# Super narrow peaks in excitation spectrum of alkali spin polarization:
non-adiabatic case of spin dynamics

E. N. Popov<EMAIL_ADDRESS>Laboratory of Quantum Processes and
Measurements, ITMO University,
199034, 3b Kadetskaya Line, Saint-Petersburg, Russia  A. A. Gaidash  A. V.
Kozubov  Laboratory of Quantum Processes and Measurements, ITMO University,
199034, 3b Kadetskaya Line, Saint-Petersburg, Russia  Department of
Mathematical Methods for Quantum Technologies, Steklov Mathematical Institute
of Russian Academy of Sciences, 119991, 8 Gubkina St, Moscow, Russia  S. P.
Voskoboynikov  Higher School of Software Engineering, Peter the Great
St.Petersburg Polytechnic University,
195251, 29 Polytechnicheskaya, Saint-Petersburg, Russia

###### Abstract

We theoretically describe the phenomenon of non-adiabatic spin dynamics, which
occurs in a gas cell filled by alkali vapor in presence of a strong
alternating magnetic field and pump light. Steep increase of the spin
polarization occurs if frequency of the magnetic field is equal to the certain
value. Although, the observable effect relies on the periodic field that
consists of two perpendicular components defined by harmonics with the same
amplitudes and different frequencies. Considered spin effect cannot be
explained by a resonance, because the own Larmor frequency of spin precession
is absent without a constant component of magnetic field. Moreover, there are
some clearly visible peaks in the excitation spectrum of spin polarization,
and they are super narrow in comparison to relaxation rate. Detailed analysis
according to proposed quantum model results in the reasoning of the effect via
qualitative properties of non-adiabatic dynamics of atomic spin.

††preprint: APS/123-QED

##  I INTRODUCTION

The atomic spin dynamics in a gas cell has been studied for over half of the
century. The main advantage of that physical model is the large lifetime of
non-excited spin states in comparison to crystals and liquids, since any atom
moves as a free particle between collisions [1, 2, 3]. Moreover, in a mixture
consisting of an alkali vapor and an inert buffer gas, only a small percentage
of collisions results in spin reversal. These factors make it possible to
observe some evident magnetic effects in an alkali vapor, which are
implemented in precise measurements for metrology and navigation [4, 5], i.e.
magnetometers [6, 7, 8, 9, 10, 11, 12, 13], gyroscopes [14, 15, 16, 17, 18,
19, 20, 21] and frequency standards [22, 23]. A rapid development in the study
of a gas cell with spin dynamics is induced by the implementation of optical
approaches to scanning and excitation of a spin polarization [24, 25, 26, 27,
28, 29, 30, 31, 32, 33, 34, 35].

In an alkali vapor, one of the most explored and utilized in the practice
magnetic effects is the electron paramagnetic resonance (EPR) [36, 37, 38]. A
necessary condition for the EPR observation is the presence of a constant
magnetic field, which defines the own Larmor frequency of the alkali spin
precession. If the frequency of periodic perturbation of the spin system is
close to or equal to the own Larmor frequency, the EPR is arising as
oscillations of spin polarization. The width of the EPR spectral line is
determined by a rate of population mixing among alkali Zeeman sublevels. The
latter is the most important parameter of the system, because the precision of
different gas cell sensors depends on lifetime of spin states [39, 40, 41, 42,
43, 44, 45, 46, 47, 48].

In this work, we theoretically explore a phenomenon in alkali vapor: its spin
polarization steeply increases near a certain frequency of an external
magnetic field in the presence of a circularly-polarized pump light. To
observe the effect, the following properties of the magnetic field should be
satisfied:

  * •

The external magnetic field is periodic with the period that is much less than
the relaxation time of a spin polarization;

  * •

The averaged over a period external magnetic field is equal to zero, thus the
constant component of the magnetic field in a gas cell with alkali vapor is
absent;

  * •

Both the magnitude and direction of the magnetic field vary;

  * •

Zeeman shift of sublevels with non-zero angular momentum projection is much
larger than broadening of these sublevels due to population mixing to
equilibrium state.

Listed above properties regard to non-adiabatic dynamics, defined by rapid
variation of an alkali spin polarization in comparison to its relaxation.
Withal, magnetic field should be periodic to analyze the steady dynamics
through its constant characteristics. It is essential, that the origin of the
effect differs from EPR induced by an externally applied periodic force.
Though, to exclude an EPR we need to disable a trend of Larmor spin
precession, therefore a constant component of magnetic field should be absent.
Hereafter in the paper, we use a term the spin effect to refer to the
phenomenon under consideration.

The listed above properties may belong to a magnetic field defined by various
temporal dependencies. Here, we consider the special case that is sufficient
for exploring the spin effect:

$\displaystyle\mathbf{B}(t)=B_{0}\mathbf{l_{z}}\cos\left(\Omega
t\right)+B_{0}\mathbf{l_{x}}\cos\left(2\Omega t\right),$ (1)
$\displaystyle\gamma B_{0}\gg\Gamma,\qquad\Omega\gg\Gamma,$ (2)

where $B_{0}$ is an amplitude of the magnetic field, $\mathbf{l_{x}}$ and
$\mathbf{l_{z}}$ are orthonormal spatial vectors, $\Omega$ is a frequency of
the magnetic field, $\gamma$ is gyromagnetic ratio of the oriented alkali
atoms, $\Gamma$ is the relaxation rate of spin polarization.

Previously, various magnetic effects has been studied in condition of the non-
adiabatic dynamics of oriented spins. There are some experimental and
theoretical works, in which non-adiabatic dynamics is essential [49, 50, 51,
52, 53, 54, 55, 56, 57, 58]. However, the cited papers do not focus on the
non-adiabatic dynamics and it’s properties. Moreover, to the best of our
knowledge, the synchronization of alkali spin motion due to periodic non-
adiabatic dynamics in presence of a strong alternating magnetic field was not
analyzed in detail. We suggest exploring a spin effect induced by properties
of non-adiabatic dynamics using a quantum model of alkali vapor.

Figure 1: The physical system for studying alkali spin dynamics in a gas cell.
To create a magnetic field, the double Helmholtz coils located around the
cell. A pump light propagates along the Z-axis and covers the cell
homogeneously. A scanning light may be directed along any axis. The atomic
spin projection onto the path of scanning light can be measured by the angle
of scanning light polarization rotation $\Delta\psi$; they are proportional to
each other

The spin effect cannot be explained in terms of spin resonance, since a
constant magnetic field is absent. Parametric resonance [59] is not a suitable
candidate for explanation of the spin effect as well, since there are no
internal parameters of the atom-field system, which determine the own
frequency of the resonance and make periodic oscillations. Though, we explain
the spin effect by the qualitative properties of non-adiabatic dynamics of
atomic spins in alkali vapor. Mathematically, the non-adiabatic case of atomic
spin motion in the presence of a strong alternating magnetic field corresponds
to a dynamic system with unpredictable behavior. Since the dynamics cannot be
described as an explicit temporal dependence, the search for physical effects
becomes a non-trivial task. Only via a numerical solution, we can explore the
phase-space trajectory of a non-adiabatic dynamic system, that is considered
in the paper.

