# Sub-Doppler spectra of sodium $D$ lines in a wide range of magnetic field:
theoretical study

Rodolphe MOMIER<EMAIL_ADDRESS>Laboratoire Interdisciplinaire
Carnot de Bourgogne, UMR CNRS 6303, Université Bourgogne Franche-Comté, 21000
Dijon, France  Institute for Physical Research, NAS of Armenia, 0203
Ashtarak-2, Republic of Armenia  Aram V. PAPOYAN  Institute for Physical
Research, NAS of Armenia, 0203 Ashtarak-2, Republic of Armenia  Claude LEROY
Laboratoire Interdisciplinaire Carnot de Bourgogne, UMR CNRS 6303, Université
Bourgogne Franche-Comté, 21000 Dijon, France

(April 2021)

###### Abstract

We compute the interaction of a sodium vapor with a static magnetic field
ranging from 0 up to 1 Tesla, which allows to obtain the behavior of all
Zeeman transitions as a function of the magnetic field for any polarization of
incident laser radiation: $\pi,\sigma^{\pm}$. We use these results combined
with a Fabry-Pérot microcavity model to describe the transmitted and reflected
signal of a sodium vapor confined in a nanometric-thin cell. This allows us to
present for the first time high resolution absorption spectra, and provide a
complete description of the Zeeman transitions, along with some important
peculiarities such as the appearance of guiding transitions and magnetically-
induced circular dichroism. The obtained theoretical results can be used in
the upcoming experiments with sodium vapor nanocells.

##  I Introduction

Alkali metal atoms are recognized as classical objects of research in atomic
physics for a number of reasons. In terms of experiment, their convenience is
due to the high density of atomic vapor at relatively low temperatures, and
the presence of strong optical transitions from the ground state ($D_{1}$ and
$D_{2}$ lines) in the visible or near-infrared region of the spectrum. From a
theoretical point of view, they are attractive due to simplicity of their
energy levels caused by the presence of a single electron on the outer shell,
which significantly facilitates the calculations, so that spectroscopic data
are now very well documented [1, 2, 3, 4, 5, 6, 7].

Magneto-optical phenomena in alkali metal vapors are of particular interest.
Complete understanding of various magneto-optical effects occurring in alkali
vapors [8] is important since they are widely used, for example in fundamental
and applied studies of electromagnetically induced transparency [9, 10],
Faraday filters [11, 12], optical magnetometry [13], laser-frequency locking
[14] and elsewhere. Such phenomena often rely on the peculiarities of the
behavior of Zeeman transitions, so that frequency separation of individual
transitions is of high importance. Meanwhile for a vapor media, the thermal
motion of atoms resulting in Doppler broadening leads to inhomogeneous
broadening and overlapping of hyperfine transitions and their Zeeman
components, thus limiting its spectroscopic applicability.

Significant sub-Doppler narrowing of atomic transitions can be attained with
the use of so-called optical nanocells [15, 16, 17]. Being enclosed in optical
cells with a thickness commensurable with the resonant wavelength, alkali
metal vapors become a powerful tool for the high-resolution atomic
spectroscopy, opening new possibilities for studying magneto-optical
processes, where spectral resolution of individual transitions between
magnetic sublevels (Zeeman transitions) is essential. Complete determination
of the behavior of all the possible individual Zeeman transitions can be done
using a well-known theoretical model first provided by Tremblay et al. [18].
Importantly, besides frequency splitting in magnetic field, Zeeman transitions
undergo also significant probability changes [17]. This allows to observe the
appearance of several peculiarities, such as Guiding Transitions (GT) and two
different types of so-called Magnetically-induced Circular Dichroism (MCD)
[19, 20]. Moreover, several efficient techniques have been developed to
enhance a spectral visibility of Zeeman transitions while preserving their
relative probability scaling, such as derivative of Selective Reflection (dSR)
[21, 13] and Second Derivative of absorption (SD) [22, 23, 24].

Most of the experimental studies was done using rubidium and cesium atomic
vapor due to the availability of relatively inexpensive cw diode lasers
operating in their resonant wavelength range. For these atoms, also the
hyperfine splitting is affordably large (the lowest frequency separation
between adjacent upper states is $\approx 30$ MHz for 85Rb).

The magneto-optical effects on sodium $D_{1}$ and $D_{2}$ lines have been much
less studied. The reason for this is the lack of convenient inexpensive lasers
operating in the region of these resonances (589.6 nm and 589.0 nm,
respectively), as well as the relatively low vapor density. Although some
information concerning Zeeman transitions of sodium exist in the literature
[25, 26, 27], the available information is far from being complete. Meanwhile,
recently interest in magneto-optical processes in sodium vapor has increased,
in particular, due to the topical problem of mesospheric sodium [28].

In this paper, we provide for the first time a comprehensive set of
theoretical sub-Doppler spectra of all Zeeman transitions for both lines of
sodium in a magnetic field, as well as a thorough description of most of the
pecularities that occur. The results we present throughout this paper are
qualitatively analogous to the ones obtained for the $D_{1}$ and $D_{2}$ line
of ${}^{87}\text{Rb}$ and ${}^{39}\text{K}$ representing similar quantum
systems, therefore we expect total agreement between them and upcoming
experiments with Na nanocells.

##  II Theoretical model

###  II.1 Interaction of an alkali vapor with a transverse magnetic field

The Hamiltonian matrix of alkali atoms interacting with a static magnetic
field is expressed as

$\mathcal{H}=\mathcal{H}_{0}+\mathcal{H}_{m}$ (1)

where $\mathcal{H}_{0}$ is the unperturbed (zero-field) Hamiltonian and
$\mathcal{H}_{m}$ is the magnetic contribution given by

$\mathcal{H}_{m}=-\frac{\mu_{B}}{\hbar}(g_{L}L_{z}+g_{S}S_{z}+g_{I}I_{z})\,.$
(2)

In relation (2), $\mu_{B}$ is the Bohr magneton, $g_{L}$, $g_{S}$ and $g_{I}$
are respectively the orbital, spin and nuclear Landé factors. $L_{z}$, $S_{z}$
and $I_{z}$ are the projection of the orbital, spin and nuclear angular
momenta along $z$ which has been chosen as the quantization axis. Due to the
influence of the magnetic field, the hyperfine energy levels $F$ split into
Zeeman sublevels denoted $\ket{F,m_{F}}$ in the coupled basis, with $m_{F}$
the magnetic quantum number varying from $-F$ to $+F$. As it has been
developed in [18], the matrix elements of $\mathcal{H}$ obey the two following
relations:

$\bra{F,m_{F}}\mathcal{H}\ket{F,m_{F}}=E_{0}(F)-\mu_{B}g_{F}m_{F}B_{z}$ (3)
$\displaystyle\bra{F-1,m_{F}}\mathcal{H}\ket{F,m_{F}}=\bra{F,m_{F}}\mathcal{H}\ket{F-1,m_{F}}$
$\displaystyle=-\frac{\mu_{B}B_{z}}{2}(g_{J}-g_{I})\left(\frac{[(J+I+1)^{2}-F^{2}][F^{2}-(J-I)^{2}]}{F}\right)^{1/2}$
(4)
$\displaystyle\times\left(\frac{F^{2}-m_{F}^{2}}{F(2F+1)(2F-1)}\right)^{1/2}\,.$

