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Atomic decay and re-absorption in inhomogeneous one-dimensional cavities: An example with an analytical solution

Martin Ligare

Published Jan 1, 2002• 0 citations
OpenAlex

Abstract

I use Laplace transforms to flnd the complete analytical solution for the quantum dynamics of a single two-level atom interacting with the quantized modes of an inhomogeneous one-dimensional multimode optical cavity. Lapleace transforms were used by Stey and Gibberd [Physica, 60, 1-26 (1972)] on several model Hamiltonians, and this paper extends their techniques to cover decay in a cavity that has regions with two difierent indexes of refraction.

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I use Laplace transforms to flnd the complete analytical solution for the quantum dynamics of a single two-level atom interacting with the quantized modes of an inhomogeneous one-dimensional multimode optical cavity. Lapleace transforms were used by Stey and Gibberd [Physica, 60, 1-26 (1972)] on several model Hamiltonians, and this paper extends their techniques to cover decay in a cavity that has regions with two difierent indexes of refraction.
Within the framework of a two-dimensional microscopic, purely quantum mechanical model, we analyse the dynamics of single-photon wave packets interacting with optical elements (beam splitters, mirrors), modelled as systems of two-level atoms. That is, we utilize a two-dimensional cavity to simulate the quantum behaviour of simple optical components and networks made thereof. The field is quantized using the canonical procedure, and only the basis states with one unit of excitation are included. This, however, covers linear optical phenomena. The field is taken to interact with localized atoms through a dipole interaction. Using different configurations of atoms, and choosing their frequencies to be resonant or off-resonance, we can model mirrors, beam splitters, focusing devices and multicomponent systems. Thus we can model arbitrary linear networks of optical components. We show the time evolution of a photon wave packet in an interferometer as an example. As the state of the field is known at each instant, spectral properties and spatial coherence can immediately be obtained from the simulations. We also know the states of the two-level atoms constituting the components, which allows us to consider their quantum behaviour. Here the decay of an excited atom into the vacuum state of the electromagnetic field in the two-dimensional cavity is studied.
Starting from first principles we simulate numerically the dynamics of a system of two-level atoms interacting with a multimode electromagnetic field in a one-dimensional cavity. In particular, we investigate the spontaneous emission of an excited two-level atom in the cavity. We show how the decay depends on the mode spectrum and on the position of the atom in the cavity. We study how the spontaneous emission is modified when the initially excited atom is surrounded by a material medium modeled as a system of two-level atoms. We also study the propagation of photon wave packets emitted by the atom within the material media. Our microscopic model provides us with a deeper understanding of the atom-field interaction and offers us a framework for a systematic investigation of the transition from a microscopic to a macroscopic (phenomenological) description of the spontaneous decay in material media (e.g., dielectrics).
The dynamics of an ensemble of two-level atoms moving through a single-mode lossless cavity is investigated in the semiclassical and rotating-wave approximations. The dynamical system for the expectation values of the atomic and field observables is considered as a perturbation to one of the following integrable versions: (i) a model with atoms moving through a spatially inhomogeneous resonant field, and (ii) a model with atoms interacting with a nonresonant eigenmode which is assumed to be homogeneous on the cavity size. We find the general exact solutions for both the models and show that they contain special solutions describing a coherent effect of population and radiation trapping. Using the Melnikov method, we prove analytically transverse intersections of stable and unstable manifolds of a hyperbolic fixed point under a small modulation of the vacuum Rabi frequency. These intersections are believed to provide the Smale horseshoe mechanism of Hamiltonian chaos. The analytical results are accompanied with direct computation of topographical maps of maximal Lyapunov exponents that give a representative image of regularity and chaos in the atom-field system in different ranges of its control parameters--the frequency detuning, the number, and the velocity of atoms.
Due to the improvement within cavity quantum electrodynamics experiments during the last decades, what was former seen as 'toy models' are today realized in laboratories. A controlled isolated coherent evolution of one or a few atoms coupled to a single mode inside a cavity is achievable. Such systems are well suited for studying purely quantum mechanical effects, and also for performing quantum gates, necessary for quantum computing. The Jaynes-Cummings model has served as a theoretical description of the interaction. However, as the experimental techniques are improved, for example, atom cooling, the use of multi-level atoms or multi-modes and driving of atoms or elds by external lasers, extensions of the original Jaynes-Cummings model are needed. In this thesis we study some of these extended models, and in particular multi-level models, time-dependent models and quantized motion models. Both analytical and numerical analysis are considered. The two-level structure of the Jaynes-Cummings model leads to applications of known solvable time-dependent two-level Schrödinger equations. In other cases, di erent forms of adiabatic approximate solutions are used, and with the analytically solvable models, the amplitudes of non-adiabatic contributions may be estimated. For higher dimensional systems, STIRAP and multi-STIRAP methods are applied. It is shown how the time-dependent models may be used for preparation of various kinds of non-classical states, and also to generate universal sets of quantum gates, both on atomic and eld qubits. When the atoms are cooled to very low temperatures, their velocities must be treated quantum mechanically, and we have studied the dynamics of such cases for di erent coupling shapes. Again numerical and analytical approaches have been used and compared, wave-packet propagations of the atom, approaching and traversing the cavity, have been performed. For periodic couplings, standing wave cavity modes, the dynamics has been described by e ective parameters; group velocity or atomic index of refraction and effective mass. Tunneling resonances for ultra cold atoms have been exhibited in the STIRAP models for certain initial conditions.
We present the complete analytical solution of the cavity QED of a two-level atom and a field mode at zero temperature. It includes both dissipation of the field due to a finite Q value of the cavity and incoherent decay mechanisms for the atom. This analytical solution is provided by a powerful method for treating general master equations that appear in quantum optics. As distinct from the usual approaches we first deal with that part of the master equation which describes the dissipative coupling of the field and the atom to their reservoirs. Rather than using number-state or dressed-state bases we expand the density operator into the eigenstates of the nonunitary parts of the master equation which model the dissipative part of the dynamics. The set of these eigenstates is the damping basis. With the aid of this expansion we find the eigenvalues and eigenstates of the total Liouville operator. The evolution of an arbitrary initial state is then known. We employ these results to give an exact solution of the dynamics of the photon field in realistic experiments with one-atom masers at very low temperatures. It includes detuning, cavity leakage effects, spontaneous decay mechanisms for the atoms, a Fizeau-type velocity distribution for the atomic beam, and a statistical parameter for the probability of the excitation of incoming atoms, covering the limits of Poissonian pumping and of regular pumping. On the same grounds one can treat the one-atom laser, consisting of a single atom which stays in permanent interaction with the field mode and which is continuously pumped by external heat baths.