1. INTRODUCTION

The interaction of classical light with quantum systems is of relevance in many fields of photonics, including the fields of laser cooling and trapping [1,2], femtosecond laser-induced electron dynamics [3], magnetic resonance imaging [4], and the physics behind qubit manipulation and quantum computing [5].

The optical Bloch equations (OBEs) provide a framework to simulate the coherent exchange of energy between a quantum system and an electromagnetic (EM) driving field in the presence of dissipative dynamics [5,6]. In general, results of this type of modeling can be used to fit experimental data in order to extract detailed information on experimental conditions and parameters. The OBEs were first proposed in 1946 [7] and have since been adapted for use in different scenarios, for example with solids [8], entangled states [9], and inclusion of stochastic relaxation processes [10]. The OBEs are of particular interest for quantum computing, where the current challenge is to minimize dissipation in order to preserve coherence over increasing time scales [6].

While the two-level and sometimes three-level OBEs for electric dipole transitions are discussed in textbooks, there is, to our knowledge, no accessible introduction to the OBEs for magnetic dipole transitions for systems with more than three levels aimed at the audience of university students or professionals in industry. This paper is aimed to meet this need.

When using trapped atoms or ions as qubits in experimental quantum systems, a long coherence time is a crucial requirement [11]. Therefore, it is important that the transition between the chosen qubit states is not an electric dipole transition with a high spontaneous decay rate. Ground state hyperfine levels that are separated by magnetic dipole transitions (i.e., electric dipole forbidden) in the microwave frequency range are desirable since they have extremely long radiative lifetimes. Hyperfine qubits have been implemented for a large number of trapped ion systems such as ${^9{{\rm Be}}^ +}$, ${^{25}{{\rm Mg}}^ +}$, ${^{43}{{\rm Ca}}^ +}$, ${^{87}{{\rm Sr}}^ +}$, ${^{138}{{\rm Ba}}^ +}$, ${^{171}{{\rm Yb}}^ +}$, ${^{173}{{\rm Yb}}^ +}$, ${^{111}{{\rm Cd}}^ +}$, and ${^{199}{{\rm Hg}}^ +}$ [11].

This paper is aimed at an application to a hyperfine qubit system of ytterbium-171 ions (${^{171}{{\rm Yb}}^ +}$) with a magnetic dipole transition between a single ground state and an excited level consisting of three states that may be degenerate or non-degenerate, depending on the presence of an external magnetic field. The system interacts with a single microwave field. If the aim of an experiment is to control the ${^{171}{{\rm Yb}}^ +}$ system so that it functions as an approximation for the ideal two-state qubit, then the four-state model is necessary to investigate under which conditions the system may deviate from the desired two-state behavior. Parameters including the polarization and amplitude of the microwave field in combination with the static magnetic field strength may result in deviation from the ideal qubit behavior. To demonstrate the potential of the four-state OBEs, we also show how it can be adapted to reproduce a three-state V-type system [12]. Additionally, we simulate examples showing how electromagnetically induced transparency and population inversion can be achieved in this four-level system interacting with a microwave field. Modeling of interactions with an EM field consisting of more than one frequency component is outside the scope of this paper, but interested readers are referred to papers by Boublil [12] and Gu [13] for guidance on how to adapt the model for this purpose.

This tutorial guides the reader through the derivation of the OBEs for a four-state system interacting with an EM field from first principles in a semi-classical framework. The system’s Hamiltonian is defined as the sum of the free atomic Hamiltonian and the magnetic dipole interaction Hamiltonian; thus, the system can be solved using the time-dependent Schrödinger equation (TDSE). To complete the model, spontaneous decay and decoherence are included in the resulting equations in their appropriate forms. Throughout the derivation, the relevant approximations, transformations, and rearrangements are discussed, enabling the reader to adapt the model to different arrangements of states and either magnetic or electric dipole transitions. The final set of coupled first-order differential equations can be used to model the time evolution of the populations and coherences of the four-state atom interacting with an EM field and can be manipulated to obtain steady state values.

We demonstrate an application of the model to the hyperfine qubit in the ${^2{S}_{1/2}}$ ground state of ${^{171}{{\rm Yb}}^ +}$ ions in an ion trapping experiment. The general model is adapted to this transition in ytterbium, and the results of numerical simulations are fitted to experimental data of Rabi oscillations to distinguish between different imperfections and loss mechanisms in the ion trapping experiment. Fundamental research on ytterbium is currently in demand as ${^{171}{{\rm Yb}}^ +}$ ions are promising qubits in quantum computing and technology. One of the first commercial ion-based quantum computers uses ${^{171}{{\rm Yb}}^ +}$ ions as qubits [14–16]. In our laboratory, the isotopes of ytterbium are used in a cost-effective experiment to realize unsharp quantum measurements [17].

In Section 2, the equations are derived. Section 3 presents simulation results that showcase the ability of the model to predict the behavior of the interacting system. Section 4 describes the experimental measurement of Rabi oscillations of ${^{171}{{\rm Yb}}^ +}$ ions and the fitting of the model results to the data. The conclusions follow in Section 5.

It is assumed that the reader knows the basics of EM fields, atomic structure, and the Zeeman effect. Textbooks aimed at the senior undergraduate level (for example, [18]) would be useful as reference material.

2. DERIVATION OF THE OPTICAL BLOCH EQUATIONS FOR A GENERAL FOUR-STATE ATOM

Consider a four-state atom with a single ground state, denoted by $|g\rangle$, and an excited level that consists of three states given by $| {-} 1\rangle$, $|0\rangle$, and $|1\rangle$. The labels ${-}1,0,1$ refer to the state’s magnetic quantum number, ${m_F}$, as explained in Appendix B. Figure 1 shows a typical energy level diagram. In the presence of an external static magnetic field, the excited level splits into three non-degenerate sub-levels due to the Zeeman effect. These sub-levels have an energy splitting, ${\delta _{{\rm Zeeman}}}$, which is dependent on the static magnetic field strength (see Section 2.C for details). In the derivation of the model, the excited states are treated as non-degenerate. However, by setting the energy splitting of the excited states to zero, the model can be applied to a degenerate system.

 

Fig. 1. Simplified energy level diagram of the qubit transition of ${^{171}{{\rm Yb}}^ +}$. The green arrows in the figure represent the photon energy (in terms of angular frequency $\omega$) of the electromagnetic field that drives the transitions.

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This model describes how the four-state atom interacts with an electromagnetic (EM) field. If the EM field’s photon energy is near the energy difference between the ground state and the excited states, the EM field drives the transition between them. A single field drives the atoms between their ground state and a linear superposition of the three excited states. This model is very general in the sense that it can be applied to any four-state atom interacting with an EM field. For instance, it can easily be adapted to a four-state atom with a degenerate ground state and singlet excited state. The model can be adapted easily for either magnetic dipole allowed transitions or electric dipole allowed transitions. Guidelines on these adaptations are given in Appendix A.

For the purpose of this tutorial, the model is applied to the magnetic dipole allowed transition between the ${^2{S}_{1/2}}\,F = 0$ and ${^2{S}_{1/2}}\,F = 1$ hyperfine levels of ${^{171}{{\rm Yb}}^ +}$ ions, where $F$ is the quantum number for the total angular momentum, driven by a microwave field. For more details on the atomic states and quantum numbers, see Appendix B. The energy level diagram of the relevant states in ytterbium is shown in Fig. 1. The transition frequencies (in units of angular frequency) between the ground state and excited states will be labeled as ${\omega _{\textit{ig}}}$ for $i = \{- 1,0,1\}$. The green arrows in the figure represent the single EM field with angular frequency $\omega$ driving the transitions. For a non-degenerate system, the field can only be resonant with at most one transition ($\omega = {\omega _{\textit{ig}}}$), while still driving the other transitions non-resonantly.

A. Derivation of the Equations Describing Driven Processes

To model this system, the TDSE must be solved. The first step is to define the system’s state and the Hamiltonian that governs the system. As the atom can be in a superposition of all four states, the system’s state can be defined as

The system is governed by the Hamiltonian $H$ given by

Here, ${B_0}$ is the amplitude of the magnetic field, and ${\hat \epsilon _B}$ is the direction of the magnetic field component of the EM field, which is perpendicular to the electric field polarization direction $\hat \epsilon$.

The system has now been defined. However, the Hamiltonian can be changed to a form that is easier to use by rewriting the magnetic dipole operator in the notation that describes its effect on the state of the system. In other words, the interaction Hamiltonian needs to be written in bracket notation. A series of transformations and definitions will follow in pursuit of converting the interaction Hamiltonian into a convenient form.