Equation (3) gives the diagonal terms of the Hamiltonian, which depend on the
projection $B_{z}$ of $\vec{B}$ along the quantization axis, the zero-field
energy $E_{0}(F)$ of the $F$ level and the Landé factor $g_{F}$ associated to
the $\ket{F,m_{F}}$ sublevel (see for example [29] or [30] for a more detailed
description). The non-diagonal terms, given by relation (4), are non-zero only
if they respect the selection rules $\Delta F=\pm 1$, $\Delta m_{F}=0$. As a
result, the Hamiltonian $\mathcal{H}$ is a block diagonal matrix, each block
corresponding to a value of $m_{F}$. It is important to note that $\mu_{B}$
has to be chosen with a negative sign for the model to be consistent, and one
can refer to [5] for further details. By diagonalizing the Hamiltonian matrix
for each value of the magnetic field, one obtains the eigenvalues which
correspond to the energy of each Zeeman sublevel $\ket{F,m_{F}}$. In the small
magnetic field limit ($B_{z}\ll B_{0}=A_{hf}\mu_{B}^{-1}\approx 63.3$ mT,
where $A_{hf}$ is the magnetic dipole constant), the coupled basis is a so-
called ”good” basis, the Zeeman sublevels split linearly according to (3). For
$B\gg B_{0}$, it is best to describe the states in the uncoupled basis
$\ket{J,m_{J},I,m_{I}}$ where $m_{J}$ and $m_{I}$ are projections of the
angular momentum $J$ and nuclear momentum $I$, see for example [31]. In
intermediate field ($B\simeq B_{0}$) one can obtain the energies of the ground
states by using the Breit-Rabi formula, as presented in [32]. All the
numerical data needed for the computations can be found in the literature, for
example in [4]. All values used for the physical constants are the ones
recommended by NIST [33]. The necessary values are given in table 1.

$\mu_{B}$ |  $1.39962449361(42)\times 10^{10}$ Hz/T
---|---
$A_{hf}$ |  $885.8130644(5)$ MHz
$g_{S}$ | $2.00231930436256(35)$
$g_{L}$ | $0.99997613$
$g_{I}$ | $-0.0008046108(8)$
Table 1: Numerical data to compute the interaction of the atomic vapor with a
magnetic field.

The new state vectors arising due to the influence of the magnetic field are
denoted by $\ket{\Psi(F,m_{F})}$. They can be expressed as linear combinations
of the set of unperturbed state vectors:

$\ket{\Psi(F,m_{F})}=\sum_{F^{\prime}}\alpha_{F,F^{\prime}}(B)\ket{F^{\prime},m_{F}}\,.$
(5)

Relation (5) is valid for both ground (denoted by the index $g$ below) and
excited states (denoted by the index $e$ below). $\alpha_{F,F^{\prime}}(B)$
are the magnetic-field-dependent mixing coefficients reflecting the coupling
of the Zeeman energy levels due to the magnetic field. The intensity $A_{eg}$
of a transition between two Zeeman sublevels $\Psi(\ket{F_{g},m_{F_{g}}})$ and
$\Psi(\ket{F_{e},m_{F_{e}}})$ is proportional to the squared of the so-called
”modified transfer coefficients”. These coefficients are given by

$\displaystyle A_{eg}$ $\displaystyle\propto
a^{2}[\ket{\Psi(F_{e},m_{F_{e}})};\Psi(F_{g},m_{F_{g}});q]$
$\displaystyle=\left(\sum_{F_{e}^{\prime},F_{g}^{\prime}}\alpha_{F_{e},F_{e}^{\prime}}a(F_{e}^{\prime},m_{F_{e}};F_{g}^{\prime},m_{F_{g}};q)\alpha_{F_{g},F_{g}^{\prime}}\right)^{2}$
(6)

where $a(F_{e},m_{F_{e}};F_{g},m_{F_{g}};q)$ are the unperturbed transfer
coefficients. They have the following form:

$\displaystyle
a(F_{e},m_{F_{e}};F_{g},m_{F_{g}};q)=(-1)^{1+I+J_{e}+F_{e}+F_{g}-m_{F_{e}}}\sqrt{2F_{e}+1}$
$\displaystyle\times$
$\displaystyle\sqrt{2F_{g}+1}\sqrt{2J_{e}+1}\begin{pmatrix}F_{e}&1&F_{g}\\\
-m_{F_{e}}&q&m_{F_{g}}\end{pmatrix}\begin{Bmatrix}F_{e}&1&F_{g}\\\
J_{g}&I&J_{e}\end{Bmatrix}\,.$ (7)

In relation (7), parenthesis represent a Wigner 3j-symbol and curly brackets a
6j-symbol. The index $q$ reflects the polarization of the laser radiation,
$q=0,\pm 1$ for $\pi,\sigma^{\pm}$. The model allows to study each invidual
Zeeman transition of the $D_{1}$ and $D_{2}$ lines of any alkali atom, but it
also works for other lines with similar structure, as presented in [34]. As a
last remark, in this work all transitions will be labelled in the basis
$\ket{F,m_{F}}$ for convenience although these quantum numbers form a ”good”
basis only for low magnetic fields.

###  II.2 Transmitted and reflected signal for a vapor of nanometric thickness

One of the possible ways to obtain sub-Doppler resolution during experiments
is to use a nano or micrometric thin-cell filled with an atomic vapor. Such
experiments have been thoroughly described in [35, 36, 37, 16, 15]. In what
follows, we recall briefly the theoretical model that can be used to compute
the fields that are reflected $S_{r}$ and transmitted $S_{t}$ through a thin
cavity (maximum a few wavelengths) seen as a Fabry-Perot interferometer. The
reader can refer to [38] for a complete description of the model. The
transmitted and reflected signals containing the information on the atomic
vapor have the following expressions

$\displaystyle S_{t}$ $\displaystyle\simeq
2t_{wc}t_{cw}^{2}E_{i}\mathrm{Re}\left\\{I_{f}-r_{w}I_{b}\right\\}/|Q|^{2}$
(8) $\displaystyle S_{r}$ $\displaystyle\simeq
2t_{cw}E_{i}\mathrm{Re}\\{r_{w}[1-\exp(-2ikL)]$
$\displaystyle\times[I_{b}-r_{w}I_{f}\exp(2ikL)]\\}/|Q|^{2}\,.$ (9)

where $t$ are transmission coefficients ($w$ stands for windows and $c$ stands
for cell), $Q=1-r_{w}^{2}\exp(2ikL)$ is the quality factor of the cavity of
length $L$, $r$ is the reflection coefficient (the two windows are here
considered identical, and all possible losses due to the windows are
neglected). $I_{f}$ and $I_{b}$ are called forward and backward integrals of
the atomic response