The paper is organized as follows. In Sec. II we describe an optical method of
spin driving inside a gas cell with vapor of alkali. Also, the simplified
scheme for an experiment is suggested. In Sec. III we propose a general
quantum model of alkali spin dynamics in presence of a strong alternating
magnetic field and relaxation. Calculations according to the model and
resulting excitation spectrum of the spin effect are demonstrated in Sec. IV.
In Sec. V the explanation of the spin effect is provided. Sec. VI concludes
the article.

##  II Physical system under consideration

In this section, we propose a principal optical scheme for the spin effect
observation, see Fig. 1. Also, a structure of Rb-87 energy levels with excited
transitions is shown in the Fig. 2. The scheme consists of a gas cell with
vapor of 87Rb and buffer gas, Helmholtz coils for generating the magnetic
field Eq. (1), a source of circularly polarized pump light (linearly polarized
laser and quarter-wave plate) and source of scanning light (low-power linearly
polarized laser). The pump light orients rubidium spins along the Z-axis, and
the scanning light detects an expectation value of the atomic spin projection
onto the path of the scanning light propagation. Note, the similar scheme of
spin perturbation was considered in the works [57] and [58] in partially-
adiabatic case.

###  II.1 Gas cell

A gas cell contains a vapor of $\mathrm{{}^{87}Rb}$ and a mixture of inert
gases. The gas mixture composes of diatomic nitrogen in order to reduce alkali
fluorescence and some noble monatomic gases as a buffer. It is important that
spin polarization can be formed by alkali atoms populating different hyperfine
ground levels. Therefore, to define a certain group of non-excited alkali
atoms by $\mathrm{Rb_{F=1}}$ and $\mathrm{Rb_{F=2}}$.

Temperature within the cell is about $80^{\circ}\,\mathrm{C}$. Under the
conditions, the concentration of the alkali is 5–6 orders of magnitude less
than the concentration of the buffer. Since an electron cloud of alkali has an
enlarged radius and anisotropic form, multiple collisions between an excited
$\mathrm{{}^{87}Rb}$ atom and $\mathrm{N}_{2}$ molecules lead to rapid non-
radiative transitions from the upper to one of the two ground levels in the
$\mathrm{D1}$ line (see Fig. 2). Moreover, since the buffer environment
freezes free motion of alkali, relaxation due to collisions with a cell’s
surface is significantly small (up to be considered absent). The latter
properties of the gas cell are necessary for the inducing of non-adiabatic
dynamics of atomic spins in an external magnetic field.

Figure 2: D1 line of Rubidium 87. Rectangles denote Zeeman sublevels in the
hyperfine structure. Degree of black filling shows its population: the darker,
the higher population. Red arrows indicate atomic transitions
$F=1\leftrightarrow F^{\prime}=2$ induced by circularly polarized pump light.
Orange arrows indicate weak perturbation on transitions $F=2\leftrightarrow
F^{\prime}=1$ and $F=2\leftrightarrow F^{\prime}=2$ excited by linearly
polarized scanning light. Symbols $F$ and $F^{\prime}$ denote total angular
momentum of ground and excited levels respectively, $m$ is a projection of
total angular momentum onto the path of the pump light propagation

###  II.2 Spin polarization pumping

As shown in the Fig. 1, a circularly polarized pump light propagates along the
Z-axis. It orients alkali spins along the path of propagation. The frequency
of a pump light is equal to the one of a transition between the levels with a
total angular momentum $F\\!=\\!1$ and $F^{\prime}\\!=\\!2$. In the presence
of an optical pump, alkali atoms populate the upper levels with one-sided
change of the angular momentum, which is projected along the path of the light
propagation. Therefore, alkali atoms accumulate non-zero angular momentum
directed along the Z-axis. Recent known data about the D1 line of
$\mathrm{{}^{87}Rb}$ can be found in [60].

According to the optical scheme shown in the Fig. 1, spins of
$\mathrm{Rb_{F=1}}$ and $\mathrm{Rb_{F=2}}$ are oriented through depopulation
and repopulation effects, respectively. Both effects create the non-
equilibrium population among Zeeman sublevels belonging to the ground levels
of the D1 line. The nature of depopulation is a selective depletion of certain
sublevels by a circularly polarized light. It should be noted, that
repopulation process is more efficient than depopulation. As presented in
[61], repopulation is based on the feature of collisional decay, that is a
nucleus spin state is not destroyed during transition of an excited alkali
atom from upper to ground levels. Therefore, spin polarization of
$\mathrm{Rb_{F=2}}$ is produced by preserved alkali nucleus spin.

###  II.3 Spin polarization scanning

Atomic ensemble with a spin polarization becomes circular birefringent due to
inducing of optical anisotropy. The magnitude of the circular birefringence is
determined by the expectation value of the atomic spin projection onto the
path of light propagation. Since refraction indices for orthogonal circular
light components are different, the linear polarization plane of the passing
light rotates around the path of propagation. Therefore, alkali spin
polarization dynamics can be measured via deviation of scanning light
polarization as follows:

$\langle\hat{F}_{\alpha}\rangle\propto\Delta\psi(t),$ (3)

where $\Delta\psi(t)$ is the angular rotation of the polarization plane shown
in the Fig. 1, $\hat{F}_{\alpha}$ is the operator of the total angular
momentum projection onto the $\alpha$-axis. By selection of the scanning light
direction, one may measure any spin polarization component of the alkali vapor
in the gas cell.

The spectral linewidth of the scanning light should be narrow enough for
resolution of the alkali hyperfine structure. It is necessary for measurement
of the spin polarization of alkali vapor only from $\mathrm{Rb_{F=2}}$. As
shown in Fig. 2, the scanning light frequency should be close to
$F=2\leftrightarrow F^{\prime}=2$ transition frequency. At the same time, it
should be far from the frequency of $F=1\leftrightarrow F^{\prime}=2$
transition. If the latter conditions are satisfied, then $\mathrm{Rb_{F=2}}$
atoms affect the polarization of the scanning light significantly stronger
than $\mathrm{Rb_{F=1}}$ atoms do. Withal, the light frequency should be
detuned from the optical resonance in order to exclude redundant depletion of
the ground level with $F=2$. Since the ground level with $F=1$ is broadened by
a pump light, the spin polarization measurements from $\mathrm{Rb_{F=1}}$ are
not advisable.

