The paper is an extension of quant-ph/9912102. The new framework is tested on the 2-photon spontaneous emission and blackbody radiation. The new effects are rather subtle. The probability of the 2-photon emission is shown to consist of a product of several terms: One which is identical to this from standard quantum optics and the remaining ones formally resembling detector inefficiencies (here arising from nontrivial vacuum structure). The blackbody distribution is indistinguishable from the Planck law for TT_{critical} the maximum of the distribution gets lowered and shifts towards higher frequencies. T_{critical} is a parameter that, in principle, should be observable. No normal ordering of operators is needed and vacuum energy is nonzero but finite. Vacuum is represented by a subspace spanned by ground states of the oscillators and is not equivalent to the cyclic vector of the GNS construction. The non-CCR algebra is discussed in more detail.
The paper is an extension of quant-ph/9912102. The new framework is tested on the 2-photon spontaneous emission and blackbody radiation. The new effects are rather subtle. The probability of the 2-photon emission is shown to consist of a product of several terms: One which is identical to this from standard quantum optics and the remaining ones formally resembling detector inefficiencies (here arising from nontrivial vacuum structure). The blackbody distribution is indistinguishable from the Planck law for TT_{critical} the maximum of the distribution gets lowered and shifts towards higher frequencies. T_{critical} is a parameter that, in principle, should be observable. No normal ordering of operators is needed and vacuum energy is nonzero but finite. Vacuum is represented by a subspace spanned by ground states of the oscillators and is not equivalent to the cyclic vector of the GNS construction. The non-CCR algebra is discussed in more detail.
A harmonic oscillator is an indefinite-frequency one if the parameter ω is replaced by an operator. An ensemble of N such oscillators may be regarded as a toy model of a bosonic quantum field. All the possible frequencies associated with a given problem are present already in a single oscillator and N can be finite. Due to the operator character of ω the resulting algebra of creation-annihilation operators is non-canonical. In the limit of large N one recovers perturbation theory formulas of the canonical quantum field theory but with form factors automatically built in. Vacuum energy of the ensemble is finite, a fact discussed in the context of the cosmological constant problem. Space of states is given by a vector bundle with Fock-type fibers. Interactions of the field with 2-level systems, including Rabi oscillations and spontaneous emission, are discussed in detail.
A new scheme of field quantization is proposed. Instead of associating with different frequencies different oscillators we begin with a single oscillator that can exist in a superposition of different frequencies. The idea is applied to the electromagnetic radiation field. Using the standard Dirac-type mode-quantization of the electromagnetic field we obtain several standard properties such as coherent states or spontaneous and stimulated emission. As opposed to the standard approach the vacuum energy is finite and does not have to be removed by any ad hoc procedure.
An earlier study of quantum-mechanical time has suggested that, from the quantum-theoretic point of view, time is a random variable. The present work explores how such a description of time can be used to bring greater detail into the analysis of the one-dimensional harmonic oscillator. Specifically, the classical equations for harmonic motion are used as a point transformation between phase space and time-energy space, permitting probability distributions in the two spaces to be different representations of the same thing. The familiar energy-quantization hypothesis is supplemented with the hypothesis that quantized energy values are means (averages) of energy distributions characteristic of each energy state. With the probability distribution of time determined from the previously cited study, the desired quantized energy distributions follow from the requirements: (1) that they should be normalized, non-negative, and with means equal to the quantized energy values; (2) that the equivalent representation of distribution in position-momentum space should have marginal densities that are proportional (except near the origin) to the densities in position and momentum, ${|\ensuremath{\phi}(q)|}^{2}$ and ${|\ensuremath{\psi}(p)|}^{2}$, which are postulated in quantum theory. Requirement (2) is a weakening of the Born postulate. In the time-energy space, the probability distributions exclude energies smaller than the mode (energy value with maximum probability). In position-momentum space this exclusion requirement becomes the dynamical restriction: $\frac{{q}^{2}m{\ensuremath{\omega}}^{2}}{2}+\frac{{p}^{2}}{2m}\ensuremath{\ge}\frac{{\ensuremath{\Lambda}}_{n}\ensuremath{\hbar}\ensuremath{\omega}}{2}$, where, for the first three energy states, ${\ensuremath{\Lambda}}_{0}=0$, ${\ensuremath{\Lambda}}_{1}=\frac{3}{2}$, ${\ensuremath{\Lambda}}_{2}=2+\ensuremath{\surd}\frac{3}{2}$. In general terms it is thus shown that the quantized harmonic oscillator can be described as a statistical ensemble of classical oscillators. Some comparisons with work on phase-space formulations of quantum mechanics (especially with the work of Wigner) are made and some aspects of the theory that should submit to experimental verification are pointed out.
