# Probe field ellipticity-induced shift in an atomic clock

V. I. Yudin<EMAIL_ADDRESS>Institute of Laser Physics SB RAS, pr. Akademika
Lavrent’eva 15 B, Novosibirsk, 630090, Russia  Novosibirsk State University,
ul. Pirogova 1, Novosibirsk, 630090, Russia  Novosibirsk State Technical
University, pr. Karla Marksa 20, Novosibirsk, 630073, Russia  A. V.
Taichenachev  Institute of Laser Physics SB RAS, pr. Akademika Lavrent’eva 15
B, Novosibirsk, 630090, Russia  Novosibirsk State University, ul. Pirogova 1,
Novosibirsk, 630090, Russia  O. N. Prudnikov  Institute of Laser Physics SB
RAS, pr. Akademika Lavrent’eva 15 B, Novosibirsk, 630090, Russia  Novosibirsk
State University, ul. Pirogova 1, Novosibirsk, 630090, Russia  M. Yu. Basalaev
Institute of Laser Physics SB RAS, pr. Akademika Lavrent’eva 15 B,
Novosibirsk, 630090, Russia  Novosibirsk State University, ul. Pirogova 1,
Novosibirsk, 630090, Russia  Novosibirsk State Technical University, pr. Karla
Marksa 20, Novosibirsk, 630073, Russia  V. G. Pal’chikov  All-Russian Research
Institute of Physical and Radio Engineering Measurements, 141570 Mendeleevo,
Moscow region, Russia  National Nuclear Research University MEPhI, Kashirskoe
sh. 31, 115409 Moscow, Russia  M. von Boehn  Physikalisch-Technische
Bundesanstalt, Bundesallee 100, D-38116 Braunschweig, Germany  T. E.
Mehlstäubler  Physikalisch-Technische Bundesanstalt, Bundesallee 100, D-38116
Braunschweig, Germany  Institut für Quantenoptik, Leibniz Universität
Hannover, Welfengarten 1, 30167 Hannover, Germany  S. N. Bagayev  Institute of
Laser Physics SB RAS, pr. Akademika Lavrent’eva 15 B, Novosibirsk, 630090,
Russia  Novosibirsk State University, ul. Pirogova 1, Novosibirsk, 630090,
Russia

###### Abstract

We investigate the probe field induced shift for atomic lattice-based and ion-
trap clocks, which can be considered as a near resonant ac-Stark shift,
connected to the Zeeman structure of atomic levels and their splitting in a dc
magnetic field. This shift arises from possible residual ellipticity in the
polarization of the probe field and uncertainty in the magnetic field
orientation. Such a shift can have an arbitrary sign and, for some
experimental conditions, can reach the fractional value of the order of
10-18-10-19, i.e., it is not negligible. Thus, it should be taken into account
in the uncertainty budgets for the modern ultra-precise atomic clocks. In
addition, it is shown that when using hyper-Ramsey spectroscopy, this shift
can be reduced to a level much lower than $10^{-19}$.

##  I Introduction

Ultra-precise atomic clocks are the forefront of modern quantum sensors and
tests of the standard model Ludlow_2015 ; Safronova_2018 ; Tanja_2018 ;
Riehle_2017 ; Delva_2017 . At present, some laboratories have demonstrated
systematic uncertainties and long-term instabilities at a fractional level of
$10^{-18}$ both for devices with neutral atoms trapped in an optical lattice
at the magic wavelength Katori_2015 ; Nicholson_2015 ; Ye_2017 ; Schioppo_2017
; McGrew_2018 ; Ye_2019 ; Bothwell_2019 , and for clocks using trapped ions
Huntemann_2016 ; Brewer_2019 . There is a recent trend to push fractional
uncertainties to the level of $10^{-19}$ Brewer_2019 ; Marti_2018 ;
Keller_2019 . However, such an extraordinary high metrological precision
requires a very thorough study of all possible frequency shifts that can
exceed (at least in principle) the value of $10^{-19}$.

Figure 1:  Scheme of light-induced transitions over Zeeman structure for an
optical transition $F_{\rm g}=F\rightarrow F_{\rm e}=F$:
(a) the ideal case of linear polarization of the probe field ${\bf E}$
directed along the magnetic field ${\bf B}$ (i.e., ${\bf E}||{\bf B}$);
(b) the general case of elliptical polarization of the probe field ${\bf E}$
under arbitrary orientation of the magnetic field ${\bf B}$, where the dotted
red lines mark the light-induced transitions leading to ac Stark shifts (II)
for the clock transitions $|m_{\rm g}=-F\rangle\rightarrow|m_{\rm
e}=-F\rangle$ and $|m_{\rm g}=+F\rangle\rightarrow|m_{\rm e}=+F\rangle$.

In this paper, we investigate the ac Stark shift due to some small residual
ellipticity of the probe field, which has not previously been widely discussed
in the scientific literature. We consider in detail the case of clock
transitions $F_{\rm g}=F\rightarrow F_{\rm e}=F$ (where $F_{\rm g}$ and
$F_{\rm e}$ are the angular momenta of the energy levels in the ground and
excited states, respectively). In particular, such a variant takes place for
strongly forbidden electronic transitions
${}^{1}\mathrm{S}_{0}\rightarrow{}^{3}$P0 for odd isotopes of alkaline earth
neutral atoms (e.g., 87Sr, 171Yb) and for some ions (e.g., with two remaining
valence electrons, such as 27Al+, 115In+). It is shown that this shift can
reach a fractional level of the order of $10^{-18}$ for some experimental
conditions and therefore, needs to be taken into account in the uncertainty
budget for modern ultra-precise atomic clocks. In addition, we propose a
method for the radical suppression of any ac Stark shift by using hyper-Ramsey
spectroscopy Ref. yudin2010 .