##  III Master equation

Here, we would like to propose the math model of alkali dynamics affected by a
monochromatic light and an alternating magnetic field. To correctly describe
the spin effect, we take into account common relaxation processes listed in
the Table 1. The model is based on the semi-classical theory of atom-field
interaction: alkali vapor is described by a density operator, and the fields
are described by classical vectors dependent on time. The density operator is
a $16\times 16$ matrix in the basis of non-perturbed Zeeman sublevels that
belong to D1 line of $\mathrm{{}^{87}Rb}$.

Table 1: Relaxation processes in the gas cell, their origins and typical rates with corresponding mapping Process | Origin of the | Rate,
---|---|---
description | process | Mapping
Mixing of ground | Alkali spin exchange | $\begin{array}[]{c}\Gamma\sim 10^{2}\>\text{Hz},\vspace{0.1cm}\\\ \hat{\rho}-\hat{\rho}_{0}\end{array}$
Zeeman sublevels | and atomic collision
population | with cell’s walls
Decay from excited | Non-ellastic collisions | $\begin{array}[]{c}\delta_{dcy}\sim 10^{8}\>\text{Hz},\vspace{0.1cm}\\\ \mathcal{R}\left\\{\hat{\rho}\right\\}\end{array}$
to ground levels | between excited alkali
without fluorescence | and buffer atoms
Inhomogeneous | Velocities mixing to | $\begin{array}[]{c}\delta_{mix}\sim 10^{9}\>\text{Hz},\vspace{0.1cm}\\\ \mathcal{M}\left\\{\hat{\rho}\right\\}\end{array}$
Doppler broadening | Maxwell-Boltzmann
of optical resonance | distribution
Decoherence of dipole | Alkali-buffer | $\begin{array}[]{c}\delta_{dec}\sim 10^{10}\>\text{Hz},\vspace{0.1cm}\\\ \mathcal{D}\left\\{\hat{\rho}\right\\}\end{array}$
oscillations on | elastic collisions
optical transitions | without decay

The master equation for alkali density matrix, that describes the certain
velocity group inside the Maxwell distribution, is as follows:

$\begin{array}[]{l}\displaystyle
i\hbar\left(\frac{\partial\hat{\rho}}{\partial
t}+kv_{z}\hat{\rho}\right)=\left[\hat{H},\hat{\rho}\right]-\Gamma\left(\hat{\rho}-\hat{\rho}_{0}\right)-\vspace{0.2
cm}\\\
\displaystyle-\delta_{mix}\mathcal{M}\left\\{\hat{\rho}\right\\}-\delta_{dcy}\mathcal{R}\left\\{\hat{\rho}\right\\}-\delta_{dec}\mathcal{D}\left\\{\hat{\rho}\right\\},\end{array}$
(4)

where $\hat{H}$ is the Hamiltonian without relaxation, $k$ is the wavenumber,
$v_{z}$ is the alkali velocity projection onto the path of the pump light,
operator $\hat{\rho}_{0}$ corresponds to the state of thermodynamic
equilibrium with mixed population among Zeeman sublevels. Operator $\hat{H}$
comprises the interaction of alkali with the pump light $\hat{V}_{E}$, the
interaction of alkali with magnetic field $\hat{V}_{B}$ and unperturbed
Hamiltonian $\hat{H}_{0}$:

$\displaystyle\hat{H}=\hat{H}_{0}+\hat{V}_{E}+\hat{V}_{B},$ (5)
$\displaystyle\hat{V}_{E}=-\left(\mathbf{\hat{d}\cdot
E}\right),\qquad\hat{V}_{B}=\sum\limits_{n=1}^{2}g_{n}\gamma_{e}\left(\mathbf{\hat{F}_{n}\cdot
B}\right),$ (6)
$\displaystyle\mathbf{E}=\frac{\mathcal{E}}{2}\,\mathbf{l}_{+}e^{-i\omega
t}+c.c.,\qquad k=\frac{\omega}{c}.$ (7)

Here $\mathcal{E}$ is a constant amplitude of the pump light, $\mathbf{l}_{+}$
is the unit vector of the circular polarization, $\omega$ is the frequency of
the pump light, the dipole operator $\mathbf{\hat{d}}$ describes all optical
transitions between Zeeman sublevels in D1 line of $\mathrm{{}^{87}Rb}$ [60],
$\gamma_{e}$ is the electron gyromagnetic ratio, $g_{n}$ is the g-factor of
the ground hyperfine levels, $\mathbf{\hat{F}}_{n}$ is the total angular
momentum operator corresponding to atoms $\mathrm{Rb_{F=n}}$, and magnetic
field $\mathbf{B}$ is defined by Eq. (1).

Below we define mappings $\mathcal{M}$, $\mathcal{D}$ and $\mathcal{R}$ from
the master equation (4):

  1. 1.

The mapping $\mathcal{M}$ determines the transition of the dependence of
density operator on alkali velocities to the uniform distribution:

$\mathcal{M}\left\\{\hat{\rho}\right\\}=\hat{\rho}-\int\limits_{-\infty}^{\infty}\hat{\rho}\>\mu(v_{z})\,\mathrm{d}v_{z},$
(8)

where $\mu(v_{z})$ is the Maxwell–Boltzmann distribution.

  2. 2.

The mapping $\mathcal{D}$ nullifies non-diagonal elements with an optical
frequency of phase rotation:

$\mathcal{D}\left\\{\hat{\rho}\right\\}=-\hat{P}_{e}\hat{\rho}\hat{P}_{g}-\hat{P}_{g}\hat{\rho}\hat{P}_{e},$
(9)

where $\hat{P}_{g}$ and $\hat{P}_{e}$ are projection operators to the space of
ground hyperfine levels $F$ and excited hyperfine levels $F^{\prime}$ in D1
line respectively, see Fig. 2.

  3. 3.

The mapping $\mathcal{R}$ decomposes the full density matrix $\hat{\rho}$ to a
tensor product of an electron $\hat{\rho}_{e}^{(0)}$ and nuclear
$\hat{\rho}^{\prime}_{n}$ density matrices:

$\displaystyle\mathcal{R}\left\\{\hat{\rho}\right\\}=-\hat{P}_{e}\hat{\rho}\hat{P}_{e}+\hat{\rho}^{\prime}_{n}\otimes\hat{\rho}_{e}^{(0)},$
(10)
$\displaystyle\hat{\rho}^{\prime}_{n}=\sum\limits_{m=-\nicefrac{{1}}{{2}}}^{\nicefrac{{1}}{{2}}}\langle{m;\,\scriptstyle
5^{2}P_{1/2}}\,|\,\hat{\rho}\,|\,m;\,{\scriptstyle 5^{2}P_{1/2}}\rangle,$ (11)
$\displaystyle\hat{\rho}_{e}^{(0)}=\sum\limits_{m=-\nicefrac{{1}}{{2}}}^{\nicefrac{{1}}{{2}}}\frac{1}{2}\,|\,m;\,{\scriptstyle
5^{2}S_{1/2}}\rangle\langle m;\,{\scriptstyle 5^{2}S_{1/2}}\,|,$ (12)

where $|m;level\rangle$ is a spin-orbital state of an external electron with a
projection $m$ of total angular momentum.