Noncanonical quantization schemes have been proposed in quantum physics long time ago. Recalling some of them I associate these investigations to the quantum mechanical im-plications of the canonically inequivalent Hamilton formula-tions of classical Newtonian dynamics. As the simplest illus-tration of such alternative Hamiltonian descriptions I consider one dimensional system of two particles interacting harmoni-cally. Description of the motion of their center of mass is stan-dard and guarantees the Galilean covariance but the internal motion is described as a noncanonical oscillator for which the Hamiltonian and the total energy are not related by a simple proportionality. Alternative Hamiltonians lead to alternative Poisson structures which may be considered as classical ana-logues of the noncanonical quantum commutation rules. Gen-Concepts of Physics, Vol. I (2004) 11 erated alternative quantizations are consistent with the same Heisenberg equations of motion, i.e., they satisfy the Wigner principle of quantization. For the one dimensional harmonic oscillator the simplest noncanonical algebraic description is given in terms of the Lie algebra so(p, q), p+ q = 3. In such a case the Hamiltonian is still a linear function while the energy is a quadratic polynomial of the main quantum number. This leads to nonstandard termodynamical properties obeyed by such a noncanonical oscillator. The 3-dimensional harmonic oscillator also has a noncanonical description given in terms of the simple Lie algebra, so(p, q), p+ q = 5 in the case under consideration. The most important physical consequence of such a description is the noncommutativity of the coordinate operators, interpreted to be an implication of the noncommu-tative geometry of the underlying spacetime.
We will begin our journey through the background of quantum optics with an elementary, but quantitative, classical theory of radiative fields.Wewill first briefly outline the electromagnetic (EM) theory of radiation, and describe how the EM radiation may be understood as a wavewhich can be represented by a set of harmonic oscillators. We shall describe how the free or non-interacting EM field may be understood as a collection of harmonic oscillators which is quantized in the standard manner, and whose energy cannot be zero as a consequence of the basic non-commutability of the canonical field variables. This chapter discusses the properties of plane EM waves and normalization of the EM field in one dimension. This is followed by a description of the Hamiltonian and the amplitudes of the EM field in terms of the annihilation and creation operators. Based on this chapter, it is possible to considerably simplify the formulation of the physical basis for the mathematical description of the major problems of quantum optics.
In this chapter we shall be using the formalism of Chapter 4 to describe some properties of the quantized free electromagnetic field, emphasizing the analogies and differences with classical electromagnetism. The analogies explain why classical optics was so successful, to the extent that, even though light was not quantized, it was able to describe almost all known optical phenomena up until the 1970s. The differences highlight typical quantum behaviour, with no classical equivalent, which lies at the heart of modern quantum optics. The chapter begins by presenting formal ways of describing the main components of quantum optics experiments, namely, photodetectors and semi-reflecting mirrors. One can then understand homodyne detection, used to measure field quadrature components, even at optical frequencies for which no detector is fast enough to follow the oscillations of the field itself. Section 5.2 deals with the quantum vacuum where, in contrast to the classical vacuum, radiation has properties, in particular, fluctuations, with which one can associate physical effects. We then discuss in Section 5.3 some quantum radiation states in the simple case where a single mode l of the field is excited (non-vacuum). We introduce the idea of a quasiclassical state, or coherent state, used to make the link with classical optics. We shall also study other states, those with a definite number of photons, or squeezed states, which exhibit typically quantum properties, quite inconceivable in classical electromagnetism, and we shall show how these properties can be understood by reference to the Heisenberg relations, with an extremely useful graphical representation.