$\displaystyle I_{f}$ $\displaystyle=I_{T}^{lin}-r_{w}I_{SR}^{lin}$ (10)
$\displaystyle I_{b}$
$\displaystyle=I_{SR}^{lin}-r_{w}\exp(2ikL)I_{T}^{lin}\,,$ (11)

where $I_{T}^{lin}$ and $I_{SR}^{lin}$ are the transmitted and reflected
signals in the linear regime whose expressions are derived in [38]. The
homogeneous linewidth of the transmitted signal $\gamma=\Gamma_{\text{nat}}/2$
is a free parameter which can be replaced by
$\gamma=\Gamma_{\text{nat}}/2+\gamma_{f}$ to account for additional
decoherence processes (for example collisional broadening). This model, valid
for a two-level system having a resonant frequency $\omega_{0}$, can be
extended to an ensemble of two-level systems in order to describe each
hyperfine transition having a resonant frequency
$\omega_{i}=\omega_{e}-\omega_{g}$ and an amplitude $C_{i}$ depending mainly
on the transition intensity $A_{eg}$, computed using relation (6) (see [30]).
All spectra presented in this paper will be computed for a cell thickness
$\lambda/2$ and an absorption linewidth of $100$ MHz. Experimentally, all
studies are typically performed around this thickness due to the appearance of
Coherent Dicke Narrowing (first observed in the microwave domain [39]).
Narrowing of the resonance occurs periodically for $L=(2p+1)\lambda/2$ with
$p\in\mathbb{N}$. Different lineshapes for the absorption and reflection
signal are presented in [38]. It is relevant to note that for other
thicknesses, one obtain assymetric lineshapes arising from different
dispersive contributions involved in the expressions of the transmitted and
reflected fields. As an example of the results provided by this theoretical
model, we present on figure 1 an absorption and dSR spectrum of $D_{2}$ line
$\pi$-transitions for an external magnetic field $B_{z}=50$ mT.

Figure 1: Sodium $D_{2}$ line spectra for $\pi$-polarized laser radiation.
Top: absorption spectrum computed for $L=\lambda/2$. Bottom: dSR spectrum
computed for $L=\lambda/2-30$ nm.

##  III Sodium $D_{1}$ line

The $D_{1}$ line of sodium corresponds to the transitions occuring between the
states $F_{g}=1,2$ of $3^{2}S_{1/2}$ and $F_{e}=1,2$ of $3^{2}P_{1/2}$. The
Zeeman manifold is thus analogous to the one of ${}^{87}\text{Rb}$ or
${}^{39}\text{K}$ $D_{1}$ line. A $\pi$-polarized incident laser will excite
$14$ transitions, whereas left (or right) circularly polarized laser will
excite only $12$ transitions. In total for sodium $D_{1}$ line, $38$
$\ket{F_{g},m_{F_{g}}}\rightarrow\ket{F_{e},m_{F_{e}}}$ Zeeman transitions are
possible. A scheme depicting all of them is presented on figure 2.

Figure 2: All possible Zeeman transitions for the $D_{1}$ line of
${}^{23}\text{Na}$ in the basis $\ket{F,m_{F}}$. a) $\sigma^{-}$-transitions.
b) $\pi$-transitions. c) $\sigma^{+}$-transitions.

Throughout this work, the computations have been performed using
$\zeta=1.7716261288(10)$ GHz the hyperfine splitting between the two ground
states $F_{g}=1$ and $F_{g}=2$. In the same manner,
$\zeta^{\prime}=188.88(26)$ MHz is the hyperfine splitting between $F_{e}=1$
and $F_{e}=2$ [4].

###  III.1 Linearly polarized incident radiation

Let us first focus on $\pi$-transitions. As said before, in this case $14$
transitions are possible. All these transitions are so-called ”allowed” since
they respect the selection rules on $F$: $\Delta F=0,\pm 1$. The transition
intensity is proportional to the square of the modified transfer coefficient.
The results obtained for
$a^{2}[\ket{\Psi(F_{e}^{\prime},m_{F_{e}})};\Psi(F_{g}^{\prime},m_{F_{g}});+1]$
are presented on figure 3. For the sake of clarity to avoid overlapping, we
separate the two following cases: transitions labelled $1$ to $6$ with ground
state $F_{g}=1$ are represented on panel a), and transitions labelled $7$ to
$14$ with ground state $F_{g}=2$ are represented on panel b).

Figure 3: (color online) Sodium $D_{1}$ line hyperfine transition intensities
for $\pi$-polarized laser radiation. a) transitions from $F_{g}=1$. b)
transitions from $F_{g}=2$. The guiding transitions $7$ and $14$ are
represented in red (dotted lines). In blue (dashed lines), we represent the
transitions whose intensities remain considerable past $B_{0}\approx 63.3$ mT,
as opposed to the orange curves (continuous lines). Magnetic field varies up
to $0.2$ T here for clarity.

Several observations can be made from figure 3. The amplitudes of transitions
$1$, $3$, $5$ (see figure 3a), $7$, $9$, $11$, $13$ and $14$ (see figure 3b)
remain considerable for $B\gg B_{0}$, thus $8$ transitions will remain present
in the spectrum when the magnetic field is high enough. These transitions will
be called r-transitions, where r stands for remaining. Here, they obey the
selection $\Delta F=0$ at low magnetic field ($F$ is not a good quantum number
at high magnetic field), and $\Delta m_{I}=\Delta m_{J}=0$ for high magnetic
field. The r-transitions $1$, $3$, $5$ occur between $F_{g}=F_{e}=1$, and the
others between $F_{g}=F_{e}=2$ . Similarly, transitions $2$, $4$, $6$, $8$,
$10$ and $12$ will be called v-transitions, where v stands for vanishing,
since their amplitude becomes negligible for $B\gg B_{0}$. One can notice that
some of the r-transitions show $a^{2}\approx 0$ for low magnetic field and
experience a huge amplitude growth as the magnetic field increases. For these
reason, we call them Magnetically Induced (MI), although they are
theoretically allowed by the selection rule $\Delta F=0,\pm 1$. Another
peculiarity is the appearance of ”transition cancellations”, occuring here at
around $15$ mT. This value depends on the different hyperfine splittings. This
phenomenon was already mentioned in [18] but no complete description was done.
Precise discussion of these cancellations is out of the scope of this paper,
however we have provided a thorough analysis and a way to determine precisely
the magnetic field values leading to dipole moment cancellation in [40, 34,
41]. Using the relations provided in these works, we can determine that
transitions $1$ and $9$ experience dipole moment cancellation for

$B=\frac{1}{\mu_{B}}\frac{3\zeta\zeta^{\prime}}{(3g_{I}-4g_{L}+g_{S})\zeta+3(g_{I}-g_{S})\zeta^{\prime}}\approx
15.32~{}\text{mT}\,.$ (12)

The last peculiarity that appears is the presence of two r-transitions ($7$
and $14$) having a constant amplitude throughout the whole range of magnetic
field. These transitions occur between uncoupled states (following the
definition of relation (5)): such states do not experience any mixing with
their neighbors (with the same $m_{F}$ value), thus leading to magnetic-field-
independent transition amplitudes. As shown on the inset of the right panel of
figure 3b, these transitions are called Guiding Transitions (GT). For further
analysis, we present on figure 4 the frequencies
$\omega_{eg}(B_{z})=(E_{e}(B_{z})-E_{g}(B_{z}))/\hbar$ of all the possible
$\pi$-transitions.