Note, the electron density matrix reduces to the one of equilibrium state,
however the nuclear density matrix does not. The partial trace in Eq. (11)
leads to a non-equilibrium distribution of population among Zeeman sublevels
of $\mathrm{Rb_{F=2}}$. Due to this feature, alkali spin polarization during
collision decay is preserved [45, 61].

Figure 3: Peaks in the $Rb_{F=2}$ spin polarization dependence on the
frequency of periodic magnetic field determined by Eq. (1); curves are defined
by Eqs. (16) and (17), correspondingly. Parameters for the evaluation are as
follows: magnetic field amplitude $B_{0}=27\ \mu T$, the relaxation rate
$\Gamma=1\ kHz$, the pump light amplitude $\mathcal{E}=100\ V/m$, the
temperature is $80^{\circ}\mathrm{C}$

##  IV Non-adiabatic spin dynamics evaluation

In order to proceed with the numerical solution of the master equation (4),
let us prior define the vectors $\mathbf{S_{[1]}}$ and $\mathbf{S_{[2]}}$ for
describing a spin polarization formed by alkali atoms $\mathrm{Rb_{F=1}}$ and
$\mathrm{Rb_{F=2}}$ respectively:

$\displaystyle\mathbf{S_{[n]}}=S_{[n],x}\mathbf{l_{x}}+S_{[n],y}\mathbf{l_{y}}+S_{[n],z}\mathbf{l_{z}},$
(13) $\displaystyle
S_{[n],\alpha}=\mathrm{Tr}\left\\{\hat{\rho}_{n}\hat{\Sigma}_{n,\alpha}\right\\},\qquad\alpha\in\\{x,y,z\\},$
(14)
$\displaystyle\hat{\rho}_{n}=\int\limits_{-\infty}^{\infty}\hat{P}_{n}\hat{\rho}\hat{P}_{n}\mu(v_{z})\,\mathrm{d}v_{z},$
(15)

where $\mathbf{l_{\alpha}}$ are orthonormal spatial vectors; $\hat{P}_{n}$ is
the operator of projection to the ground hyperfine level with the total
angular momentum $F=n$; symbol $\hat{\Sigma}_{n,\alpha}$ denotes a Pauli
matrix equivalent to a spin-n particle; value $S_{[n],\alpha}$ is a component
of spin polarization formed by $\mathrm{Rb}_{F=n}$ atoms; cropped density
matrix $\hat{\rho}_{n}$ belongs to a subspace of $F=n$ hyperfine level, its
rank is equal to $2n+1$ according to the number of Zeeman sublevels. Note, the
cropped density matrix is averaged over alkali velocities.

As the vectors above have been defined, we come to a solution of the master
equation (4) with satisfied conditions of non-adiabatic dynamics. The solution
is density matrix dependent on time for any frequency $\Omega$ of the magnetic
field defined by Eq. (1). For the last period of the solution, we calculate
spin polarization components via Eqs. (13) – (15) for a wide range of
frequencies $\Omega$. Finally, frequency dependence of two convolutions are
presented in the Fig. 3, the latter demonstrates some peaks of spin
polarization:

$\displaystyle\mathcal{C}_{1}(\Omega)=\max\limits_{t\in
T_{last}}\,\left|\mathbf{S_{[2]}}(t)\right|=\mathrm{sup}\,\left|\>\mathbf{S_{[2]}}\>\right|,$
(16)
$\displaystyle\mathcal{C}_{2}(\Omega)=\left|\>\frac{1}{T}\\!\\!\\!\int\limits_{t\in
T_{last}}\\!\\!\\!\\!\\!\mathbf{S_{[2]}}(t)\,\mathrm{d}t\>\right|=\left|\>\overline{\mathbf{S_{[2]}}}\>\right|.$
(17)

Regarding the selective scanning of alkali spin polarization (see Fig. 2), we
observe only $\mathbf{S_{[2]}}$ vector.

These convolutions give general information about spin dynamics. The first one
describes the radius of the sphere that contains a trajectory of the vector
$\mathbf{S_{[2]}}$ moving in the three-dimensional space of spin polarization
components. The larger the $\mathcal{C}_{1}$, the more oriented alkali spins
are. The second one describes alkali vapor magnetization, determined by the
mean spin polarization over time. It can be measured by rotation of the
linearly polarized scanning light, i.e. angle $\Delta\psi$ in the Fig. 1 and
Sec. II.3.

According to Fig. 3, the highest peak is the second from the right, it appears
near the frequency $\Omega\approx 33.2\ kHz$. There are no additional peaks in
the region above $50\ kHz$. Low frequencies are not considered in the work,
since corresponding dynamics are close to adiabatic case there.

###  IV.1 Width of the highest peak

Relaxation of a spin polarization occurs due to population mixing among the
ground Zeeman sublevels, see the Table 1. This process is defined by the
relaxation rate $\Gamma$, which determines the form of peaks.

In the Fig. 3, the second from the right (at $\approx 33.2\ kHz$) peak of the
convolution $\mathcal{C}_{2}$ is of interest. The high altitude of the peak is
more important than its narrow width from the practical point of view: the
less altitude, the lower signal-to-noise ratio. It is so since approximately
only one alkali atom in a hundred creates spin polarization, hence detection
of the effect may be challenging. Also, the $\mathcal{C}_{2}$ is more contrast
compare to the convolution $\mathcal{C}_{1}$.

Figure 4: The second from the right peak in the Fig. 3 calculated for the
different relaxation rates $\Gamma$. There is only convolution Eq. (17) in the
picture

In Fig. 4 one may observe dependence of its shape on different rates of
$\Gamma$. Note, that the peak altitude and relaxation rate $\Gamma$ are not in
inverse proportion. Therefore, a the spin effect behavior differs from a
driven damped oscillator in condition of the resonance, when an amplitude of
oscillations and the damping ratio are inverse proportional. Furthermore, the
results are not well approximated by the Lorentz function as for a damped
oscillator. Regarding the latter, we consider steep increase of spin
polarization as non-adiabatic effect of spin dynamics, though its physical
nature is different from the EPR.

