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Frontiers in Optics

Modelling Trapped Ion Qubits: Optical Bloch Equations of Magnetic Dipole Transitions in a Four State System

Siann Bester, Christine M. Steenkamp

Published Jan 1, 2022• 0 citations
Publisher

Abstract

An optical Bloch equation model is derived for a four state system interacting with electromagnetic radiation via magnetic dipole transitions. Applying the model to Rabi oscillation data of trapped ytterbium-171 ions contributes to experimental characterization.
DOI
10.1364/fio.2022.fth3c.8

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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.

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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.
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 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.
Abstract Numerical simulations are used to study unsharp measurements in a system of trapped ytterbium ions to monitor resonantly driven Rabi oscillations of a target ion by weak coupling to an auxiliary ion. The simulation combines the positive operator-valued measure formalism and optical Bloch equations to model the state of the target ion during the implementation of the unsharp measurement protocol. State monitoring is done using an estimation scheme that is updated with the unsharp measurement results. The model expands on previous work by explicitly including spontaneous emission into the calculations. We propose a method to determine convergence of the estimation scheme, useful in an experimental setting, which does not rely on using the target state. The simulation results show the feasibility conditions for implementing unsharp measurements experimentally and demonstrate the robustness of the estimation scheme in the presence of experimental noise and when the qubit Rabi frequency is not precisely known. We find that the timescale of transitions that change the motional state is the main technical challenge for experimental implementation.
This paper provides the first experimental test of the Bloch equations for an optical transition in a low-temperature impurity-ion solid, Pr3+{\mathrm{Pr}}^{3+}Pr3+: LaF3{\mathrm{F}}_{3}F3​, where the condition ${T}_{1}\ensuremath{\gg}{T}_{2}$ applies. Intensity-dependent optical free-induction-decay measurements are performed by use of an ultrastable cw dye laser (linewidth, 300 Hz) and reveal an $\mathrm{exp}(\ensuremath{-}\frac{2t}{{T}_{2}})$ decay law at low Rabi frequency ($\ensuremath{\chi}$), as predicted by the Bloch equations, and an $\mathrm{exp}(\ensuremath{-}\ensuremath{\chi}t)$ decay at high $\ensuremath{\chi}$, which violates the Bloch prediction but agrees with Redfield's characterization of NMR saturation in solids.
The quantum Rabi model describes the interaction between a two-level quantum system and a single bosonic mode. We propose a method to perform a quantum simulation of the quantum Rabi model, introducing an implementation of the two-level system provided by the occupation of Bloch bands in the first Brillouin zone by ultracold atoms in tailored optical lattices. The effective qubit interacts with a quantum harmonic oscillator implemented in an optical dipole trap. Our realistic proposal allows one to experimentally investigate the quantum Rabi model for extreme parameter regimes, which are not achievable with natural light-matter interactions. When the simulated wave function exceeds the validity region of the simulation, we identify a generalized version of the quantum Rabi model in a periodic phase space.