# Coherence-preserving cooling of nuclear spin qubits in a weak magnetic field

Xiao-Feng Shi  School of Physics, Xidian University, Xi’an 710071, China

###### Abstract

Nuclear spin memories of divalent neutral atoms can allow spin-preserving
resolved-sideband cooling in a strong magnetic field [Phys. Rev. Lett. 99,
123001 (2007)]. We present a theory for cooling 87Sr nuclear-spin qubits in a
weak magnetic field. The theory depends on laser excitation of
$5s5p~{}^{1}P_{1}$ to a nearby state which results in $m_{J}$-dependent AC
Stark shifts large compared to the hyperfine interaction. This effectively
suppresses the nuclear-spin mixing due to the hyperfine interaction. Sideband
cooling via the clock state quenched by the AC Stark-shifted ${}^{1}P_{1}$
state leads to nuclear-spin-preserving spontaneous emission back to the ground
state. More than being compatible with low magnetic fields, the theory is
applicable when the nuclear spin qubits are defined by the two lowest Zeeman
substates.

##  I introduction

Long-lived quantum registers provide a favorable setting for large-scale
quantum computing [1]. Physical systems studied for this purpose include
superconducting circuits and trapped ions, where the number of qubits can be
up to about 50 in one register [2, 3, 4], and the number of typical gate
operations (such as Bell-state creation) within the qubit lifetime is on the
order of $10^{3}$ and $10^{6}$ for superconducting circuits and trapped ions,
respectively [5].

Recently, quantum registers with over 200 neutral-atom qubits [6, 7, 8, 9]
were experimentally realized for coherent quantum control. The long lifetime
[10, 11] of atomic qubits and fast entangling operations [12] suggest that
neutral atoms are leading candidates for quantum memories. For the widely used
alkali-metal atoms where qubits are encoded in hyperfine states [5], however,
heating effects inevitably require recooling of the atoms. Standard laser
cooling methods will destroy the quantum information stored [13, 14] limiting
the total number of quantum gates that can be executed within the register
lifetime. To prolong the memory lifetime effectively, coherence-preserving
cooling of alkali-metal atoms was proposed by resorting to superfluid
immersion [15], cavity QED [16], or coupling qubits to auxiliary atoms [17].

When the qubits are defined by the nuclear spin states of alkaline-earth-like
(AEL) atoms, including alkaline-earth metals, some lanthanides [18] and some
transition metals, resolved-sideband cooling may preserve the nuclear spin
coherence in the presence of sufficiently strong magnetic fields [19]. With
$I$ and $m_{I}$ the nuclear spin and its projection along the quantization
axis, numerical analyses in Ref. [19] showed that for qubits defined with $\pm
m_{I}$ in the ground state of 87Sr and 117Yb, where $0<m_{I}\leq I$,
spontaneous emission during cooling can preserve the qubit-state coherence
with a fidelity over 0.99 in a strong magnetic field. The B-field is about
$10$ mT to achieve a fidelity over 0.99 for 87Sr.

In this paper we propose resolved-sideband cooling of 87Sr atoms in a weak
magnetic field while preserving the coherence of nuclear spin qubits.
Following Ref. [18], we consider a cooling cycle in which the ground state is
driven to the vibrational sideband of the clock state, which is further driven
to the $(5s5p)^{1}P_{1}$ state, which decays rapidly back to ground. In our
theory, the hyperfine-interaction-induced mixing of different nuclear spin
states in $|[5s5p^{1}P_{1}]m_{J},m_{I}\rangle$ is effectively suppressed by
coupling it to a nearby state which causes $m_{J}$-dependent AC Stark shifts.
In particular, the Stark shift is large compared to the hyperfine interaction,
so that the nuclear spin mixing due to the hyperfine interaction becomes
negligible. This mechanism is not dependent on the Zeeman shift, which leads
to two features. First, a weak magnetic field is applicable, which is
compatible with recent nuclear-spin-qubit experiments, where a B-field of $11$
G [20], $4.11$ G [21], or a value in the range $(0,~{}18]$ G [22] was used
with 87Sr [20] or 171Yb [21, 22]. Second, the theory is for qubits defined
with the two lowest nuclear spin Zeeman substates, which was commonly used in
experiments, such as in the experiment of Ref. [20]. Numerical simulations
with feasible parameters show that nuclear spin coherence can be preserved
with a fidelity over $0.999$. This theory brings opportunities for coherent
control of nuclear-spin quantum memories [23, 24, 25, 26, 27, 28].

Figure 1: Cooling scheme illustrated with atomic levels and vibrational states
in (a) and (b), respectively; the inset of (b) shows a simplified process
about removing one vibrational quantum number. Altogether five energy levels
are involved in the nuclear-spin-preserving sideband cooling of 87Sr atoms,
namely, the ground state, the clock state,
$5\text{s}5\text{p}~{}^{1}\text{P}_{1}$,
$5\text{s}6\text{s}~{}^{1}\text{S}_{0}$, and
$5\text{s}15\text{d}~{}^{1}\text{D}_{2}$. The directions of the arrows do not
indicate polarization of photons. Cooling starts from a narrow-line laser
excitation of the clock transition from the ground state to
$5\text{s}5\text{p}~{}^{3}\text{P}_{0}$ when the vibrational quantum number
reduces by one. A two-photon transition via an intermediate state (not shown
here) transfers the state $5\text{s}5\text{p}~{}^{3}\text{P}_{0}$ to
$5\text{s}5\text{p}~{}^{1}\text{P}_{1}$ which decays back to the ground state
rapidly. The hyperfine interaction in $5\text{s}5\text{p}~{}^{1}\text{P}_{1}$
mixes nuclear spins by nature; by inducing a transition between
$5\text{s}5\text{p}~{}^{1}\text{P}_{1}$ and
$5\text{s}6\text{s}~{}^{1}\text{S}_{0}$ with a strong Rabi frequency which is
large compared to the hyperfine interaction, the nuclear spin mixing is
suppressed, so that polarization resolution is removed in the spontaneous
emission to the ground state. A small diagonal-hyperfine-interaction induced
energy difference between the two nuclear spin states is compensated by the AC
Stark shift via off-resonantly exciting
$5\text{s}5\text{p}~{}^{1}\text{P}_{1}$ to
$5\text{s}15\text{d}~{}^{1}\text{D}_{2}$, which removes the frequency
resolution in the spontaneous emission.

The remainder of this paper is organized as follows. In Sec. II, we discuss
sideband cooling when ignoring the hyperfine interaction as a warm-up. In Sec.
III, we present the theory of using AC Stark shifts to suppress the hyperfine
interaction. Section IV shows the detail with a concrete model and presents
numerical results of the cooling. Section V discusses the influence from
fluctuation of laser frequency, intensity, and polarization on the cooling.
Section VI gives a discussion especially on the possibility to apply the
cooling scheme with other elements, and a brief conclusion is given in Sec.
VII.

##  II When there is no hyperfine interaction

With 87Sr as an example, the essence of nuclear-spin-preserving resolved
sideband cooling in a weak magnetic field is understood by first ignoring the
hyperfine interactions in $(5s5p)^{1}P_{1}$. The full treatment of hyperfine
interaction will be shown in Sec. III.

The cooling consists of three steps.