##  II Theoretical model

Let us consider an electric-dipole ($E1$) clock transition $F_{\rm
g}=F\rightarrow F_{\rm e}=F$ with an unperturbed frequency $\omega_{0}$, where
$F_{\rm g}$ and $F_{\rm e}$ are the angular momenta of the energy levels in
the ground and excited states, respectively. As noted above, this type of
transition takes place in atomic clocks based on the strongly forbidden
1S${}_{0}\rightarrow{}^{3}$P0 intercombination transition in alkaline earth
(and similar) atoms (Mg, Ca, Sr, Yb, Hg), as well as for some ions (Al+, In+).
For these elements, the value of $F$ is determined by the nuclear spin, which
is a half-integer (odd isotopes). Under the presence of an external magnetic
field ${\bf B}$, the Zeeman splitting of the levels in magnetic sublevels
$|m_{\rm g}\rangle$ and $|m_{\rm e}\rangle$ occurs due to the nonzero magnetic
moment, where $|m_{\rm g,e}|\leq F$ (see Fig. 1). To eliminate the linear
magnetic sensitivity in atomic clocks, the following measurement procedure is
usually used. A probe light field is chosen with a linear polarization vector
${\bf E}$ parallel to the dc magnetic field, ${\bf E}||{\bf B}$. For this
case, possible light-induced transitions between Zeeman sublevels are shown in
Fig. 1(a). Next, by pumping atoms in turn to the extreme Zeeman sublevels of
the ground state $|m_{\rm g}=\pm F\rangle$, two frequencies $\omega_{-F-F}$
and $\omega_{+F+F}$ are successively measured on the optical clock transitions
$|m_{\rm g}=-F\rangle\rightarrow|m_{\rm e}=-F\rangle$ and $|m_{\rm
g}=+F\rangle\rightarrow|m_{\rm e}=+F\rangle$, respectively [see Fig. 1(a)].
Each of these frequencies separately experiences a linear Zeeman shift with
respect to the unperturbed frequency $\omega_{0}$:

$\displaystyle\omega_{-F-F}$ $\displaystyle=$
$\displaystyle\omega_{0}-(\Delta_{\rm e}-\Delta_{\rm g})F\,,$
$\displaystyle\omega_{+F+F}$ $\displaystyle=$
$\displaystyle\omega_{0}+(\Delta_{\rm e}-\Delta_{\rm g})F\,,$ (1)

where

$\Delta_{\rm g}=g_{\rm g}\mu_{B}|{\bf B}|/\hbar\,,\quad\Delta_{\rm e}=g_{\rm
e}\mu_{B}|{\bf B}|/\hbar\,,$ (2)

are the Zeeman splittings in the ground and excited states, respectively [see
Fig. 1(a)]; $\mu_{B}$ is the Bohr magneton; and $g_{\rm g}$ and $g_{\rm e}$
are the $g$-factors in the ground and excited states, respectively. However,
for the superposition frequency

$\omega_{\rm clock}=\frac{\omega_{-F-F}+\omega_{+F+F}}{2}=\omega_{0}\,,$ (3)

considered as a clock frequency, there is no linear sensitivity to a magnetic
field. The residual quadratic sensitivity to the magnetic field, $\propto|{\bf
B}|^{2}$, is due to the non-resonant interaction between different energy
levels induced by the magnetic-dipole interaction operator
$-({\hat{\bf\mu}{\bf B}})$, where $\hat{\bf\mu}$ is the operator of the
magnetic moment of an atom (ion). For example, for the above mentioned
elements, this occurs as a result of the magnetic-dipole interaction between
the 3P0 and 3P1 fine structure states.

However, the above ideal picture of interaction [see Fig. 1(a)] in reality can
be violated due to the presence of an uncontrolled ellipticity of the probe
field, and also due to an uncertainty in the direction of the magnetic field
vector. In the general case of arbitrary orientation of the magnetic field
${\bf B}$ relative to the polarization ellipse [see Fig. 2(a)], the electric
vector of the probe resonant field (with frequency $\omega$) has the following
form

$\displaystyle{\bf E}(t)={\rm Re}\\{Ee^{-i\omega t}{\bf a}\\}=(Ee^{-i\omega
t}{\bf a}+c.c.)/2\,,$ $\displaystyle{\bf a}=\sum_{q=0,\pm 1}a^{(q)}{\bf
e}_{q}\,,\quad\sum_{q=0,\pm 1}|a^{(q)}|^{2}=1\,,$ (4)

where $E$ is the scalar amplitude; $a^{(q)}$ are the contravariant components
of the unit complex vector of polarization ${\bf a}$ (i.e., $|{\bf a}|=1$) in
the cyclic basis ${\bf e}_{0}={\bf e}_{z}$, ${\bf e}_{\pm 1}=\mp({\bf
e}_{x}\pm i{\bf e}_{y})/\sqrt{2}$ (where ${\bf e}_{x}$, ${\bf e}_{y}$, ${\bf
e}_{z}$ are unit basis vectors of the Cartesian coordinate system, for which
the quantization axis $Oz$ is directed along the magnetic vector ${\bf B}$).
In the case of the electric-dipole interaction operator $-(\hat{{\bf d}}{\bf
E})$, the general picture of light-induced transitions near the extreme Zeeman
sublevels is shown in Fig. 1(b), where the following expressions hold for the
corresponding Rabi frequencies (their absolute values):

$|\Omega_{0}|=\frac{|d_{\rm
eg}Ea^{(0)}|\sqrt{F}}{\hbar\sqrt{F+1}}\,,\;\;|\Omega_{\pm}|=\frac{|d_{\rm
eg}Ea^{(\pm 1)}|}{\hbar\sqrt{F+1}}\,,$ (5)

where $d_{\rm eg}=\langle F_{\rm e}||\hat{{\bf d}}||F_{\rm g}\rangle$ is the
reduced matrix element of the dipole moment operator $\hat{{\bf d}}$ for the
clock transition $F_{\rm g}=F\rightarrow F_{\rm e}=F$.