Figure 4: (color online) $\pi$-transition frequencies of sodium $D_{1}$ line
as a function of the magnetic field. The color code corresponds to the one
used in figure 3. The zero frequency is set to the transition
$F_{g}=1\rightarrow F_{e}=1$.

From figure 4, one can notice the following things: the two guiding
transitions (represented in red dotted lines) occur between uncoupled states
having for only difference their $m_{F}$ values ($m_{F_{g}}=m_{F_{e}}=\pm 2$),
their frequency shifts are perfectly linear with respect to $B_{z}$ and only
differ by sign. Going back to the model of section 1, the uncoupled states
give rise to $1\times 1$ blocks in $\mathcal{H}$ and thus their Zeeman
splitting is perfectly described by relation (3), hence it is quite
straightforward to show that their slope is

$s_{\pm,GT}^{D_{1}^{(\pi)}}\approx\pm\frac{2\mu_{B}}{3}\,,$ (13)

under the approximation $g_{S}\approx 2$, $g_{L}\approx 1$ and $g_{I}\ll
g_{J}$. Numerically, by using the values provided in table 1, we obtain
$s_{\pm,GT}^{D_{1}^{(\pi)}}=\pm 0.935269$ MHz/G. Each group of r-transitions
is thus driven by one of the GTs, more precisely r-transitions $1$, $3$ and
$5$ are driven by the GT labelled $7$ having a slope
$s_{-,GT}^{D_{1}^{(\pi)}}=-2\mu_{B}/3$, and r-transitions $9$, $11$ and $13$
are driven by the GT labelled $14$ having a slope
$s_{+,GT}^{D_{1}^{(\pi)}}=2\mu_{B}/3$, with $\mu_{B}$ chosen to be negative as
mentioned earlier. Driven means here that each individual transition belonging
to a given group is bounded in amplitude by the one the Guiding Transition,
and all the frequency shifts asymptotically tend to a linear behavior when
$B\gg B_{0}$ is high enough, with the slope being
$s_{\pm,r}^{D_{1}^{(\pi)}}=s_{\pm,GT}^{D_{1}^{(\pi)}}$, as written in relation
(13). Due to the absence of coupling between states, the modified transfer
coefficient of a GT occuring between two uncoupled states
$\ket{F_{g},m_{F_{g}}}\rightarrow\ket{F_{e},m_{F_{e}}}$ is exactly equal to
the unperturbed transfer coefficient
$a^{2}(F_{e},m_{F_{e}};F_{g},m_{F_{g}},q)$. For transitions $7$ and $14$, we
obtain

$a(2,\pm 2;2,\pm 2;q=0)=\pm\frac{1}{\sqrt{3}}\rightarrow a^{2}=\frac{1}{3}\,.$
(14)

Past $B_{0}$, the 8 r-transitions remain in the spectrum, which is a
manifestation of the hyperfine Paschen-Back regime. This behavior has been
observed experimentally for cesium, rubidium and potassium (for example in
[42]). As mentioned before, experimental results are harder to provide due to
the lack of diode lasers operating in the wavelength range of sodium. However,
a sodium nanocell has been developed and is available in our laboratory [24].
A set of theoretical absorption spectra for the all $\pi$-transitions of
sodium $D_{1}$ line is presented on figure 5 for visualization.

Figure 5: (color online) Theoretical absorption spectra of the $D_{1}$ line of
sodium for $\pi$-polarized incident laser radiation. The magnetic field
intensity varies from $0$ to $0.21$ T with a step of $30$ mT. The shadow lines
on the $(x,y)$ plane are the same transition frequencies shown on figure 4.
Each component has a Full Width at Half Maximum (FWHM) of
$\omega_{\text{FWHM}}\approx 100$ MHz. The zero frequency is set to the
transition $F_{g}=1\rightarrow F_{e}=1$.

All the previous statements are visible on figure 5. The r-transitions can be
separated in two groups all tending to the same amplitude and a complete
symmetry between the red and blue wings of the spectrum appears when $B$ is
high enough. Although becoming invisible, the v-transition frequency shifts
also exhibit a linear behavior at high magnetic fields, tending to a slope
twice bigger than for the r-transitions. Precisely,
$s_{\pm,v}^{D_{1}^{(\pi)}}=2s_{\pm,r}^{D_{1}^{(\pi)}}\approx\pm 4\mu_{B}/3$
using the same approximations. It is usually estimated that complete Hyperfine
Paschen-Back regime is reached when a magnetic field $B>10B_{0}$ is applied.
This correspond here to a magnetic field $B_{z}\approx 0.63$ T. As a
comparison, for rubidium 87, $B_{0}\approx 0.24$ T [2] and for cesium,
$B_{0}\approx 0.16$ T [3]. Theoretically, this makes sodium, along with
potassium 39 [30, 6] more convenient than cesium or rubidium for the
experimental study of magnetic-optical processes occuring in the complete
hyperfine Paschen-Back since it is reached for a much weaker value of external
magnetic field. However, since the natural linewidth of sodium $D_{1}$ line
$\Gamma_{\text{nat}}/2\pi=9.765$ MHz [4] is nearly twice bigger than for the
$D_{1}$ lines of cesium or rubidium, one expects to obtain significantly
broader lines when studying sodium. The linewidth is also affected by the
inhomogeneous Doppler broadening $\Gamma_{D}$ given by

$\Gamma_{D}=\omega_{0}\sqrt{\frac{8k_{B}T\ln 2}{mc^{2}}}\,,$ (15)

(see [43]) bigger for the $D_{1}$ line of sodium than in the case of rubidium
or cesium. In relation (15), $\omega_{0}$ is the central angular frequency of
the transition, $m$ is the atomic mass, $T$ is the temperature of vapor, $c$
is the speed of light and $k_{B}$ is the Boltzmann constant. However,
performing experiments with nanocells allows to reach sub-Doppler spectral
resolution [15]. On figure 6, we present a theoretical absorption spectrum of
sodium $D_{1}$ $\pi$-transitions, computed with the same numerical parameters
as before, as long with the second derivative spectrum (SD).

Figure 6: (color online) Theoretical absorption spectra of the $D_{1}$ line of
sodium for $\pi$-polarized incident laser radiation in the hyperfine Paschen-
Back regime ($B=0.9$ T). Top: transmitted signal. Bottom: second derivative
(SD). Inset: ”mirror” overlap of the two groups when the high-frequency group
is computed as a function of $-\omega$. The top curve has been normalized so
that the maximum amplitude is $1$.

