# Characterization and Coherent Control of Spin Qubits
with Modulated Electron Beam and Resonator

Soheil Yasini  School of Electrical and Computer Engineering, College of
Engineering, University of Tehran, Iran  Zahra Shaterzadeh-Yazdi
<EMAIL_ADDRESS>School of Engineering Science, College of
Engineering, University of Tehran, Iran  Mahmoud Mohammad Taheri  School of
Electrical and Computer Engineering, College of Engineering, University of
Tehran, Iran

###### Abstract

The coherent dynamics and control of spin qubits are essential requirements
for quantum technology. A prominent challenge for coherent control of a spin
qubit in a set of qubits is the destructive effect of the applied magnetic
field on the coherent dynamics of neighboring qubits due to its spatial
extension. We propose a novel scheme to characterize the coherent dynamics of
these quantum systems and to coherently control them using a magnetic field.
Our scheme consists of a resonator that encompasses the desired quantum system
and a modulated electron beam that passes through the resonator in close
proximity to the quantum system of interest. The dynamics of the system is
obtained by solving the Lindblad master equation. To verify the reliability of
our model, we tested the model on a Potassium atom, 41K and NV- center in
Diamond. The results show that by properly controlling the parameters of the
resonator and the electron beam, the coherence and decoherence rates of these
quantum systems can be improved. Our model has the potential to be used for
characterizing different types of spin-based quantum systems, and implementing
quantum logic gates for quantum computation.

††preprint: APS/123-QED

##  I Introduction

Qubits are the primary component in many areas of quantum science and
technology, including quantum computation [1, 2], data encryption, safe
information transmission [3, 4], and quantum metrology and imaging [5, 6]. To
effectively utilize qubits, it is necessary to characterize these two-level
quantum systems and optimize their characteristics [7, 8]. Additionally,
controlling each individual qubit with minimum effect on the coherent dynamics
of the neighboring ones is a necessary requirement for a wide range of
applications, from investigating fundamental quantum phenomena to quantum
information computation [9].

Among different types of qubits, spin qubits are prominent candidates for
quantum technology. A common approach for individually controlling a spin
qubit is by employing a proper magnetic field. However, due to the spatial
extension of the magnetic field effect and the difficulty of localizing the
field in a small volume of space, individual control of these quantum systems
in a set of spin qubits is a considerable challenge [10].

In this paper, we propose a scheme for tackling this challenge, characterizing
spin qubits, and controlling their coherent dynamics. As shown in Fig. 1, our
proposed scheme consists of a resonator encompassing a set of two-level
quantum systems and a modulated electron beam that passes through a near
distance $h$ from the qubit that is considered to be controlled. The desired
qubit is located in the near-field region of the magnetic field generated by
the electron beam. By adjusting the parameters of the resonator and the
electron beam, one can characterize the dynamics of the quantum system and
properly control it.

Figure 1: The proposed scheme, consisting of a modulated electron beam,
generated by a klystron lamp, and a resonator with a resonance frequency
$\omega_{r}$ that is encompassing a set of two-level quantum systems with
coherence frequency $\omega_{0}$.

We were inspired by previous experimental and theoretical studies on measuring
tunnelling rate and decoherence in double-quantum-dot systems in the microwave
and terahertz regimes [11, 12, 13]. Based on these studies, the qubit of
interest would resonate by using a proper frequency provided by the resonator,
concomitantly with employing a proper magnetic field strength provided by the
electron beam. If the frequency of the resonator is off-resonant with the
coherence rate of the quantum system, it would have a small perturbative
effect on the quantum system. However, if the off-resonance effect is
compensated by the magnetic-field control of the electron beam, it causes
resonance in the dynamics of the quantum system. The results of measurement
should reveal the coherence rate and some properties of decoherence. Also, it
would allow one to control over the dynamics of the quantum system by tuning
the parameters of the resonator and the electron beam.