Figure 2:  (a) the general case of an arbitrary orientation of dc magnetic
field ${\bf B}$ with respect to the polarization ellipse ${\bf a}$, where
$\xi$ is the angle between the magnetic field vector and the plane of the
polarization ellipse;
(b) the geometric parametrization of the elliptical polarization ${\bf a}$
according to Eq. (13); ${\bf n}^{\rm(ell)}_{\rm B}$ is the projection of the
unit vector ${\bf n}_{\rm B}={\bf B}/|{\bf B}|$ onto the plane of the
polarization ellipse, and $\varphi$ is the angle between ${\bf
n}^{\rm(ell)}_{\rm B}$ and the major axis of the polarization ellipse.

Let us consider the effect of transitions caused by the presence of two
circular components $a^{(\pm 1)}$ of the polarization vector ${\bf a}$ [see
Eq. (II)], which correspond to the Rabi frequencies $|\Omega_{\pm}|$ [see Fig.
1(b)]. Since, during atomic clock operation, the probe field frequency
$\omega$ is stabilized to the extreme transitions $|m_{\rm
g}=-F\rangle\rightarrow|m_{\rm e}=-F\rangle$ and $|m_{\rm
g}=+F\rangle\rightarrow|m_{\rm e}=+F\rangle$, the presence of two circular
components $a^{(\pm 1)}$ will lead to ac Stark shifts of the lower ($|m_{\rm
g}=\pm F\rangle$) and upper ($|m_{\rm e}=\pm F\rangle$) Zeeman sublevels due
to non-resonant interaction with the neighboring magnetic sublevels $|m_{\rm
g}=\pm F\mp 1\rangle$ and $|m_{\rm e}=\pm F\mp 1\rangle$. As a result, instead
of Eq. (II) we get the following expressions for the frequencies
$\omega_{-F-F}$ and $\omega_{+F+F}$:

$\displaystyle\omega_{-F-F}$ $\displaystyle=$
$\displaystyle\omega_{0}-(\Delta_{\rm e}-\Delta_{\rm
g})F+\bar{\delta}_{-F-F}\,,$ $\displaystyle\omega_{+F+F}$ $\displaystyle=$
$\displaystyle\omega_{0}+(\Delta_{\rm e}-\Delta_{\rm
g})F+\bar{\delta}_{+F+F}\,,$ (6)

where $\bar{\delta}_{-F-F}$ and $\bar{\delta}_{+F+F}$ are the corresponding ac
Stark shifts:

$\displaystyle\bar{\delta}_{-F-F}$ $\displaystyle=$
$\displaystyle\frac{|\Omega_{+}|^{2}}{4\Delta_{\rm
e}}-\frac{|\Omega_{-}|^{2}}{4\Delta_{\rm g}}\,,$
$\displaystyle\bar{\delta}_{+F+F}$ $\displaystyle=$
$\displaystyle\frac{|\Omega_{+}|^{2}}{4\Delta_{\rm
g}}-\frac{|\Omega_{-}|^{2}}{4\Delta_{\rm e}}\,.$ (7)

The expressions (II) and (II) are derived under the condition

$|\Delta_{\rm e,g}|\gg|\Omega_{\pm}|\,,$ (8)

which holds well in real experiments and corresponds to a non-resonant
interaction for circular components $Ea^{(\pm 1)}{\bf e}_{\pm 1}$.

Thus, the clock frequency

$\omega_{\rm
clock}=\frac{\omega_{-F-F}+\omega_{+F+F}}{2}=\omega_{0}+\bar{\delta}^{\text{(el-
ind)}}_{\rm ac}\,,$ (9)

becomes shifted relatively to the unperturbed frequency $\omega_{0}$ by the
value

$\displaystyle\bar{\delta}^{\text{(el-ind)}}_{\rm ac}$ $\displaystyle=$
$\displaystyle\frac{\bar{\delta}_{-F-F}+\bar{\delta}_{+F+F}}{2}=\frac{\Delta_{\rm
g}+\Delta_{\rm e}}{8\Delta_{\rm g}\Delta_{\rm
e}}(|\Omega_{+}|^{2}-|\Omega_{-}|^{2})=$ (10) $\displaystyle\frac{|d_{\rm
eg}E|^{2}}{\hbar^{2}(F+1)}\frac{\Delta_{\rm g}+\Delta_{\rm e}}{8\Delta_{\rm
g}\Delta_{\rm e}}(|a^{(+1)}|^{2}-|a^{(-1)}|^{2})\,,$

which we will denote as the ellipticity-induced shift. The nature of which is
an ac Stark shift through interaction with neighboring Zeeman sublevels. Using
the expression for $|\Omega_{0}|$ in Eq. (5), the formula (10) can be
rewritten as

$\bar{\delta}^{\text{(el-ind)}}_{\rm ac}=|\Omega_{0}|^{2}\frac{\Delta_{\rm
g}+\Delta_{\rm e}}{8F\Delta_{\rm g}\Delta_{\rm
e}}\frac{|a^{(+1)}|^{2}-|a^{(-1)}|^{2}}{|a^{(0)}|^{2}}\,,$ (11)

which we will analyze further.

