We show that the interaction between a movable mirror with a quantized field tha t interacts with a two-level atom may be simplified via a transformation that involves Susskind-Glogower operator s (SGO). By using this transformation it is easy to show that we can cast the Hamiltonian, after a unitary transformation, into a Hamiltonian that is equivalent to the ion-laser Hamiltonian. We would like to stress that the transformation in terms of SGO already simplifies e nough the Hamiltonian in the sense that, in an exact way, it eliminates one of the three-subsystems, namely the quantized fie ld.
The basic tool in quantum optics for the description of atomic systems interacting with coherent and incoherent radiation fields is the atomic density matrix governed by the master equation. In the present paper we apply standard condensed-matter nonequilibrium many-body methods to the problem of a two-level atom driven by an external laser field and relaxing by interaction with the radiation field. We recover the well-known dressed-atom picture and, in particular, we present an alternative derivation of the master equation for the atomic density matrix which allows simple generalizations.
The interaction of a propagating pulse of quantum radiation with a localized quantum system can be described by a cascaded master equation with a distinct initially populated input and a finally populated output field mode [Kiilerich and M\o{}lmer, Phys. Rev. Lett. 123, 123604 (2019)]. By transformation to an appropriate interaction picture, we break the cascaded nature of the master equation and recover an effective time-dependent interaction with a lossless single mode and a supplementary lossy mode. The former closely represents the traveling pulse, while the latter constitutes a non-Markovian component in the exchange of quanta between the scatterer and the quantized field. The transformed master equation offers important insights into the system dynamics, and it permits numerically efficient solutions.
This book provides exact analytical solutions to some of the most challenging and exciting problems in quantum optics, ranging from the theory of multi-level atom-radiation field interaction to multi-photon parametric processes in non-linear quantum optics. The problems are generally reduced to two-state dynamics for which detailed reader-friendly solution procedures displaying every important mathematical step are presented. The exact discrete and equivalent continuous variable time evolution operators generate explicitly specified time evolving spin / helicity coherent state vectors and Schroedinger wave functions describing the general quantum states of light, including dissipative dynamics. Photon polarization state dynamics under semi-classical and fully quantized parametric interactions is clearly presented. All the results in fully quantized parametric processes in non-linear quantum optics are new and are appearing for the first time in the literature. The book is particularly useful for graduate level teaching and as a resource text for research in linear and non-linear quantum optics.
In a previous paper we showed how time-dependent projection operators may be employed to enable the Nakajima-Zwanzig projection-operator technique for deriving exact master equations to deal efficiently with two or more coupled classical or quantum systems, neither of which is reservoir like. We considered in detail the case where the relevant part of the classical-system probability-density function (PDF) or quantumsystem density operator (DO) is a product of the PDF's or DO's for the separate subsystems, and we applied the techniques developed to problems in quantum optics and in the kinetic theory of dilute nonideal gases. In this paper we make the time-dependent projection-operator approach useful for a greater variety of systems by allowing the relevant part of the PDF or DO to include correlations between two of the interacting subsystems. This extension allows us to describe well the dynamics of strongly interacting systems in a low degree of approximation while avoiding the use of infinite resummations. We derive exact generalized master equations in this manner for the same two cases as in our earlier work, namely a classical gas of
N molecules interacting via a two-body potential and a quantum-optical system of
N two-level atoms interacting with an electromagnetic field. In the former case, the relevant part of the PDF contains two-body correlations, and we obtain two exact coupled master equations for the singlet and doublet PDF's
F1(t) and
F2(t). In the latter case, the relevant part of the DO contains atom-field correlations, and the result is three exact coupled equations for the single-atom DO ${\ensuremath{\rho}}_{1}(t)$, the field DO
R(t), and the DO for one atom plus the field $\ensuremath{\chi}(t)$. From the exact equations we derive approximate equations by making simplifying assumptions. In the case of the gas we carry out a straightforward density expansion of the ${\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{F}}_{2}$ master equation and obtain a set of kinetic equations for dense gases, which have been derived previously by Frieman, Goldman, and Dorfman and which describe well effects due to the finite mean free path. In the quantum-optical case we derive kinetic equations for a single-mode laser by treating the $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\chi}}$ equation in the Born-Markoff approximation. The equations describe gain saturation and other effects on the field dynamics which are of infinite order in the atom-field coupling constant. It is shown that these equations reduce to previously derived laser master equations if atom-atom correlations can be neglected. Finally, we mention possible generalizations of the time-dependent projection-operator approach used in this paper and briefly discuss its application to other problems, including the kinetics of liquids, collisional line broadening, superradiance, and amplified spontaneous emission.