Sergey I. Kryuchkov

The minimum-uncertainty squeezed states for atoms and photons in a cavity

We describe a multi-parameter family of the minimum-uncertainty squeezed states for the harmonic oscillator in nonrelativistic quantum mechanics. They are derived by the action of the corresponding maximal kinematical invariance group on the standard ground state solution. We show that the product of the variances attains the required minimum value 1/4 only at the instances that one variance is a minimum and the other is a maximum, when the squeezing of one of the variances occurs. The generalized coherent states are explicitly constructed and their Wigner function is studied. The overlap coefficients between the squeezed, or generalized harmonic, and the Fock states are explicitly evaluated in terms of hypergeometric functions and the corresponding photon statistics are discussed. Some applications to quantum optics, cavity quantum electrodynamics and superfocusing in channelling scattering are mentioned. Explicit solutions of the Heisenberg equations for radiation field operators with squeezing are found.

On the Problem of Electromagnetic-Field Quantization

We consider the radiation field operators in a cavity with varying dielectric medium in terms of solutions of Heisenberg’s equations of motion for the most general one-dimensional quadratic Hamiltonian. Explicit solutions of these equations are obtained and applications to the radiation field quantization, including randomly varying media, are briefly discussed.

Complex Form of Classical and Quantum Electrodynamics

We consider a complex covariant form of the macroscopic Maxwell equations, in a moving medium or at rest, following the original ideas of Minkowski. A compact, Lorentz invariant, derivation of the energy-momentum tensor and the corresponding differential balance equations are given. Conservation laws and quantization of electromagnetic field will be discussed in this covariant approach elsewhere.

The Pauli–Lubański vector, complex electrodynamics, and photon helicity

We critically analyze the concept of photon helicity and its connection with the Pauli?Luba?ski vector from the viewpoint of the complex electromagnetic field, sometimes attributed to Riemann but studied by Weber, Silberstein, and Minkowski. To this end, a complex covariant form of Maxwell?s equations is used. Weyl?s two-component wave equation for massless neutrinos is also briefly discussed.

Degenerate parametric amplification of squeezed photons: Explicit solutions, statistics, means and variances

In the Schrodinger picture, we find explicit solutions for two models of degenerate parametric oscillators in the case of multi-parameter squeezed input photons. The corresponding photon statistics and Wigners function are also derived in coordinate representation. Their time evolution is investigated in detail. The unitary transformation and an extension of the squeeze/evolution operator are briefly discussed.

The role of the Pauli-Lubański vector for the Dirac, Weyl, Proca, Maxwell and Fierz-Pauli equations

We analyze basic relativistic wave equations for the classical fields, such as Diracs equation, Weyls two-component equation for massless neutrinos, and the Proca, Maxwell, and Fierz-Pauli equations, from the viewpoint of the Pauli-Lubanski vector and the Casimir operators of the Poincare group. In general, in this group-theoretical approach, the above wave equations arise in certain overdetermined forms, which can be reduced to the conventional ones by a Gaussian elimination. A connection between the spin of a particle/field and consistency of the corresponding overdetermined system is emphasized in the massless case.