First of all, using vector notations, we represent in an invariant form the
following expression

$\frac{|a^{(+1)}|^{2}-|a^{(-1)}|^{2}}{|a^{(0)}|^{2}}=\frac{i([{\bf
a}\times{\bf a}^{\ast}]\cdot{\bf n}_{\rm B})}{|{\bf a}\cdot{\bf n}_{\rm
B}|^{2}}\,,$ (12)

where $[{\bf a}\times{\bf a}^{\ast}]$ denotes the cross product of two vectors
${\bf a}$ and ${\bf a}^{\ast}$, and ${\bf n}_{\rm B}={\bf B}/|{\bf B}|$ is the
unit orientation vector of the magnetic field ${\bf B}$. As shown in Fig.
2(b), the degree of ellipticity of the polarization vector ${\bf a}$ can be
parametrized by the angular parameter $\varepsilon$

${\bf a}=\cos(\varepsilon){\bf e}^{\prime}_{x}+i\sin(\varepsilon){\bf
e}^{\prime}_{y}\,,$ (13)

where ${\bf e}^{\prime}_{x,y}$ are the Cartesian unit vectors oriented along
the axes of the polarization ellipse. In this case we have:

$i[{\bf a}\times{\bf a}^{\ast}]=\sin(2\varepsilon){\bf n}_{\rm k}\,,$ (14)

where ${\bf n}_{\rm k}={\bf e}^{\prime}_{z}$ is the unit vector orthogonal to
the plane of the polarization ellipse, i.e. directed along the wave vector of
the probe field. This, in turn, leads to

$i([{\bf a}\times{\bf a}^{\ast}]\cdot{\bf n}_{\rm B})=\sin(2\varepsilon)({\bf
n}_{\rm k}\cdot{\bf n}_{\rm B})=\sin(2\varepsilon)\sin(\xi)\,,$ (15)

where $\xi$ is the angle between the vector ${\bf B}$ and the plane of the
polarization ellipse ${\bf a}$ [see Fig. 2(a)]. In addition, using Fig. 2(b),
we obtain an expression for $|{\bf a}\cdot{\bf n}_{\rm B}|^{2}$

$\displaystyle|{\bf a}\cdot{\bf n}_{\rm B}|^{2}=|{\bf a}\cdot{\bf
n}^{\rm(ell)}_{\rm B}|^{2}=$
$\displaystyle\left[\cos^{2}(\varepsilon)\cos^{2}(\varphi)+\sin^{2}(\varepsilon)\sin^{2}(\varphi)\right]\cos^{2}(\xi)\,,$
(16)

where ${\bf n}^{\rm(ell)}_{\rm B}$ is the projection of the unit vector ${\bf
n}_{\rm B}$ onto the plane of the polarization ellipse, and $\varphi$ is the
angle between ${\bf n}^{\rm(ell)}_{\rm B}$ and the major axis of the
polarization ellipse.

Thus, using (15) and (II) in Eq. (12), the shift (11) can be calculated by the
formula

$\bar{\delta}^{\text{(el-ind)}}_{\rm ac}=|\Omega_{0}|^{2}\frac{\Delta_{\rm
g}+\Delta_{\rm e}}{8F\Delta_{\rm g}\Delta_{\rm
e}}\frac{\sin(2\varepsilon)\sin(\xi)}{|{\bf a}\cdot{\bf n}_{\rm B}|^{2}}\,.$
(17)

Based on this expression, one can find three conditions where the shift
vanishes, $\bar{\delta}^{\text{(el-ind)}}_{\rm ac}=0$:
1) purely linear polarization, $\varepsilon=0$, for any value of $\xi$;
2) $\xi=0$ for any ellipticity $\varepsilon$, when the magnetic field vector
${\bf B}$ lies in the plane of the polarization ellipse;
3) $g_{\rm g}=-g_{\rm e}$ (when $\Delta_{\rm g}+\Delta_{\rm e}=0$) for any
$\varepsilon$ and $\xi$. However, this “exotic” variant does not occur for
real atomic transitions $F_{\rm g}=F\rightarrow F_{\rm e}=F$.

Figure 3:  Illustration of a series of optical resonances $|m_{\rm
g}=m\rangle\rightarrow|m_{\rm e}=m\rangle$ ($-F\leq m\leq F$) in a linearly
polarized field for $F_{\rm g}=F\rightarrow F_{\rm e}=F$ transitions, using
$F=7/2$ as an example. The frequency distance between adjacent resonances is
defined as $\Delta_{Z}=|\Delta_{\rm g}-\Delta_{\rm e}|$.

Regarding to atomic clocks, the residual shift $\bar{\delta}^{\text{(el-
ind)}}_{\rm ac}$ appears due to some uncontrollability of small values
$\varepsilon$ and $\xi$. Thus, the signs of the quantities $\varepsilon$,
$\xi$, and therefore the sign of $\bar{\delta}^{\text{(el-ind)}}_{\rm ac}$
should also be considered as uncontrolled. Therefore, we will be interested in
the absolute value $|\bar{\delta}^{\text{(el-ind)}}_{\rm ac}|$. In addition,
consider the complete picture of the resonance series over all Zeeman
sublevels shown in Fig. 3, where the distance between adjacent resonances is
equal to $\Delta_{Z}=|\Delta_{\rm g}-\Delta_{\rm e}|$. This allows us to
represent the shift in the form

$\displaystyle|\bar{\delta}^{\text{(el-ind)}}_{\rm ac}|$ $\displaystyle=$
$\displaystyle\frac{|\Omega_{0}|^{2}}{\Delta_{Z}}\frac{|\Delta^{2}_{\rm
g}-\Delta^{2}_{\rm e}|}{8F|\Delta_{\rm g}\Delta_{\rm
e}|}\frac{|\sin(2\varepsilon)\sin(\xi)|}{|{\bf a}\cdot{\bf n}_{\rm B}|^{2}}=$
(18) $\displaystyle\frac{|\Omega_{0}|^{2}}{\Delta_{Z}}\frac{|g^{2}_{\rm
g}-g^{2}_{\rm e}|}{8F|g_{\rm g}g_{\rm
e}|}\frac{|\sin(2\varepsilon)\sin(\xi)|}{|{\bf a}\cdot{\bf n}_{\rm
B}|^{2}}\,.$

Typically for atomic clocks, the experimental conditions are close to the
ideal case of linear probe field polarization ${\bf E}$ that is parallel to
the magnetic field ${\bf B}$. Therefore, under small non-idealities, the
condition of small value for all three angular parameters
$|\varepsilon,\xi,\varphi|\ll 1$ can be assumed, which allows us to use the
following approximations