Due to the linewidth $\omega_{\text{FWHM}}\approx 100$ MHz, even for a
magnetic field $B_{z}\gg 10B_{0}$, the peaks are not totally resolved hence
resulting in a variation of the peaks’ amplitude. Experimentally, as
demonstrated on the bottom panel of figure 6, SD would allow to obtain
completely resolved peaks, all having the same amplitude. DSR would also be
suitable here. These techniques allow to obtain much narrower resonances (two
to three times, see [30]). One can also see that all the peaks are evenly
spaced, which is a manifestation of the linear behavior of the frequency
shifts mentioned above and a good sign HPB regime is reached. Moreover, as
shown by the inset, the red and blue wing of the spectrum are completely
symmetric. The frequency detuning $\Delta\omega_{r}^{D_{1}^{(\pi)}}$ between
the two groups of $4$ r-transitions (their center of gravity, see the dashed
blue lines on figure 6) for $B_{z}\gg B_{0}$ can be estimated roughly by
performing the difference between the two GTs, thus

$\Delta\omega_{r}^{D_{1}^{(\pi)}}\approx\left|\frac{4\mu_{B}B_{z}}{3}\right|\,.$
(16)

For a magnetic field $B_{z}=0.9$ T, we obtain
$\Delta\omega_{r}^{D_{1}^{(\pi)}}\approx 16.8$ GHz which is in excellent
agreement with the value $16.9$ GHz computed numerically and shown on figure
6.

###  III.2 Circularly polarized incident radiation

All the possible $\sigma^{-}$ (resp. $\sigma^{+}$) transitions of the $D_{1}$
line of sodium are schematized on panel a) (resp. panel b)) of figure 2, and
their associated intensities are presented on figure 7.

Figure 7: (color online) Sodium $D_{1}$ Zeeman transition intensities for
circularly polarized laser radiation. a) $\sigma^{-}$-transitions. b)
$\sigma^{+}$-transitions. Labelling is provided only for the non-vanishing
transitions (dashed blue lines) and is the same as on panels a) and c) of
figure 2.

As a first remark, no $\sigma^{\pm}$ has a constant amplitude throughout the
whole magnetic field range, which is due to the fact that no transition occurs
between uncoupled states. At best, only the excited state is uncoupled
(transitions $1$ and $6$ for $\sigma^{-}$, and their ”symmetric” transitions
$5$ and $12$ for $\sigma^{+}$), but this is not enough to avoid the magnetic-
field dependence of the transition intensities, thus there is no guiding
phenomenon as it happens for $\pi$-transitions. Although being presented in
arbitrary units, the amplitude scale used for figure 7 is the same as for the
$\pi$-transitions, allowing to compare the three cases. Each circular
polarization gives rise to a group of four r-transitions, all tending to an
amplitude ($\approx 0.66$ a. u.) twice bigger than the amplitude of the
$\pi$-GTs ($\approx 0.33$ a. u.). The r-transitions obey the selection rules
$\Delta m_{I}=0$, $\Delta m_{J}=\pm 1$ depending on the sign of the incident
polarization. A set of absorption spectra is presented on figure 8.

Figure 8: (color online) Theoretical absorption spectra of sodium $D_{1}$ line
for a magnetic field varying from $0$ to $0.21$ T with a step of $0.03$ T in
case of simultaneous $\sigma^{+}$ and $\sigma^{-}$ excitation. Blue and orange
shadow lines in the $(x,y)$ plane represent the transition frequencies. The
zero detuning is set to the transition $F_{g}=1\rightarrow F_{e}=1$. As
before, each peak has a FWHM of approximately $100$ MHz.

Due to the absence of GTs for circularly polarized incident radiation, less
pecularities occur in this case. One can also note the absence of dipole
moment cancellation, as was observed in [40, 34]. However, some observations
can still be made from figure 8. Here, the behavior of the r-transitions is a
bit different than before: for $\sigma^{+}$ polarization, the frequency shifts
all tend to a linear behavior of slope twice bigger than for the
$\pi$-r-transitions. For example, we take a look at transition $11$
($\sigma^{-}$) occuring between the ground state $\ket{2,2}$ and the excited
state $\ket{1,1}$ Using relation (3), we can easily determine that

$E_{g}=\zeta-\mu_{B}B_{z}\,.$ (17)

For the excited state, we only compute the block corresponding to
$m_{F_{e}}=1$ using relations (3) and (4). This leads to

$H_{e}=\begin{pmatrix}X&-\sqrt{3}X\\\
-\sqrt{3}X&\zeta^{\prime}-X\end{pmatrix}$ (18)

where we denoted $X=\mu_{B}B_{z}/6$. Diagonalizing $H_{e}$ and performing an
asymptotic expansion allows to obtain the energy $E_{e}$ of state $\ket{1,1}$
for $B_{z}\gg B_{0}$:

$\displaystyle E_{e}$ $\displaystyle=\frac{\zeta^{\prime}+[\zeta^{\prime
2}-4\zeta^{\prime}X+16X^{2}]^{1/2}}{2}$
$\displaystyle\underset{X\rightarrow+\infty}{\sim}2X+\frac{\zeta^{\prime}}{4}+\frac{3\zeta^{\prime
2}}{64X}+\mathcal{O}\left(\frac{1}{X^{2}}\right)\,.$ (19)

The difference between (19) and (17) leads to
$8X+\zeta^{\prime}/4-\zeta+\mathcal{O}{(1/X)}$, thus we obtain the asymptotic
slope of the frequency shift of transition $11$ ($\sigma^{-}$) and by
extension of all $\sigma^{\pm}$ r-transitions:

$s_{\pm,r}^{D_{1}^{(\sigma^{\pm})}}\approx\mp\frac{4\mu_{B}}{3}\,.$ (20)

This method was used just before for the $D_{1}$ $\pi$-transitions and will be
used throughout all this paper. As we can see on figure 8, it can be
demonstrated that the v-transitions (orange continuous lines) that are not
overlapped with any r-transition (blue dashed lines) have a frequency shift of
slope twice smaller than the value presented in relation (20). We present two
spectra for $B\gg B_{0}$ on figure 9.

Figure 9: (color online) Theoretical absorption spectrum of the $D_{1}$ line
of sodium for simultaneous $\sigma\pm$ polarized incident laser radiation in
the hyperfine Paschen-Back regime ($B=0.9$ T). Top: transmitted signal.
Bottom: second derivative (SD). Inset: ”mirror” overlap of the two groups when
the high-frequency group is computed as a function of $-\omega$. The top curve
has been normalized so that the maximum amplitude is $1$.

The numerical parameters used for the computation of the spectra are the same
as the ones used before in the case of $\pi$-transitions. On the top panel,
the peaks are overlapped due to the linewidth. However, clear manifestation of
the hyperfine Paschen-Back regime is visible on the bottom panel where all the
peaks are evenly spaced and have the same amplitude. Due to the fact that the
slope of the frequency shifts corresponding to the two groups of r-transitions
is twice bigger, the frequency detuning between them can also be roughly
estimated by

$\Delta\omega_{r}^{D_{1}^{(\sigma)}}\approx\left|\frac{8\mu_{B}B_{z}}{3}\right|\,.$
(21)

Here, we obtain $\Delta\omega_{r}^{D_{1}^{(\sigma)}}\approx 33.59$ GHz which
is in perfect agreement with the value $33.6$ GHz measured on figure 9. In
this section, we have provided a complete description of the influence of the
magnetic field on the behavior of the Zeeman transitions of sodium $D_{1}$
line for the three main types of incident laser radiation. We will now analyze
deeply what happens for the $D_{2}$ line, where much more transitions and
different phenomena arise.