First, a narrow-line $698$ nm laser field coherently excites the ground state
to the clock state when the vibrational quantum number reduces by one. See
Fig. 1. The electron-nuclear spin state and the vibration state of a ground-
state 87Sr atom is denoted by

$\displaystyle|[5\text{s}^{2}~{}^{1}\text{S}_{0}]m_{I}\rangle\otimes|n+1\rangle,$

where $m_{I}$ is the nuclear spin projection along the quantization axis
(specified by an external magnetic field $B\mathbf{z}$), and $|n+1\rangle$
denotes the vibrational state of the atom in the trap with $n+1$ vibrational
quantum number. We suppose that the clock state and the ground state are
simultaneously trapped in a trap of magic wavelength [29, 20], so that the
vibration states of the atom in the ground and clock states can be denoted by
the same set of vibrational states $|n\rangle$. The linewidth for the atomic
electric dipole transition from $5\text{s}^{2}~{}^{1}\text{S}_{0}$ to
$5\text{s}5\text{p}~{}^{3}\text{P}_{0}$ is about $2\pi\times 0.001$ Hz [30],
while the (radial) frequency of the trap can be significantly larger than the
transition linewidth; for example, it was $2\pi\times 95$ kHz and $2\pi\times
260$ kHz in the experiment of Ref. [20] and Ref. [29], respectively. As a
result, the sideband of the vibrational states can be well resolved in the
laser excitation of the clock transition

$\displaystyle|[5\text{s}^{2}~{}^{1}\text{S}_{0}]m_{I}\rangle\otimes|n+1\rangle\rightarrow|[5\text{s}5\text{p}~{}^{3}\text{P}_{0}]m_{F}=m_{I}\rangle\otimes|n\rangle$
(1)

via a $\pi$ polarized laser field. The atomic state
$|[5\text{s}5\text{p}~{}^{3}\text{P}_{0}]m_{F}\rangle$ can be written as
$|[5\text{s}5\text{p}~{}^{3}\text{P}_{0}]m_{I}\rangle$ because the hyperfine
and spin-orbit coupling result in a state
$|[5\text{s}5\text{p}~{}^{3}\text{P}_{0}]m_{F}=m_{I}\rangle\approx|[5\text{s}5\text{p}~{}^{3}\text{P}_{0}]m_{I}\rangle+\eta|\text{hyper-
so}\rangle$, where the value of $m_{F}$ is equal to that of $m_{I}$,
$|\eta|^{2}\approx 4\times 10^{-8}$, and $|\text{hyper-so}\rangle$ is a
superposition of $|[5\text{s}5\text{p}~{}^{3}\text{P}_{1}]m_{F}\rangle$,
$|[5\text{s}5\text{p}~{}^{3}\text{P}_{2}]m_{F}\rangle$, and
$|[5\text{s}5\text{p}~{}^{1}\text{P}_{1}]m_{F}\rangle$ [30]. The tiny $|\eta|$
indicates that the hyperfine-interaction-induced nuclear spin decoherence in
the transition between the ground state and the clock state can be ignored.

Second, a two-photon Raman transition between
$|[5\text{s}5\text{p}~{}^{3}\text{P}_{0}]m_{F}=m_{I}\rangle\otimes|n\rangle$
and $|[5\text{s}5\text{p}~{}^{1}\text{P}_{1}]m_{F}-1\rangle\otimes|n\rangle$
via an intermediate state is realized with the total change of angular
momentum projection equal to -1. The intermediate state should have both
singlet and triplet components for which we have two choices. One choice is a
high-lying $5\text{s}n\text{s}$ Rydberg state in which the hyperfine
interaction can induce mixing between the ${}^{3}S_{1}$ and ${}^{1}S_{0}$
states [31, 32]. The other choice is a low-lying $5\text{s}n\text{d}$ Rydberg
state in which the spin-orbit coupling can induce strong mixing between the
${}^{1}D_{2}$ and ${}^{3}D_{2}$ states. For the first choice, the hyperfine
interaction in the intermediate state can lead to different Rabi frequencies
for the two nuclear-spin states, and it demands efforts to tune the laser
frequencies and detunings for addressing the hyperfine-split intermediate
state or use multiple laser fields to achieve equal excitation Rabi
frequencies for the two nuclear-spin states. For the second choice, sizable
singlet-triplet mixing can occur for $5\text{s}n\text{d}$ states of principal
quantum number from $n=10$ to $25$ [33]. For example, the components of
${}^{1}D_{2}$ and ${}^{3}D_{2}$ in the ${}^{3}D_{2}$-dominated wavefunction
for $n=11$ have a ratio of about $5.5$ (see Fig. 3 of Ref. [33]). The
transition from pure ${}^{3}P_{0}$ to ${}^{1}D_{2}$ or ${}^{3}D_{2}$ states is
difficult, but the wavefunction of the clock state has an overlap coefficient
$-2\times 10^{-4}$ with the pure ${}^{3}P_{1}$ state [30], which makes it
possible to couple the clock state to the intermediate $5\text{s}n\text{d}$
state. The hyperfine splittings in the $5\text{s}n\text{d}$ state with $n=11$
are small (see Figs. 6 and 7 of Ref. [33]), so that when we use a Raman
transition with the detuning at the intermediate $5\text{s}n\text{d}$ state of
the two-photon transition large compared to the hyperfine interaction, the
hyperfine structure of the intermediate state is indiscernible. This means
that by the second choice with $n\lesssim 11$, the Raman transitions for the
two nuclear-spin qubit states can have the same Rabi frequencies as required
by our theory. The state $5\text{s}5\text{p}~{}^{1}\text{P}_{1}$ has a
linewidth $2\pi\times 32$ MHz [34, 35], and is not trapped by the optical
trap. When there is no hyperfine interaction for this state (the full theory
with hyperfine interaction is in Sec. III), a two-photon $\sigma^{-}$
transition via an intermediate state can lead to

$\displaystyle|[5\text{s}5\text{p}~{}^{3}\text{P}_{0}]m_{F}=m_{I}\rangle\otimes|n\rangle$
$\displaystyle\rightarrow|[5\text{s}5\text{p}~{}^{1}\text{P}_{1}]m_{J}=-1,m_{I}\rangle\otimes|n\rangle,$
(2)

where we preserve the vibrational state when the atom is in
$5\text{s}5\text{p}~{}^{1}\text{P}_{1}$ following Ref. [19]. To understand
this, we note that the time for the atom to stay in
$5\text{s}5\text{p}~{}^{1}\text{P}_{1}$ is about $5$ ns, while the vibration
period is over 10 $\mu$s for a radial trap frequency $2\pi\times 95$ kHz [20].
Note that the effective motional temperature of the atom was below $5~{}\mu$K
in recent experiments with ytterbium [22] or strontium [36, 37, 38], and we
can assume that the atomic temperature is on the order of $10~{}\mu$K at the
beginning of the sideband cooling. At this temperature, the r.m.s. speed of
the atom $\sqrt{k_{\text{B}}T/m}$ is on the order of 0.02 nm$/$ns, which means
that the atom moves by $\lesssim 0.1$nm during the $5$-ns transient time
staying at $5\text{s}5\text{p}~{}^{1}\text{P}_{1}$. To good approximation, the
vibrational state of the atom remains during the transient at
$5\text{s}5\text{p}~{}^{1}\text{P}_{1}$.

Third, the fast spontaneous decay rate of the state
$5\text{s}5\text{p}~{}^{1}\text{P}_{1}$ causes an incoherent transition

$\displaystyle|[5\text{s}5\text{p}~{}^{1}\text{P}_{1}]m_{J}=-1,m_{I}\rangle\otimes|n\rangle\rightsquigarrow
5\text{s}^{2}~{}^{1}\text{S}_{0}|m_{I}\rangle\otimes|n\rangle,$ (3)

which returns the state back to the ground state. The transitions in Eqs. (1),
(2), and (3) involve a state with a common $m_{I}$, and similar transitions
can happen with a superposition state of different $m_{I}$-eigenstates. So,
the vibrational quantum number is lowered by one following Eqs. (1), (2), and
(3). As along as the atom is in a state with the vibrational quantum number
$n$ larger than zero, the three-step cooling can proceed following Eqs. (1),
(2), and (3) until the atom reaches the ground.