$\sin(2\varepsilon)\approx 2\varepsilon,\;\;\sin(\xi)\approx\xi,\;\;|{\bf
a}\cdot{\bf n}_{\rm B}|^{2}\approx 1-\varepsilon^{2}-\varphi^{2}-\xi^{2},$
(19)

which, in turn, leads to the main approximation for the ellipticity-induced
shift:

$|\bar{\delta}^{\text{(el-ind)}}_{\rm ac}|\approx
A\frac{|\Omega_{0}|^{2}}{\Delta_{Z}}|\varepsilon\xi|\,,$ (20)

where the coefficient

$A=\frac{|g^{2}_{\rm g}-g^{2}_{\rm e}|}{4F|g_{\rm g}g_{\rm e}|}\,,$ (21)

is a characteristics of the specific clock transition, and the angular
parameters $\varepsilon$ and $\xi$ in Eq. (20) are defined in radian units. We
remind that $|\Omega_{0}|$ is the Rabi frequency for clock transitions
$|m_{\rm g}=-F\rangle\rightarrow|m_{\rm e}=-F\rangle$ and $|m_{\rm
g}=+F\rangle\rightarrow|m_{\rm e}=+F\rangle$ between the extreme Zeeman
sublevels.

Table 1: Values of the coefficient $A$ [see Eq. (21)] for clock transitions $F_{\rm g}=F\rightarrow F_{\rm e}=F$ in atoms and ions for which the strongly-forbidden transition 1S${}_{0}\rightarrow{}^{3}$P0 currently used for clock operation. We used the values of $g$-factors from Refs. Porsev_2004 ; Lu_2018 ; Rosenband_2007 ; Becker_2001 . |  171Yb |  173Yb |  87Sr |  27Al+ |  115In+
---|---|---|---|---|---
$F$ | 1/2 | 5/2 | 9/2 | 5/2 | 9/2
$A$ | 0.85 | 0.18 | 0.053 | 0.209 | 0.045

Table 1 presents the values of the parameter $A$ for the optical clock
transitions in some atoms and ions. Note that, as follows from Eq. (21), the
sensitivity to the ellipticity-induced shift for $F_{\rm g}=F\rightarrow
F_{\rm e}=F$ transitions decreases with increasing of $F$.

Let us now estimate the magnitude of possible values of the angular parameters
$\varepsilon$ and $\xi$. Under typical experimental conditions, after the
passage of light through vacuum windows, a degree of ellipticity can be
estimated as (e.g., see Refs. Nemitz_2019 ; Mehlstaubler )

$\frac{I_{x^{\prime}}}{I{}_{y^{\prime}}}=\tan^{2}(\varepsilon)\approx\varepsilon^{2}\sim
0.01\,,\;\Rightarrow\;|\varepsilon|\sim 0.1~{}\text{rad}\,,$ (22)

using the ratio of the intensities of the orthogonal components
$I_{x^{\prime}}$ and $I{}_{y^{\prime}}$ along the main axes of the
polarization ellipse [see Fig. 2(b)]. We also assume that the possible
uncertainty of the magnetic field orientation ${\bf n}_{B}$ with respect to
the polarization vector ${\bf a}$ can be several angular degrees, i.e.,
$|\xi|\sim 0.1$ rad. Thus, we obtain the following general estimate for the
angular parameters:

$|\varepsilon\xi|\sim 0.01\,,$ (23)

which we will use in further evaluations.

##  III Total ac Stark shift

Since the presented ellipticity-induced shift is proportional to the square of
the electric field, $\bar{\delta}^{\text{(el-ind)}}_{\rm ac}\propto|E|^{2}$,
it can be considered as an additional, transition internal ac Stark shift.
Simultaneously, there always exists the well-known standard ac Stark shift,
$\bar{\delta}^{\text{(off-res)}}_{\rm ac}\propto|E|^{2}$, due to the
interaction of the probe field with other far-off-resonant atomic levels, and
which can be represented as

$\bar{\delta}^{\text{(off-res)}}_{\rm ac}=\alpha|\Omega_{0}|^{2},$ (24)

where $\alpha$ is some proportionality factor. Therefore, to estimate the
uncertainty in atomic clocks, one must always consider the total ac Stark
shift

$\bar{\delta}^{\rm(tot)}_{\rm ac}=\bar{\delta}^{\text{(el-ind)}}_{\rm
ac}+\bar{\delta}^{\text{(off-res)}}_{\rm ac}=|\Omega_{0}|^{2}K\,,$ (25)

where the parameter $K$ in our case is defined as

$\displaystyle K$ $\displaystyle=$ $\displaystyle\frac{\Delta_{\rm
g}+\Delta_{\rm e}}{8F\Delta_{\rm g}\Delta_{\rm
e}}\frac{\sin(2\varepsilon)\sin(\xi)}{|{\bf a}\cdot{\bf n}_{\rm
B}|^{2}}+\alpha\approx$ (26) $\displaystyle\frac{\Delta_{\rm g}+\Delta_{\rm
e}}{4F\Delta_{\rm g}\Delta_{\rm e}}\,\varepsilon\xi+\alpha\,,$

according to the formula (17), as well as the conditions $|\varepsilon,\xi|\ll
1$ and $|{\bf a}\cdot{\bf n}_{\rm B}|^{2}\approx 1$.

Unlike the ellipticity-induced shift, $\bar{\delta}^{\text{(off-res)}}_{\rm
ac}$ has a fixed sign. For example, in the case of the clock transition
1S0$\to$3P0, the quantity $\bar{\delta}^{\text{(off-res)}}_{\rm ac}$ depends
very weakly on the ellipticity parameter $\varepsilon$. Therefore, the
absolute value of the total ac Stark shift can take two values

$|\bar{\delta}^{\rm(tot)}_{\rm ac}|=||\bar{\delta}^{\text{(off-res)}}_{\rm
ac}|\pm|\bar{\delta}^{\text{(el-ind)}}_{\rm ac}||\,,$ (27)

where the sign ($\pm$) can change depending on the signs of $\varepsilon$ and
$\xi$ [see Eq. (17)]. However, if we assume that a small degree of ellipticity
$\varepsilon$ appears in experiments in an uncontrolled way (as well as
$\xi$), then the maximal value should be used for metrological estimates

$\bar{\delta}^{\rm(met)}_{\rm ac}=|\bar{\delta}^{\text{(off-res)}}_{\rm
ac}|+|\bar{\delta}^{\text{(el-ind)}}_{\rm ac}|\,,$ (28)

which is more conservative (from a metrological viewpoint). It is also
interesting to note that, based on Eq. (26), one can choose such values of
$\varepsilon$, $\xi$ and $|{\bf B}|$ for which $K=0$, i.e., the total ac Stark
shift vanishes, $\bar{\delta}^{\rm(tot)}_{\rm ac}=0$.