##  IV Sodium $D_{2}$ line

The $D_{2}$ line of sodium corresponds to the transitions occuring between the
states $3^{2}S_{1/2}$ and $3^{2}P_{3/2}$. Due to the bigger value of $J_{e}$,
much more transitions are possible than for the $D_{1}$ line. In total, $68$
transitions are possible. Since natural sodium is only composed of one isotope
of nuclear spin $I=3/2$, the hyperfine manifold is still simpler than the one
of natural rubidium (two isotopes: ${}^{85}\text{Rb}$ ($I=5/2$) and
${}^{87}\text{Rb}$ ($I=3/2$)) or cesium (only one isotope but $I=7/2$), thus
leading to spectra which are simpler to analyze. For the numerical
computations, we considered the hyperfine splittings
$\alpha^{\prime}=15.809(80)$ MHz between $F_{e}=0$ and $F_{e}=1$,
$\beta^{\prime}=34.344(69)$ MHz between $F_{e}=1$ and $F_{e}=2$, as well as
$\delta^{\prime}=58.326(43)$ MHz between $F_{e}=2$ and $F_{e}=3$.

###  IV.1 Linearly polarized incident radiation

$24$ $\pi$-transitions are possible for the $D_{2}$ line of sodium. A complete
scheme as well as the transition intensities are presented on figure 10. In
this case, $F_{g}=1,2$ and $F_{e}=0,1,2,3$ [4].

Figure 10: (color online) a) scheme of all possible $\pi$-transitions for the
$D_{2}$ line of sodium. b) transitions from $F_{g}=1$. c) transitions from
$F_{g}=2$. Respective behaviors for very low magnetic field are shown in the
insets. The v-transitions are represented in orange (continuous lines) and the
r-transitions are represented in blue (dashed lines).

The behavior of the $\pi$-transitions of sodium $D_{2}$ line is quite
different from the $D_{1}$ line. One quickly notices the absence of
transitions between uncoupled states thus the absence of guiding transitions.
This is due to absence of states such that $F_{g}=3$. Overall, the general
shape is different, and this is due to the fact that the states experience
”more coupling” (for a given spectroscopic state in the case of $D_{1}$ line,
the magnetic field can couple at most levels two $F$-levels, whereas for the
$D_{2}$ line the coupling goes up to three $F$ levels. Many transition
intensities are very small throughout the whole range of magnetic field and
most of them quickly vanish, as we can notice on the insets of figure 10.
Using the same denomination, we call r-transitions the transitions $2$, $5$,
$8$, $12$, $15$, $18$, $21$ and $23$ since they obey the selection rules
$\Delta m_{I}=\Delta m_{J}=0$, and all the others will be denoted
v-transitions. It is worth noticing that for the $D_{1}$ line, we could
exhibit the fact that all r-transitions obeyed $\Delta F=0$. No such rule in
the uncoupled basis can be exhibited here. In the previous part, we called MI
the transitions whose intensity was $0$ for a very small magnetic field. On
figure 10, this phenomenon is clearly visible for r-transitions $5$ and $18$.
For the sake of clarity, labelling is not presented here for the v-transitions
although it is completely possible to associate each curve to a given
transition. However, this notion of Magnetically Induced transitions can be
completed. Here, we will also denote transitions $3$, $7$, $10$ and $16$ as
Magnetically Induced. All of them are v-transitions and their amplitude tends
to $0$ for $B_{z}\rightarrow 0$, except for transition $3$
($\ket{1,-1}\rightarrow\ket{3^{\prime},-1}$) which experiences a huge increase
in amplitude. They are called magnetically induced due to the coupling of
states by $B_{z}$ making so-called ”forbidden” transitions ($\Delta F=\pm 2$)
possible. Here, we have $3$ r-transitions arising from $F_{g}=1$ and $5$ from
$F_{g}=2$ (transitions $12$ and $23$ are exactly overlapped on figure 10). The
total number of $\pi$-r-transitions is the same as for the $D_{1}$ line
meaning there are much more v-transitions here (16 compared to 6). On figure
11, we present a set of absorption spectra along with the transition
frequencies to observe their behavior as the magnetic field increases.

Figure 11: (color online) Theoretical absorption spectra of the $D_{2}$ line
of sodium for $\pi$-polarized incident laser radiation. The magnetic field
intensity varies from $0$ to $0.21$ T with a step of $30$ mT. The zero
frequency is set to the transition $F_{g}=1\rightarrow F_{e}=0$. The inset
shows the SD of the three peaks $F_{g}=1\rightarrow F_{e}=0,1,2$.

Since the hyperfine splittings of the states $3^{2}P_{3/2}$ are much smaller
for sodium than for rubidium or cesium [4, 2, 1, 3], the peaks are in general
much closer here. At zero-field the 6 peaks remain completely overlapped.
Using SD or dSR is a good way to recover the spectral information, as
demonstrated on the inset of figure 11 where the relative transition
intensities are in perfect agreement with the oscillator strengths presented
in [4] ($S_{10}=1/6$, $S_{11}=S_{12}=5/12$).

At first, for the same values of magnetic field as before, one notices that
all the transitions experiencing the biggest frequency shift with respect to
the magnetic field are the 16 v-transitions. They can be divided into three
groups of same frequency shift (in absolute value). The frequency shifts of
the two groups of r-transitions tend asymptotically to a linear behavior of
slope

$s_{\pm,r}^{D_{2}^{(\pi)}}\approx\pm\frac{\mu_{B}}{3}=\frac{s_{\pm,r}^{D_{1}^{(\pi)}}}{2}\,.$
(22)

All the slopes are derived using similar procedures used in relations (17) to
(20). The v-transitions undergo much bigger frequency shifts and can be
divided into three groups having asymptotically the same slope in absolute
value. Under the approximations $g_{S}\approx 2$, $g_{L}\approx 1$ and
$g_{I}\ll g_{J}$, the first group of v-transitions (in terms of proximity with
the r-transitions) experiences a frequency shift of asymptotic slope
$\pm\mu_{B}$. The frequency shifts of the second group of v-transitions have
an asymptotic slope $\pm 5\mu_{B}/3$. This slope for the last group is $\pm
3\mu_{B}$. This is coherent with figure 11 where we can see the last group of
v-transitions experience the biggest shifts (more than $4$ MHz/G, 9 times more
than the r-transitions). Numerically, we obtain respectively $\pm 1.40061$,
$\pm 2.29035$ and $\pm 4.15417$ MHz/G. A spectrum for $B_{z}\gg B_{0}$ is
presented on figure 12.

Figure 12: (color online) Theoretical absorption spectrum of the $D_{2}$ line
of sodium for $\pi$-polarized excitation in the hyperfine Paschen-Back regime
($B=0.9$ T). Top: transmitted signal. Bottom: second derivative (SD). Inset:
”mirror” overlap of the two groups when the high-frequency group is computed
as a function of $-\omega$. The top curve has been normalized so that the
maximum amplitude is $1$.