Coherent control of an individual quantum system, with spatial resolution and
on an atomic scale, becomes possible taking into account that the electron-
beam characteristics are related to de-Broglie wavelength of the electrons
[9], which is in the order of Angstroms and comparable to the atomic spacing.
Electron-beam control of a quantum system can be achieved when the length of
the wave packets associated with individual electrons is much shorter than the
modulated wavelength. Such electron-beam current density modulation is easily
achievable in the microwave frequency range and has wide applications in
electronic technologies [14].

To investigate the dynamics of the desired quantum system, we first solve the
Lindblad master equation, governing the Hamiltonian of the proposed structure
and obtain the density state of the system. Quantum system’s coherence and
decoherence rates are then extracted from the time evolution of the density
state. In addition, we evaluate our model by considering two quantum systems,
i.e. Potassium atom (41K) as a representative of Alkali atoms [15], and
Nitrogen vacancy defect centres in Diamond (NV-) [16]. By solving the
equations governing the dynamics of these quantum systems, we show that their
coherence properties are improved at some frequencies of the resonator.

This paper is organized as follows. In Sec. II, we provide a detailed
explanation of the constituent elements of our proposed scheme. We describe
the setup of the resonator and the modulated electron beam. In Sec. III, we
introduce the model that we use in order to simulate the dynamics of the
quantum system in the proposed scheme. We provide a detailed explanation of
the Lindblad master equation governing the Hamiltonian of the system and how
we use it to obtain the density state of the system. In Sec. IV, we present
the results of our simulations for two quantum systems, i.e. Potassium atom
and Nitrogen vacancy defect centres in diamond. Then, we discuss the
implications of our results in Sec. V. Finally, in Sec. VI, we summarize the
achievements of our work and suggest some future works to further develop and
improve the proposed scheme.

##  II Background: Constituent Elements of the Proposed Scheme

###  II.1 Klystron Lamp and the Electron Beam

One of the key elements of the proposed scheme is the modulated electron beam.
The electron beam is generated by a cathode, and then modulated by a klystron
lamp. Klystron is a type of radio frequency amplifier that modulates the speed
of the electron beam passing through it [17, 18]. Klystrons are proposed in
different dimensions and output powers, according to the need for accelerating
the electrons. Also, they can operate in different operating frequencies
ranging from microwave to terahertz regime [19, 20, 21].

Figure 2: A klystron lamp, composed of a buncher cavity and a catcher cavity.
Passage of the electron beam, produced by a cathode, through the buncher
cavity with frequency $\omega_{0}$ causes the electron beam to get modulated
to a fixed modulation wavelength, $\lambda_{0}$. The diameter of the cross
section of the electron beam is assumed to be $2a$. The wave function
attributed to each electron in the electron beam is assumed to be in Gaussian
form with a length $\Delta z$ much smaller than the modulation wavelength of
the electron beam.

Structure of a Klystron is schematically shown in Fig. 2. Klystron consists of
two cavities, which are located at a certain distance from each other, along
the length of the klystron. The first cavity is the buncher cavity and the
second one is the catcher cavity. The electron beam is first produced by a
cathode, and then moves along the length of the klystron and first passes
through the buncher cavity.

Passing through the buncher cavity with frequency $\omega_{0}$, causes the
electron beam to become modulated with a fixed modulation wavelength
$\lambda_{0}$. The electrons are accelerated due to the applied potential
difference $V_{0}$ between the two klystron cavities, consequently the kinetic
energy of the electron beam increases. The electron beam with high kinetic
energy in the output cavity produces a signal with a fixed modulation
wavelength $\lambda_{0}$ and with a higher field amplitude. Therefore, the
output signal of the klystron lamp is the same as the amplified input
radiofrequency signal with a fixed modulation wavelength $\lambda_{0}$ [18].

###  II.2 Resonator

A resonator is usually a cavity with metal shields [22] or a dielectric
waveguide with high permittivity [23]. According to its size, a resonator
possesses a number of resonant frequencies that are related to different modes
of electromagnetic fields.

To model a resonator, we assume that, in its resonant-frequency mode, the
resonator is equivalent to a quantum system in the excited state. Similarly,
for off-resonance mode, the resonator is equivalent to the quantum system in
its ground state. Therefore, a resonator with a certain resonant frequency
behaves similar to a two-level quantum system, where the energy difference
between its two energy levels is equivalent to the resonant frequency of the
resonator.