##  III nuclear-spin-preserving cooling

###  III.1 Hyperfine interaction mixes the nuclear spins

The hyperfine interaction in $5\text{s}5\text{p}~{}^{1}\text{P}_{1}$ is not
considered above. In practice, hyperfine interaction causes Eq. (2) to become

$\displaystyle|[5\text{s}5\text{p}~{}^{3}\text{P}_{0}]m_{F}=m_{I}\rangle\otimes|n\rangle\rightarrow|[5\text{s}5\text{p}~{}^{1}\text{P}_{1}]m_{F}-1\rangle\otimes|n\rangle,$
(4)

where $|[5\text{s}5\text{p}~{}^{1}\text{P}_{1}]m_{F}-1\rangle\otimes|n\rangle$
is a hyperfine eigenstate that mixes nuclear spin states with $m_{I},m_{I}\pm
1,m_{I}\pm 2$, where the mixing coefficients are determined by the detail of
the hyperfine interaction. To understand this coupling, we note that in the
presence of a magnetic field $B\mathbf{z}$, the Hamiltonian including the
hyperfine interaction between the valence electrons and the nuclear spin is
described by

$\displaystyle\hat{H}_{\text{hf}}$ $\displaystyle=$ $\displaystyle
A\hat{\mathbf{I}}\cdot\hat{\mathbf{J}}+Q\frac{3(\hat{\mathbf{I}}\cdot\hat{\mathbf{J}})^{2}+1.5\hat{\mathbf{I}}\cdot\hat{\mathbf{J}}-IJ(I+1)(J+1)}{2IJ(2I-1)(2J-1)}$
(5) $\displaystyle+g_{J}\mu_{\text{B}}\hat{\mathbf{J}}\cdot
B\mathbf{z}-g_{I}\mu_{\text{n}}\hat{\mathbf{I}}\cdot B\mathbf{z}.$

Here, $A$ and $Q$ are the nuclear magnetic dipole and electric quadrupole
interaction constants, respectively, $\hat{\mathbf{I}}$ and $\hat{\mathbf{J}}$
are the nuclear spin and electron orbital angular momentum operators (divided
by the reduced Planck constant), respectively, $g_{J}$ and $g_{I}$ are the
electron and nuclear $g$-factors, respectively [39], and $\mu_{\text{B}}$ and
$\mu_{\text{n}}$ are the Bohr magneton and the nuclear magnetic moment,
respectively. According to the measurement in Ref. [40], the hyperfine
constants are $(A,Q)/2\pi=(-3.4,~{}39)$ MHz for
$5\text{s}5\text{p}~{}^{1}\text{P}_{1}$, and the measured value for
$\mu_{\text{n}}$ reported in Ref. [41] is $-1.0924\mu_{\text{N}}$, where
$\mu_{\text{N}}$ is the nuclear magneton.

The hyperfine interaction in Eq. (5) couples states with equal $m_{J}+m_{I}$,
as shown in Appendix A so that there can be decoherence in the nuclear spin
state during the cooling if we do not introduce extra schemes. To uncouple the
electron state and the nuclear spin state, strong magnetic fields about 10 mT
can be used so that the nuclear-spin coherence can be preserved with a 99$\%$
fidelity when qubits are defined with nuclear spin projections $\pm m_{I}$ as
studied in Ref. [19].

In our theory, nuclear spin qubits are defined with $m_{I}=-I$ and $1-I$, so
that the state
$|[5\text{s}5\text{p}~{}^{1}\text{P}_{1}]-1,-I\rangle\otimes|n\rangle$ can’t
be coupled to another state by hyperfine interaction, but
$|[5\text{s}5\text{p}~{}^{1}\text{P}_{1}]-1,1-I\rangle\otimes|n\rangle$ is
coupled with
$|[5\text{s}5\text{p}~{}^{1}\text{P}_{1}]0,-I\rangle\otimes|n\rangle$. To
remove the nuclear spin mixing by the hyperfine interaction, we propose to use
$\pi$-polarized laser excitation of an electric dipole transition between
$5\text{s}5\text{p}~{}^{1}\text{P}_{1}$ and a nearby
$5\text{s}n\text{s}~{}^{1}\text{S}_{0}$ state. An electric dipole transition
directly couples two states with the change of $m_{J}$ equal to that of the
angular momentum of the photon of the laser field. As a result,
$5\text{s}n\text{s}~{}^{1}\text{S}_{0}$ can be coupled with the state
$|[5\text{s}5\text{p}~{}^{1}\text{P}_{1}]0,-I\rangle$, but can be coupled with
neither $|[5\text{s}5\text{p}~{}^{1}\text{P}_{1}]-1,1-I\rangle$ nor
$|[5\text{s}5\text{p}~{}^{1}\text{P}_{1}]-1,-I\rangle$. When this coupling is
strong, a large AC Stark shift can arise in
$|[5\text{s}5\text{p}~{}^{1}\text{P}_{1}]0,-I\rangle$. When the AC Stark shift
is large compared to the hyperfine interaction, the hyperfine-interaction-
induced state mixing between
$|[5\text{s}5\text{p}~{}^{1}\text{P}_{1}]-1,1-I\rangle$ and
$|[5\text{s}5\text{p}~{}^{1}\text{P}_{1}]0,-I\rangle$ is suppressed. The
questions is, is there a $5\text{s}n\text{s}~{}^{1}\text{S}_{0}$ state
sufficient near to $5\text{s}5\text{p}~{}^{1}\text{P}_{1}$ so that a large
electric dipole matrix element exists?

###  III.2 The method

The suppression of the hyperfine interaction by AC Stark shifts of laser
excitation requires that a state near to
$5\text{s}5\text{p}~{}^{1}\text{P}_{1}$ should have a large electric dipole
transition matrix element, so that a large Rabi frequency can arise for the
transition. There are several candidates satisfying this condition, among
which $5s4d~{}^{1}D_{2}$ and $5s6s~{}^{1}S_{0}$ are nearest. The reduced
dipole matrix element between $5s5p~{}^{1}P_{1}$ and $5s4d~{}^{1}D_{2}$ is
$1.92ea_{0}$ [37], where $e$ is the elementary charge and $a_{0}$ is the Bohr
radius; however, the transition between $5s5p~{}^{1}P_{1}$ and
$5s4d~{}^{1}D_{2}$ requires a 6.5 $\mu$m [34] laser, which can be challenging
since lasers with such wavelength may not be immediately available [42, 43].
On the other hand, the transition from $5s5p~{}^{1}P_{1}$ to
$5s6s~{}^{1}S_{0}$ has a wavelength $1124$ nm [44] for which a laser is
readily available. The reduced dipole matrix element for this transition is
about $2.09ea_{0}$ (see Appendix B), and with a laser of field intensity about
$17$W$/$cm2, a Rabi frequency $\Omega/2\pi=300$ MHz can be achieved for the
transition between $|[5\text{s}5\text{p}~{}^{1}\text{P}_{1}]0,-I\rangle$ and
$|[5\text{s}6\text{s}~{}^{1}\text{S}_{-}]0,-I\rangle$ as shown in Appendix B.
By the AC Stark shift, the hyperfine coupling between
$|[5\text{s}5\text{p}~{}^{1}\text{P}_{1}]-1,1-I\rangle$ and
$|[5\text{s}5\text{p}~{}^{1}\text{P}_{1}]0,-I\rangle$ is suppressed, resulting
in the suppression of the polarization resolution in the spontaneous emission
from $5\text{s}5\text{p}~{}^{1}\text{P}_{1}$ to the ground state.