First, the EM field can be rewritten as the sum of oppositely rotating terms,

The magnetic dipole operator can be expanded in terms of the states of the system by using the identity, $I$, defined as

Therefore, the magnetic dipole operator can be rewritten as

If this equation is expanded, it will consist of 16 terms where each term describes the transition between a state $|i\rangle$ and a state $|j\rangle$. However, only a few of these terms are significant, with the rest either representing forbidden transitions or terms that are negligible for other reasons. If we were considering electric dipole allowed transitions with an electric dipole moment ${{\boldsymbol \mu}_{\boldsymbol e}}$, then it could be argued that only the terms of the form $\langle i|{{\boldsymbol \mu}_{\boldsymbol e}}|g\rangle$ when $i \in \{- 1,0,1\}$ are significant, assuming that the excited state $|i\rangle$ and the ground state $|g\rangle$ are of opposite parity (see Appendix A for more details). However, in this case, we are dealing with a magnetic dipole transition that is allowed when the magnitude or the direction of the atomic magnetic dipole moment changes during the transition. Therefore, terms with the factor of the form $\langle i|{\boldsymbol \mu}|i\rangle$ when $i \in \{g, - 1,0,1\}$ are zero. However, terms of the form $\langle g|{\boldsymbol \mu}|i\rangle$ and $\langle i|{\boldsymbol \mu}|j\rangle$ for $i,j \in \{- 1,0,1\}$ and $i \ne j$ represent magnetic dipole allowed transitions and, thus, should remain in the expression for ${\boldsymbol \mu}$. Note that we are making an assumption that these terms are real and, hence, $\langle i|{\boldsymbol \mu}|j\rangle = \langle j|{\boldsymbol \mu}|i\rangle$. Therefore, the magnetic dipole operator can be simplified as follows:

We will show that terms with the form $\langle i|{\boldsymbol \mu}|j\rangle$ for $i,j \in \{- 1,0,1\}$ and $i \ne j$ can also be neglected in the rotating wave approximation, finally yielding a similar result to the electric dipole case.

To further simplify the magnetic dipole operator, raising and lowering operators are defined as

With this notation, the magnetic dipole operator can be written as follows:

Following from Eq. (8), the interaction Hamiltonian can be written as

When the magnetic dipole operator [Eq. (16)] is substituted into Eq. (17) and we multiply out, the result will contain many terms, each consisting of two rotating parts, namely a raising or lowering operator and a magnetic field part. The magnetic field terms rotate at the driving frequency of the EM field $\omega$, which in practice is near the resonance frequency between the ground and excited states, ${\omega _{\textit{ig}}}$ for all $i \in \{- 1,0,1\}$. In the case of ytterbium, the resonance frequency ${\omega _{\textit{ig}}}$ is approximately 12 GHz. In the experiment done by Matjelo [20], whose experimental data are used for comparison with our model, they introduced a static magnetic field of 3.4 gauss over the ions in order to split the hyperfine states and eliminate degeneracy among these levels. More details can be found in Appendices B and C. The Zeeman splitting for the $F = 1$ level for the magnetic field of 3.4 gauss is then ${\omega _{\textit{ij}}} = 4.67\;{\rm MHz} $ when $i \ne j$. Thus, the Zeeman levels are split by a frequency, which is almost 4 orders of magnitude smaller than the resonance frequency. We know that ${\sigma _i}$, ${\sigma _{\textit{ij}}}$, and ${{\boldsymbol B}^{(+)}}(t)$ rotate in the same direction since their time dependencies have a negative sign in the power, while $\sigma _i^\dagger$, $\sigma _{\textit{ij}}^\dagger$, and ${{\boldsymbol B}^{(-)}}(t)$ all rotate in the opposite direction. This is useful as the rotating wave approximation can be invoked to simplify this equation.

In the rotating wave approximation, fast rotating terms are considered to average to zero on the time scales of the slower rotations; therefore, the fast rotating terms are neglected. Thus, in our expression, terms such as ${\sigma _i}{B^{(+)}}(t)$ and $\sigma _i^\dagger {B^{(-)}}(t)$ contain two fast rotations in the same direction so that the frequencies add to $({\omega _{\textit{ig}}} + \omega) \approx 24\,{\rm GHz}$. Terms such as ${\sigma _i}{B^{(-)}}(t)$ and $\sigma _i^\dagger {B^{(+)}}(t)$ contain two fast rotations in opposite directions so that the frequencies subtract to $({\omega _{\textit{ig}}} - \omega) \,\approx\def\LDeqbreak{} 5\;{\rm MHz} $. Terms such as ${\sigma _{\textit{ij}}}{B^{(+)}}(t)$, ${\sigma _{\textit{ij}}}{B^{(-)}}(t)$, $\sigma _{\textit{ij}}^\dagger {B^{(+)}}(t)$, and $\sigma _{\textit{ij}}^\dagger {B^{(-)}}(t)$ rotate at $({\omega _{\textit{ij}}} \pm \omega) \approx 12\;{\rm GHz} $. Therefore, by making the rotating wave approximation, we can neglect all the terms oscillating on the GHz time scale and only keep the terms that oscillate on the MHz time scale. However, this then introduces certain constraints on the system such as it then requires that the frequency of the EM field driving the system should never be far detuned from the ${\omega _{\textit{ig}}}$ resonance frequencies (${\Delta _{\textit{ig}}} = \omega - {\omega _{\textit{ig}}} \ll {\omega _{\textit{ig}}}$). This also implies that the Zeeman splitting should be much smaller than the ${\omega _{\textit{ig}}}$ resonance frequencies (${\omega _{\textit{ij}}} \ll {\omega _{\textit{ig}}}$). The rotating wave approximation is consistent with the choice of not including any other energy levels of the atom in the model, as within this limit other energy levels will always be far off resonance ([19], page 151).

Thus, in the rotating wave approximation, the interaction Hamiltonian is simplified to

By substituting the definitions of both the magnetic field as well as the raising and lowering operators into Eq. (18), what remains is

The Rabi frequency is defined as follows:

The Rabi frequency, for the transition between the ground state $|g\rangle$ and excited state $|i\rangle$, is the expectation value of the scalar product between the magnetic dipole moment vector of the atom and the magnetic field vector of the EM field. In the presence of an additional static magnetic field, as would be the case for our application, this static field direction defines the quantization axis (usually denoted as the $z$ axis) relative to which the orientation of the magnetic dipole moments of the excited states are described. If the static magnetic field is the only field present and there has not been an oscillating field present for a long time compared to the leading dissipative time scale of the atom, the magnetic dipole moment of the atom in each state has a well-defined orientation relative to the static magnetic field. The direction of the oscillating magnetic field with respect to the static field has to be taken into account to determine the relative magnitudes of ${\Omega _{- 1}}$, ${\Omega _0}$, and ${\Omega _1}$. This is discussed further in Section 2.C.

Finally, the full Hamiltonian can be expressed as the sum of both the atomic Hamiltonian and the simplified atom-field interaction Hamiltonian,

The Hamiltonian contains three types of terms. The first line represents the time-independent terms that describe the energies of the four states in our system, namely, the three excited states and the ground state that is taken to have zero energy. The second and third lines describe the time-dependent terms that arise due to the presence of the oscillating EM field. The second line represent the terms that describe stimulated emission, while the third line describes absorption. Here, the Rabi frequencies determine the coupling strength for each transition. This Hamiltonian only includes stimulated processes and not phenomena such as decoherence and spontaneous emission. These will be addressed in Section 2.B.

The time evolution of this system is obtained by solving the TDSE,

The full Hamiltonian in Eq. (21) and the state defined in Eq. (1) are inserted into the TDSE. The left hand side of the equation, after simplification, can be expressed as

Since the coefficients of the states contain all the time dependence, the right hand side of the equation can be written as

By taking an inner product with $\langle i|$ for each $i \in \{g, - 1,0,1\}$, respectively, the TDSE results in the following four equations:

These are the equations that describe the time evolution of the expansion coefficients for each of the states and, hence, describe the four-state system. This is an abstract way of visualizing the system’s evolution as the expansion coefficients have no direct physical description.

A more intuitive description is provided by the density matrix formalism. In this formalism, the time evolution of the population of each state and of the coherences between states are described. The population of each state is a physical attribute of that state and describes the probability of finding the atom in that state in the case of a single atom, or when considering an ensemble of atoms it gives the fraction of the total number of atoms that are in that particular state. The coherence provides information about the relative phase between different states. If two states are coherent, then they have a well-defined relative phase.

To map the differential equations for the expansion coefficients [Eqs. (25)–(28)] into the density matrix formalism, a transformation into a rotating frame is made. The frame rotates at the frequency $\omega$ of the EM field. This is done by defining slowly varying expansion coefficients in the rotating frame for each of the excited states $|j\rangle$ with $j \in \{- 1,0,1\}$,

The time derivatives of the expansion coefficients in the rotating frame are calculated using the product rule for differentiation. Taking ${\tilde c_0}$ as example,

By substituting Eq. (27) into Eq. (30), the corresponding differential equation for ${\tilde c_0}$ in the rotating frame is obtained,

The resulting differential equations for the four-state system in the rotating frame are:

To describe the dynamics of the populations and coherences of the system, the density matrix formalism is introduced. It is assumed that the system is in a pure state; therefore, the density matrix operator in the rotating frame is defined as

The matrix elements of the density matrix can then be defined as

Therefore, the density matrix for this system has the following form:

In this matrix, elements with two identical subscripts represent the population of that state, while elements with different subscripts represent the coherence between those two states. Note that ${\tilde \rho _{\textit{ii}}} = {\tilde c_i}\tilde c_i^* = {c_i}{e^{i\omega t}}c_i^*{e^{- i\omega t}} = {c_i}c_i^* = {\rho _{\textit{ii}}}$ for populations and that the rotation falls away for coherences between two excited states, ${\tilde \rho _{\textit{ij}}} = {\tilde c_i}\tilde c_j^* = {c_i}{e^{i\omega t}}c_j^*{e^{- i\omega t}} = {c_{\!j}}c_{\!j}^* = {\rho _{\textit{ij}}}$, where $i,j \in \{- 1,0,1\}$.

