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arXiv:1905.04766

Non-uniqueness of second quantization of electromagnetic field: Stationary States of Jaynes-Cummings model in free space

Atom Zh. Muradyan, G. A. Muradyan

Published May 12, 2019• 0 citations
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Abstract

On the example of stationary states of a system consisting of an atom and a quantized electromagnetic field (the Jaynes-Cummings model in free space), it is shown that the physical characteristics of the system (as the energy and the probability distribution of finding the center of mass of the atom) explicitly depend on the choice of the basis of quantization of the electromagnetic field. Therefore, the secondary quantization procedure is not independent of the choice of the quantization basis in the interaction of the field with the material medium.

DOI
10.48550/arxiv.1905.04766

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On the example of stationary states of a system consisting of an atom and a quantized electromagnetic field (the Jaynes-Cummings model in free space), it is shown that the physical characteristics of the system (as the energy and the probability distribution of finding the center of mass of the atom) explicitly depend on the choice of the basis of quantization of the electromagnetic field. Therefore, the secondary quantization procedure is not independent of the choice of the quantization basis in the interaction of the field with the material medium.
The Jaynes-Cummings model describing the interaction of a single linearly- polarized mode of the quantized electromagnetic field with an isolated two- level atom is generalized to the case of atomic levels degenerate in the projections of the angular momenta on the quantization axis, which is a usual case in the experiments. This generalization, like the original model, obtains the explicit solution. The model is applied to calculate the dependence of the atomic level populations on the angle between the polarization of cavity field mode and that of the laser excitation pulse in the experiment with one-atom micromaser.
The problem of the interaction of a single two-level atom with a single mode of the quantized electromagnetic field in a coherent state in an ideal resonator in the resonance case is considered. The evolution in time of the atomic inversion, represented by the Jaynes-Cummings sum, is studied. On the basis of the application of the theorem on the approximation of a trigonometric sum by a shorter one (ATS), a new efficient method for approximating the Jaynes-Cummings sum is constructed. New asymptotic formulas for the atomic inversion are found, which approximate it on various time intervals, defined by relation between the atom-field coupling constant and the average photon number in the resonator field before the interaction of the field with the atom. The asymptotics that we obtain give the possibility to predetermine the details of the process of the inversion in the Jaynes-Cummings model depending on the field characteristics.
The eigenstate problem of the Jaynes-Cummings model on the basis of complete Hamiltonian, including the center-of -mass kinetic energy operator, is treated. The energy spectrum and wave functions in standing-wave (SW)- and counterpropagating waves (CPW)- cases are calculated and compared with each other. It is shown that in CPW-case i) the atomic momentum distribution is asymmetric and somewhat narrower in general; ii) the concept of quasimomentum is not applicable and instead the ordinary momentum concerns the problem; iii) atomic and photonic state distributions are self-consistent, and, in consequence iiii) mean number of photons in the counterpropagating traveling waves and mean atomic momentum match. Explicit analytic expressions for energy eigenvalues and eigenfunctions are found in Tavis -Cummings-type approximation [Phys. Rev. 170, 379(1968)] and is pointed, that it implies only the bounded-like states for atomic center-of-mass motion. It is also shown that if the recoil energy is taken into account, the Doppleron resonance is split into two branches, one of which diverges to Bragg-like resonance in the high-order range.
The Jaynes-Cummings model of early quantum optics is examined from the point of view of a discrete-space discrete-time formulation of the electromagnetic field (the FDTD formulation) excited by an elementary quantum Hertzian dipole associated to a single edge of the discrete-space lattice. The quantum Hertz dipole is a natural 2-level atomic system, and the quantum dynamics of the electromagnetic field in the discrete space surrounding the dipole, described by functional integrals over field space, is essentially a stochastic (Monte-Carlo) algorithm for solving the FDTD equations by searching for the field distribution about which the field action Lagrangian functional S is stationary.
The eigenstate problem of the Jaynes–Cummings model on the basis of a complete Hamiltonian, including the centre-of-mass kinetic energy operator, is considered. The energy spectrum and wavefunctions in the standing-wave (SW) and counterpropagating-wave (CPW) cases are calculated and compared with each other. It is shown that in the CPW case: (i) the atomic momentum distribution is asymmetric; (ii) the concept of quasimomentum is not applicable and instead the problem concerns the ordinary momentum; (iii) atomic and photonic state distributions are self-consistent; and as a consequence (iv) the mean number of photons in counterpropagating travelling waves and mean atomic momentum are matched. Explicit analytic expressions are proposed to describe the transition from counterpropagating quantized waves to a standing wave. It is also shown that, if the recoil energy is taken into account, the Doppleron resonance is split into two branches, one of which diverges to Bragg-like resonance in high-order range.