The spontaneous decay from the states
$|[5\text{s}5\text{p}~{}^{1}\text{P}_{1}]-1,1-I\rangle$ and
$|[5\text{s}5\text{p}~{}^{1}\text{P}_{1}]-1,-I\rangle$ back to the ground
state should not be frequency resolved so as to preserve the coherence of the
nuclear spin qubit. In a B-field of Gauss scale, the Zeeman shift between the
two nuclear spin states is negligible. However, there is still a MHz-scale
energy difference between
$|[5\text{s}5\text{p}~{}^{1}\text{P}_{1}]-1,1-I\rangle$ and
$|[5\text{s}5\text{p}~{}^{1}\text{P}_{1}]-1,-I\rangle$ due to the diagonal
hyperfine interaction (see Appendix A). To remove this energy difference, a
highly detuned laser field of wavelength 424 nm for the transition between
$5\text{s}5\text{p}~{}^{1}\text{P}_{1}$ and
$5\text{s}15\text{d}~{}^{1}\text{D}_{2}$ can be employed. We choose the state
$5\text{s}15\text{d}~{}^{1}\text{D}_{2}$ for it has a strong hyperfine
interaction [33], so that when the laser is tuned near to one of its $F$
states, $|[5\text{s}5\text{p}~{}^{1}\text{P}_{1}]-1,1-I\rangle$ and
$|[5\text{s}5\text{p}~{}^{1}\text{P}_{1}]-1,-I\rangle$ can obtain different AC
Stark shifts due to the different coupling strengths determined by the angular
momentum selection rule. As shown in Appendix C with data from Ref. [33], the
two states with $F=I+2$ and $F=I+1$ of
$5\text{s}15\text{d}~{}^{1}\text{D}_{2}$ are separated by about $2\pi\times
1.3$ GHz, so that when a left-hand polarized laser field is tuned near to,
e.g., the $F=I+1$ state, the AC Stark shift for the state
$|[5\text{s}5\text{p}~{}^{1}\text{P}_{1}]-1,-I\rangle$ is negligible for it
can only couple with the $F=I+2$ state, while AC Stark shifts can readily
appear for other relevant $5\text{s}5\text{p}~{}^{1}\text{P}_{1}$ states. As a
result, the frequency resolution in the spontaneous emission from
$5\text{s}5\text{p}~{}^{1}\text{P}_{1}$ to the ground state can be avoided.

Compared to the theory of Ref. [19] which needs a magnetic field over 10 mT,
the method of cooling here is with a B-field on the order of 1 G. The sideband
cooling with a low B-field is compatible with setups used in recent
experiments with nuclear-spin qubits, where a B-field equal to $11$ G [20],
$4.11$ G [21], or a value in the range $(0,~{}18]$ G [22] was used with 87Sr
[20] or 171Yb [21, 22].

##  IV Master equation analysis of cooling dynamics

We consider a nuclear spin qubit defined by the two maximal spin projections
along the quantization axis, namely, the two lowest in energy [20],
$\lvert\uparrow\rangle=|m_{I}=1-I\rangle$, and
$\lvert\downarrow\rangle=|m_{I}=-I\rangle$; this type of qubit can be
initialized with a bias magnetic field, while qubits defined with other
nuclear spin states require extra fields [25]. To simplify the notation, we
label the intrinsic state of, e.g., the ground-state atom, by
$|[5s^{2}~{}^{1}S_{0}]0,\uparrow(\downarrow)\rangle$, where $0$ denotes the
value of $m_{J}$ for the state and the arrow denotes the nuclear spin. With
$\pi$ polarized laser fields employed for the atomic transitions, a general
qubit state in the ground state

$\displaystyle(\alpha|[5s^{2}~{}^{1}S_{0}]0,\uparrow\rangle+\beta|[5s^{2}~{}^{1}S_{0}]0,\downarrow\rangle)\otimes|n+1\rangle,$

is excited to

$\displaystyle(\alpha|[5s5p~{}^{3}P_{0}]0,\uparrow\rangle+\beta|[5s5p~{}^{3}P_{0}]0,\downarrow\rangle)\otimes|n\rangle,$
(6)

where $|\alpha|^{2}+|\beta|^{2}=1$ and one vibrational quantum is removed in
the above transition. In principle, there can be a change of the relative
phase between the two spin components in the above transition, which can be
amended by first exciting the state from the ground to the clock states,
applying single-qubit phase gate to nuclear-spin qubits in the clock state,
and then exciting the clock state to the ${}^{1}P_{1}$ state; an alternative
is to design a compensating relative phase in the Rabi frequencies for the two
nuclear spin states in the Raman transition between the clock and the
${}^{1}P_{1}$ states. Via an intermediate state with a two-photon $\sigma^{-}$
polarized transition, the state in Eq. (6) is excited to

$\displaystyle\Big{[}\alpha(\alpha_{0}|[5s5p~{}^{1}P_{1}]0,\downarrow\rangle+\alpha_{-}|[5s5p~{}^{1}P_{1}]-1,\uparrow\rangle)$
$\displaystyle+\beta|[5s5p~{}^{1}P_{1}]-1,\downarrow\rangle\Big{]}\otimes|n\rangle,$
(7)

where the coefficients $\alpha_{0},\alpha_{-}$ are determined by the hyperfine
interaction that couples states with equal $m_{J}+m_{I}$ (see Appendix A). We
note that Eq. (7) is shown for illustration; in practice, Eq. (7) is split
into several different states with different energies in the presence of
hyperfine interaction.

To shift away the transitions with the nuclear spin flip in Eq. (7), a
$\pi$-polarized laser field is used to excite $5s5p~{}^{1}P_{1}$ to
$5s6s~{}^{1}S_{0}$, where the angular momentum selection rule allows the
transition between $|[5s5p~{}^{1}P_{1}]0,\downarrow\rangle$ and
$|[5s6s~{}^{1}S_{0}]0,\downarrow\rangle$, while the other two state components
in Eq. (7) are not excited. As a result, the state component
$|[5s5p~{}^{1}P_{1}]0,\downarrow\rangle$ can obtain an AC Stark shift large
compared to the hyperfine interaction, leading to suppression of the hyperfine
interaction, i.e., $\alpha_{0}\rightarrow 0$ in Eq. (7).

Once the nuclear-spin flip state $|[5s5p~{}^{1}P_{1}]0,\downarrow\rangle$ is
suppressed in Eq. (7), polarization resolution in the spontaneous emission is
suppressed. However, there is a MHz-scale energy difference between
$|[5s5p~{}^{1}P_{1}]-1,\downarrow\rangle$ and
$|[5s5p~{}^{1}P_{1}]-1,\uparrow\rangle$ mainly from the diagonal hyperfine
interaction, leading to frequency resolution of the spontaneous emission. To
remove this energy difference, one can excite $5s5p~{}^{1}P_{1}$ to a certain
hyperfine substate of a ${}^{1}D_{2}$ state. A useful ${}^{1}D_{2}$ state for
this purpose shall possess a large hyperfine interaction, ensuring different
$F$ states being well separated. Then, tuning the frequency of a
$\sigma^{-}$-polarized laser near to, e.g., the $F=I+1$ substate of
${}^{1}D_{2}$, the state $|[5s5p~{}^{1}P_{1}]-1,\downarrow\rangle$ barely
acquires any AC Stark shift for it can’t be excited to the $F=I+1$ state with
the $\sigma^{-}$ laser field, while $|[5s5p~{}^{1}P_{1}]-1,\uparrow\rangle$
can acquire an AC Stark shift to compensate the energy difference between it
and $|[5s5p~{}^{1}P_{1}]-1,\downarrow\rangle$. As discussed in Sec. III, we
choose 5s15d 1D2 for it has a large hyperfine interaction.