Using the relationship between the expansion coefficients and the populations and coherences, we obtain the differential equations for the density matrix elements. The time derivative of each density matrix element is calculated in terms of the expansion coefficients as follows:

The differential equations for the expansion coefficients [Eqs. (34)–(37)] are substituted into Eq. (42). The resulting equations are known as the OBEs for this system:

Note that Eqs. (47)–(52) each have a complex conjugate that is not written out. The complex conjugates are related by

B. Optical Bloch Equations Including Decay Parameters

The OBEs [Eqs. (43)–(52)] describe all the stimulated processes occurring in the system, such as stimulated emission and absorption—however, they do not include incoherent decay processes such as spontaneous emission or decoherence.

To include the incoherent decay processes present within the system, it is accepted practice to add phenomenological decay terms to the differential equations for the density matrix elements, Eqs. (43)–(52), ([19], page 177). These phenomenological terms include the spontaneous decay of the excited states down to the ground state by the spontaneous emission of a photon, which occurs at a decay rate $\Gamma$, as well as spontaneous decay among the excited states, at the decay rate ${\Gamma _0}$ [13]. In our application, both of these spontaneous decay processes are only possible by means of magnetic dipole transitions; therefore, the transition rates are slower by a factor of approximately ${(\frac{{{\mu _B}/c}}{{e{a_0}/Z}})^2} \approx 0.07$ compared with that of electric dipole transitions ([18], Appendix C). Spontaneous decay rates are proportional to ${\omega ^3}|\mu {|^2}$; therefore, the decay between the different excited states is approximately ${10^{10}}$ times smaller than the decay from an excited state to the ground state. This means that ${\Gamma _0} \ll \Gamma$. Within this system, the spontaneous decay rates of the excited states to the ground state, $\Gamma$, are approximately the same for all three excited states and are treated as equal in the model. The decay rates between excited states, ${\Gamma _0}$, are also considered to be the same between any two of the excited states. The dephasing of the coherences between the ground state and excited states are described by a decay rate $\gamma$, while the dephasing of the coherences between two excited states occurs at a rate ${\gamma _ \bot}$. Finally, the collisional rate is given by the decay rate ${\gamma _{\rm{col}}}$. These five parameters represent the incoherent decay processes to be included in the model.

The decay rates $\Gamma$ and ${\Gamma _0}$ are characteristics of a particular atom; hence, once the system is defined with a specific atom in mind, these decay rates are determined. The collisional rate ${\gamma _{\rm{col}}}$ is determined by the physical conditions of the systems such as the pressure, temperature, and density of atoms. The collisions do not change the state of the atoms but instead change the phase relation. Therefore, the collisional rate can be thought of as adding to the rate of dephasing between two states, which causes the coherence between these states to decay.

Using the method described in an article by Boublil et al. [12], the dephasing decay rates can be calculated. The total decay rate of the population of a particular state (${\rho _{\textit{ii}}}$ where $i \in \{g, - 1,0,1\}$) is defined as ${\Gamma _{i,{\rm Total}}}$, which is the sum of all the decay paths from that particular state to the other three states. The ground state does not decay spontaneously,

The dephasing rate ${\gamma _{i,j}}$ of the coherences (${\rho _{\textit{ij}}}$, where $i,j \in \{g, - 1,0,1\}$ and $i \ne j$) is given by Boublil [12] as

Therefore, the coherence between an excited state and the ground state decays at a rate of

Including all of these terms into the model and defining the detuning of the laser frequency from the atomic frequency as ${\Delta _{\textit{ig}}} = \omega - {\omega _{\textit{ig}}}$, the OBEs are given below. This set of 16 first-order coupled differential equations [Eqs. (59)–(74)] represents the general OBEs for the four-state system under consideration, interacting with a monochromatic constant EM field,

Solving these equations numerically yields the transient time-dependent values of the populations (${\rho _{\textit{gg}}}$, ${\rho _{- 1 - 1}}$, ${\rho _{00}}$, ${\rho _{11}}$) showing Rabi oscillations and of the coherences (${\tilde \rho _{g0}}$, ${\tilde \rho _{g1}}$, ${\rho _{01}}$, etc.). The damping terms cause the transient behavior to eventually decay to steady state values on time scales longer than the inverse of the coherence decay rate $\gamma$. The steady state values can be calculated by setting each rate ${\partial _t}{\rho _{\textit{ij}}}$ in the equations above to zero and solving the resulting equations numerically.

C. Adaption of the Model to the ${^2{S}_{1/2}}$$F = 0 \to F = 1$ Transition in ${^{171}{{\rm Yb}}^ +}$

In this section, we will make a few simplifications to the model in order to adapt it to the transition in the [Xe] $4{{\rm f}^{14}}6{\rm s}{^2}{S_{1/2}}$ hyperfine levels in ${^{171}{{\rm Yb}}^ +}$ ions. More details on the atomic levels and quantum numbers can be found in Appendix B. The ground state has its quantum number $F = 0$ while the excited level has $F = 1$. The excited level has three states with ${m_F} = \{- 1,0,1\}$. These states can be degenerate or non-degenerate depending on whether there is an external magnetic field present. The presence of a static external magnetic field lifts the degeneracy of the excited states due to the Zeeman effect. The Zeeman splitting, ${\delta _{{\rm Zeeman}}}$, between neighboring excited states and given in angular frequency (${\rm rad \cdot {\rm s}^{- 1}}$) is given by

The transition between the ground and excited states is driven by a constant continuous microwave field, which is generated by a precision signal generator, amplified, and emitted from a microwave horn. The frequency of these microwaves is set to the resonance frequency of the transition between the ground state and the excited state with ${m_F} = 0$ and has a precise frequency of $\omega = 12.6428\;{\rm GHz} $. This value can be altered to detune the field from this resonance. The microwave horn emits linearly polarized radiation in a solid angle, and the horn can be oriented such that the polarization drives the appropriate transitions within the atoms. The propagation direction of the microwave field is oriented perpendicular to the direction of the static magnetic field (a typical choice). The component of the microwave field that is polarized perpendicularly to the static magnetic field (meaning that its magnetic field oscillates parallel to the static magnetic field) drives the $\pi$ transition (${m_F} = 0 \to {m_F} = 0$). The component of the microwave field that is polarized parallel to the static magnetic field (meaning that its magnetic field oscillates perpendicular to the static magnetic field) drives the $\sigma$ transitions (${m_F} = 0 \to {m_F} = \pm 1$). Note that the polarization direction is the direction of the electric field component of the EM field; hence, the magnetic field component is perpendicular to the polarization. The polarization directions driving the $\pi$ and $\sigma$ components of a magnetic dipole transition are exactly opposite to those driving the $\pi$ and $\sigma$ components of an electric dipole transition (compare [21] to [18], Section 1.8).

If the magnetic field component of the field emitted by the microwave horn is oriented at an angle $\theta$ from the static magnetic field direction, then it will have magnetic field components both parallel and perpendicular to the static magnetic field. Therefore, this angled field can drive both $\pi$ and $\sigma$ transitions. The angle, $\theta$, determines the relative magnitudes of the Rabi frequencies, ${\Omega _{- 1}}$, ${\Omega _0}$, and ${\Omega _1}$. For the purpose of the simulations, we would like all the Rabi frequencies to have the same magnitude; hence, $|{\Omega _{- 1}}| = |{\Omega _0}| = |{\Omega _1}| = \Omega$. To meet this condition, the angle of the magnetic component of the EM field relative to the static field direction should be $\theta \approx {55^ \circ}$. This angle agrees with experimental results showing that the $\pi$ component of the magnetic dipole emission along the transverse direction is double as intense as the $\sigma$ component [21]. Combined with the relation between the Rabi frequency and emission rate in steady state [19], this gives the condition $2{I_\pi} = {I_\sigma}$. At this orientation, the magnitudes of the Rabi frequencies are all equal. This will be the scenario explored in the simulations, unless otherwise specified.

The model can be implemented with different Rabi frequencies for the $\sigma$ and $\pi$ transitions ($|{\Omega _{- 1}}| = |{\Omega _1}|$ but differing from $|{\Omega _0}|$), meaning experimentally the microwave horn has been rotated to change the orientation of the polarization of the microwaves.

For convenience let us define a single detuning, $\Delta$, to be the frequency difference between the microwave field frequency and the frequency of the $|g\rangle \to |0\rangle$ transition, ${\omega _{0g}}$. Therefore,

Using these simplifications, the transitions between the hyperfine states of the ${^2{S}_{1/2}}$ level in ${^{171}{{\rm Yb}}^ +}$ ions can be simulated. We coded these 16 coupled differential equations in Mathematica where it is then possible to solve the system of equations numerically, resulting in the time evolution of all populations and coherences. An example of a typical Mathematica workbook used in the simulations is available as Code 1, Ref. [22] .

3. SIMULATION RESULTS

A. Simulation Results for a Degenerate Atom

In order to illustrate the basic behavior of the system, consider an ensemble of atoms with a degenerate excited level, interacting with a continuous monochromatic EM field. This means that, for this simulation, the static magnetic field was switched off; thus, the Zeeman splitting was zero, ${\delta _{{\rm Zeeman}}} = 0$. Each simulation below was done by solving the OBEs numerically using NDSolve in Mathematica. The relevant simulation parameters are listed in the figure captions (more detail on each simulation’s parameters can be found in Table 3 in Appendix D).