Below we will carry out a comparative analysis of three different
spectroscopic schemes (see Fig. 4) with the same total interrogation time
$t_{int}$.

Figure 4:  Scheme of three variants of spectroscopy with the same total
interrogation time $t_{int}$:
(a) $\pi$-pulse in standard Rabi spectroscopy;
(b) the standard Ramsey sequence with two $\pi/2$-pulses, $T=(t_{int}-2\tau)$
is the free evolution time;
(c) hyper-Ramsey sequence yudin2010  including composite pulse,
$T=(t_{int}-4\tau)$ is the free evolution time.

##  IV Rabi spectroscopy

As applied to Rabi spectroscopy with a single $\pi$-pulse [see Fig. 4(a)], the
frequency shift of the clock transition $\bar{\delta}^{\rm(Rabi)}_{\rm ac}$ is
determined by the shift $\bar{\delta}^{\rm(tot)}_{\rm ac}$ [see Eq. (25)],
which can be represented as:

$\bar{\delta}^{\rm(Rabi)}_{\rm ac}=\bar{\delta}^{\rm(tot)}_{\rm
ac}\approx\frac{\pi^{2}}{t^{2}_{int}}K\,,$ (29)

taking into account the condition $\Omega_{0}t_{int}=\pi$ in Eq. (20).

Analyzing various papers, we found that the frequency shift
$|\bar{\delta}^{\rm(Rabi)}_{\rm clock}|$ under certain experimental conditions
can reach several mHz (when using Rabi spectroscopy), which noticeably exceeds
the fractional level of $10^{-18}$. Thus, this shift must always be estimated,
and further, depending on this estimate, should be included/excluded in the
budget of uncertainties for modern ultra-precise atomic clocks.

For example, consider the experiment Ref. Kobayashi_2020 , where the clock
transition $F_{\rm g}=1/2\rightarrow F_{\rm e}=1/2$ in 171Yb was used under
$t_{int}=40$ ms Rabi $\pi$-pulse and dc magnetic field of 65 $\mu$T. In this
case, we have $|\Omega_{0}|/2\pi=12.5$ Hz and $\Delta_{Z}/2\pi=260$ Hz. Taking
into account the value $|\varepsilon\xi|\sim 0.01$ of Eq. (23), we obtain a
possible shift $|\bar{\delta}^{\text{(el-ind)}}_{\rm ac}|\sim 5$ mHz, which
corresponds to the fractional value of $9.6\times 10^{-18}$ for the 578 nm
clock transition. At the same time, the standard ac Stark shift
$|\bar{\delta}^{\text{(off-res)}}_{\rm ac}|$ is estimated in Ref.
Kobayashi_2020  at the level of $4\times 10^{-18}$. Thus, according to Eq.
(28), the total ac Stark shift $|\bar{\delta}^{\rm(Rabi)}_{\rm ac}|$ can
potentially reach the value of $1.4\times 10^{-17}$.

##  V Standard Ramsey spectroscopy

Let us consider standard Ramsey spectroscopy with two identical $\pi/2$ pulses
separated by a free evolution time $T$ [see Fig. 4(b)], for which the central
Ramsey resonance will be used for frequency stabilization in atomic clocks.
Therefore, the frequency shifts $\bar{\delta}_{-F-F}$ and
$\bar{\delta}_{+F+F}$ due to the ellipticity of the light, as well as the
standard ac Stark shift $\bar{\delta}^{\text{(off-res)}}_{\rm ac}$, will occur
only during Ramsey pulses of duration $\tau$, while during the free interval
$T$ these shifts are absent. In this case, the shift of the central Ramsey
resonance $\bar{\delta}^{\rm(Rams)}_{\rm ac}$ is no longer equal to
$\bar{\delta}^{\rm(tot)}_{\rm ac}$ (as for Rabi spectroscopy), but defined as
(see Ref. yudin2010 ; tai09 )

$\bar{\delta}^{\rm(Rams)}_{\rm ac}\approx\frac{2\bar{\delta}^{\rm(tot)}_{\rm
ac}}{2+|\Omega_{0}|T}\,.$ (30)

Using the expression (25) and the condition $|\Omega_{0}|\tau=\pi/2$, the
shift (30) can be represented in the form

$\bar{\delta}^{\rm(Rams)}_{\rm ac}\approx\frac{\pi^{2}}{2\tau^{2}[2+\pi
T/(2\tau)]}K\,.$ (31)

Comparing Eq. (31) and Eq. (29), it is easy to show the inequality

$|\bar{\delta}^{\rm(Rams)}_{\rm ac}|>|\bar{\delta}^{\rm(Rabi)}_{\rm ac}|\,.$
(32)

Indeed, taking into account that $T=(t_{int}-2\tau)$, consider the ratio of
the quantities (31) and (29) for the same interrogation time $t_{int}$:

$\frac{\bar{\delta}^{\rm(Rams)}_{\rm ac}}{\bar{\delta}^{\rm(Rabi)}_{\rm
ac}}=\frac{1}{2(\tau/t_{int})^{2}\left[2-\pi+0.5\pi(\tau/t_{int})^{-1}\right]}\,,$
(33)

where $\tau<0.5t_{int}$.