In the cooling process, the excitation from the ground to the clock state can
be achieved with high accuracy because of the long lifetime of the clock
state, so that we analyze the cooling fidelity by starting from a state like
Eq. (6). In particular, we would like to see how precise we can map the state
in Eq. (6) to

$\displaystyle(\alpha|[5s^{2}~{}^{1}S_{0}]0,\uparrow\rangle+\beta|][5s^{2}~{}^{1}S_{0}]0,\downarrow\rangle)\otimes|n\rangle,$

during which the vibrational quantum number does not change, so that we can
omit it when writing the state vectors in the analysis. The time dynamics is
described by

$\displaystyle\frac{d\hat{\rho}}{dt}$ $\displaystyle=$ $\displaystyle
i(\hat{\rho}\hat{H}-\hat{H}\hat{\rho})+\sum_{i=0}^{8}\left[2\hat{c}_{i}\hat{\rho}\hat{c}_{i}^{\dagger}-\hat{c}_{i}^{\dagger}\hat{c}_{i}\hat{\rho}-\hat{\rho}\hat{c}_{i}^{\dagger}\hat{c}_{i}\right]/2.$

Here, $\hat{\rho}$ is the density matrix of the atomic state, and the
Hamiltonian is

$\displaystyle\hat{H}_{\text{hf}}+\bigg{\\{}\frac{\Omega_{\text{eff}}}{2}\sum_{s_{\text{z}}\in\\{\uparrow,\downarrow\\}}|[5s5p~{}^{1}P_{1}]-1,s_{\text{z}}\rangle\langle[5s5p~{}^{3}P_{0}]0,s_{\text{z}}|$
$\displaystyle+\frac{\Omega_{\text{ps}}}{2}|[5s6s~{}^{1}S_{0}]0,\downarrow\rangle\langle[5s5p~{}^{1}P_{1}]0,\downarrow|+\frac{\Omega_{\text{pd}}}{2}$
$\displaystyle\cdot\bigg{(}|[5s15d~{}^{1}D_{2}]F=\frac{13}{2},m_{F}=-\frac{13}{2}\rangle\langle[5s5p~{}^{1}P_{1}]-1,\downarrow|$
$\displaystyle+\xi_{0}|[5s15d~{}^{1}D_{2}]F=\frac{13}{2},m_{F}=-\frac{11}{2}\rangle\langle[5s5p~{}^{1}P_{1}]0,\downarrow|$
$\displaystyle+\xi_{1}|[5s15d~{}^{1}D_{2}]F=\frac{13}{2},m_{F}=-\frac{11}{2}\rangle\langle[5s5p~{}^{1}P_{1}]-1,\uparrow|$
$\displaystyle+\xi_{2}|[5s15d~{}^{1}D_{2}]F=\frac{11}{2},m_{F}=-\frac{11}{2}\rangle\langle[5s5p~{}^{1}P_{1}]0,\downarrow|$
$\displaystyle+\xi_{3}|[5s15d~{}^{1}D_{2}]F=\frac{11}{2},m_{F}=-\frac{11}{2}\rangle$
$\displaystyle~{}~{}~{}\langle[5s5p~{}^{1}P_{1}]-1,\uparrow|\bigg{)}+\text{H.c.}\bigg{\\}}$
$\displaystyle+(\Delta+\Delta_{\text{pd}})\sum_{k=0}^{2}|S_{k}\rangle\langle
S_{k}|+\Delta\sum_{k=3}^{6}|S_{k}\rangle\langle S_{k}|,$ (9)

where $\hat{H}_{\text{hf}}$ is in a rotating frame derived from Eq. (5),
$\Omega_{\text{eff}}$ is the effective two-photon Rabi frequency between
$5s5p~{}^{1}P_{1}$ and $5s5p~{}^{3}P_{0}$, $\Omega_{\text{ps}}$ is the $1124$
nm infrared-laser Rabi frequency between $5s5p~{}^{1}P_{1}$ and
$5s6s~{}^{1}S_{0}$, $\Omega_{\text{pd}}$ is the $424$ nm UV-laser Rabi
frequency between $5s5p~{}^{1}P_{1}$ and $5s15d~{}^{1}D_{2}$, the factors
$\\{\xi_{j},j=0-3\\}$ are angular momentum factors shown in Appendix C,
$|S_{k}\rangle$ with $k=0-2$ are the three $5s15d$ states in the bracket
$(\cdots)$ of Eq. (9), $|S_{k}\rangle$ with $k=3-6$ are the states including
$|[5s6s~{}^{1}S_{0}]0,\downarrow\rangle$ and the three $5s5p$ states in the
bracket $(\cdots)$ of Eq. (9), $\Delta_{\text{pd}}$ is the detuning for the
dipole transition of the UV laser, and $\Delta$ is a detuning set to tune
resonance for the transition between
$|[5s5p~{}^{1}P_{1}]-1,s_{\text{z}}\rangle$ and
$|[5s5p~{}^{3}P_{0}]0,s_{\text{z}}\rangle$; this latter detuning is added for
when we use the UV laser field to induce AC Stark shifts in
$|[5s5p~{}^{1}P_{1}]-1,s_{\text{z}}\rangle$, the states
$|[5s5p~{}^{1}P_{1}]-1,\uparrow\rangle$ and
$|[5s5p~{}^{1}P_{1}]-1,\downarrow\rangle$ can finally acquire a common,
nonzero energy $-\Delta$ in the rotating frame. To simplify the numerical
simulation, the hyperfine interaction in $5s15d~{}^{1}D_{2}$ is included with
hyperfine eigenstates [when laser detuning is accounted for, see Eq. (37)]. As
analyzed in Appendix C, one can estimate by the measured hyperfine constants
in Ref. [33] that the $F=\frac{13}{2}$ state of $5s15d~{}^{1}D_{2}$ is lower
by about $2\pi\times 1.3$ GHz than the $F=\frac{11}{2}$ state.

In Eq. (9), we ignore the coupling between
$|[5s5p~{}^{1}P_{1}]1,\downarrow\rangle$ and $5s15d~{}^{1}D_{2}$ because
$|[5s5p~{}^{1}P_{1}]1,\downarrow\rangle$ is populated neither directly nor
indirectly (via hyperfine interaction) from $5s5p~{}^{3}P_{0}$, though it can
be populated via spontaneous emission from $5s6s~{}^{1}S_{0}$. However,
$5s6s~{}^{1}S_{0}$ only has a negligible population when it is coupled to
$|[5s5p~{}^{1}P_{1}]0,\downarrow\rangle$. In the present cooling scheme,
$|[5s5p~{}^{1}P_{1}]0,\downarrow\rangle$ is barely populated, least to say how
negligible the population in $|[5s5p~{}^{1}P_{1}]1,\downarrow\rangle$ is via
the higher-order process. This is why we can ignore the energy shift of
$5s15d~{}^{1}D_{2}$ induced by the coupling between
$|[5s5p~{}^{1}P_{1}]1,\downarrow\rangle$ and $5s15d~{}^{1}D_{2}$.

In Eq. (5), the collapse operators for $5s5p~{}^{1}P_{1}$ are

$\displaystyle\hat{c}_{0}$ $\displaystyle=$
$\displaystyle\sqrt{\Gamma_{\text{p}}/3}\Big{\\{}|[5s^{2}~{}^{1}S_{0}]0,\uparrow\rangle\langle[5s5p~{}^{1}P_{1}]-1,\uparrow|$
$\displaystyle+|[5s^{2}~{}^{1}S_{0}]0,\downarrow\rangle\langle[5s5p~{}^{1}P_{1}]-1,\downarrow|\Big{\\}},$
$\displaystyle\hat{c}_{1}$ $\displaystyle=$
$\displaystyle-\sqrt{\Gamma_{\text{p}}/3}|[5s^{2}~{}^{1}S_{0}]0,\downarrow\rangle\langle[5s5p~{}^{1}P_{1}]0,\downarrow|,$
$\displaystyle\hat{c}_{2}$ $\displaystyle=$
$\displaystyle\sqrt{\Gamma_{\text{p}}/3}|[5s^{2}~{}^{1}S_{0}]0,\downarrow\rangle\langle[5s5p~{}^{1}P_{1}]1,\downarrow|,$
(10)

