time-dependent analytic solutions for the wave functions with vortex lines. Moreover, the method is general enough to allow for various initial configurations of the vortex lines.

Space–time description of squeezing

We present a field-theoretical description of the squeezed states of the electromagnetic field. A definition of the squeezed state is introduced that is a natural generalization of the definition for one or two modes. We show that the squeezing produced by the medium with space- and time-dependent material coefficients ∊ and μ is directly related to the photon-pair production. In Appendix A we describe the time evolution of the Gaussian squeezed states in terms of the field variables.

Why photons cannot be sharply localized

Physicists have pondered over the problem of photon localization and the related problem of the photon wave function for almost 80 years now, beginning with the work of Landau and Peierls 1. An extensive review of the photon localization problem was recently presented by Keller 2. The problem of photon localization is closely related to the widely studied problem of the photon position operator. In this paper we introduce an operational definition of partial localization based on the measurements of correlation functions for electric or magnetic fields. For a want of better names, we shall use the terms electric localization and magnetic localization even though this might erroneously suggest the presence of some electric or magnetic devices that confine the photons. Since a sharp localization of photons according to our operational definition of localization is not possible, a photon position operator compatible with this definition does not exist. Earlier studies of the photon localization emphasized the mathematical aspects see, for example, 3‐5. In this paper we emphasize the physical properties of the electromagnetic field. In particular, we exhibit the role of the photon helicity and the symmetry between the electric and magnetic fields. We proceed in the footsteps of Glauber 6‐9 who was the first to recognize the significance of the space-dependent creation and annihilation operators. This approach was recently summarized and expanded in an extensive paper by Smith and Raymer 10. In our work we concentrate on the analysis of the photon localization in terms of the electric and magnetic field operators. We treat these fields after smearing over space-time regions as bona fide observables. We put emphasis on the field aspect that is complementary to the particle aspect. It might have a weaker connection with experiments usually based on photon counting as highlighted by Glauber, but it is more precise as was explained in detail by Bohr and Rosenfeld 11,12 The standard method of quantization of the free electromagnetic field based on the decomposition into monochromatic modes is not well suited for the discussion of localizability because the monochromatic mode functions are not localized. To overcome this problem we further developed an alternative method of quantization that does not require a mode decomposition. The essential mathematical tools in our analysis are the Riemann-Silberstein RS vector and the helicity operator. This formulation has some merits of its own, and it can also be used to study other general properties of

Relativistic Electron Wave Packets Carrying Angular Momentum

There are important differences between the nonrelativistic and relativistic description of electron beams. In the relativistic case the orbital angular momentum quantum number cannot be used to specify the wave functions and the structure of vortex lines in these two descriptions is completely different. We introduce analytic solutions of the Dirac equation in the form of exponential wave packets and we argue that they properly describe relativistic electron beams carrying angular momentum.

Uncertainty relation for photons.

The uncertainty relation for the photons in three dimensions that overcomes the difficulties caused by the nonexistence of the photon position operator is derived in quantum electrodynamics. The photon energy density plays the role of the probability density in configuration space. It is shown that the measure of the spatial extension based on the energy distribution in space leads to an inequality that is a natural counterpart of the standard Heisenberg relation. The equation satisfied by the photon wave function in momentum space which saturates the uncertainty relations has the form of the Schrödinger equation in coordinate space in the presence of electric and magnetic charges.

Reconstruction of the Wavefunction from the Photon Number and Quantum Phase Distributions

Abstract The problem of a reconstruction of relative phases in the one-mode photon wavefunction from a given photon number distribution and the modulus of the wavefunction in the phase representation is solved for the case of a finite superposition of Fock states. The solution involves two independent numerical algorithms, both based on a Fourier transform from the occupation number representation to the phase domain.

Exponential beams of electromagnetic radiation

We show that in addition to well-known Bessel, Hermitte–Gauss and Laguerre–Gauss beams of electromagnetic radiation, one may also construct exponential beams. These beams are characterized by a fall-off in the transverse direction described by an exponential function of ρ. Exponential beams, like Bessel beams, carry definite angular momentum and are periodic along the direction of propagation, but unlike Bessel beams they have a finite energy per unit beam length. The analysis of these beams is greatly simplified by an extensive use of the Riemann–Silberstein vector and the Whittaker representation of the solutions of the Maxwell equations in terms of just one complex function. The connection between the Bessel beams and the exponential beams is made explicit by constructing the exponential beams as wave packets of Bessel beams.