The result of a general simulation is the time evolution of the total excited state population, given by the sum, ${\rho _{- 1 - 1}} + {\rho _{00}} + {\rho _{11}}$. The total population (${\rho _{\textit{gg}}} + {\rho _{- 1 - 1}} + {\rho _{00}} + {\rho _{11}}$) is normalized to 1 so that the values given for the population of any state correspond to a fraction of the total population of the ensemble of atoms. An alternative interpretation of the data is to consider a single atom rather than an ensemble of atoms. In this case, the density matrix elements ${\rho _{\textit{ii}}}$ for $i = \{g, - 1,0,1\}$ are interpreted as the probability to find the atom in state $|i\rangle$. However, in the descriptions that follow, we will consider an ensemble of atoms.

Figure 2 illustrates the effect of a driving field with different field amplitudes, resulting in different Rabi frequencies interacting with degenerate atoms. The frequency of the field is on resonance with the ground to excited state transition, and the collisional rate is set to zero. The field drives the total excited state population to oscillate between zero and one. These oscillations are named Rabi oscillations and describe the stimulated emission and absorption of light when the atoms are driven by a coherent field. Something to notice is that the set Rabi frequency (given in the legend) and the observed Rabi frequency (obtained from the oscillation periods of the curves) do not match. For example, the light blue curve with a set Rabi frequency of $\,\Omega = 50\; {\rm rad \cdot {\rm s}^{- 1}}$ should have a period, which is

 

Fig. 2. Effect that varying the Rabi frequency has on the time evolution of the total excited state population.

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However, the observed period of oscillation for the light blue curve is ${T_{{\rm observed}}} = 0.073 \;{\rm s}$, which corresponds to an observed Rabi frequency of $86.1 \;{\rm rad \cdot{\rm s}^{- 1}}$. Thus, the set Rabi frequency is approximately $1.72 \approx \sqrt 3$ times smaller than the observed Rabi frequency. This factor of $\sqrt 3$ arises due to the fact that the three excited states are degenerate. Therefore, the field drives a single transition from the ground state to an excited state, which is an equal linear combination of the three excited states, $(| - 1\rangle + |0\rangle + |1\rangle)/\sqrt 3$. This discrepancy between the set Rabi frequency and the observed Rabi frequency is not present when the three excited states are separated by the presence of a static magnetic field and the field only populates one of the excited states (since the other two are far-detuned), and then the observed Rabi frequency will equal the set Rabi frequency.

When the detuning of the EM field is changed, the field will not drive the transition as efficiently as it would when it is on resonance. In Fig. 3, the effect of varying the detuning is illustrated, and two observations can be made. First, as the detuning increases, so does the frequency of the observed Rabi oscillations even though the set Rabi frequency is constant. Second, the amplitude of the Rabi oscillations decreases with an increasing detuning. This agrees with the notion of a generalized Rabi frequency $\tilde \Omega$ in the case of non-zero detuning for a two-state system ([19], page 160),

 

Fig. 3. Effect that varying the detuning of the driving field $\Delta$ has on the time evolution of the total excited state population.

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This means that, when the transition is driven by a strong field near resonance, the amplitude is 1 [19]. Note that both Eqs. (78) and (79) were derived for a two-state, two-level system; however, in this case, we have a two-level, degenerate four-state system. Noting this, we would like to compare the observed Rabi frequency that can be identified from the simulation result with the theoretical values that can be calculated using Eqs. (78) and (79), in order to identify whether we can model the change of the Rabi frequency and the Rabi amplitude using these equations. Table 1 summarizes these findings. It shows the agreement of the Rabi frequency and oscillation amplitude observed in simulation and calculated by Eqs. (78) and (79). Thus, it is shown that the equations do apply to a degenerate four-state system of our kind.

Table 1. Rabi Frequencies Observed in the Simulations Compared to the Generalized Rabi Frequencies Which Were Calculated from Eq. (78)^a

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Figure 4 illustrates the effect of introducing collisions into the model by varying the collisional rate ${\gamma _{{\rm col}}}$ from zero to 10 collisions per second. When the collisional rate is set to zero (light green curve), the system undergoes Rabi oscillations between an excited state population of 0 and 1. However, as the collisional rate is increased to $1\;{{\rm s}^{- 1}}$ (blue curve), the Rabi oscillations decay slowly. By increasing the collisional rate to $10\;{{\rm s}^{- 1}}$ (orange curve), the Rabi oscillations decay more rapidly to an approximately constant population after 0.8 s. This constant population is the steady state value to which the total excited state population decays. The steady state value is 0.75 due to the degeneracy of 3 in the excited state. Reaching this steady state population of 0.75 is what happens in the scenario of strong pumping or “saturation” of the transition. This saturation steady state is plotted in Fig. 4 as the dashed black line at 0.75. This is the same steady state excited state population that would be achieved if the same system was modeled with the classical rate equation model under saturation conditions. This agrees with the concept that increasing the collisional rate leads to decoherence so that the system tends to the classical description. However, if the pumping was below saturation, meaning that the Rabi frequency is greatly reduced, then as the collisional rate increases, the Rabi oscillations will decay in the same fashion; however, the steady state excited state population will be smaller than 0.75.

 

Fig. 4. Effect that varying the collisional rate ${\gamma _{{\rm col}}}$ (with $\Delta = 0$) has on the total excited state population in time.

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B. Simulation Results for a Non-Degenerate Atom

In the experimental setup of Matjelo [20], ${^{171}{{\rm Yb}}^ +}$ ions are exposed to a microwave field of 12.64 GHz, and the excited states are non-degenerate due to Zeeman splitting in the presence of an external static magnetic field. The external magnetic field is set at 3.4 gauss ($3.4 \times {10^{- 4}}$ tesla) and hence the Zeeman shift can be calculated using Eq. (75). For the ${^2{S}_{1/2}}\,F = 1$ level in ${^{171}{{\rm Yb}}^ +}$ ions, ${g_F} = 1$ and thus the splitting between neighboring states is approximately 4.76 MHz (see Appendix B for details). When, for instance, the microwave radiation is resonant with the ground to excited state with ${m_F} = 0$ transition, then the ${m_F} = - 1$ transition sees radiation that is detuned by ${\Delta _{- 1g}} = {\delta _{{\rm Zeeman}}}$, while the ${m_F} = 1$ transition sees radiation that is detuned by ${\Delta _{1g}} = - {\delta _{{\rm Zeeman}}}$.

An example of a simulation where the microwave radiation was detuned from the central transition is shown in Fig. 5. In this simulation, the detuning of the radiation was set to $\Delta = 0.003{\delta _{{\rm Zeeman}}}$ such that the field is only slightly off resonance with the transition between the ground state and the excited state with ${m_F} = 0$. Due to this detuning, the field does not drive the transition as efficiently as a resonant drive frequency, and the excited state population of the $|0\rangle$ state experiences Rabi oscillations with a maximum population, which is less than 1. Only the $|0\rangle$ state excited state population undergoes Rabi oscillations, and the excited state populations for the ${m_F} = \pm 1$ states are negligible throughout the simulation. This implies that, even though we are treating this system as a four-state system interacting with the field, two of the three transitions are not being driven by the field as the field is too far-detuned from these transition resonances. Therefore, the population evolution within the system would be approximately the same if it was modeled by a two-state model including only the ground state and the excited state with ${m_F} = 0$ and the relevant decay processes.

 

Fig. 5. Excited state population evolution for a simulation where the detuning of the driving field is set to $\Delta = 0.003{\delta _{{\rm Zeeman}}}$. The Rabi frequency was set to $\Omega = 1.81 \times {10^5}\; {\rm rad \cdot{\rm s}^{- 1}}$, and the magnetic field was set to 3.4 G.

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However, this is not always the case. In order to identify when it is sufficient to model the system as a two-state system or when it is necessary to include all three excited states in the model, a simulation was done where the detuning of the field was scanned over the three transitions. For each detuning value, a simulation was run, and the maximum value of the total excited state population reached was saved and plotted against the detuning for that simulation, as shown in Fig. 6. In the figure, the detuning is expressed relative to the Zeeman shift, which in this case is 4.76 MHz. As the detuning is scanned, three resonances are identified by the peaks situated at $\Delta = 0, \pm {\delta _{{\rm Zeeman}}}$. These correspond to the three transitions between the ground state and each of the excited states.

 

Fig. 6. Maximum value of the total excited state population for a range of detuning values calculated at different Rabi frequencies.

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The Rabi frequency was also varied, as shown by the three different colored curves. As the Rabi frequency increases (corresponding to pumping with a higher intensity field), the peaks broaden. This is called power broadening and is caused by the saturation of the transition by the on-resonance field, while the excited state population continues to increase for off-resonance fields. This highlights that, when a transition is driven by an off-resonant field, a larger maximum total excited state population can be achieved by increasing the Rabi frequency.

When the system is pumped with a high intensity, as in the case where the Rabi frequency is $\Omega = 10 \times 1.81 \times {10^5}\; {\rm rad \cdot{\rm s}^{- 1}}$, the resonance peaks of neighboring transitions overlap substantially, meaning that all three transitions will be driven by an EM field tuned to one of the peaks. From these results, it is concluded that under strong pumping the two-state model will not describe our system adequately and the use of the four-state model is required due to the peak broadening and overlap.

