Vol 39, No 4 (2018)

The models we use, habitually, to describe quantum nonlinear optical processes have been remarkably successful, yet, with few exceptions, they each contain a mathematical flaw. We present this flaw, show how it can be fixed, and, in the process, suggest why we can continue to use our favored Hamiltonians.

The Janszky representation constructs quantum states of a field mode as a superposition of coherent states on a line in the complex plane. We show that this provides a natural Schrödinger picture description of the interference between a pair of modes at a beam splitter.

In the Hilbert space of a light mode (harmonic oscillator), we construct a representation, in which an arbitrary state vector is expanded using Bargmann states ‖α〉 with real parameters α being in an infinitesimal vicinity of zero. The complete Hilbert-space structure is represented in the one- and multimode cases as well, making the representation able to deal with problems of continuous-variable quantum information processing.

We review some concepts and reasonings regarding the notion of no-signaling and its relation to quantum mechanics in bipartite Bell-type scenarios. We recapitulate the no-signaling property of joint conditional probability distributions in geometrical and information theoretic terms. We summarize the reasons why quantum mechanics does not enable instantaneous communication. We make some comments on quantum field theoretic aspects.

We consider a quantum charged particle moving in the xy plane under the action of a time-dependent magnetic field described by means of the linear vector potential A = H(t) [−y(1 + β), x(1 − β)] /2 with a fixed parameter β. The systems with different values of β are not equivalent for nonstationary magnetic fields due to different structures of induced electric fields, whose lines of force are ellipses for |β| < 1 and hyperbolas for |β| > 1. Using the approximation of the stepwise variation of the magnetic field H(t), we obtain explicit formulas describing the evolution of the principal squeezing in two pairs of noncommuting observables: the coordinates of the center of orbit and relative coordinates with respect to this center. Analysis of these formulas shows that no squeezing can arise for the circular gauge (β = 0). On the other hand, for any nonzero value of β, one can find the regimes of excitations resulting in some degree of squeezing in the both pairs. The maximum degree of squeezing can be obtained for the Landau gauge (|β| = 1) if the magnetic field is switched off and returns to the initial value after some time T, in the limit T → ∞.

We obtain the tomogram of squeezed correlated states of a quantum parametric damped oscillator in an explicit form. We study the damping within the framework of the Caldirola–Kanai model and chose the parametric excitation in the form of a very short pulse simulated by a δ-kick of frequency; the squeezing phenomenon is reviewed for both models. The cases of strong and weak damping are investigated.