Resonant frequency of a resonator depends on the shape, dimension, and the
material of the resonator. Therefore, the value of the resonant frequency
$\omega_{m}$ is determined, if the resonator is designed and characterized
[24]. For a rectangular-cubic resonator with metal walls, the value for the
resonant frequency of the first excited mode of the resonator, $f$, is
obtained from,

$\displaystyle
f=\frac{c}{2\sqrt{\epsilon_{r}\mu_{r}}}\sqrt{\left(\frac{m}{l_{1}}\right)^{2}+\left(\frac{n}{l_{2}}\right)^{2}+\left(\frac{q}{l_{3}}\right)^{2}},$
(1)

where $l_{i}$ for $i\in\\{1,2,3\\}$ are resonators 3D dimensions, $c$ is the
speed of light in the vacuum, $\epsilon_{r}$ is the relative dielectric
permittivity coefficient, and $\mu_{r}$ is the relative magnetic permeability
coefficient of the material inside the cavity [22]. Parameters $m,n,$ and $q$
are the electromagnetic mode’s arithmetic numbers, and they are not zero
altogether. For higher values of these parameters, the frequency of the
higher-order modes of the resonator can be obtained.

It should be noted that in resonators, made of metal shields with vacuum
inside, for high frequency electromagnetic fields, the loss resulting from the
dielectric property of the vacuum is small. In such cases, the electrical
losses are mostly related to the currents flowing in the cavity walls. The
walls are usually coated with silver or gold to increase electrical
conductivity and reduce energy losses. Unlike copper, which oxidizes quickly,
silver and gold are safe from oxidation [23].

##  III Modelling the Proposed Scheme

###  III.1 Basic Assumptions

To model the proposed scheme, assumptions are considered for the electron
beam, the quantum system, and the resonator. As shown in Fig. 2, for the
electron beam, it is assumed that the wavefunction attributed to each electron
is a Gaussian wave packet of length $\Delta z$ and the wavelength of the wave
packet, attributed to each electron, is much smaller than the modulation
wavelength of the electron beam ($\Delta z\ll\lambda_{0}$). It is also assumed
that the electron beam current and, accordingly, the radius of the cross-
section of the electron beam ($a$) is such that the distance between the
center of the electron beam and the desired quantum system $h$ is large enough
compared to the width of the cross-section of the beam ($2a\ll h$).

With this assumption, the magnetic field or the non-radiative electric field
caused by this modulated electron beam can be obtained from the classical
relations and the interaction of the quantum system with the field caused by
the electron beam can be considered semi-classical [9].

Regarding the quantum system, we assume that the dimension of the desired
quantum system is small compared to the wavelength of the electron beam
modulation ($\lambda_{0}$) as well as the distance of the quantum system from
the center of the electron beam ($h$). We also assume that the quantum system
in question, has only two energy levels, i.e. the ground state and the first
excited state, and the probability of transition from the ground state to
higher excited states is almost zero and can be ignored.

The distance between the electron beam and the quantum system is such that the
magnetic field, produced by the beam, affects the system in the near field
region, thus enabling the control of the dynamics of the quantum system by the
electron beam. We consider the interaction of the electron beam with the
quantum system to be of a weak type, so its effect on causing decoherence in
the dynamics of the quantum system can be ignored.

The resonator is assumed to be a Fabry-Perot type, and the quantum system is
located inside this resonator. The resonator is used to drive the quantum
system. For this purpose, the resonant frequency of the resonator is assumed
to be close to the frequency associated with the energy difference between the
ground and excited energy levels of the quantum system. If the frequency
difference between the resonant frequency of the resonator and the quantum
system is compensated by an electron beam, then the quantum system resonates
and its state changes coherently between the energy levels of its ground state
and excited state.

According to the above assumptions, in the following section, we provide a
suitable theoretical model to investigate the dynamics of the desired system.