Lastly, the effect of varying the collisional rate within the system was investigated. This simulation was done with a Rabi frequency of $\Omega = 1.81 \times {10^5}\; {\rm rad \cdot{\rm s}^{- 1}}$. Since changing the collisional rate causes the amplitude of the Rabi oscillations to decay over time, the steady state value that the total excited state population reaches after the oscillations have decayed was calculated. Figure 7 shows the steady state value of the total excited state population over a range of detunings. The figure shows four simulations that have different collisional rates, namely, ${\gamma _{{\rm col}}} = 0,1,20$ and $1.25 \times {10^3}\; {{\rm s}^{- 1}}$. The steady state excited state population for a collisional rate of zero is included in this figure as a reference for the steady state behavior of the standard three resonance peaks.

 

Fig. 7. Steady state excited states population for a range of detuning values calculated at different collisional rates.

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As the collisional rate increases, the resonance peaks broaden, and the maximum steady state value increases until it reaches 0.75. This is what is expected in the case of strong pumping and has been shown before in Fig. 4. An interesting feature that is shown in this simulation is that, when the collisional rate is high, as in the case of ${\gamma _{{\rm col}}} = 20$ and $1.25 \times {10^3}\; {{\rm s}^{- 1}}$, the steady state excited state population does not taper off quickly beyond the ${m_F} = \pm 1$ resonances but tends to zero very slowly. This means that even when the microwave frequency is far off resonance, the steady state excited state population is non-zero. If one considers the OBEs for the coherences between the ground and excited states [Eqs. (63)-(68)] for the case when the collisional rate, which is included in $\gamma$, is much larger than the detuning $\Delta$, then $\gamma$ dominates the behavior of the evolution of the coherences; thus, the steady state excited state population decreases more slowly with detuning. However, the rotating wave approximation becomes invalid in the case of significant detuning; thus, our model cannot accurately describe this scenario of a large detuning. In experimental control of qubits, such conditions are avoided; hence, it is not necessary to model their behavior.

C. Simulations at Variable External Magnetic Field Strengths

The simulations in the previous section highlight how the $^{171}{{\rm Yb}^ +}$ system deviates from the ideal two-state system when the transitions are strongly driven or when there is a large collisional rate, even when the Zeeman splitting of the $|\! - \!1\rangle$ and $|1\rangle$ states is in the MHz frequency range. The deviations become more significant if the Zeeman splitting is smaller, and this could be the case in an experimental setup due to technical limitations on the magnetic field. The effect of coherence on these types of systems is very sensitive to the magnetic field parameters [23]. Therefore, we simulated scenarios in which the magnetic field is varied.

First, we consider the scenario of using the ${^{171}{{\rm Yb}}^ +}$ system as a qubit; hence, the aim is to have an ideal two-state system. The system is driven by microwaves resonant to the $|g\rangle \to |0\rangle$ transition, and its polarization direction is oriented appropriately so that the resonant qubit transition ($\pi$ transition) is pumped 10 times more efficiently than the $|g\rangle \to |\! \pm \!1\rangle$ transitions ($\sigma$ transitions), meaning that $\frac{1}{{10}}{\Omega _0} = {\Omega _1} = {\Omega _{- 1}}$. The collisional rate and detuning are zero, and the Rabi frequency is small enough to avoid power broadening. This section investigates the effect of varying the strength of the static magnetic field on the behavior of the populations and coherences, both in their steady state and transient forms. Note that the static magnetic field strength affects the Zeeman splitting and, hence, changing the field strength causes the excited states to split more or less.

In Fig. 8, the steady state populations are plotted in dependence of the static magnetic field ${B_{{\rm ext}}}$. At ${B_{{\rm ext}}} = 0$ (the degenerate case), it is expected and observed that both the populations of the ground and $|0\rangle$ states, ${\rho _{\textit{gg}}}$ and ${\rho _{00}}$, are close to 0.5. In contrast, ${\rho _{- 1 - 1}}$ and ${\rho _{11}}$ have small values in the steady state due to being driven 10 times weaker than the $\pi$ transition, yet still having the same decay rates as $|0\rangle$. In the limit of a large external magnetic field, ${\rho _{\textit{gg}}}$ and ${\rho _{00}}$ tend to 0.5, and ${\rho _{- 1 - 1}}$ and ${\rho _{11}}$ tend to 0 due to the increasingly large detuning of the driving field to the $\sigma$ transition resonances. The dip (peak) in ${\rho _{\textit{gg}}}$ and ${\rho _{00}}$ (${\rho _{- 1 - 1}}$ and ${\rho _{11}}$) at $\approx 0.01$ gauss has been first described by Boublil et al. [12] for a V-type three-state system. When the driving field is symmetrically detuned, in this case symmetrically relative to the $|\! - \!1\rangle$ and $|1\rangle$ states, and the Rabi frequency is resonant with the total splitting between the states ($2{\delta _{{\rm Zeeman}}} = {\Omega _0}$ in this simulation), then the rate equation prediction of equal populations in all states is obtained. Boublil et al. describe this as a saturation resonance [12]. We will call the strength of the external static magnetic field at which this saturation resonance takes place ${B_{{\rm ext,res}}}$.

 

Fig. 8. Steady state populations of each state in the model as a function of the static external magnetic field. In this simulation, the detuning was set to zero, $\Delta = 0\,{\rm rad \cdot{\rm s}^{- 1}}$, and the $|g\rangle \to |0\rangle$ transition was driven 10 times stronger than the other two transitions. Therefore, ${\Omega _0} = 1.81 \times {10^5}\;{\rm rad \cdot{\rm s}^{- 1}}$ and ${\Omega _{- 1,1}} = 1.81 \times {10^4}\; {\rm rad \cdot{\rm s}^{- 1}}$.

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In the context of the OBEs, the rate equation limit is the limit where effects of coherence vanish (see Steck [19], page 189). Under conditions of sufficiently intense pumping (saturation) and equal spontaneous decay rates, the rate equation limit predicts equal distribution of population over all states in the steady state (see Steck [19], page 95).

This $2{\delta _{{\rm Zeeman}}} = {\Omega _0}$ resonance behavior has a significant impact on the time evolution of the transient populations of the qubit states. The time evolution of the populations is plotted in Fig. 9(a) for the resonance condition $2{\delta _{{\rm Zeeman}}} = {\Omega _0}$ at ${B_{{\rm ext,res}}} = 0.0104$ gauss and in Fig. 9(b) for ${B_{{\rm ext}}} = 0.0167$ gauss. In contrast to the ideal two-state qubit, where the total population will continue to oscillate between states $|g\rangle$ and $|0\rangle$, the Rabi oscillations in Fig. 9(a) are strongly modulated. For an ideal qubit, any number of complete Rabi oscillations will bring the quantum system back to its original state. However, in Fig. 9(a), the total population flows from the $|g\rangle$ and $|0\rangle$ states to the $| \!-\! 1\rangle$ and $|1\rangle$ states and back on the time scale of approximately 14 Rabi oscillations. The saturation resonance peak is sharp; therefore, at ${B_{{\rm ext}}} = 0.0167$ gauss, the modulation of the Rabi oscillations is much smaller and faster. The four-state model can, thus, predict the deviation from ideal qubit behavior and be used to determine the minimum magnetic field needed for the required qubit quality.

 

Fig. 9. Time evolution of the populations of each state in the model at an external magnetic field (a) ${B_{{\rm ext}}} = 0.0104$ gauss and (b) ${B_{{\rm ext}}} = 0.0167$ gauss. The detuning was set to $\Delta = 0 \;{\rm rad \cdot{\rm s}^{- 1}}$, and all the Rabi frequencies were equal, ${\Omega _{- 1}} = {\Omega _0} = {\Omega _1} = 1.81 \times {10^5}\; {\rm rad \cdot{\rm s}^{- 1}}$.

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To further investigate the dependence of the saturation resonance on the absolute Rabi frequency and to confirm its origin in the case of the four-state system, a simulation was done with all of the Rabi frequencies set to be equal to one another (${\Omega _0} = {\Omega _1} = {\Omega _{- 1}}$). Figure 10(a) confirms that the resonance condition is $2{\delta _{{\rm Zeeman}}} = {\Omega _0}$ since the magnetic field at which the saturation resonance takes place shifts linearly with ${\Omega _0}$. Figure 10(b) shows the populations of all four states as a function of the magnetic field, confirming that all the populations reach the rate equation prediction of equal populations at the resonance.

 

Fig. 10. (a) Ground state population in the steady state for a range of magnetic fields highlighting the effect of the Rabi frequency on the saturation resonance. (b) Steady state populations of each state in the model as a function of the static external magnetic field and for a Rabi frequency of $\Omega$. For both simulations, the detuning was set to $\Delta = 0\; {\rm rad \cdot{\rm s}^{- 1}}$, and all the Rabi frequencies were equal, ${\Omega _{- 1}} = {\Omega _0} = {\Omega _1}$ and $\Omega = 1.81 \times {10^5}\; {\rm rad \cdot{\rm s}^{- 1}}$.