Another observable parameter is a width of the peak dependent on relaxation
rate $\Gamma$, we specify it by the half width at half maximum (HWHM). We
would like to compare the width of the peak dependence on relaxation rates
$\Gamma$ with the corresponding dependence in case of the EPR under the same
conditions of pumping and relaxation. The data for EPR is obtained by solving
the master equation (4) implying magnetic field is defined as follows (instead
of Eq. (1) for the spin effect):

$\mathbf{B}=B_{dc}\mathbf{l_{z}}+B_{ac}\mathbf{l_{x}}\cos\left(\Omega
t\right),\qquad B_{dc}\gg B_{ac},$ (18)

where the constant magnetic field $B_{dc}$ determines the resonance frequency,
and the alternating magnetic field with the amplitude $B_{ac}$ induces the
resonance dynamics of alkali spin. In the Fig. 5 one may observe calculated
HWHMs for a given relaxation rates $\Gamma$ as in Fig. 4. As a result, we
estimate that the HWHM of the observed peak for the spin effect is $3.5$ times
less than the one of the EPR. Note, that HWHM and relaxation rate are
proportional in both cases.

Figure 5: Comparison of the peak’s width in the Fig. 4 with the width of the
EPR calculated under the same conditions of pumping and relaxation. When the
EPR was modeling, a magnetic field was defined by Eq. (18) with $B_{dc}=10\
\mu T$ and $B_{ac}=10\ nT$. These parameters are chosen to match the frequency
of the peak ($\approx 33.2\ kHz$)

###  IV.2 Spin dynamics near peak polarization

Further we explore the steady non-adiabatic dynamics of the spin polarization
created by atoms $\mathrm{Rb}_{F=2}$, when the frequency of the magnetic
field, defined by Eq. (1), maximizes
$\left|\overline{\mathbf{S_{[2]}}}\right|$, i.e. $\Omega\approx 33.2\ kHz$,
see the Fig. 3. We consider the three components of the spin polarization,
which can be measured by scanning light: $S_{[2],x}(t)$, $S_{[2],y}(t)$, and
$S_{[2],z}(t)$. The trajectory of the vector $\mathbf{S_{[2]}}(t)$ in three-
dimension space of spin polarization components and its parametric
representation are plotted in the Fig. 6. It is mirror symmetric about two
planes: $S_{x}S_{z}$ and $S_{y}S_{z}$. A little deviation from the mirror
symmetry about $S_{x}S_{z}$ the plane occurs due to location of the magnetic
field in the $XZ$ plane, see Fig. 1. At the same time, the trajectory is not
mirror symmetric about the $S_{x}S_{y}$ plane. We assume that the axis of the
general shift of the spin polarization is correlated to the direction of pump
light.

The average radius and the Z-shift of the trajectory is described by two
convolutions $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$, defined in the Eqs. (16)
and (17). So alteration of these trajectory characteristics may be estimated
by the observation of the latter convolutions dependence on frequency
$\Omega$, shown in Fig. 3. If the frequency deviates from the value at the
point of maximum, the trajectory collapses to the axis-origin and reshapes to
isotropic tangle clew. Reshaping of the trajectory explains the higher rate of
the collapse of $\mathcal{C}_{2}$ convolution compare to the one of
$\mathcal{C}_{1}$ convolution. Note, the spin polarization trajectories are
different for each of four peaks shown in the Fig. 3. Despite we provide the
figure for only one, all other trajectories exhibit similar properties: two
mirror symmetries and “protuberant” along the path of pump light propagation.
To observe correlations, time dependencies $S_{\alpha}(t)$ in the Fig. 6 are
plotted together with the magnetic field dynamics. Surprisingly, non-adiabatic
dynamics of spin polarization involves strong higher harmonics, which are
absent in the dynamics of the magnetic field.

Figure 6: The purple parametric 3d-curve drawn by three temporal dependencies
of the spin polarization components $S_{[2],\alpha}$, where
$\alpha\in\\{x,y,z\\}$. There is a steady dynamics of spin polarization shown
in the picture. Blue curves demonstrate two components of the periodic
magnetic field, defined by Eq. (1). Note, the index $[2]$ in $S_{[2],\alpha}$
is dismissed for the sake of visual simplicity. The purple curves are
calculated for the frequency $\Omega\approx 33.2\ kHz$, which matches the
maximum of the peak in the Fig. 3

##  V Analysis

We investigate the nature of steep increase of an alkali spin polarization
under non-adiabatic dynamics in presence of the strong alternating magnetic
field. As discussed in the previous section, the spin effect appears to be not
a resonance. We claim that the spin effect is a phenomenon of unpredictable
periodic spin dynamics, that lasts while spin state decays. It occurs only in
the non-adiabatic case. Then, the main condition of the spin effect to be
observed is a synchronous spin motion inside an alkali vapor. If atomic spins
are not synchronized, the ensemble of atoms tends to a mixed spin state with
equilibrium distribution, and the spin polarization is vanished. Note, that
observed effect does not require the same spin dynamics for all atoms.

In case of EPR, one component of the vapor magnetization approaches to a
saturated state and then does not vary; this component is often called
longitudinal. At the same time, both transverse components rapidly oscillate,
but in most cases its magnitudes are small in comparison to the longitudinal
component. Thus, EPR cannot be interpreted as a non-adiabatic process.

In order to explain the synchronous spin motion, let us describe an alkali
atom $\mathrm{Rb}_{L=2}$, that is moving as free particle between two time
points: the moment of initial spin state creation due to pump photon
absorption, and the moment of spin state decay due to collision with the cell
walls or with another alkali atom. Time interval of the free path of motion is
estimated by inverse of relaxation rate $\Gamma$. As the first order
approximation, we can use the Pauli equation to model a free alkali atom in an
alternating magnetic field:

$\displaystyle
i\hbar\dot{\varphi}=\gamma\hbar\left(\hat{\sigma}\cdot\mathbf{B}(t)\right)\varphi,$
(19)
$\displaystyle\hat{\sigma}=\frac{1}{2}\left(\hat{\sigma}_{x}\mathbf{l_{x}}+\hat{\sigma}_{y}\mathbf{l_{y}}+\hat{\sigma}_{z}\mathbf{l_{z}}\right),$
(20) $\displaystyle\varphi(t)=\hat{U}(t,t_{0})\varphi_{0},$ (21)

where $\varphi$ is the spinor, $\gamma$ is the gyromagnetic ratio,
$\hat{\sigma}_{\alpha}$ are Pauli matrices, where $\alpha\in\\{x,y,z\\}$,
$\varphi_{0}$ is the spinor state at the initial time $t_{0}$,
$\hat{U}(t,t_{0})$ is the unitary time-evolution operator.

