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Physical Review A

Quantum effects versus the semiclassical approximation

Luca Bonci, Paolo Grigolini A

Published Oct 1, 1992• 5 citations
Publisher

Abstract

We study a two-level quantum system interacting through a coupling strength g with a single-mode radiation field. We show that the rotating-wave-approximation (RWA) Hamiltonian produces quantum-mechanical effects more persistent than those generated by the full Hamiltonian. Within the full Hamiltonian the semiclassical prediction is recovered upon increase of the radiation field intensity, whereas with the RWA Hamiltonian no reconciliation with the predictions of the semiclassical approximation is realized by fields of any intensity, if the interaction strength g is kept unchanged.

DOI
10.1103/physreva.46.4445

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We study a two-level quantum system interacting through a coupling strength g with a single-mode radiation field. We show that the rotating-wave-approximation (RWA) Hamiltonian produces quantum-mechanical effects more persistent than those generated by the full Hamiltonian. Within the full Hamiltonian the semiclassical prediction is recovered upon increase of the radiation field intensity, whereas with the RWA Hamiltonian no reconciliation with the predictions of the semiclassical approximation is realized by fields of any intensity, if the interaction strength g is kept unchanged.
Recalling that the rotating wave approximation (RWA) is only valid in the weak coupling regimes, the purpose of this paper is to study the Hamiltonian dynamics describing the full quantum mechanical approach of the interaction between various configurations of three-level atoms Lambda, V and ladder distinctly with a two-mode radiation field, while the RWA is not considered; the counter-rotating terms (CRTs) are taken into account. Generally, the presence of CRTs in the Hamiltonian prevents one to achieve an analytical solution. Moreover, as we will show in the present work, using the perturbation theory, analytical solvable Hamiltonians can be successfully obtained. According to our calculations, the contribution of CRTs within the ordinary Hamiltonian is equivalent to the replacement of the constant detuning with a specific intensity dependent detuning in the first order, and the constant atom field coupling with a particular intensity dependent (f deformed) coupling in the second order of the associated perturbation parameter. Moreover, noticing that according to the initial expression of the Hamiltonian, each mode of the field interacts only with a specific pair of the allowed transitions of each type of the three-level atom, it is surprisingly observed that via applying the mentioned approach, the obtained intensity dependent coupling functions depend on both modes of the field, f(na,nb). In this way, it is seen that the CRTs are removed with the price of arriving at some intermixed intensity dependent atom field coupling functions of the two modes of the field. In this way, the obtained final Hamiltonians are analytically solvable. At last, by determining the time evolution of the atom field wave function, we study the effects of CRTs on a few nonclassical properties of the state of the system, including the atomic population inversion and photon statistics.
We study the interaction between a two-level system and a harmonic oscillator without the rotating-wave approximation (RWA). In quantum optics, this model (sometimes called the spin-boson model) is traditionally thought of as a two-Ievel atom coupled to a single-mode radiation field, but it could also describe (1) a two-level ion in a trap coupled to the motion of its own center of mass or (2) a two-level particle in a solid coupled to a phonon mode. For sufficiently strong coupling, this model is known to exhibit chaos in a variety of semi-classical approximations.1
In the preceding chapters concerning the interaction of a radiation field with matter, we assumed the field to be classical. In many situations this assumption is valid. There are, however, many instances where a classical field fails to explain experimentally observed results and a quantized description of the field is required. This is, for example, true of spontaneous emission in an atomic system which was described phenomenologically in Chapter 5. For a rigorous treatment of the atomic level decay in free space, we need to consider the interaction of the atom with the vacuum modes of the universe. Even in the simplest system involving the interaction of a single-mode radiation field with a single two-level atom, the predictions for the dynamics of the atom are quite different in the semiclassical theory and the fully quantum theory. In the absence of the decay process, the semiclassical theory predicts Rabi oscillations for the atomic inversion whereas the quantum theory predicts certain collapse and revival phenomena due to the quantum aspects of the field. These interesting quantum field theoretical predictions have been experimentally verified. In this chapter we discuss the interaction of the quantized radiation field with the two-level atomic system described by a Hamiltonian in the dipole and the rotating-wave approximations. For a single-mode field it reduces to a particularly simple form. This is a very interesting Hamiltonian in quantum optics for several reasons. First, it can be solved exactly for arbitrary coupling constants and exhibits some true quantum dynamical effects such as collapse followed by periodic revivals of the atomic inversion.
We consider the description of the quantum dynamics of a two-level system interacting with a single Glauber coherent mode of the radiation field. An exact solution (within the rotating wave and electric dipole approximations) is provided for both the time-independent and time-dependent Green’s functions for the coupled matter–radiation system. The atom (molecular) transition probabilities are shown to be Poisson averages of the corresponding semiclassical problem when the latter is likewise treated in the rotating wave approximation. At low radiation intensities, the quantum and semiclassical results differ, but at high intensities, they become the same when the average photon number n0 in the field satisfies the condition n01/2≫1. A similar behavior is displayed for the expectation value of the electric field of the coherent mode in the presence of the atom. The spectrum of the radiation, resonantly scattered by the atom, is shown to coincide with the results of Mollow.
We use the path-integral technique to solve the quantum-mechanical problem of interaction of a single two-level atom with a single-mode quantized field without the rotating-wave approximation (RWA). We calculate the full spin-Bose propagator in the form of a matrix as an expansion around a path corresponding to the RWA. The effect of energy-nonconserving terms on the atomic population inversion is studied, and we show that even under the conditions in which the RWA is considered to be valid there is a significant effect on the atomic inversion due to these terms.