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The expression $|{\rho _{- 10}}{|^2}$ quantifies the coherence between two of the excited states. The steady state absolute value squared of this coherence is plotted in Fig. 11 and shows that there is a minimum coherence between the excited states at the saturation resonance. This minimum coherence provides us with a reason for why the populations tend to the rate equation prediction at this point. This agrees with the definition of the rate equation limit. The coherence is largest when ${B_{{\rm ext}}} = 0$, which is where the excited states are degenerate.

 

Fig. 11. Steady state absolute value squared of the coherence ${\rho _{- 10}}$ for a range of magnetic fields. In this simulation, all three of the transitions were equally driven, $\Omega = 1.81 \times {10^5}\; {\rm rad \cdot{\rm s}^{- 1}}$. The detuning of the field was set to zero, $\Delta = 0$.

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In Fig. 12, the imaginary parts of ${\tilde \rho _{g - 1}}$ and ${\tilde \rho _{g0}}$ are plotted as these are proportional to the one-photon absorption associated with the $|g\rangle \to |\!-\!1\rangle$ and $|g\rangle \to |0\rangle$ transitions, respectively. Due to symmetry, the imaginary parts of ${\tilde \rho _{g - 1}}$ and ${\tilde \rho _{g1}}$ are equal; hence, Fig. 12(a) also represents the imaginary component of the coherence ${\tilde \rho _{g1}}$. In the absence of resonance effects, the one photon absorption in the $|g\rangle \to | \!-\! 1\rangle$ transition is expected to peak at ${B_{{\rm ext}}} = 0$ as this is when the transition frequency is on resonance with the microwave field frequency. As ${B_{{\rm ext}}}$ increases, the transition frequency is tuned further from the microwave frequency; thus, one would expect the coherence ${\tilde \rho _{g - 1}}$ to decrease. However, the peak (of which only half is seen in the graph) in Fig. 12(a) shows a dip (increased transmission) at ${B_{{\rm ext}}} = 0$ and an absorption peak at the saturation resonance. This is a typical result of EM-induced transparency [24]. The absorption behavior of the $|g\rangle \to |0\rangle$ transition, seen in Fig. 12(b), is different as this transition stays on resonance with the microwave field and is affected by mechanisms of optical pumping and saturation. Its absorption would have been unaffected by ${B_{{\rm ext}}}$ (that means constant) if the $| {\pm} 1\rangle$ states were not pumped, meaning that the broad dip at small ${B_{{\rm ext}}}$ values is caused by the coupling to the other excited states. These are, to our knowledge, new results for a four-state system.

 

Fig. 12. Steady state value of the imaginary component of the coherence (a) ${\tilde \rho _{g - 1}}$ and (b) ${\tilde \rho _{g0}}$ for a range of magnetic fields. All three of the transitions were equally driven, $\Omega = 1.81 \times {10^5}\; {\rm rad \cdot{\rm s}^{- 1}}$. The detuning of the field was set to zero, $\Delta = 0$.

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Finally, a simulation that has little relevance for qubit manipulation, but shows the possibility of obtaining population inversion in a four-state system, is shown in Fig. 13. Population inversion was shown to be achievable in V-type three-state systems [12]. In the three-state system, the condition for achieving population inversion is that the symmetry between the two excited states must be broken either by making the decay rates of the excited states different or by applying asymmetrical detunings of the driving fields from the two states [12]. In the four-state system, it is possible to use the ground state and two of the excited states (say $| \!-\! 1\rangle$ and $|0\rangle$) as an effective three-state system by setting ${\Omega _1} = 0$ and, thus, switching the fourth state ($|1\rangle$) off. With this adjustment, our model reproduced the relevant results from Boublil et al. [12] on the three-state V-type system perfectly (specifically Figs. 8–10 of Boublil’s paper). The natural question was whether switching the $|1\rangle$ transition back on would serve to break the symmetry and cause population inversion, while the two conditions above hold. This simulation was implemented with symmetric detuning with respect to the $| {-} 1\rangle$ and $|0\rangle$ states, meaning that the microwave frequency was tuned to halfway between $| {-} 1\rangle$ and $|0\rangle$. The Rabi frequencies of all the transitions are set to be equal, ${\Omega _{- 1}} = {\Omega _0} = {\Omega _1}$, and the decay rates of the three excited states were equal as usual. Figure 13 shows the populations of the four states as a function of the static magnetic field strength, ${B_{{\rm ext}}}$. In a region near the saturation resonance ${B_{{\rm ext,res}}} \approx 0.02$ gauss, the ground state population ${\rho _{\textit{gg}}}$ decreases below 0.30 while ${\rho _{- 1 - 1}}$ peaks above 0.36, meaning that population inversion has been achieved under conditions that would not have allowed it in a three-state system.

 

Fig. 13. Steady state populations of each state in the model as function of the static external magnetic field. The detuning was set to $\Delta = {\delta _{{\rm Zeeman}}}/2\; {\rm rad \cdot{\rm s}^{- 1}}$. All the Rabi frequencies were equal, ${\Omega _{- 1}} = {\Omega _0} = {\Omega _1} = 1.81 \times {10^5}\; {\rm rad \cdot{\rm s}^{- 1}}$.

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4. EXPERIMENTAL RABI OSCILLATION MEASUREMENT

This section of the tutorial is aimed at using the model we have developed to fit experimental data. The model was fitted to the experimentally recorded Rabi oscillations of a cloud of trapped ${^{171}{{\rm Yb}}^ +}$ ions obtained by the ion trapping group at Stellenbosch University. More information on the experiment and a reproduction of the data can be found in the dissertation of Matjelo [20]. The shaded region in Fig. 14 is the envelope of the raw experimental data, with the upper and lower bounds representing the minima and maxima of the error bars of the data as smoothed functions. One can interpret this as the envelope of the time evolution of the excited state population, which is driven by a microwave field over a period of approximately 260 µs.

 

Fig. 14. Experimental data showing the excited state population evolution of a cloud of ${^{171}{{\rm Yb}}^ +}$ ions undergoing Rabi oscillations [20]. The shaded region shows the envelope of the error bars on the experimental data. The solid curve shows the four-state optical Bloch model, which was fitted to the experimental data. The best fit parameters for the model can be found in Table 2, in the second column.

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In his work, Matjelo fitted a two-level optical Bloch model to the experimental data. His model did not include any decoherence or damping terms into the equations for the two qubit states. However, he did include an exponential decay factor that was multiplied to his excited state population solution for reasons that will be explained later. We aim to use our four-state model, with all the decay and decoherence terms, to improve on the results of Matjelo’s model such that we can better identify the experimental conditions. Before fitting our model to the data, it is important to understand the experiment itself in order to identify any adaptions or additional terms that should be included into the model.

In this experiment, ytterbium-171 atoms are ionized, trapped, and cooled in a linear Paul ion trap within a vacuum chamber. The cooled ions are prepared in their ${^2{S}_{1/2}}\,F = 0$ ground state. The ions are then exposed to a microwave field for a precisely measured period of time given by the time coordinate of each data point. During this time, the ions are driven by the microwave field and undergo Rabi oscillations. After the interaction with the microwaves, a measurement of fluorescence is used to detect in which state the ions are in. Each ion in the cloud is a quantum particle in a superposition of the ground and three excited states; however, when measured, the ion’s state collapses into one of these states. If you only had one ion, you would need to repeat the experiment multiple times in order to build up a probability distribution to identify which superposition the ion was in prior to the measurement. However, in a cloud of hundreds of ions, one measurement of fluorescence intensity provides us with a quantity that can be normalized to a value between 0 and 1, which gives us the probabilistic measurement in one go. This process of cooling, state preparation, exposure to the driving microwave field, and fluorescence measurement is repeated, and every data point on the graph is the average of 50 such measurements.

In the data shown in Fig. 14, one can see eight clear peaks that correspond to eight Rabi oscillations. Initially, the cloud of ions starts in a superposition. The excited state is initially partially populated with an excited state population of about 0.2. During the first Rabi oscillation period, the ions are driven into the excited state, but the maximum excited state population is only about 0.75 and not 1. In the experimental data, the amplitude of the Rabi oscillations decays over the duration of the experiment. In addition, this behavior does not match the decay seen for a non-zero collisional rate, as shown in Fig. 4, because the data tends to a value below the initial minimum of the oscillation, which should not be the case if the initial population in the experiment is conserved.

There are many explanations for this decay in the amplitude of the Rabi oscillations, and it is important to note that more than one decay mechanism can play a role. In the cloud of ytterbium ions, it is possible that the ions are colliding with one another or at least interacting with their neighbors as they oscillate in the cloud due to the trapping parameters. These interactions and/or collisions could be a source of decoherence. However, we have already noted that the decay in the excited state population time evolution does not follow the population decay behavior seen for a non-zero collisional rate. A second cause of decay could be that the total population of ions within the trap is decreasing or ions are being lost from the trap. Ions escaping the trap correspond to the total population decaying.

One way in which ions can escape the trap is if they have enough kinetic energy to overcome the potential well keeping them trapped. Within the vacuum chamber, the cloud of ytterbium ions is not alone. There are background gas particles. The pressure in the vacuum system in this experiment was on the order of ${10^{- 9}}\; {\rm Torr}$ (whereas the recommended pressure for ion trapping is ${10^{- 11}}\; {\rm Torr}$), meaning a density of $3 \times {10^7}$ atoms per cubic cm with speeds most likely larger than 200 m/s. Collisions with background gas particles could mean loss of ions from the trap, and this is not modeled by the collisional rate term in the optical Bloch model. Another reason for the loss of ions from the trap is if the ions are not cooled well enough and, thus, have excess kinetic energy. The lasers used to do the Doppler cooling were not frequency stabilized during the measurement, and it is possible that, over the course of the experiment, the frequency of the cooling laser drifted so that the efficiency of the cooling decreased.