###  III.2 Theoretical Model for the Dynamics of the Quantum System

We consider the quantum system and the resonator as the main system, which is
affected by the electron beam. The Hamiltonian governing the whole system is
defined as

$\displaystyle\hat{H}_{\text{s}}=\hat{H}_{0}+\hat{H}_{\text{int}},$ (2)

where the first term, $\hat{H}_{0}$, is the Hamiltonian governing the main
system including the two-level quantum system and the resonator, and the
second term $\hat{H}_{\text{int}}$ shows the interaction between all the
components of the proposed system.

The energy levels of the quantum system are considered to be $|g\rangle$ for
the ground state and $|e\rangle$ for the excited state, and the energy
difference between these two levels is equal to $\Delta E=\hbar\omega_{0}$. We
consider the role of the resonator equivalent to quantized electromagnetic
waves with the attributed energy $E_{\text{ph}}=\hbar\omega_{\text{M}}$, where
$\omega_{\text{M}}$ is the resonant frequency of the resonator. Then,
$\hat{H}_{0}$ is defined by

$\displaystyle\hat{H}_{0}=\hbar\omega_{0}\frac{\hat{\sigma}_{z}}{2}+\hbar\omega_{\text{M}}\hat{a}_{\text{n}}^{\dagger}\hat{a}_{\text{n}}.$
(3)

The operator $\hat{\sigma}_{z}=|g\rangle\langle g|-|e\rangle\langle e|$
represents the transition between the two levels of the quantum system, and
the operators $\hat{a}_{\text{n}}$ and $\hat{a}_{\text{n}}^{\dagger}$ are,
respectively, the annihilation and creation operators of the quantized
electromagnetic field, between the primary state $|\text{m}\rangle$ and the
secondary state $|\text{m}\rangle$, attributed to the resonator.

The second term, in Eq. (2), is composed of the mutual interaction between the
quantum system, the resonator and the electron beam and is defined by,

$\displaystyle\hat{H}_{\text{int}}=\hat{H}_{\text{int1}}+\hat{H}_{\text{int2}}+\hat{H}_{\text{int3}}.$
(4)

In the above equation, the first term is the interaction between the magnetic
field $B$, caused by the electron beam, and the magnetic dipole moment of the
two-level quantum system $\hat{\bf{\mu}}$, which is defined as [9],

$\displaystyle\hat{H}_{\text{int1}}=-\hat{\bf{\mu}}.\textbf{B}.$ (5)

We assume that the direction of the magnetic field is perpendicular to the
electron beam propagation, so the mentioned interaction only happens in the
$z$ direction.

The second term in Eq. (4), $\hat{H}_{\text{int2}}$, expresses the interaction
between the resonator and the quantum system, which is defined by

$\displaystyle\hat{H}_{\text{int2}}=\gamma_{\text{n}}\left(\hat{\sigma}+\hat{\sigma}^{\dagger}\right)\left(\hat{a}_{\text{n}}+\hat{a}_{\text{n}}^{\dagger}\right),$
(6)

where $\hat{\sigma}=|g\rangle\langle e|$ and
$\hat{\sigma}^{\dagger}=|e\rangle\langle g|$, represent the transition of the
two-level quantum system from the ground state to the excited state, and from
the excited state to the ground state, respectively. Parameter
$\gamma_{\text{n}}$ represents the interaction coefficient between the
resonator and the quantum system.

The third term in Eq. (4) represents the interaction between the magnetic
field of the electron beam and the resonator, which is defined as,

$\displaystyle\hat{H}_{\text{int3}}=\gamma\left(\hat{a}_{\text{n}}^{\dagger}+\hat{a}_{\text{n}}\right)\left(\hat{a}_{\text{e}}^{\dagger}+\hat{a}_{\text{e}}\right)$
(7)

where $\gamma$ is the interaction coefficient between the magnetic field of
the electron beam and the resonator. We consider a weak interaction between
the magnetic field and the quantum system as well as the resonator. Therefore,
we consider classical behavior for the applied magnetic field and the
relations for these two parts are semi-classical.