Table 2: The highest four frequencies, that lead to periodic solution of the Pauli equation (19), and corresponding solution properties Frequency | Initial spin | Spin components
---|---|---
from the set | direction | averaged over a period ($\times 10^{3}$)
$\Omega_{\mathcal{A}}/\gamma B_{0}$ | $\langle\varphi_{0}|\hat{\sigma}|\varphi_{0}\rangle$ | $\overline{\langle\varphi|\hat{\sigma}_{x}|\varphi\rangle}$ | $\overline{\langle\varphi|\hat{\sigma}_{y}|\varphi\rangle}$ | $\overline{\langle\varphi|\hat{\sigma}_{z}|\varphi\rangle}$
$0.099$ | $=\mathbf{l_{x}}/2$ | -48.4 | 0 | -26.4
$=\mathbf{l_{y}}/2$ | 0 | -5.8 | 0
$=\mathbf{l_{z}}/2$ | -22.6 | 0 | 56.3
$0.126$ | $=\mathbf{l_{x}}/2$ | 1.5 | 0 | -56.9
$=\mathbf{l_{y}}/2$ | 0 | 31.7 | 0
$=\mathbf{l_{z}}/2$ | 1.6 | 0 | 57.5
$0.175$ | $=\mathbf{l_{x}}/2$ | -4.7 | 0 | 196.8
$=\mathbf{l_{y}}/2$ | 0 | -35.6 | 0
$=\mathbf{l_{z}}/2$ | -52.5 | 0 | -17.6
$0.259$ | $=\mathbf{l_{x}}/2$ | -44.3 | 0 | 5.5
$=\mathbf{l_{y}}/2$ | 0 | -82.2 | 0
$=\mathbf{l_{z}}/2$ | 4.2 | 0 | 58.4

The Pauli equation (19) with the alternating magnetic field $\mathbf{B}(t)$,
defined by Eq. (1), generates a set $\mathcal{A}$ of frequencies, which lead
to a periodic solution independent of the spinor state at the initial time:

$\forall\varphi(t):\quad\varphi(t+2\pi/\Omega_{\mathcal{A}})=\varphi(t),$ (22)

where $\Omega_{\mathcal{A}}\in\mathcal{A}$. The set $\mathcal{A}$ is discrete
and upper-bounded by a maximal frequency. A sequence of the highest
frequencies in the set $\mathcal{A}$ corresponds to coordinates of the peaks
in the Fig. 3. Note, that if a frequency from the absolute complement of
$\mathcal{A}$ is substituted into the Pauli equation, the solution can be
periodic only for the certain initial state.

In the Table 2 we present the four highest frequencies from the set
$\mathcal{A}$ divided by $\gamma B_{0}$ and corresponding properties of the
Pauli equation solutions. Although, periodicity does not depend on an initial
state for the frequency $\Omega_{\mathcal{A}}$, the solution of the Pauli
equation (19) does. To describe a behavior of the spin dynamics, we define
spin components averaged over a period. The latter corresponds to the second
convolution expressed by Eq. (17). It is evaluated for a given three
(orthogonal) initial states and four different frequencies
$\Omega_{\mathcal{A}}$.

Data in the Table 2 shows us, that maximal magnetization occurs when frequency
$\Omega$ is equal to the second highest $\Omega_{\mathcal{A}}$ ($\approx 33.2\
kHz$ and corresponds to $\Omega_{\mathcal{A}}/\gamma B_{0}\approx 0.175$). The
averaged spin polarization is quasi-collinear with the Z-axis, which matches
with the path of the pump light propagation.

Based on the arguments above, we propose the following statements that explain
the studied spin effect:

  1. 1.

Permanent relaxation and pump of the alkali spin polarization leads to jumps
of atomic spin between different quantum states;

  2. 2.

If the dynamics is non-adiabatic and periodic, alkali spins can synchronously
move starting from a random the initial spin state;

  3. 3.

Condition of the synchronous moving is the certain frequency/amplitude rate of
an alternating magnetic field, which is determined by corresponding temporal
profile;

  4. 4.

The effect of synchronized and repetitive motion of the spins appears as steep
increase of an averaged spin polarization over a period.

##  VI Conclusion

We have considered a non-adiabatic spin polarization dynamics, which occurs in
alkali vapor under optical spin orientation and in the presence of a strong
alternating magnetic field. By solving the master equation for a density
matrix, we have revealed the effect that resembles a resonance. If the
frequency of a periodic magnetic field without a constant component is equal
to a certain value, an alkali spin polarization steeply increases. However,
with a certainty we claim, that it is not an electron spin resonance, as it is
revealed by the analysis of the peaks that may be observed in the dependence
of the spin polarization on the magnetic field frequency. We interpret the
effect as a fundamental property of non-adiabatic spin polarization dynamics
in presence of the external magnetic field, when spins that belong to an
alkali ensemble move synchronously only for a certain frequency.
Mathematically, it can be explained by the existence of Pauli equation
solutions with periodic behavior, if periodic magnetic field has the certain
profile.

The important result from the practical point of view is that the width of one
of the found peaks is significantly narrower than the width of EPR induced in
the same medium with similar conditions, see Fig. 5. Therefore, the effect may
find its place in the field of precise sensing.

## Funding

This work was financially supported by Russian Ministry of Education (Grant
No. 2019-0903)

###### Acknowledgements.

We thank our colleagues A.D. Kiselev and G.P. Miroshnichenko for fruitful
discussions during the research.