those for $5s6s~{}^{1}S_{0}$ are

$\displaystyle\hat{c}_{3}$ $\displaystyle=$
$\displaystyle\sqrt{\Gamma_{\text{s}}}|[5s5p~{}^{1}P_{1}]0,\downarrow\rangle\langle[5s6s~{}^{1}S_{0}]0,\downarrow|,$
$\displaystyle\hat{c}_{4}$ $\displaystyle=$
$\displaystyle\sqrt{\Gamma_{\text{s}}}|[5s5p~{}^{1}P_{1}]1,\downarrow\rangle\langle[5s6s~{}^{1}S_{0}]0,\downarrow|,$
$\displaystyle\hat{c}_{5}$ $\displaystyle=$
$\displaystyle\sqrt{\Gamma_{\text{s}}}|[5s5p~{}^{1}P_{1}]-1,\downarrow\rangle\langle[5s6s~{}^{1}S_{0}]0,\downarrow|,$
(11)

and those for the $5s15d~{}^{1}D_{2}$ states are

$\displaystyle\hat{c}_{k}$ $\displaystyle=$
$\displaystyle\sqrt{\Gamma_{\text{d}}}|\mathscr{A}\rangle\langle S_{k}|,$ (12)

where $k=6-8$ and $|S_{k}\rangle$ runs through the three
$|[5s15d~{}^{1}D_{2}]F,m_{F}\rangle$ states in Eq. (9) and
$|\mathscr{A}\rangle$ is a virtual reservoir state that does not respond to
the laser excitation. Here, $\Gamma_{\text{p}}/2\pi=32$ MHz [34, 35],
$\Gamma_{\text{s}}/2\pi=3.0$ MHz [45], and $\Gamma_{\text{d}}/2\pi=0.47$ MHz
[46]. Note that in principle the decay rates in $\hat{c}_{k}$ with $k\in[6,8]$
should be smaller than the linewidth of the state $5s15d~{}^{1}D_{2}$ due to
angular momentum selection rules, but a larger decay rate is employed so as to
give a lower bound for the cooling fidelity.

Figure 2: State dynamics in the cooling simulated by the master equation in
Eq. (LABEL:Lindblad1) with
$(\Omega_{\text{eff}},\Omega_{\text{pd}},\Omega_{\text{ps}},\Delta_{\text{pd}},\Delta,\Gamma_{\text{p}},\Gamma_{\text{s}},\Gamma_{\text{d}})/2\pi=(1,144.27,300,-1700,3.8826,32,3,0.47)$
MHz, $B=1$ G, and initial state $|\psi_{0}\rangle$. The hyperfine constants
$(A,Q)/2\pi$ are $(-3.4,~{}39)$ MHz for
$5\text{s}5\text{p}~{}^{1}\text{P}_{1}$ and $(-194,-75)~{}$MHz for
$5\text{s}15\text{d}~{}^{1}\text{D}_{2}$. (a) The solid, long-dashed, dash-
dotted, short-dashed, and dotted curves show the populations of log10 scale in
the states
$\lvert\perp\rangle=(|[5s^{2}~{}^{1}S_{0}]0,\uparrow\rangle-|[5s^{2}~{}^{1}S_{0}]0,\downarrow\rangle)\otimes|n\rangle/\sqrt{2}$,
$\lvert\mathscr{A}\rangle$, $5s5P~{}^{1}P_{1}$, $5s15d~{}^{1}D_{2}$, and
$5s6s~{}^{1}S_{0}$, respectively. $\lvert\perp\rangle$ is the state
perpendicular to the final target state
$|\psi_{\text{f}}\rangle=(|[5s^{2}~{}^{1}S_{0}]0,\uparrow\rangle+|[5s^{2}~{}^{1}S_{0}]0,\downarrow\rangle)\otimes|n\rangle/\sqrt{2}$.
The final populations in $\lvert\perp\rangle$ and $\lvert\mathscr{A}\rangle$
are $1.0\times 10^{-4}$ and $2.9\times 10^{-4}$, respectively. (b) Evolution
of the population of the states
$|\psi_{0}\rangle=(|[5s5p~{}^{3}P_{0}]0,\uparrow\rangle+|[5s5p~{}^{3}P_{0}]0,\downarrow\rangle)\otimes|n\rangle/\sqrt{2}$
and $|\psi_{\text{f}}\rangle$. The populations in $|\psi_{0}\rangle$ and
$|\psi_{\text{f}}\rangle$ at $20~{}\mu$s are $7\times 10^{-6}$ and $0.9996$,
respectively, and the main population loss is in the reservoir state and the
state perpendicular to $|\psi_{\text{f}}\rangle$ as shown in (a).

The laser parameters are chosen with the following considerations. (i) First,
the effective Rabi frequency $\Omega_{\text{eff}}$ for the transition between
the clock state and $5s5p~{}^{1}P_{1}$ is via a highly off-resonant
intermediate state, so that it is in general small and we use
$\Omega_{\text{eff}}/2\pi=1$ MHz in the numerical example. (ii) Second, the
$\pi$-polarized laser for the transition between $5s5p~{}^{1}P_{1}$ and
$5s6s~{}^{1}S_{0}$ is relevant for $|[5s5p~{}^{1}P_{1}]0,\downarrow\rangle$.
$|[5s5p~{}^{1}P_{1}]0,\downarrow\rangle$ can be populated via hyperfine
interaction, and our purpose is to suppress its population via the AC stark
effect. To induce a large shift, the laser is tuned resonant for the
transition between $|[5s5p~{}^{1}P_{1}]0,\downarrow\rangle$ and
$|[5s6s~{}^{1}S_{0}]0,\downarrow\rangle$, and the laser Rabi frequency
$\Omega_{\text{ps}}$ shall be much larger than the hyperfine interaction.
Therefore, we can set, as an example, $\Omega_{\text{ps}}/2\pi=300$ MHz, for
which $|[5s5p~{}^{1}P_{1}]0,\downarrow\rangle$ can exhibit a shift
$\Omega_{\text{ps}}/2$ that is much larger than the hyperfine interaction
strength. Larger $\Omega_{\text{ps}}$ can work for the theory but requires
stronger laser power. (iii) Third, the excitation of $5s15d~{}^{1}D_{2}$ is
for balancing the energies of $|[5s5p~{}^{1}P_{1}]-1,\uparrow\rangle$ and
$|[5s5p~{}^{1}P_{1}]-1,\downarrow\rangle$, which have frequencies
$2\pi\times(-10.05,-6.95)$ MHz mainly from the diagonal hyperfine interaction
$2\pi\times(-8.65,-5.55)$ MHz in a B-field of 1 G. Because the AC Stark shift
in a highly off-resonant dressing is $-\frac{1}{4}($Rabi${}^{2}/$detuning$)$
[47], one can tune the $\sigma^{-}$-polarized laser to the blue side of the
transition
$|[5s5p~{}^{1}P_{1}]-1,\downarrow\rangle\rightarrow|[5s15d~{}^{1}D_{2}]F=13/2,m_{F}=-13/2\rangle$
with a detuning $\Delta_{\text{pd}}<0$. Because the $F=11/2$ state is higher
than the $F=13/2$ state by about $E_{\text{hf}}=2\pi\times 1.3$ GHz, the
transition
$|[5s5p~{}^{1}P_{1}]-1,\uparrow\rangle\rightarrow|[5s15d~{}^{1}D_{2}]F=11/2,m_{F}=-11/2\rangle$
has a detuning $\Delta_{\text{pd}}+E_{\text{hf}}$. For
$\Delta_{\text{pd}}+E_{\text{hf}}<0$, the AC Stark shift of
$|[5s5p~{}^{1}P_{1}]-1,\uparrow\rangle$ is also positive. The detuning of
$|[5s5p~{}^{1}P_{1}]-1,\uparrow\rangle$ is smaller and it can obtain a larger
shift compared to $|[5s5p~{}^{1}P_{1}]-1,\downarrow\rangle$, so that the
energy difference between $|[5s5p~{}^{1}P_{1}]-1,\downarrow\rangle$ and
$|[5s5p~{}^{1}P_{1}]-1,\uparrow\rangle$ can be compensated. In order to have
negligible population in 5s15d, the dressing detuning shall be much larger
than the dressing Rabi frequency. As an example, we choose
$\Delta_{\text{pd}}/2\pi=-1.7$ GHz, i.e., about 400 MHz over the F=$11/2$
state. The value of $\Omega_{\text{pd}}$ can be searched for achieving the
same energies for $|[5s5p~{}^{1}P_{1}]-1,\downarrow\rangle$ and
$|[5s5p~{}^{1}P_{1}]-1,\uparrow\rangle$, which we numerically found
$\Omega_{\text{pd}}/2\pi=144.27$ MHz; note that different $\Delta_{\text{pd}}$
will lead to different $\Omega_{\text{pd}}$. With these parameters, the final,
common frequency $\nu$ for $|[5s5p~{}^{1}P_{1}]-1,\downarrow\rangle$ and
$|[5s5p~{}^{1}P_{1}]-1,\uparrow\rangle$ is in general nonzero. We have
$\nu=2\pi\times(-3.8826)$ MHz with the above parameters, which means that we
shall have a detuning $-\nu$ in the two-photon transition between the clock
and the $5s5p~{}^{1}P_{1}$ states so as to recover the resonant condition.