In order to determine the relative importance of collisional decoherence and loss of ions from the trap on the decay behavior, the model was adapted to take the ion loss into account and then fitted to the experimental data. The ion loss is modeled as the exponential decay of the total population. In order to obtain the modeled excited state population evolution, the standard simulation is run, and the final result is multiplied by an exponential decay term with decay rate $\tau$. The same approach was taken in the fit of Matjelo’s two-state model [20].

 

Fig. 15. Sum of squares error (SSE) at specific Rabi frequencies ($\Omega$) as a function of the collisional decay rate (${\gamma _{\rm{col}}}$) and the ion loss rate ($\tau$). The parameter values where the SSE is minimized are $\Omega = 1.807 \times {10^5}\; {\rm rad \cdot{\rm s}^{- 1}}$, ${\gamma _{\rm{col}}} = 1.50 \times {10^3}\; {{\rm s}^{- 1}}$, and $\tau = 4.68 \times {10^3}\; {{\rm s}^{- 1}}$.

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We identified three fitting parameters within the model, which were the Rabi frequency ($\Omega$), the collisional decay rate (${\gamma _{\rm{col}}}$), and the ion loss rate ($\tau$). The detuning of the microwave radiation was zero, while the spontaneous decay rate was assumed to be very small; hence, we set it to $\Gamma = 0.01\; {{\rm s}^{- 1}}$. In order to identify the best fit for each of the fitting parameters, a program was designed that varied each of these parameter values around their best estimates. Each simulation result of the excited state population evolution was compared to the experimental data by calculating the sum of squares error (SSE) between the model and the data. In this way, we were able to determine which of these simulations, and thus which fitting parameter values, best suited the data. By repeating and minimizing the SSE throughout this process, we were able to extract the best fit Rabi frequency, collisional rate, and ion loss rate. The minimum value of the SSE was 0.223, which occurred when the Rabi frequency was $\Omega = 1.807 \times {10^5}\; {\rm rad \cdot{\rm s}^{- 1}}$, the collisional rate was ${\gamma _{\rm{col}}} = 1.50 \times {10^3}\; {{\rm s}^{- 1}}$, and the ion loss rate was $\tau = 4.68 \times {10^3}\; {{\rm s}^{- 1}}$. For comparison, the SSE for the fit done by Matjelo was 0.248.

Figure 15 shows the effect of varying the fitting parameters on the value of the SSE. The collisional rate (${\gamma _{\rm{col}}}$) and the ion loss rate ($\tau$) are plotted on the horizontal axes, while the two curved surfaces (in orange and blue) represent simulations done with different Rabi frequencies. This figure only shows two different Rabi frequencies, but in the simulation many more were included. This is shown in Fig. 16, where the minimum value of the SSE is plotted for each Rabi frequency used within the simulation.

 

Fig. 16. Minimum SSE for each Rabi frequency simulation.

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It was considered relevant to quantify how sensitive the fit is for a variation of the values of the three fitting parameters, $\Omega ,{\gamma _{\rm{col}}}$, and $\tau$. For that purpose, we chose a particular value, $\delta {\rm SSE}$, by which we increased the SSE from its minimum value (${{\rm SSE}_{{\min}}}$). Starting with the best fit values of all three parameters, and varying the value of one parameter at a time, we calculated by what amount the value of the parameter has to change in order for the SSE of the fit to increase to ${{\rm SSE}_{{\min}}} + \delta {\rm SSE}$. The choice of the value $\delta {\rm SSE}$ is arbitrary, and we chose it to be 1% of ${{\rm SSE}_{{\min}}}$. Using this value for $\delta {\rm SSE}$, the results for the changes in $\Omega ,{\gamma _{\rm{col}}}$ and $\tau$, as given in Table 2, show that the fit is most sensitive for a change in the Rabi frequency, as a change of only 0.13% increases the SSE by the chosen amount. This is expected as varying the Rabi frequency will alter the central position of each peak and, hence, will have a large effect of the fit of the model. Looking at the two decay parameters, we see that the fit is almost 20 times more sensitive for a change in the ion loss rate than for a change in the collisional rate. This means that, while the ion loss rate is sensitive and varying it changes the model drastically, varying the collisional rate by a large amount does not have a large effect on the model. This is because, on the time scale of this measurement, the collisions cause the Rabi oscillations to decay only slightly but would have a larger effect on the steady state population (long times). The effect that each decay parameter has on the population evolution can be interpreted as follows: the ion loss rate determines the general structure of the evolution and, thus, the fit, while the collisional rate is a fine tuning parameter to optimize the fit.

Table 2. Values of the Fitting Parameters $\Gamma ,{\gamma _{\rm{col}}}$ and $\tau$ Tabulated for the Case of Best Fit, and the Change in the Parameter Value Required to Increase the SSE Value to ${{\rm SSE}_{{\min}}} + \delta {\rm SSE}$

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The final result of the best fitting model is plotted against the experimental data in Fig. 14. The total excited state population ${\rho _{\textit{ee}}}$ is dominated by ${\rho _{00}}$, and the simulated value for the populations ${\rho _{11}}$ and ${\rho _{- 1 - 1}}$ are approximately zero throughout the simulation. The model in Fig. 14 is calculated using the best fit parameter values shown in Table 2.

From the simulation, we can identify the effects of the independent parameters that govern the decoherence of ions and the loss of ions from the trap. It is clear that the loss of ions from the trap is the dominant parameter that causes the decay of the Rabi oscillation amplitude; however, decoherence does have a non-negligible effect on the population evolution. The decoherence rates in the model, $\gamma$ and ${\gamma _ \bot}$, both have very small contributions from the spontaneous emission rates for the magnetic dipole allowed transitions, and their values are dominated by the value of ${\gamma _{\rm{col}}}$. To reduce the collisional decoherence, the temperature and density of ions in the trap must be reduced. The loss of ions from the trap should be countered by frequency locking the cooling laser and improving the vacuum conditions in the chamber.

This model, thus, comes to a similar conclusion as the simple two-level model from Matjelo’s result [20]; however, it goes beyond the scope of his model and provides us with a collisional decay rate as well as a more accurate Rabi frequency and ion loss rate.

5. CONCLUSION

This tutorial follows the step-by-step derivation of the OBEs describing the magnetic dipole interaction between a four-state atom and an EM field. The example of an atom with a single ground state and an excited level with three states, which can be degenerate or non-degenerate, is discussed. This model is applied to the hyperfine qubit within ${^{171}{{\rm Yb}}^ +}$ ions. The model can be adapted for a different atomic structure by retracing the derivation steps.

Simulations are done with both a degenerate and non-degenerate atomic system, which highlight various aspects of the model and show how the system responds to changes in the Rabi frequency, detuning, and collision rate. The model was also used to show that, under specific conditions, this particular system can be modeled by a two-state model; however, this will not always be the case, and the four-state model is often better suited to describe the time evolution of the system.

The behavior of the system under variable external magnetic field strengths shows a saturation resonance when the Rabi frequency is equal to the $|\! -\! 1\rangle \to |1\rangle$ splitting, associated with extrema in the steady state populations and coherence matrix elements, as well as modulation of the transient populations that is relevant for qubit manipulation. Examples of electromagnetically induced transparency and population inversion in the four-state system are presented.

The usefulness of the model is highlighted in Section 4, where we show how the model was used to fit experimental data of ${^{171}{{\rm Yb}}^ +}$ ions undergoing Rabi oscillations. The model was used to quantify the rates of population decay caused by decoherence and ion loss. This was vital in further developing the trapped ion experiment by Matjelo, as different techniques are used to reduce the different decay processes.

APPENDIX A: ADAPTING THE MODEL FOR ELECTRIC DIPOLE ALLOWED TRANSITION

In this tutorial, the interaction between the atom and the field was modeled as a magnetic dipole interaction. In the example used, this is necessary since the transition between the hyperfine qubit levels of ${^{171}{{\rm Yb}}^ +}$ ions is a magnetic dipole transition. However, the model can easily be adapted to describe an electric dipole transition. For instance, it can be adapted to describe the electric dipole allowed but spin-forbidden transition in zinc atoms between the ground state (${^1{S}_0}$) and the metastable degenerate excited states (${^3{P}_1}$). This transition has a single ground state and three states in the excited level and, thus, suits the model [25].

The derivation of the optical Bloch equations for an electric dipole transition is very similar to the derivation described in Section 2, and the differences or changes will be mentioned below.

First, the atom-field interaction Hamiltonian for an electric dipole transition is defined as

(A1)$${H_{\textit{AF}}} = - {{\boldsymbol \mu}_{\textbf e}} \cdot {\boldsymbol E},$$

where ${{\boldsymbol \mu}_{\textbf e}}$ is the electric dipole moment and ${\boldsymbol E}$ is the electric field vector defined as

(A2)$${\boldsymbol E}(t) = \hat \epsilon \frac{{{E_0}}}{2}({e^{- i\omega t}} + {e^{i\omega t}}).$$

Here, $\hat \epsilon$ is the electric field polarization direction and ${E_0}$ is the amplitude of the electric field. The rest of the derivation continues as it does for the magnetic dipole transition. However, for the expansion of the dipole operator done in Eq. (16), the electric dipole operator equation can be simplified immediately without the need for two sets of raising and lowering operators. An electric dipole transition requires that the configuration of the state changes, in other words $\Delta l = \pm 1$. This rule strictly forbids transitions among the excited states. Therefore, terms in the expansion containing the factor $\langle i|{\boldsymbol \mu}|j\rangle$ for $i,j \in \{- 1,0,1\}$ and $i \ne j$ are zero.