## References

  * Balabas  _et al._ [2010a] M. V. Balabas, T. Karaulanov, M. P. Ledbetter, and D. Budker, Polarized alkali-metal vapor with minute-long transverse spin-relaxation time, Phys. Rev. Lett. 105, 070801 (2010a).
  * Balabas  _et al._ [2010b] M. V. Balabas, K. Jensen, W. Wasilewski, H. Krauter, L. S. Madsen, J. H. Müller, T. Fernholz, and E. S. Polzik, High quality anti-relaxation coating material for alkali atom vapor cells, Opt. Express 18, 5825 (2010b).
  * Chi  _et al._ [2020] H. Chi, W. Quan, J. Zhang, L. Zhao, and J. Fang, Advances in anti-relaxation coatings of alkali-metal vapor cells, Applied Surface Science 501, 143897 (2020).
  * Goldfarb and Stoll [2018] D. Goldfarb and S. Stoll,  _EPR Spectroscopy: Fundamentals and Methods_ (John Wiley and Sons, 2018).
  * Weil and Bolton [2007] J. A. Weil and J. R. Bolton,  _Electron Paramagnetic Resonance: Elementary Theory and Practical Applications_ (John Wiley and Sons, 2007).
  * Pomerantsev  _et al._ [1972] N. M. Pomerantsev, V. M. Ryzhkov, and G. V. Skrotskii,  _Physical Principles of Quantum Magnetometry_ (Nauka, Moscow, 1972) p. 448.
  * Farr and Otten [1974] W. Farr and E. W. Otten, A rb-magnetometer for a wide range and high sensitivity, Applied physics 3, 367 (1974).
  * Aleksandrov  _et al._ [1999] E. B. Aleksandrov, M. V. Balabas, A. K. Vershovskii, and A. S. Pazgalev, Multiple-quantum radio-frequency spectroscopy of atoms: Application to the metrology of geomagnetic fields, Technical Physics 44, 1025 (1999).
  * Budker and Romalis [2007] D. Budker and M. Romalis, Optical magnetometry, Nature Physics 3, 227 (2007).
  * Bison  _et al._ [2003] G. Bison, R. Wynands, and A. Weis, A laser-pumped magnetometer for the mapping of human cardiomagnetic fields, Applied Physics B 76, 325 (2003).
  * Budker and Kimball [2013] D. Budker and D. F. J. Kimball,  _Optical Magnetometry_ (Cambridge University Press, 2013).
  * Aleksandrov and Vershovskii [2009] E. B. Aleksandrov and A. K. Vershovskii, Modern radio-optical methods in quantum magnetometry, Phys.-Usp. 52, 573 (2009).
  * Zhivun  _et al._ [2014a] E. Zhivun, A. Wickenbrock, B. Patton, and D. Budker, Alkali-vapor magnetic resonance driven by fictitious radiofrequency fields, Applied Physics Letters 105, 10.1063/1.4902028 (2014a).
  * Woodman  _et al._ [1987] K. F. Woodman, P. W. Franks, and M. D. Richards, The nuclear magnetic resonance gyroscope: a review, Journal of Navigation 40, 366–384 (1987).
  * Donley [2010] E. A. Donley, Nuclear magnetic resonance gyroscopes, 2010 IEEE Sensors , 17 (2010).
  * Vershovskii  _et al._ [2018] A. K. Vershovskii, Y. A. Litmanovich, A. S. Pazgalev, and V. G. Peshekhonov, Nuclear magnetic resonance gyro: Ultimate parameters, Gyroscopy and Navigation 9, 162 (2018).
  * Kornack  _et al._ [2005] T. W. Kornack, R. K. Ghosh, and M. V. Romalis, Nuclear spin gyroscope based on an atomic comagnetometer, Phys. Rev. Lett. 95, 230801 (2005).
  * Walker and Larsen [2016] T. G. Walker and M. S. Larsen, Chapter eight - spin-exchange-pumped nmr gyros (Academic Press, 2016) pp. 373–401.
  * Popov  _et al._ [2018a] E. N. Popov, K. A. Barantsev, N. A. Ushakov, A. N. Litvinov, L. B. Liokumovich, A. N. Shevchenko, F. V. Sklyarov, and A. V. Medvedev, Behavior of signal from optical circuit of quantum rotation sensor based on nuclear magnetic resonance, Gyroscopy and Navigation 9, 183 (2018a).
  * Zhang  _et al._ [2020] K. Zhang, N. Zhao, and Y.-H. Wang, Closed-loop nuclear magnetic resonance gyroscope based on rb-xe, Scientific Reports 10, 2258 (2020).
  * Gao  _et al._ [2024] G. Gao, J. Hu, F. Tang, W. Liu, X. Zhang, B. Wang, D. Deng, M. Zhu, and N. Zhao, Stability improvement of nuclear magnetic resonance gyroscope with self-calibrating parametric magnetometer, Phys. Rev. Appl. 21, 014042 (2024).
  * Camparo and Frueholz [1987] J. Camparo and R. Frueholz, A comparison of various alkali gas-cell atomic-frequency standards, IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control 34, 607 (1987).
  * Jau and Carter [2020] Y.-Y. Jau and T. Carter, Vapor-cell-based atomic electrometry for detection frequencies below 1 khz, Phys. Rev. Appl. 13, 054034 (2020).
  * Happer [1972] W. Happer, Optical pumping, Rev. Mod. Phys. 44, 169 (1972).
  * Happer  _et al._ [2010] W. Happer, Y.-Y. Jau, and T. Walker,  _Optically Pumped Atoms_ (John Wiley and Sons, 2010).
  * Kubo [1972] T. Kubo, Optical detection of magnetic resonance in alkali-metal, vapor using d(1) and d(2) radiation simultaneously, Appl Opt. 11, 1521 (1972).
  * Aleksandrov and Zapasskii [1981] E. B. Aleksandrov and V. S. Zapasskii, Magnetic resonance in the faraday-rotation noise spectrum, JETP 54, 64 (1981).
  * Happer and Van Wijngaarden [1987] W. Happer and W. A. Van Wijngaarden, An optical pumping primer, Hyperfine Interactions 38, 435 (1987).
  * Suter [2020] D. Suter, Optical detection of magnetic resonance, Magnetic Resonance 1, 115 (2020).
  * Školnik  _et al._ [2009] G. Školnik, N. Vujičić, and T. Ban, Optical pumping of the zeeman components in the rubidium vapor, Optics Communications 282, 1326 (2009).
  * Kozlov  _et al._ [2016] A. N. Kozlov, S. A. Zibrov, A. A. Zibrov, A. V. Yudin, V. I. Taichenachev, V. P. Yakovlev, A. S. Tsygankov, E. A. Zibrov, V. V. Vassiliev, and V. L. Velichansky, Structure of magnetic resonance in 87rb atoms, JETP 122, 823 (2016).
  * Talker  _et al._ [2019] E. Talker, P. Arora, Y. Barash, D. Wilkowski, and U. Levy, Efficient optical pumping of alkaline atoms for evanescent fields at dielectric-vapor interfaces, Opt. Express 27, 33445 (2019).
  * Franzen and Emslie [1957] W. Franzen and A. G. Emslie, Atomic orientation by optical pumping, Phys. Rev. 108, 1453 (1957).
  * Kastler [1963] A. Kastler, Displacement of energy levels of atoms by light, J. Opt. Soc. Am. 53, 902 (1963).
  * Kastler [1966] A. Kastler, Les methodes optiques pour l’etude des resonances hertziennes, in  _Preprint les Prix Nobel en 1966_ (Conference Nobel, Stockholm, 1966).
  * Zavoisky [1945] E. Zavoisky, Spin-magnetic resonance in paramagnetics, J. Phys. USSR , 211 (1945).
  * Dehmelt [1957] H. G. Dehmelt, Modulation of a light beam by precessing absorbing atoms, Phys. Rev. 105, 1924 (1957).
