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.
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.
One of the simplest models involving the atom-field interaction is the coupling of a single two-level atom with single-mode optical field. Under the rotating wave approximation, this problem is reduced to a form that can be solved exactly. But the approximation is only valid when the two levels are resonant or nearly resonant with the applied electromagnetic radiation. Here we present an analytical solution without the rotating wave approximation and applicable to general atom-field interaction far away from the resonance. We find that there exists remarkable influence of the initial phase of optical field on the Rabi oscillations and Rabi splitting, and this issue cannot be explored in the context of the rotating wave approximation. Due to the retention of the counter-rotating terms, higher-order harmonic appears during the Rabi splitting. The analytical solution suggests a way to regulate and control the quantum dynamics of a two-level atom and allows for exploring more essential features of the atom-field interaction.
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.
In this article, by using the perturbation theory, we analytically solve the eigenvalue problem for the Hamiltonian describing the interaction of a Λ -type three-level atom with a single-mode radiation field without the rotating wave approximation (RWA). For this purpose, the atom–field interaction Hamiltonian, which contains the counter-rotating terms (CRTs), is transformed to an analytically solvable Hamiltonian by applying three successive unitary transformations. According to our calculations, the contribution of CRTs within the transformed Hamiltonian is in fact replaced by transforming the 'constant detuning' with the 'intensity-dependent detuning' in the first order, and the 'constant atom–field coupling' with the intensity-dependent coupling in the second order of the perturbation parameters. Then, by solving the eigenvalue problem for the transformed Hamiltonian, the eigenvector of the considered atom–field Hamiltonian is obtained analytically. Finally, after achieving the state vector of the atom–field system at an arbitrary time, a few nonclassical properties of the system state are investigated numerically. Meanwhile, we compare our results with the presence of RWA, from which the role of CRTs will be established.
The interaction of a nonrelativistic, ``two-level'' atom with a single-frequency, quantized electromagnetic field is usually analyzed in the dipole approximation by applying the rotating-wave approximation (RWA) to obtain the exactly solvable Jaynes-Cummings model (JCM). In this paper, it is shown that for a spin-1/2 particle, at rest in a constant magnetic field and for a system that obeys the electric dipole \ensuremath{\Delta}m=\ifmmode\pm\else\textpm\fi{}1 selection rules, a choice of a perturbing field that consists of circularly polarized photons can make the RWA unnecessary for the solution of the quantum-electrodynamic (QED) problem. For these cases, the only approximation necessary to obtain an exact solution of the QED problem is to neglect the spontaneous emission of photons with a sense of polarization opposite that of the perturbing field. The matrix elements corresponding to these ``virtual'' processes are independent of the number of photons n in the applied field and become relatively small, compared with those that correspond to energy-conserving processes, as n becomes large.However, a two-level system that obeys the \ensuremath{\Delta}m=0 selection rule will require the RWA in order to reduce it to the JCM, regardless of the choice of polarization of the perturbing field. For this case, the emission of the photons that do not conserve energy is an induced process that corresponds to matrix elements that increase as \ensuremath{\surd}n+1 . It is thus concluded that, provided the right choice of selection rules and polarization of the applied field is made, the JCM provides a more accurate description of some systems than the accuracy of the RWA. Furthermore, it is concluded that an experimental test of the validity of the RWA can be made by performing the same experiments on an electric dipole \ensuremath{\Delta}m=\ifmmode\pm\else\textpm\fi{}1 transition, first with circularly polarized light and then with linearly polarized light of the same intensity. Any difference in response of the atomic system for these two experiments can be identified with terms that are neglected in the RWA. This experimental technique may prove useful for observing the optical Bloch-Siegert shift and for investigating the failure of the RWA and the chaotic quantum behavior predicted by Milonni, Ackerhalt, and Galbraith [Phys. Rev. Lett. 50, 966 (1983)].
We propose a general perturbative approach to quantum-optical models without the rotating-wave approximation. We show that a generic Hamiltonian describing interaction between two subsystems can be represented as a series of operators corresponding to different transitions between bare energy levels of the whole system. Under certain relations between frequencies of interacting subsystems one of these transitions becomes resonant. The rotating-wave approximation leads to separation of the resonant transition and to appearance of the integral of motion, which makes the problem exactly solvable in this approximation. (Different resonance conditions lead to different integrals of motion.) All of the other terms in these expansion can be considered as a perturbation. They result in dynamic Stark shifts and small corrections to the integrals of motion. All possible resonances are classified, and approximate integrals of motion are found for each resonance. Examples of field-field, field-atom, and atom-atom interactions are considered.
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.