As before, the bottom panel shows that SD is convenient to obtain complete
resolution of peaks. Hyperfine Paschen-Back is reached since all the peaks are
evenly spaced and tend to the same amplitude. It is again possible to estimate
the frequency detuning between the two groups of r-transitions as

$\Delta\omega_{r}^{D_{2}^{(\pi)}}\approx\left|\frac{2\mu_{B}B_{z}}{3}\right|\,.$
(23)

Here, we obtain $\Delta\omega_{r}^{D_{2}^{(\pi)}}\approx 8.40$ GHz (by
numerical computation, we obtain $8.46$ GHz. The small variation is due to the
fact that among all approximations, the influence of the hyperfine splittings
is neglected.

###  IV.2 Circularly polarized incident laser radiation

$44$ Zeeman transitions are possible in case of circularly polarized radiation
for the $D_{2}$ line of sodium. All these possible $\sigma^{\pm}$-transitions
are schematized on figure 13.

Figure 13: All possible $\sigma^{\pm}$ Zeeman transitions for the $D_{2}$ line
of sodium in the basis $\ket{F,m_{F}}$. a) $\sigma^{-}$-transitions. b)
$\sigma^{+}$-transitions.

One directly sees on the manifold presented on figure 13 that, due to the
selection rule $\Delta m_{F}=\pm 1$, transitions are again possible between
uncoupled states. These transitions are labelled $10$ ($\sigma^{-}$) and $22$
($\sigma^{+}$). They occur between states $\ket{2,\pm 2}$ and $\ket{3,\pm 3}$
depending on the polarization and obey $\Delta F=1$. They are again called
guiding transitions and their constant amplitude for any magnetic field is
clearly visible on figure 14.

Figure 14: (color online) Sodium hyperfine transition intensities for
circularly polarized laser radiation. a) $\sigma^{-}$-transitions. b)
$\sigma^{+}$-transitions. Past $B_{0}\approx 0.063$ T, two different groups of
transitions have an amplitude that remains considerable. These two different
groups are represented in blue (dashed lines) and purple (dash-dotted), and
the vanishing transitions are represented in orange. Labelling is consistent
with figure 13.

For each different polarization appears a group of r-transitions ($1$, $3$,
$6$ and $10$ for $\sigma^{-}$ and $16$, $19$, $21$, $22$ for $\sigma^{+}$)
denoted r1-transitions. Transition $10$ ($\sigma^{-}$) acts a guiding
transition meaning it is an amplitude bound for all its associated
r1-transitions. For $\sigma^{+}$, $16$, $19$ and $21$ are all bounded in
amplitude by transition $22$. As we did in relation (14), we can prove that
for both of these transitions, the squared modified transfer coefficients are
always equal to 1. Two other groups of r-transitions, denoted r2-transitions,
represented in purple (dash-dotted lines), appear and all tend to an amplitude
equal to a third the amplitude of the r1-transitions. For both polarization,
r2-transitions tend to reach their maximum amplitude much faster than
r1-transitions. In orange are represented the vanishing transitions
(v-transitions) for each polarizations. Apart from transitions $11$ and $13$
($\sigma^{-}$) and transitions $4$ and $7$ ($\sigma^{+}$), all the
v-transitions have overall smaller amplitudes than all r1 and r2-transitions.

Figure 15: (color online) Theoretical absorption spectra of the $D_{2}$ line
of sodium for $\sigma^{-}$-polarized incident laser radiation. The magnetic
field intensity varies from $0$ to $0.21$ T with a step of $30$ mT. The zero
frequency is set to the transition $F_{g}=1\rightarrow F_{e}=0$.

The $22$ possible $\sigma^{-}$-transitions are visible on figure 15. The GT
(labelled $10$ on figure 13a) has a constant amplitude and the three other
r1-transitions are also driven by the GT in terms of frequency shift. The
guiding transition has a perfectly linear frequency shift of slope

$s_{GT}^{D_{2}^{(\sigma^{-})}}\approx\mu_{B},$ (24)

under the usual approximations, the numerical value being $\approx-1.39958$
MHz/G while $\mu_{B}=-1.39962$ MHz/G. The r2-transitions, having an amplitude
three times smaller, experience a frequency shift of approximately
$5\mu_{B}/3$ for $B_{z}$ high enough. Numerically, we obtain approximately
$-2.331$ MHz/G. As for the previous cases, the v-transitions experience much
bigger frequency shifts, reaching again as much as approximately
$3\mu_{B}=-4.199$ MHz/G (numerically, $-4.202$ MHz/G). The behavior of the
$\sigma^{+}$ is reversed compared to $\sigma^{-}$, as we can see on figure 16.

Figure 16: (color online) Theoretical absorption spectra of the $D_{2}$ line
of sodium for $\sigma^{+}$-polarized incident laser radiation. The zero
frequency is set to the transition $F_{g}=1\rightarrow F_{e}=0$.

The guiding transition ($22$ in this case) experiences a frequency shift
$s_{GT}^{D_{2}^{(\sigma^{+})}}=-s_{GT}^{D_{2}^{(\sigma^{-})}}$ and so do all
the r1-transitions when $B$ is high enough so that the frequency shift becomes
linear. The r2-transitions will also analogously experience a linear frequency
shift $-5\mu_{B}/3$ for $B_{z}$ (numerically, $\approx 2.331$ MHz/G). When the
hyperfine Paschen-Back regime is reached ($B_{z}>10B_{0}$), only the r1 and
r2-transitions of each polarization remain visible in the spectrum ($16$ peaks
in total). This is clearly demonstrated on figure 17 where the spectrum is
presented in case of simultaneous $\sigma^{\pm}$ excitation for $B_{z}=0.9$ T.

Figure 17: (color online) Theoretical absorption spectrum of the $D_{2}$ line
of sodium for simultaneous $\sigma^{\pm}$ polarized excitation in the
hyperfine Paschen-Back regime ($B=0.9$ T). Top: transmitted signal. Bottom:
second derivative (SD).

With the slopes exhibited before, we can estimate that the frequency detuning
between the two groups of r1-transitions is approximately

$\Delta\omega_{r1}^{D_{2}^{(\sigma)}}\approx\left|2\mu_{B}B_{z}\right|\,.$
(25)

leading to $\Delta\omega_{r1}^{D_{2}^{(\sigma)}}\approx 25.19$ GHz, perfectly
consistent with the value $25.15$ GHz on the top panel of figure 17, the
difference coming for the fact that hyperfine splittings are neglected.
Similarly, we can estimate $\Delta\omega_{r2}^{D_{2}^{(\sigma)}}$ to be

$\Delta\omega_{r2}^{D_{2}^{(\sigma)}}\approx\left|\frac{10\mu_{B}B_{z}}{3}\right|$
(26)

that is to say approximately $41.98$ GHz, to compare with the value $42.07$
GHz of figure 17. Reaching hyperfine Paschen-Back regime combined with sub-
Doppler spectroscopic techniques allows to form narrow resonances far-detuned
from the resonant frequency of the transitions. Choosing the right alkali for
such studies is a key point since $B_{0}$ varies a lot. Hyperfine splittings
and natural linewidths are also to take into account depending on the desired
resolution. We will now focus on the so-called forbidden transitions, obeying
$\Delta F=\pm 2$. Their intensities are plotted on figure 18.

