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Exploring the Solvability of the Jaynes-Cummings and Jaynes-Cummings-like Models: Implementing Quantum Control

Austen Couvertier

Published Jan 1, 2017• 0 citations
OpenAlex

Abstract

In this paper we aim to explore the dynamics and overall solvability of the Jaynes-Cummings \& Jaynes-Cummings-Like models. As a lens to understand these dynamics, we focused on cases where the parameters of the system were made time-dependent. All previous work on solving the dynamics of the Jaynes-Cummings models has relied heavily on the use of differential methods and setting the parameters as time-independent constants which were zero or one. To account for this, we utilized the Wei-Norman method which allowed us to analytical solve the time-dependent Hamiltonian. Through the use of this method, we can understand the more general characteristics of the dynamics associated with the Jaynes-Cummings model. This research related back to the field of Quantum information and control as the model highlights interactions with qubits and our time-dependent parameters can force certain transitions.

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In this paper we aim to explore the dynamics and overall solvability of the Jaynes-Cummings & Jaynes-Cummings-Like models. As a lens to understand these dynamics, we focused on cases where the parameters of the system were made time-dependent. All previous work on solving the dynamics of the Jaynes-Cummings models has relied heavily on the use of differential methods and setting the parameters as time-independent constants which were zero or one. To account for this, we utilized the Wei-Norman method which allowed us to analytical solve the time-dependent Hamiltonian. Through the use of this method, we can understand the more general characteristics of the dynamics associated with the Jaynes-Cummings model. This research related back to the field of Quantum information and control as the model highlights interactions with qubits and our time-dependent parameters can force certain transitions.
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.
Jaynes–Cummings (JC) models are by far the most widely used to study the dynamics of two-level systems, such as atoms or molecules under suitable conditions, coupled to quantized electromagnetic fields. Nowadays, analytical solutions of JC models are more useful than ever for understanding and controlling radiation-matter interactions relevant to quantum information processing, quantum computing, and other quantum technologies. This study presents the analysis and generalization of a method for the solution of the dynamical problem in the time-independent JC model (Carbonaro et al., 1979) that enables an exact description of the atom-field dynamics in the resonant time-dependent JC model and in a nontrivial class of off-resonance time-dependent JC models. Our analysis produces the ready-to-use forms of the evolution operator in the resonant and non-resonant cases, which are then used to derive the evolution of the atomic population inversion for some types of atom-field coupling. The central idea behind the proposed approach offers opportunities for its extension to other time-dependent cases and its generalization to JC-related models.
The interaction between an atom and a one mode external driving field is an ubiquitous problem in many branches of physics and is often modeled using the Rabi Hamiltonian. In this paper we present a series of analytically solvable Hamiltonians that approximate the Rabi Hamiltonian and compare our results to the Jaynes-Cummings model which neglects the so-called counter-rotating term in the Rabi Hamiltonian. Through a unitary transformation that diagonalizes the Jaynes-Cummings model, we transform the counter-rotating term into separate terms representing several different physical processes. By keeping only certain terms, we can achieve an excellent approximation to the exact dynamics within specified parameter ranges.