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Journal of the Optical Society of America B

Optical Bloch modeling of magnetic dipole transitions in a four-state system and its application in ion trapping: tutorial

Siann Bester, Christine M. Steenkamp

Published Feb 21, 2023• 4 citations
Publisher

Abstract

This tutorial covers the derivation of the optical Bloch equations for a four-state atomic system interacting with an electromagnetic field via magnetic dipole transitions. The derivation of the equations is discussed at length and from first principles such that any student or graduate with a basic quantum mechanics background could follow. The tutorial shows numerical simulations of the model under different conditions, rounding off the understanding of the dependencies of the model. Finally, the functionality of the model is shown by fitting the model to experimental data of ytterbium-171 ions undergoing Rabi oscillations.

DOI
10.1364/josab.481238

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This tutorial covers the derivation of the optical Bloch equations for a four-state atomic system interacting with an electromagnetic field via magnetic dipole transitions. The derivation of the equations is discussed at length and from first principles such that any student or graduate with a basic quantum mechanics background could follow. The tutorial shows numerical simulations of the model under different conditions, rounding off the understanding of the dependencies of the model. Finally, the functionality of the model is shown by fitting the model to experimental data of ytterbium-171 ions undergoing Rabi oscillations.
This thesis describes work on numerical modelling of the 43Ca+ ion in a Paul trap using the optical Bloch equations. This is a challenging system to study, with many states involved in the internal dynamics. A major outcome is the development of a cooling scheme for the 146.09 gauss atomic clock transitions which makes use of a dark resonance. It is much more effective than methods that avoid coherent effects. The scheme is realised in experiment. Complicated fluorescence data is modelled very well, and predictions for the ion's motional temperature show good agreement with measured values. Data and fits for an ion that has been Doppler cooled below the Doppler limit are presented. I describe GLOBES, a set of routines that simulates an arbitrary ion in the presence of an arbitrary system of laser beams using the optical Bloch equations. Techniques used to efficiently calculate the steady state, analyse fluorescence scans and solve time-dependent problems for small and large times are discussed. A new routine SILVER IMPER that leapfrogs over the initial dynamics to model the approach to the steady state is introduced. Doppler cooling in 40Ca+ is analysed and two extensions made to the basic theory. The ‘excursion method’ of calculation takes account of the non-linear variation with velocity of the scattering rate. The ‘dynamic method’ allows for the fact that the ion may not be in equilibrium with the incident radiation during its oscillations, a necessity as the timescale of the external motion is of order the characteristic timescale of the internal motion for standard secular frequencies. This 'dynamic effect' is a general property of trapped ion systems and is also observed in a two-state system. A two-variable fluorescence scan taken from a four-laser, five-level system in 40Ca+ is analysed. Techniques to fit large data sets and automatically resolve resonant features are demonstrated. A general treatment of resonant behaviour in three, four and five level pump/probe systems is used to describe the data. This is verified by a second, tailor-made set of scans.
This book provides a thorough introduction to the interaction of atoms and atomic ions with optical and magnetic fields. Particular emphasis is placed on the wealth of important multilevel effects, where atomic vapours exhibit anisotropic behaviour. As well as covering the classic two-level atom approach to light-atom interactions, a general multi-level formalism is also described in detail, and used to discuss optical pumping, two-dimensional spectroscopy and nonlinear optical dynamics. The final chapter deals with the mechanical effects of light, including the cooling and trapping of atoms. With full theoretical and experimental coverage, and over 250 illustrations, the book will be of great interest to graduate students of laser spectroscopy, quantum electronics and quantum optics, and to researchers in these fields.
We derive a set of optical Bloch equations (OBEs) directly from the minimal-coupling Hamiltonian density of the bound-state quantum electrodynamics (bound-state QED). Such optical Bloch equations are beyond the former widely-used ones due to that there is no electric dipole approximation (EDA) on the minimal-coupling Hamiltonian density of the bound-state QED. Then our optical Bloch equations can describe a two-level atom interacting with a monochromatic light of arbitrary wavelength, which are suitable to study the spectroscopy and the Rabi oscillations of two-level atoms in X-ray laser beams since that the wavelength of X-ray is close to an atom to make the electric dipole approximation (EDA) invalid.
This tutorial article provides a detailed mathematical overview of the Optical Bloch equation describing the interaction of the two level atom with the Electromagnetic radi- ation field. The equations have been derived in the weak atom field coupling limit. The notion of the optical Bloch equations has been introduced in a systematic process starting from the interaction of the two level atom with the semiclassical radiation field which cannot completely the idea of the Spontaneous emission process later which was included theoretically by,Weigner-Weisskopf theory of spontaneous emission .In the later part of the article the optical Bloch equations has been derived exactly by treating the radiation field quantum mechanically. The radiation field has been quantized using the free field quantization techniques in the Coloumb Gauge with which the Lindblad Master equation has been derived systematically in the weak coupling limit. The optical Bloch equations has been derived for the resonantly driven two level atom from the Lindblad type master equation and later those equations has been solved in the steady state limit. The article also discuss the analytical aspects of describing the theory of atom matter interaction using full-fledged quantum mechanical approach.
Quantum Simulation of Spin Models with Trapped Ions C. Monroe, W. C. Campbell, E. E. Edwards, R. Islam, D. Kafri, S. Koren- blit, A. Lee, P. Richerme, C. Senko, and J. Smith Joint Quantum Institute University of Maryland Department of Physics College Park, MD 20742, USA Summary. — Laser-cooled and trapped atomic ions form an ideal standard for the simulation of interacting quantum spin models. Effective spins are represented by appropriate internal energy levels within each ion, and the spins can be measured with near-perfect efficiency using state-dependent fluorescence techniques. By ap- plying optical fields that exert optical dipole forces on the ions, their Coulomb inter- action can be modulated in ways that give rise to long-range and tunable spin-spin interactions that can be reconfigured by shaping the spectrum and pattern of the laser fields. Here we review the theory behind this system, recent experimental data on the adiabatic prepration of complex ground states and dynamical studies with small collections of ions, and speculate on the near future when the system becomes so complex that its behavior cannot be modeled with conventional computers. PACS 03.67 – a. PACS 37.10 – Ty. PACS 75.25 – j. Introduction The advent of individual atomic control with external electromagnetic fields, both involving internal states through optical pumping and external states through laser cool- c Societ` a Italiana di Fisica