Figure 5:  Dependence of the ratio $\bar{\delta}^{\rm(Rams)}_{\rm
ac}/\bar{\delta}^{\rm(Rabi)}_{\rm ac}$ on $\tau/t_{int}$.

Fig. 5 shows the dependence of the ratio (33) on $\tau/t_{int}$, from which
the validity of the inequality (32) follows, since
$\bar{\delta}^{\rm(Rams)}_{\rm ac}/\bar{\delta}^{\rm(Rabi)}_{\rm ac}>1$. We
see that the shift $|\bar{\delta}^{\rm(Rams)}_{\rm ac}|$ increases with
decreasing $\tau$. For example, in the case of $\tau<0.1t_{int}$,
$|\bar{\delta}^{\rm(Rams)}_{\rm ac}|>3.4|\bar{\delta}^{\rm(Rabi)}_{\rm ac}|$.
Note that under the condition $\tau\ll t_{int}$, which is typical for Ramsey
spectroscopy, the expressions (30)-(31) can be represented as

$\bar{\delta}^{\rm(Rams)}_{\rm
ac}\approx\frac{2}{T}\frac{\bar{\delta}^{\rm(tot)}_{\rm
ac}}{|\Omega_{0}|}\approx\frac{\pi}{\tau T}K\,.$ (34)

Thus, we can assert that using standard Ramsey spectroscopy, any shift
(including the ellipticity-induced shift) proportional to the the probe field
intensity (i.e. $\propto|E|^{2}$) will be noticeably larger compared to Rabi
spectroscopy. Moreover, since the value of the coefficient $K$ in Eq. (25) did
not play any role in proving, this statement about ac Stark shift is general
when comparing Ramsey spectroscopy and Rabi spectroscopy for any one-photon
transitions.

In this context, it is interesting to consider the experiment in Ref.
McGrew_2018 , where the clock transition $F_{\rm g}=1/2\rightarrow F_{\rm
e}=1/2$ in 171Yb was used under $t_{int}=560$ ms Rabi $\pi$-pulse and dc
magnetic field of 0.1 mT. In this case, we have $|\Omega_{0}|/2\pi=0.9$ Hz and
$\Delta_{Z}/2\pi=400$ Hz. Taking into account the value of Eq. (23), we obtain
a possible shift of $|\bar{\delta}^{\text{(el-ind)}}_{\rm ac}|\sim 0.017$ mHz,
which corresponds to the fractional value of $3.3\times 10^{-20}$ for the 578
nm clock transition. At the same time, the standard ac Stark shift
$\bar{\delta}^{\text{(off-res)}}_{\rm ac}$ is estimated in Ref. McGrew_2018
as $2\times 10^{-20}$. Therefore, according to Eq. (28), the total ac Stark
shift $\bar{\delta}^{\rm(Rabi)}_{\rm ac}$ is estimated at the fractional level
of $5.3\times 10^{-20}$. On the other hand, Ref. McGrew_2018  also presents an
experiment with $T=510$ ms free-evolution-time Ramsey spectroscopy, when the
total interrogation time for Ramsey spectroscopy is $t_{int}=560$ ms, i.e. the
same as for Rabi spectroscopy Ludlow . In this case, we have $\tau=25$ ms
duration of each Ramsey $\pi/2$ pulse, which gives the ratio
$\tau/t_{int}=0.0446$. Then, according to Eq. (33), we find the relationship
$\bar{\delta}^{\rm(Rams)}_{\rm ac}=7.38\,\bar{\delta}^{\rm(Rabi)}_{\rm ac}$.
As a result, the total ac Stark clock shift $|\bar{\delta}^{\rm(Rams)}_{\rm
ac}|$ in Ramsey spectroscopy can potentially reach the fractional value of
$4\times 10^{-19}$, which is no longer negligible for the uncertainty budget
in Ref. McGrew_2018 .

Note also that when using Rabi spectroscopy or standard Ramsey spectroscopy,
the ellipticity-induced shift $\bar{\delta}^{\text{(el-ind)}}_{\rm ac}$ can be
reduced by increasing the magnetic field [i.e., increasing the value of
$\Delta_{Z}$ in the denominator in Eq. (20)]. However, in this case the
quadratic Zeeman shift and its fluctuations can increase significantly, while
the standard ac Stark shift $\bar{\delta}^{\text{(off-res)}}_{\rm ac}$ will
not change.

##  VI hyper-Ramsey spectroscopy

Let us now consider the hyper-Ramsey spectroscopy, which was proposed in Ref.
yudin2010  and first experimentally implemented in Ref. hunt12 . This method
consists of using a sequence of pulses, of which one is a composite pulse with
inverted phase [see Fig. 4(c)]. As applied to our case, the main advantage of
this hyper-Ramsey sequence is that the frequencies of the central Ramsey
resonances $\omega_{-F-F}$ and $\omega_{+F+F}$ between the extreme Zeeman
sublevels will be shifted by values

$\displaystyle\bar{\delta}^{\text{(h-Rams)}}_{-F-F}\approx\frac{\pi}{T}\left(\frac{\bar{\delta}_{-F-F}+\bar{\delta}^{\text{(off-
res)}}_{\rm ac}}{|\Omega_{0}|}\right)^{3},$
$\displaystyle\bar{\delta}^{\text{(h-Rams)}}_{+F+F}\approx\frac{\pi}{T}\left(\frac{\bar{\delta}_{+F+F}+\bar{\delta}^{\text{(off-
res)}}_{\rm ac}}{|\Omega_{0}|}\right)^{3},$ (35)

respectively, as follows from Ref. yudin2010 . For standard experimental
conditions with $F_{\rm g}=F\rightarrow F_{\rm e}=F$ clock transitions in odd
isotopes, the following inequality is usually satisfied

$\left|\frac{\bar{\delta}_{\pm F\pm F}+\bar{\delta}^{\text{(off-res)}}_{\rm
ac}}{\Omega_{0}}\right|\sim\left|\frac{\bar{\delta}^{\rm(tot)}_{\rm
ac}}{\Omega_{0}}\right|<0.001\,.$ (36)

Therefore, due to the cubic dependence in Eq. (VI) on small values [see Eq.
(36)], the ac Stark shift for the hyper-Ramsey scheme

$\bar{\delta}^{\text{(h-Rams)}}_{\rm
ac}=\frac{\bar{\delta}^{\text{(h-Rams)}}_{-F-F}+\bar{\delta}^{\text{(h-Rams)}}_{+F+F}}{2}\,,$
(37)

becomes significantly lower than $10^{-19}$ relative to the clock frequency
$\omega_{0}$.