To check the dynamics of the desired system, we consider the quantum system
and the resonator together, as an open system interacting with the electron
beam and assume that this interaction is weak. In order to investigate the
temporal evolution of the density matrix of an open quantum system with weak
interaction, we use the Lindblad equation, which is one of the well-known
theories for use in this type of systems [25, 26, 27, 28, 29]. This equation
for our desired system turns to be

$\displaystyle\frac{\partial\rho}{\partial t}=$ $\displaystyle-$
$\displaystyle\frac{i}{\hbar}\left[\hat{H}_{\text{s}},\rho\right]+\frac{\gamma}{2}\left(2\hat{\sigma}\rho\hat{\sigma}^{\dagger}-\hat{\sigma}^{\dagger}\hat{\sigma}\rho-\rho\hat{\sigma}^{\dagger}\hat{\sigma}\right)$
(8) $\displaystyle+$
$\displaystyle\frac{\gamma_{\text{n}}}{2}\left(2\hat{a}_{\text{n}}\rho\hat{a}_{\text{n}}^{\dagger}-\hat{a}_{\text{n}}^{\dagger}\hat{a}_{\text{n}}\rho-\rho\hat{a}_{\text{n}}^{\dagger}\hat{a}_{\text{n}}\right).$

In the above formula, $\hat{H}_{\text{s}}$ is the Hamiltonian of the whole
system given in Eq. (2). Also, the density state of the quantum system is
considered as $\rho=\rho_{1}\otimes\rho_{2}$, where $\rho_{1}$ is the density
matrix of the quantum system and $\rho_{2}$ is the density matrix of the
resonator. Taking into account the assumptions, mentioned earlier, the
corresponding matrix gets the following form

$\displaystyle\rho=\begin{pmatrix}\rho_{\text{gg}}\rho_{\text{mm}}&\rho_{\text{gg}}\rho_{\text{mn}}&\rho_{\text{ge}}\rho_{\text{mm}}&\rho_{\text{ge}}\rho_{\text{mn}}\\\
\rho_{\text{gg}}\rho_{\text{nm}}&\rho_{\text{gg}}\rho_{\text{nn}}&\rho_{\text{ge}}\rho_{\text{nm}}&\rho_{\text{ge}}\rho_{\text{nn}}\\\
\rho_{\text{eg}}\rho_{\text{mm}}&\rho_{\text{eg}}\rho_{\text{mn}}&\rho_{\text{ee}}\rho_{\text{mm}}&\rho_{\text{ee}}\rho_{\text{mn}}\\\
\rho_{\text{eg}}\rho_{\text{nm}}&\rho_{\text{eg}}\rho_{\text{nn}}&\rho_{\text{ee}}\rho_{\text{nm}}&\rho_{\text{ee}}\rho_{\text{nn}}\end{pmatrix}.$
(9)

From Eq. (9), 16 differential equations are obtained for all the possible
states of the system. Solutions of these equations expresses the time
evolution of different states of the system.

By simulating the equations governing the density state of the quantum system,
using finite-difference time-domain method (FDTD), and optimizing the time
response of these states according to different values of the existing
parameters for tuning the system, we can improve the time characteristics and
the coherent dynamics of the quantum system. In the following section, we
apply this model to a couple of known quantum systems, and show the
reliability of our model.

##  IV Results

In this section, the results of numerical simulation for the dynamics
associated with temporal evolution of the proposed system, are examined. We
assume that the system is initially in one of the two states
$\rho_{\text{gg}}\rho_{\text{nn}}$ or $\rho_{\text{ee}}\rho_{\text{mm}}$.
These two states correspond to the cases where the quantum system is in its
ground state $|g\rangle$ and the resonator is in its excited state
$|n\rangle$, or vice versa, the quantum system is in the excited state
$|e\rangle$ and the resonator is in its ground state $|m\rangle$. In fact, the
quantum system evolves between its ground state and excited state, due to
receiving and returning energy to the resonator.

We consider two quantum systems, i.e. Potassium atom 41K, as a representative
of Alkali atoms, and NV- centers in Diamond. By using the available parameters
for these two quantum systems, which are given in the following, the results
obtained from the simulation are compared with the results presented in
reference [9].