Figure 18: (color online) Transition intensities obeying $\Delta F=\pm 2$. a)
$\sigma^{-}$-transitions. b) $\sigma^{+}$-transitions. The insets show the
very-low-field behavior of transitions having a very small intensity.
Labelling is consistent with figure 13.

At first glance, what can be observed on figure 18 is the overall difference
in intensity between the two polarizations: while v-transition $2$
($\sigma^{-}$, hereafter denoted $2^{-}$) peaks at approximately $0.16$,
v-transition $4^{+}$ peaks at nearly 0.6 (both panels have the same y-scale).
These transitions arise from the same ground state $\ket{1,-1}$, and $2^{-}$
(resp. $4^{+}$) goes to $\ket{3,-2}$ (resp. $\ket{3,0}$) due to the selection
rule on $\Delta m_{F}$ and obey $\Delta F=+2$. The inverse behavior can be
observed for v-transitions $16^{-}$ and $13^{+}$: their ground states are
different, respectively $\ket{2,-1}$ and $\ket{2,+1}$, but they have the same
excited state $\ket{0,0}$. They obey $\Delta F=-2$. Transition $16^{-}$
reaches a little more than $0.14$, while transition $13^{+}$ only reaches
$5\times 10^{-3}$. To observe this behavior more clearly, we will plot the
ratios $I(4^{+})/I(2^{-})$ and $I(16^{-})/I(13^{+})$. They are presented on
figure 19. We call group G2+ the transitions obeying $\Delta F=+2$ and group
G2- the transitions obeying $\Delta F=-2$.

Figure 19: (color online) Intensity ratio between the two strongest MI-
transitions of their group. Top panel: MI-transitions obeying $\Delta F=+2$
(group G2+, $I(4^{+})/I(2^{-})$). Bottom panel: MI-transitions obeying $\Delta
F=-2$ (group G2-, $I(16^{-})/I(13^{+})$).

The ratio $I(4^{+})/I(2^{-})$ has a maximum of approximately $4$, occuring for
$B_{z}\approx 6.12$ mT, whereas on the bottom panel $I(16^{-})/I(13^{+})$
reaches as much as $1.25\times 10^{14}$ for $B_{z}\approx 38.41$ mT. This
phenomenon is referred to as MI-Circular Dichroism (MCD) or frequency-
controllable Circular Dichroism: for $\Delta F=+2$ the strongest $\sigma^{+}$
transition intensity is more than $4$ times higher than the strongest
$\sigma^{-}$ one and for $\Delta F=-2$, $\sigma^{-}$ transitions are stronger
than $\sigma^{+}$. By varying the laser frequency and its polarization, it is
thus possible to choose whether $\sigma^{+}$ or $\sigma^{-}$ MI-transition
will dominate. The figure of merit coefficient allows, for each group, to
better visualize the dichroism phenomenon. It is defined as

$C_{\text{MCD}}=\frac{I^{+}-I^{-}}{I^{+}+I^{-}}$ (27)

and is represented for the aforementioned transitions on figure 20.

Figure 20: (color online) Figure of merit coefficient $C_{\text{MCD}}$ for
each group of transitions. Blue (dashed) line:
$(I(4^{+})-I(2^{-}))/(I(4^{+})+I(2^{-}))$. Orange (continuous) line:
$(I(13^{+})-I(16^{-}))/((13^{+})+I(16^{-}))$.

The coefficient defined in relation (27) can be interpreted as follows:
$C_{\text{MCD}}=0$ represents equality between the intensities of the two
strongest MI-transitions (one per polarization) of each group.
$C_{\text{MCD}}>0$ (resp. $C_{\text{MCD}}<0$) means that the strongest
transition occurs for $\sigma^{+}$ (resp. $\sigma^{-}$) thus leading to an
asymmetry. $C_{\text{MCD}}=-1$, which is the case for the orange curve of
figure 20 (transitions belonging to group G2-), means complete vanishing of
the $\sigma^{+}$ transition and is reached for $B_{z}>3$ mT. This type of MCD
is referred to as Type 1 MCD and has been thoroughly described and observed
for other alkali (Rb and Cs, mainly) in [30] and [20].

Moreover, it was recently demonstrated in [19] (theoretically and
experimentally, by using a Cs-filled nanocell) that even among the strongest
MI-transitions arising from different ground states for each circular
polarization, the strongest MI-transition obeying $\Delta F=+2$ is always
bigger than the strongest MI-transitions obeying $\Delta F=-2$. This is
verified here: transition $4^{+}$ ($\ket{1,-1}\rightarrow\ket{3,0}$) is
stronger than $16^{-}$ ($\ket{2,1}\rightarrow\ket{0,0}$). Numerically, here is
it verified for the range $2.4$ mT - $1$ T. As mentioned in [19], calculations
suggest that Type 1 and Type 2 MCD could be observed for other first
fundamental series of $nS\rightarrow nP$ $D_{2}$ lines, where MI-transitions
also occur.

##  V Conclusion

In this paper, we have performed a complete theoretical description of the
behavior of a sodium vapor confined in a nanometric-thin cell for a wide range
of magnetic field (varying up to $1$ Tesla) and for three incident laser
polarizations (linear, left and right-circular). While the Zeeman structure of
sodium is identical to the one of ${}^{87}\text{Rb}$ or ${}^{39}\text{K}$,
several changes can be reported such as $B_{0}$ and the hyperfine splittings
(leading to slight changes in the behavior of transition intensities and
transition frequencies). As mentioned before, the natural linewidth
$\Gamma_{\text{nat}}$ of sodium is twice bigger than for other alkalis and
leads to much broader absorption lines. However, for the same temperature the
vapor pressure of sodium is much smaller than heavier alkali atoms ($\approx
10^{-7}$ torr, whereas for cesium it is $\approx 6\times 10^{-4}$ torr and
$\approx 2.5\times 10^{-4}$ torr for ${}^{85}\text{Rb}$). This leads to
smaller collisional broadening, but also makes the transmitted and reflected
signal smaller thus either a bigger temperature or more sensitive detectors
would be required to record the signal when performing experiments. As an
addition, we provided in this paper rough estimates of the detuning (with
respect to the resonant frequency of the hyperfine transitions
$F_{g}=1\rightarrow F_{e}=1$ (for $D_{1}$ line) and $F_{g}=1\rightarrow
F_{e}=0$ (for $D_{2}$ line)) between the various groups of transitions that
remain present in the spectra when the hyperfine Paschen-Back regime is
reached. We highlighted for the first time the appearance of Type 1 and Type 2
Circular Dichroism in sodium vapors and provided high-resolution absorption
spectra of sodium Zeeman transitions depending on the value of the external
magnetic field. Complete description and understanding of all these magneto-
optical processes are of utmost importance for further applications, for
example in optical magnetometry. Upcoming experiments involving nanocells are
planned at the Institute for Physical Research to provide an experimental
verification of these results. Complete agreement between experiments and
theory is expected as it was proven for all other alkalis (except lithium, for
which it is very hard to fabricate a nano-cell).

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