The last significant change is in the definition of the Rabi frequency. In the case of an electric dipole transition, the Rabi frequency is defined as

(A3)$${\Omega _i} = - \langle g|{{\boldsymbol \mu}_{\textbf e}} \cdot \hat \epsilon |i\rangle \frac{{{E_0}}}{\hbar}.$$

The resulting optical Bloch equations for an electric dipole transition have the exact same form as Eqs. (59)–(74) with a different definition for the Rabi frequency.

APPENDIX B: THE ZEEMAN EFFECT IN THE HYPERFINE ENERGY LEVELS OF ${^{171}{{\rm Yb}}^ +}$

The electronic ground state of the ${^{171}{{\rm Yb}}^ +}$ ion has an electron configuration of [Xe] $4{{\rm f}^{14}}6{\rm s}{^2}{S_{1/2}}$, where [Xe] is the electron configuration of xenon [26]. The single valence electron in this state has angular momentum quantum number $L = 0$, spin quantum number $S = \frac{1}{2}$, and total electronic angular momentum quantum number $J = \frac{1}{2}$.

The 171 isotope has nuclear spin with quantum number $I = \frac{1}{2}$. The hyperfine interaction between the total electronic angular momentum and nuclear spin produces two total angular momentum eigenstates associated with different values of the total angular momentum quantum number, $F$. The lowest energy state (the hyperfine ground state) has its quantum numbers $F = 0$, ${m_F} = 0$. The excited level has $F = 1$ and three states with ${m_F} = - 1,0,1$.

In the presence of a weak static external magnetic field, sufficiently weak to be treated as a perturbation to the hyperfine interaction, the Zeeman effect causes an ${m_F}$ dependent change in the energies of the excited states with ${m_F} = - 1,0,1$. The energy splitting, selection rules, and relative line strengths of the Zeeman line components of a magnetic dipole transition are identical to those of an electric dipole transition, but the field polarization that drives the transitions with different $\Delta {m_F}$ values is opposite for the magnetic dipole case [21].

The frequency difference, ${\delta _{{\rm Zeeman}}}$, in angular frequency (${\rm rad \cdot{\rm s}^{- 1}}$, between a state with quantum number ${m_F}$ and the state with ${m_F} = 0$) due to the Zeeman effect is defined as

(B1)$${\delta _{{\rm Zeeman}}} = {g_F}{\mu _B}{B_{{\rm ext}}}{m_F},$$

where

(B2)$$\begin{split}{g_F} &= \left({\frac{3}{2} + \frac{{S(S + 1) - L(L + 1)}}{{2J(J + 1)}}} \right)\\&\quad\times\left({\frac{{F(F + 1) + J(J + 1) - I(I + 1)}}{{2F(F + 1)}}} \right),\end{split}$$

${\mu _B}$ is the Bohr magneton in the units of ${\rm rad \cdot{\rm s}^{- 1}\cdot {\rm T}^{- 1}}$ ($2\pi \times 1.399624 \times {10^{10}} \;{\rm rad \cdot{\rm s}^{- 1}.{\rm T}^{- 1}}$), and ${B_{{\rm ext}}}$ is the magnitude of the external static magnetic field (in tesla) [18].

APPENDIX C: DETAILS OF THE FIELDS WITHIN THE EXPERIMENT

It is important to distinguish between the different fields that are present within the experiment and model. There are two magnetic fields that are often referred to within the paper. These include the magnetic field component of the EM field, which drives the transitions within the model, and then the static magnetic field, which splits the hyperfine levels in the excited state such that they are non-degenerate.

The magnetic field component of the EM field drives the transition from the ground state to the excited states. This is a time-dependent field and takes the form of a sinusoidal wave. In the ytterbium experiment described by Matjelo [20], this field is a microwave field with a resonance frequency of 12.6428 GHz. In the experiment, a microwave horn is used to create this field. First, a signal is produced by a precise and narrowband signal generator. This signal then goes through an amplifier and from there to the microwave horn itself. The horn emits the radiation in a solid angle and, depending on where the ions are within this cone of radiation and how far away the ions are, the intensity of the field at the ions differs. The intensity is rather difficult to estimate since there are a few losses that occur from the horn to the ions, for instance, going through a vacuum chamber window before reaching the ions. However, one can do a few crude calculations to get an estimate for the amplitude of the electric and magnetic fields at the ions.

On the microwave signal generator, a power in dBm is selected, and this signal is then amplified. The amplification of the power will depend on the specific amplifier; thus, what you eventually want to know is the power of the signal at the horn. The horn has a given efficiency that decreases the output power of the microwaves emitted. The radiation is emitted into a solid angle; thus, at each distance from the horn, you can calculate the intensity of the field using the power and the area at that distance within the solid angle. Using this intensity, it is simple to get to the magnetic field amplitude at the ions using the equations below:

(C1)$$I = \frac{1}{2}c \epsilon {E_{0}^{2}},$$

(C2)$${B_0} = \frac{{{E_0}}}{c}.$$

Another factor to consider is that the microwave horn emits partially polarized radiation. By changing the orientation of the horn, one can drive different transitions between the ground and excited states. This is discussed in more detail in Section 2.C.

The other magnetic field that is referred to is the static magnetic field, which is used to separate the hyperfine states in ytterbium. This static magnetic field acts as the perturbation to the hyperfine interaction, which is referred to in Appendix B. Therefore, this field causes the hyperfine levels to split by the Zeeman effect. This field is experimentally introduced into the setup using a Helmholtz coil. In the experiment by Matjelo [20], the coil is powered with a current of $I = 1 \;{\rm A}$, has a radius of $R = 0.1012\;{\rm m} $, and has $n = 38$ turns, thus producing a magnetic field of

(C3)$${B_{{\rm static}}} \def\LDeqtab{}= {\left(\frac{4}{5}\right)^{3/2}}\frac{{{\mu _0}nI}}{R}$$

(C4)$$\def\LDeqtab{}= {\left(\frac{4}{5}\right)^{3/2}}\frac{{4\pi \times {{10}^{- 7}} \times 38 \times 1}}{{0.10122}}$$

(C5)$$\def\LDeqtab{}\approx 3.4 \times {10^{- 4}}\;{\rm tesla}$$

(C6)$$\def\LDeqtab{}= 3.4\,{\rm gauss}.$$

APPENDIX D: SIMULATION DETAILS FOR THE FIGURES

Table 3 shows the simulation parameters used for each figure. Note that not all figures correspond to a simulation; hence, only those which do are listed.

The decision of having $\Gamma = 0.01\; {{\rm s}^{- 1}}$ for the simulations was to appropriately simulate a long-lived excited state, which was a key element to having non-zero coherences between the states. In ${^{171}{{\rm Yb}}^ +}$ ions, this value for the spontaneous decay rate from the level with $F = 1$ down to $F = 0$ is not well known, but it is known that it has been long compared to the time scales of the experiment, which take place on the scale of nanoseconds to a few microseconds. Hence, this value of 0.01 decays per second is reasonable. Recent studies have shown that this value could be much smaller; however, a precise value is not known, and only a lower limit is identified in a paper by Wang et al. [27].

Table 3. Simulation Parameters for Each Figure within the Text

View Table | View all tables in this article

The value of the Rabi frequency varies throughout the simulations. In the general simulations, this value was chosen in order to have the general behavior visible on the time scale of 1 s, whereas, in the simulations involving ytterbium, the Rabi frequency was set close to the experimental Rabi frequency. Note that the Rabi frequency values in the table are the values set in the simulation and may differ from the observed frequency of the Rabi oscillations in the case of a degenerate excited state or a non-zero detuning.

A value of 3.4 gauss is used for the external magnetic field since it matches the field strength used in Matjelo’s experimental setup. The various detunings and collisional rates were chosen in order to see the different behaviors of the excited state population when these parameters were varied.

Funding

Council for Scientific and Industrial Research, South Africa; Universiteit Stellenbosch; National Research Foundation, South Africa; Department of Science and Innovation, South Africa.

Acknowledgment

The authors thank Prof. Hermann Uys as the initiator of the ion trapping project, as well as Nancy Payne and Dr Naleli Matjelo for their helpful discussion and experimental data. The CSIR National Laser Centre through the Rental Pool Programme provided financial support for this project. This material is based upon work supported financially in part by the National Research Foundation. Any opinion, findings, and conclusions or recommendations expressed in this material are those of the author(s), and, therefore, the NRF does not accept any liability in regard thereto. S. Bester received bursaries from the NRF-DAAD collaboration as well as Stellenbosch University’s Postgraduate Scholarship Program and currently has a bursary from the Department of Science and Innovation, South Africa.

Disclosures

The authors declare no conflicts of interest.

Data availability

An example Mathematica workbook is available as Code 1, Ref. [22]. The experimental data used are available on request to authors.

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