The transition frequency between two energy levels of an individual 41K atom
and similarly an NV- center in Diamond, is equal to $254$ MHz and $2.78$ GHz,
respectively [9]. These frequencies are equivalent to $\omega_{0}=1.60\times
10^{9}$ rad/s and $\omega_{0}=17.34\times 10^{9}$ rad/s, respectively. The
coefficients $\gamma$ and $\gamma_{n}$ in Eq. (9) are taken equal to $150$ Hz.
The frequency of the resonator in transaction with the two-level quantum
system is assumed to be $\omega_{\text{M}}=1.13\times 10^{9}$ rad/s for the
41K atom, and $\omega_{\text{M}}=12.26\times 10^{9}$ rad/s for the NV- center.

The electron-beam current is chosen to be $100~{}\mu$A and the radius of the
cross section of the electron beam is taken to be $25~{}\mu$m. The magnetic
field resulting from such electron beam at the location of the quantum system
is chosen to be $B=3\times 10^{-9}$ T. The relationship between the magnetic
field resulting from the electron beam current and the distance of the beam
from the quantum system is given by the Ampere’s law [30]. Hence, the distance
of the beam to the quantum system, $h$, in order to obtain a magnetic field of
$3\times 10^{-9}$ T in the vicinity of the quantum system, is obtained to be
$6.7$ mm. Alternatively, for a current of $50$ nA, in order to obtain the same
amount of magnetic field, $h$ should be equal to $3.3~{}\mu$m.

Figure 3: Time evolution of the 41K atom, when (a) the atom is initially in
the excited state $|e\rangle$ and (b) the atom is initially in the ground
state $|g\rangle$. In the inset of each diagram, the initial quantum state of
the atom is shown. (c) shows the difference between the two states and (d)
demonstrates the sum of the two states.

To obtain an electron beam with the desired current value and electron-beam
cross section, we need to know the voltage that is applied to the klystron
cavities, in order to evaluate the energy of the electron-beam output from the
klystron lamp. We assume that the electrons, with initial speed $u_{0}$, have
a kinetic energy equal to $18,000$ eV. According to the relation between the
potential difference $V_{0}$, between the two cavities of the Klystron, and
the initial speed of the electron beam $u_{0}$, it is possible to estimate the
value of $V_{0}$, necessary to produce the desired kinetic energy, based on
the law of conservation of energy [18]. Therefore, to produce electrons with
the assumed initial kinetic energy, the value of the applied voltage should be
equal to $V_{0}=15.750$ Kv, taking into account the relativistic conditions of
the speed of these particles. For these calculations, we used CST software
[24].

In Fig. 3, the time-evolution diagrams for the two states, $\rho_{\text{ee}}$
and $\rho_{\text{gg}}$, of the 41K atom are presented. The initial quantum
state of the 41K atom is shown in the inset of each diagram. In Figs. 3a (3b),
it is assumed that the atom is in the excited state $|e\rangle$ (the ground
state $|g\rangle$), at the initial time $t=0$. In Figs. 3c and 3d, the
difference and sum of the two states $\rho_{\text{ee}}$ and $\rho_{\text{gg}}$
are shown, respectively, as a function of time. Similar calculations were
performed for the case of NV-, where the results are presented in Fig. 4. In
both cases, the diagrams display valuable points about the dynamical behavior
of the entire system, which will be discussed in the next section.

Figure 4: Time evolution of the NV- center in Diamond for the case where (a)
the NV- is initially in its excited state $|e\rangle$ and (b) the NV- center
is initially in its ground state $|g\rangle$. In the inset of each diagram,
the initial quantum state of the NV- center is shown. (c) shows the difference
between the two states and (d) the sum of the two states.

##  V Discussion

Comparison between Figs. 3c and 3d for the 41K atom, and similarly Figs. 4c
and 4d for the NV- center, shows that for the case
$\rho_{\text{ee}}-\rho_{\text{gg}}$, the time evolution of the desired quantum
system has an oscillatory behavior, whereas
$\rho_{\text{ee}}+\rho_{\text{gg}}$ decreases exponentially. According to the
$\pm$ sign between the states $\rho_{\text{ee}}$ and $\rho_{\text{gg}}$, these
results can be an indication that if the time evolution of the state of the
resonator has a $\pi$-phase difference with the time evolution of the state of
the quantum system, then the energy exchange between the resonator and the
quantum system occurs effectively. Consequently, the resonator can control the
state of the two-level quantum system and be the driving factor for the
dynamics of the quantum system.