By using Eq. (LABEL:Lindblad1) with
$(\Omega_{\text{eff}},\Omega_{\text{pd}},\Omega_{\text{ps}},\Delta_{\text{pd}},\Delta)/2\pi=(1,144.27,300,-1700,3.8826)$
MHz and the decay rates shown around Eq. (12), we numerically simulated the
time evolution of the system by QuTip [48, 49]. We found that two eigenstates
$|e_{\uparrow}\rangle$ and $|e_{\downarrow}\rangle$ of the Hamiltonian driven
by the hyperfine interaction and laser excitation highly overlap with two pure
spin states, namely, we found

$\displaystyle\langle[5s5p~{}^{1}P_{1}]-1,\uparrow|e_{\uparrow}\rangle$
$\displaystyle=$ $\displaystyle 0.99409,$
$\displaystyle\langle[5s5p~{}^{1}P_{1}]-1,\downarrow|e_{\downarrow}\rangle$
$\displaystyle=$ $\displaystyle 0.99910.$ (13)

The states $|e_{\uparrow}\rangle$ and $|e_{\downarrow}\rangle$ have a common
eigenenergy $2\pi\times(-3.8826)$ MHz, which is why we set
$\Delta/2\pi=3.8826$ MHz. Though $|e_{\uparrow}\rangle$ and
$|e_{\downarrow}\rangle$ have populations in other state components, their
decay rates are much smaller than that of $5s5p~{}^{1}P_{1}$, so that when the
cooling starts, the spontaneous emission takes the atom back to the ground
state with a high fidelity.

$\alpha/\beta$ | $1/10$ | $1/3$ | $1/2$ | 2 | 3 | 10
---|---|---|---|---|---|---
Fidelity | 99.99$\%$ | 99.99$\%$ | 99.98$\%$ | 99.93$\%$ | 99.92$\%$ | 99.91$\%$
Table 1: Fidelity of the cooling,
$\langle\psi_{\text{f}}|\rho(t)|\psi_{\text{f}}\rangle$, at $t=20~{}\mu$s
starting from the initial state
$|\psi_{0}\rangle=(\alpha|[5s5p~{}^{3}P_{0}]0,\uparrow\rangle+\beta|[5s5p~{}^{3}P_{0}]0,\downarrow\rangle)\otimes|n\rangle$,
where
$|\psi_{\text{f}}\rangle=(\alpha|[5s^{2}~{}^{1}S_{0}]0,\uparrow\rangle+\beta|[5s^{2}~{}^{1}S_{0}]0,\downarrow\rangle)\otimes|n\rangle$.
The parameters used in the simulation are the same to those in Fig. 2.

As discussed previously, the excitation from the ground state to the clock
state can proceed with a high fidelity, so that we start from the initial
state in the clock-state space,
$|\psi_{0}\rangle=(|[5s5p~{}^{3}P_{0}]0,\uparrow\rangle+|[5s5p~{}^{3}P_{0}]0,\downarrow\rangle)\otimes|n\rangle/\sqrt{2}$,
from which a two-photon transition via an intermediate state can excite it to
the $5s5p~{}^{1}P_{1}$ state which rapidly decays back to the ground state.
The desired final state is
$|\psi_{\text{f}}\rangle=(|[5s^{2}~{}^{1}S_{0}]0,\uparrow\rangle+|[5s^{2}~{}^{1}S_{0}]0,\downarrow\rangle)\otimes|n\rangle/\sqrt{2}$.
As shown in the numerical result in Fig. 2, the fidelity to cool the initial
state
$(|[5s^{2}~{}^{1}S_{0}]0,\uparrow\rangle+|[5s^{2}~{}^{1}S_{0}]0,\downarrow\rangle)\otimes|n+1\rangle/\sqrt{2}$
to the desired final state $|\psi_{\text{f}}\rangle$ is about $99.96\%$. The
final population loss is mainly in the reservoir state
$\lvert\mathscr{A}\rangle$, and in the state
$\lvert\perp\rangle=(|[5s^{2}~{}^{1}S_{0}]0,\uparrow\rangle-|[5s^{2}~{}^{1}S_{0}]0,\downarrow\rangle)\otimes|n\rangle/\sqrt{2}$.
The cooling fidelity shows a weak dependence on the value of $|\alpha/\beta|$
in Eq. (6). However, Table 1 shows that the fidelity decreases slowly when
$|\alpha/\beta|$ increases. This decrease is due to that the state
$|e_{\uparrow}\rangle$ has a smaller overlap with the correct state component
shown in Eq. (13). However, we found that the fidelity is $99.909\%$ even with
$|\alpha/\beta|=100$. This means that the theory can easily have a coherence-
preserving cooling fidelity over $99.9\%$ in a weak magnetic field, and much
higher fidelity can be achieved with stronger laser fields for suppressing the
hyperfine interaction.

##  V influence from laser parameters

###  V.1 Fluctuation of Rabi frequency and detuning

The fluctuation of the power and frequency of the lasers has a minor influence
on the cooling. We don’t discuss the transition from the ground to the clock
state for it doesn’t involve change of angular momentum and it can be realized
with a high fidelity. Beside this laser, there are three sets of lasers, one
for the transition $5s5p~{}^{3}P_{0}\rightarrow 5s5p~{}^{1}P_{1}$, one for
$5s5p~{}^{1}P_{1}\rightarrow 5s6s~{}^{1}S_{0}$, and one for
$5s5p~{}^{1}P_{1}\rightarrow 5s15d~{}^{1}D_{2}$. Below, we discuss the
influence on the cooling from the fluctuation of Rabi frequency and detuning
of the lasers.

(i) The fluctuation of the Rabi frequency $\Omega_{\text{eff}}$ and detuning
$\Delta$ for $5s5p~{}^{3}P_{0}\rightarrow 5s5p~{}^{1}P_{1}$ can slow down the
cooling but barely impacts the fidelity. The cooling can proceed when the
population is transferred from $5s5p~{}^{3}P_{0}$ to $5s5p~{}^{1}P_{1}$. So,
larger $\Omega_{\text{eff}}$ and the resonant condition $\Delta+\nu=0$ can
facilitate the population transfer, while smaller $\Omega_{\text{eff}}$ or
off-resonant condition can lead to slower cooling. For example, with
$\Omega_{\text{eff}}/2\pi=2$ MHz while other parameters are the same as in
Fig. 2, simulation shows that the cooling can reach the final fidelity of Fig.
2 at a much earlier time $5~{}\mu$s. But with $\Delta=0$, i.e., off-resonant
with a detuning $\nu$, while other parameters are the same as in Fig. 2,
simulation shows that the cooling reaches a fidelity $0.9991$ at $20~{}\mu$s,
and the fidelity $0.9996$ is achieved at a later time $26~{}\mu$s. However,
the above discussion is based on that the ratio of the Rabi frequencies for
the two nuclear spin states does not change.