Thus, the use of hyper-Ramsey spectroscopy Ref. yudin2010  in ultra-precise
atomic clocks is a simple and effective method to radically solve the problem
of any ac Stark shift (including the ellipticity-induced shift
$\bar{\delta}^{\text{(el-ind)}}_{\rm ac}$ as well as the standard ac Stark
shift $\bar{\delta}^{\text{(off-res)}}_{\rm ac}$) without changing other
experimental conditions and for arbitrary clock transition.

## CONCLUSION

We considered a systematic shift in atomic optical clocks due to both some
uncontrolled ellipticity of the probe field and some uncertainty in the
direction of the dc magnetic field vector (ellipticity-induced shift). Using
the example of the clock transitions $F_{\rm g}=F\rightarrow F_{\rm e}=F$, it
is shown that, in the presence of an uncontrolled ellipticity of the probe
field and at a few degrees misalignment of the dc magnetic field, this shift
can reach a fractional level in the range of $10^{-18}$-$10^{-19}$ for Rabi
spectroscopy or standard Ramsey spectroscopy. Therefore, in case the alignment
of the dc magnetic field and probe field polarisation is not well known, it
needs to be taken into account in the uncertainty budget for modern ultra-
precise atomic clocks.

In addition, it is shown that when using hyper-Ramsey spectroscopy Ref.
yudin2010 , the total ac Stark shift (including the ellipticity-induced shift)
can be suppressed to a level significantly lower than $10^{-19}$. This is of
particular importance for mobile devices, in which the interrogation time of
atoms is relatively short, which requires the use of a higher probe field
intensity. Also, in mobile devices, a high level of control of the probe field
ellipticity and the magnetic field orientation is difficult, that leads to an
increase in the value of $|\varepsilon\xi|$ in Eq. (20) and, therefore, to an
increase of ellipticity-induced shift $\bar{\delta}^{\text{(el-ind)}}_{\rm
ac}$. Moreover, hyper-Ramsey spectroscopy allows to reduce the dc magnetic
field in order to noticeably reduce the quadratic Zeeman shift and its
fluctuations. All of these combined can improve the accuracy and long-term
stability of mobile devices (compared to using Rabi spectroscopy or standard
Ramsey spectroscopy).

Our approach can also be adapted to other cases. For example, in Appendix A,
we considered the clock transitions of the type $F_{\rm g}=F\rightarrow F_{\rm
e}=F+1$. Further variants can be considered in a similar way, when for an
arbitrary transition $F_{\rm g}\rightarrow F_{\rm e}$ not only extreme, but
also intermediate Zeeman sublevels are used. In this case, the polarization of
the probe field can be either linear or circular (using two opposite circular
polarizations). For the case of circularly polarized probe fields, one needs
to consider a possible ac Stark shift due to some non-ideal circular
polarization and some deviation of the magnetic field orientation from the
wave vector direction of the probe field.

###### Acknowledgements.

We thank P. Schmidt, C. Lisdat and U. Sterr for useful discussions.

Figure 6:  Scheme of light-induced transitions over Zeeman structure for an
optical transition $F_{\rm g}=F\rightarrow F_{\rm e}=F+1$:
(a) the ideal case of linear polarization of the probe field ${\bf E}$
directed along the magnetic field ${\bf B}$ (i.e., ${\bf E}||{\bf B}$);
(b) the general case of elliptical polarization of the probe field ${\bf E}$
under arbitrary orientation of the magnetic field ${\bf B}$, where the dotted
lines mark the light-induced transitions leading to ac Stark shifts (II) for
the clock transitions $|m_{\rm g}=-F\rangle\rightarrow|m_{\rm e}=-F\rangle$
and $|m_{\rm g}=+F\rangle\rightarrow|m_{\rm e}=+F\rangle$.

## Appendix A

In addition to transitions of the type $F_{\rm g}=F\rightarrow F_{\rm e}=F$,
there can, in principle, be the case of the transitions $F_{\rm
g}=F\rightarrow F_{\rm e}=F+1$, for which the clock frequency is also
determined from the transitions between the extreme Zeeman sublevels $|m_{\rm
g}=-F\rangle\rightarrow|m_{\rm e}=-F\rangle$ and $|m_{\rm
g}=+F\rangle\rightarrow|m_{\rm e}=+F\rangle$ in a linearly polarized field
[see Fig. 6(a)]. This ideal picture of interaction becomes disturbed due to
the presence of some uncontrolled ellipticity of the probe field, and also due
to some uncertainty of the direction of the magnetic field vector ${\bf B}$
[see Fig. 6(b)]. In this case, a residual shift $\bar{\delta}^{\text{(el-
ind)}}_{\rm ac}$ appears, which is also formally described by the expression
(20). However, the coefficient $A$ is now defined by

$A=|g_{\rm g}-g_{\rm e}|\left|\frac{F}{4g_{\rm
g}}+\frac{2F^{2}+3F+2}{4(2F+1)g_{\rm e}}\right|\,.$ (38)

As can be seen, for the $F_{\rm g}=F\rightarrow F_{\rm e}=F+1$ transitions,
the sensitivity to the considered shift increases with $F$ [in contrast to the
transitions $F_{\rm g}=F\rightarrow F_{\rm e}=F$, see Eq. (21)].

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