Furthermore, the $\rho_{\text{ee}}-\rho_{\text{gg}}$ time-response diagrams
for 41K atom and NV- center are compared with the corresponding diagrams
presented in reference [9] for the same quantum systems and in the same time
interval. We considered the same values for the used parameters so that the
results can be compared. One can see that for both of the quantum systems, the
number of Rabi oscillations in our proposed scheme has been almost doubled as
compared to the number of Rabi oscillations in reference [9]. Hence, by
employing our proposed scheme, we have been able to double the coherence rate
in the quantum systems of interest (i.e. the 41K and NV- center), and
significantly improve the dynamic behavior of these quantum systems. In fact,
these results show the effect of the presence of the resonator and its key
role in driving the quantum system of interest.

In addition, for evaluation of the decoherence rate, an exponential function
is fitted to the damping of the oscillating graphs. For the results presented
in the reference article [9], this function is proportional to
$f(t)\propto\text{e}^{-57.34t}$ for the 41K atom and
$f(t)\propto\text{e}^{-923t}$ for the NV- center, whereas the exponential
function fitted to our graphs is first proportional to
$g(t)\propto\text{e}^{-4332t}$, in the descending part of the function, and
then it is proportional to $g(t)\propto\text{e}^{58.35t}$, in the ascending
part of the function for the 41K atom. Similarly, the exponential functions
fitted to the graph associated with the NV- center, in our proposed scheme, is
proportional to $g(t)\propto\text{e}^{-11770t}$, in the descending part of the
function, and then it is proportional to $g(t)\propto\text{e}^{209.4t}$, in
the ascending part of the function.

Comparison between the two functions $f(t)$ and $g(t)$, for each quantum
system of interest, shows that the decoherence rate of the 41K atom and the
NV- center in our proposed scheme, has decreased by more than $70$ times and
$12$ times, respectively, as compared to the decoherence rates given in
reference [9]. Therefore, the presence of the resonator in our proposed
scheme, not only results in the increment of the coherence rate of the quantum
systems, but also significantly decreases the decoherence rate. Both of these
factors are important requirements for a quantum system to be recognized as a
_well-characterized_ qubit to be used, for instance, in fault tolerant quantum
computation.

##  VI Conclusion

In this study, we proposed a scheme for characterization and coherent control
of spin-based two-level quantum systems. We showed that proper use of a
resonator and a modulated electron beam lead to a better time response of the
quantum states of a desired system. The modulated electron beam, generated by
a Klystron lamp, is a suitable tool for individual magnetic control of a spin-
based quantum system in a set of quantum systems. Also, we showed that placing
a quantum system inside a proper resonator can improve the performance and
characteristics of the quantum system, especially improving the coherence and
decoherence rates of the system.

To model the proposed scheme, we developed a theory using the Lindblad master
equation. By numerically solving the obtained equations, for the 41K atom, as
a representative of Alkali atoms, and for the NV- centers in diamond, we
derived the time evolution of the state of these quantum systems and examined
the dynamics of the states. The results showed that specific resonance
frequencies can be provided by the resonator for each quantum system, which
improves the coherence dynamics of the desired system.

Improving the coherent dynamics of a quantum system can have many
applications. For instance, one can design various quantum gates for the
purpose of fault tolerant quantum computation [31, 32, 33, 34, 35]. In fact,
by controlling the tuning parameters provided by the resonator and the
electron beam, suitable quantum gates can be designed and applied.
Furthermore, due to the unique control that the electron beam provides for the
individual access of spin-type quantum systems, it is possible to use a set of
quantum systems at close proximity of each other and engineer such a compound
system by electron beams for the purpose of quantum simulations and
computations. Therefore, the proposed model and the equations governing its
dynamics can be generalized for more than one quantum system and also for more
than one resonator.

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