  * Bell and Bloom [1957] W. E. Bell and A. L. Bloom, Optical detection of magnetic resonance in alkali metal vapor, Phys. Rev. 107, 1559 (1957).
  * Hawkins [1961] W. B. Hawkins, Cesium transition probabilities for optical pumping, Phys. Rev. 123, 544 (1961).
  * Anderson and Ramsey [1963] L. W. Anderson and A. T. Ramsey, Study of the spin-relaxation times and the effects of spin-exchange collisions in an optically oriented sodium vapor, Phys. Rev. 132, 712 (1963).
  * Gamblin and Carver [1965] R. L. Gamblin and T. R. Carver, Polarization and relaxation processes in ${\mathrm{he}}^{3}$ gas, Phys. Rev. 138, A946 (1965).
  * Aymar  _et al._ [1967] M. Aymar, M. Bouchiat, and J. Brossel, Relaxation of optically pumped rb in collisions against kr atoms, Physics Letters A 24, 753 (1967).
  * Bouchiat  _et al._ [2003] M. A. Bouchiat, J. Brossel, and L. C. Pottier, Evidence for Rb‐Rare‐Gas Molecules from the Relaxation of Polarized Rb Atoms in a Rare Gas. Experimental Results, The Journal of Chemical Physics 56, 3703 (2003).
  * Bernabeu and Tornos [1985] E. Bernabeu and J. Tornos, Relaxation of quadrupole orientation in an optically pumped alkali vapour, Il Nuovo Cimento D 5, 315 (1985).
  * Franz and Franz [1966] F. A. Franz and J. R. Franz, Excited-state mixing in the optical pumping of alkali-metal vapors, Phys. Rev. 148, 82 (1966).
  * Bouchiat and Brossel [1966] M. A. Bouchiat and J. Brossel, Relaxation of optically pumped rb atoms on paraffin-coated walls, Phys. Rev. 147, 41 (1966).
  * Breault [1964] R. O. Breault,  _A Measurement of the Spin Relaxation Time of Optically Pumped Rubidium Vapor_ , Master’s thesis, Western Michigan University (1964).
  * Cutler  _et al._ [2020] T. Cutler, W. Hamlyn, J. Renger, K. Whittaker, D. Pizzey, I. Hughes, V. Sandoghdar, and C. Adams, Nanostructured alkali-metal vapor cells, Phys. Rev. Appl. 14, 034054 (2020).
  * Appelt  _et al._ [1995] S. Appelt, G. Wäckerle, and M. Mehring, A magnetic resonance study of non-adiabatic evolution of spin quantum states, Zeitschrift für Physik D 34, 75 (1995).
  * Gaitan [1999] F. Gaitan, Berry’s phase in the presence of a non-adiabatic environment with an application to magnetic resonance, Journal of Magnetic Resonance 139, 152 (1999).
  * Zhivun  _et al._ [2014b] E. Zhivun, A. Wickenbrock, B. Patton, and D. Budker, Alkali-vapor magnetic resonance driven by fictitious radiofrequency fields, Applied Physics Letters 105, 192406 (2014b), https://pubs.aip.org/aip/apl/article-pdf/doi/10.1063/1.4902028/13802950/192406_1_online.pdf .
  * Liu  _et al._ [2021] G. Liu, V. Guarrera, S. Gu, and S. Zhang, Response dynamics of an alkali-metal–noble-gas hybrid trispin system, Phys. Rev. A 104, 032827 (2021).
  * Titimbo  _et al._ [2023] K. Titimbo, D. C. Garrett, S. S. Kahraman, Z. He, and L. V. Wang, Numerical modeling of the multi-stage stern–gerlach experiment by frisch and segrè using co-quantum dynamics via the bloch equation, Journal of Physics B: Atomic, Molecular and Optical Physics 56, 205004 (2023).
  * Giuseppe  _et al._ [2021] B. Giuseppe, B. Valerio, D. Yordanka, F. Alessandro, and V. Antonio, Spin dynamic response to a time dependent field, Applied Physics B 127, 128 (2021).
  * Zhang  _et al._ [2017] R. Zhang, Z.-G. Wang, X. Peng, W.-H. Li, S.-J. Li, and H. Guo, Spin dynamics of magnetic resonance with parametric modulation in a potassium vapor cell*, Chinese Physics B 26, 030701 (2017).
  * Gong  _et al._ [2007] F. Gong, Y.-Y. Jau, and W. Happer, Magnetic resonance reversals in optically pumped alkali-metal vapor, Phys. Rev. A 75, 053415 (2007).
  * Su  _et al._ [2023] S. Su, Z. Xu, X. He, C. Yin, M. Kong, X. Zhang, Y. Ruan, K. Li, and Q. Lin, An integrated single-beam three-axis high-sensitivity magnetometer, Sensors 23, 10.3390/s23063148 (2023).
  * Chen  _et al._ [2022] L. Chen, M. Yao, Y. Shi, J. Duan, S. Kang, P. Wang, Z. Yu, S. Guo, and X. Liu, Microwave magnetic field strength imaging based on Rabi resonance with alkali-atom vapor cell, Applied Physics Letters 120, 234001 (2022), https://pubs.aip.org/aip/apl/article-pdf/doi/10.1063/5.0098071/16449633/234001_1_online.pdf .
  * Weigert [2002] S. Weigert, Quantum parametric resonance, Journal of Physics A: Mathematical and General 35, 4169 (2002).
  * Steck  _et al._ [2001] D. A. Steck, M. Gehm, and T. Tiecke, Alkali d-line data, https://steck.us/alkalidata/ (2001).
  * Popov  _et al._ [2018b] E. N. Popov, V. A. Bobrikova, S. P. Voskoboinikov, K. A. Barantsev, S. M. Ustinov, A. N. Litvinov, A. K. Vershovskii, S. P. Dmitriev, V. A. Kartoshkin, A. S. Pazgalev, and M. V. Petrenko, Features of the formation of the spin polarization of an alkali metal at the resolution of hyperfine sublevels in the 2s1/2 state, JETP Letters 108, 513 (2018b).
  * Preston [1996] D. W. Preston, Doppler‐free saturated absorption: Laser spectroscopy, American Journal of Physics 64, 1432 (1996).
  * Happer  _et al._ [1984] W. Happer, E. Miron, S. Schaefer, D. Schreiber, W. A. van Wijngaarden, and X. Zeng, Polarization of the nuclear spins of noble-gas atoms by spin exchange with optically pumped alkali-metal atoms, Phys. Rev. A 29, 3092 (1984).
  * Zeng  _et al._ [1985] X. Zeng, Z. Wu, T. Call, E. Miron, D. Schreiber, and W. Happer, Experimental determination of the rate constants for spin exchange between optically pumped k, rb, and cs atoms and ${}^{129}\mathrm{Xe}$ nuclei in alkali-metal–noble-gas van der waals molecules, Phys. Rev. A 31, 260 (1985).
  * Knize  _et al._ [1988] R. Knize, Z. Wu, and W. Happer, Optical pumping and spin exchange in gas cells (Academic Press, 1988) pp. 223–267.
  * Walker [1989] T. G. Walker, Estimates of spin-exchange parameters for alkali-metal–noble-gas pairs, Phys. Rev. A 40, 4959 (1989).
  * Walker and Happer [1997] T. G. Walker and W. Happer, Spin-exchange optical pumping of noble-gas nuclei, Rev. Mod. Phys. 69, 629 (1997).
  * Happer, W. [1985] Happer, W., Spin exchange, past, present and future, Ann. Phys. Fr. 10, 645 (1985).
  * Sinatra [2022] A. Sinatra, Noble-gas and alkali spins exchange excitations, Nature Physics 18, 487 (2022).