(ii) The cooling is not sensitive to small fluctuation of power and frequency
of the laser for the transition $5s5p~{}^{1}P_{1}\rightarrow
5s6s~{}^{1}S_{0}$. Because the $\pi$ polarization of the laser, only the state
$|[5s5p~{}^{1}P_{1}]0,\downarrow\rangle$ is excited while the other two states
$|[5s5p~{}^{1}P_{1}]-1,\uparrow\rangle$ and
$|[5s5p~{}^{1}P_{1}]-1,\downarrow\rangle$ are not. The numerical example of
Fig. 2 and Table 1 assumed $\Omega_{\text{ps}}/2\pi=300$ MHz, a little
deviation from this value alters the AC Stark shift, but as long as the shift
is large compared to the hyperfine interaction, the hyperfine-interaction-
induced spin mixing is suppressed. For example, with
$\Omega_{\text{ps}}/2\pi=250$ MHz, i.e., smaller by $1/6$, while other
parameters are the same as in Fig. 2, we numerically found that a cooling
fidelity $0.9996$ can still be achieved at $20~{}\mu$s. Likewise, by adding a
dtuning, e.g., $2\pi\times 10$ MHz, to this laser while preserving all the
parameters as in Fig. 2, a fidelity $0.9996$ is still achieved at $20~{}\mu$s.

(iii) The fluctuation of the laser parameters for the transition
$5s5p~{}^{1}P_{1}\rightarrow 5s15d~{}^{1}D_{2}$ can influence the cooling
fidelity. This is because when the Rabi frequency or detuning is not set so as
to have the same frequency for the two states in Eq. (13), decoherence will
occur due to the frequency resolution. For one example, with
$\Omega_{\text{pd}}/2\pi=140$ MHz while other parameters kept the same as used
in Fig. 2, the cooling fidelity is $0.99946$ at $20~{}\mu$s, and it can only
reach $0.99947$ even at $30~{}\mu$s due to a relatively large population
$2.6\times 10^{-4}$ in $\lvert\perp\rangle$; the final population in
$\lvert\perp\rangle$ is $1.0\times 10^{-4}$ for the case simulated in Fig. 2.
For another example, with $\Delta_{\text{pd}}/2\pi=-1.75$ GHz while other
parameters are the same as in Fig. 2, the cooling fidelity is $0.9988$ at or
beyond $20~{}\mu$s. This small cooling fidelity is due to that the two states
in Eq. (13) have a relatively large energy separation $2\pi\times 0.42$ MHz
which leads to a large final population $9.2\times 10^{-4}$ in
$\lvert\perp\rangle$.

###  V.2 Influence from laser polarization impurity

The cooling depends on high purity in laser polarization. We follow Ref. [27]
and use polarization intensity impurity $\chi$ for this discussion, where
$\chi=0$ denotes perfect polarization. We discuss the laser polarization
impurity of the three sets of laser fields discussed in Sec. V.1 about their
influence on the cooling.

(i) The transition $5s5p~{}^{3}P_{0}\rightarrow 5s5p~{}^{1}P_{1}$ is a two-
photon process via a highly detuned intermediate state. If each of the two
lasers in the two-photon process has a polarization intensity impurity $\chi$,
the effective Rabi frequency becomes $(1-\chi)\Omega_{\text{eff}}$ for the
desired transitions
$\lvert[5s5p~{}^{3}P_{0}]0,\uparrow(\downarrow)\rangle\rightarrow\lvert[5s5p~{}^{1}P_{1}]-1,\uparrow(\downarrow)\rangle$.
That the effective Rabi frequency decreases can slow down the cooling as
discussed in the last paragraph. But the wrong polarization can excite
$\lvert[5s5p~{}^{3}P_{0}]0,\uparrow(\downarrow)\rangle$ to
$\lvert[5s5p~{}^{1}P_{1}]m_{J},m_{I}\rangle$ where $m_{J}\neq-1$ or $m_{I}$ is
not equal to the correct value even when $m_{J}=-1$. There is some chance to
accidentally lead to the correct state transition. For example, if the
polarization should be $\sigma^{-}+\pi$ in the transition
$5s5p~{}^{3}P_{0}\rightarrow$intermediate$\rightarrow 5s5p~{}^{1}P_{1}$, a
wrong polarization $\pi+\sigma^{+}$ in the two corresponding lasers can result
in the correct state transfer. To estimate the worse case, we assume all
polarization errors result in wrong state transfer, which means that there is
a Rabi frequency $\chi\Omega_{\text{eff}}$ to create population loss to the
correct ground state. The data in Fig. 2 shows that
$\lvert\psi_{\text{f}}\rangle$ has a population over 0.5 beyond the time
$2.4\pi/\Omega_{\text{eff}}$, and we can estimate that the total chance to
have incorrect spontaneous emission due to wrong population in
$5s5p~{}^{1}P_{1}$ is around or below $\sin^{2}(2.4\pi\chi)$ when $\chi\ll 1$.
For $\chi=0.01$, it means that the cooling fidelity decreases by about $0.6\%$
due to the polarization impurity for this transition.

(ii) The laser for $5s5p~{}^{1}P_{1}\rightarrow 5s6s~{}^{1}S_{0}$ is assumed
$\pi$ polarized. A wrong polarization with a small $\chi$ barely alters the AC
Stark shift for shifting $\lvert[5s5p~{}^{1}P_{1}]0,\downarrow\rangle$, but
can excite $\lvert[5s5p~{}^{1}P_{1}]-1,\uparrow(\downarrow)\rangle$ to
$\lvert[5s6s~{}^{1}S_{0}]0,\uparrow(\downarrow)\rangle$. In the worst case
when the wrong polarization is fully $\sigma^{+}$, the Rabi frequency is
$-\sqrt{\chi}\Omega_{\text{ps}}$ for the transition
$\lvert[5s5p~{}^{1}P_{1}]-1,m_{I}\rangle\rightarrow\lvert[5s6s~{}^{1}S_{0}]0,m_{I}\rangle$
where the minus sign is from the Clebsch-Gordan coefficient. This will result
in common energy shifts to
$\lvert[5s5p~{}^{1}P_{1}]-1,\uparrow(\downarrow)\rangle$ which does not hamper
the cooling fidelity directly though it adds detuning to the two-photon
excitation $5s5p~{}^{3}P_{0}\rightarrow 5s5p~{}^{1}P_{1}$ that will slow down
the cooling as discussed above. However, when $-\sqrt{\chi}\Omega_{\text{ps}}$
is much larger than $\Omega_{\text{eff}}$, the transition
$\lvert[5s5p~{}^{1}P_{1}]-1,m_{I}\rangle\rightarrow\lvert[5s6s~{}^{1}S_{0}]0,m_{I}\rangle$
can lead to half population in $5s5p~{}^{1}P_{1}$ while the other half in
$5s6s~{}^{1}S_{0}$, and $5s6s~{}^{1}S_{0}$ decays with a rate
$\Gamma_{\text{s}}$ that is about a tenth of the decay rate of
$5s5p~{}^{1}P_{1}$. This means that the condition
$|\sqrt{\chi}\Omega_{\text{ps}}/\Omega_{\text{eff}}|\gg 1$ can reduce the
cooling fidelity by about 0.1, which is significant. The condition
$\chi<|\Omega_{\text{eff}}/\Omega_{\text{ps}}|^{2}$ can resolve this issue but
it is difficult to realize since the parameters used in Sec. IV require
$\chi<10^{-5}$. A solution to this problem is to add a detuning to the laser
field so that incorrect polarization in the field can only drive the
transition
$\lvert[5s5p~{}^{1}P_{1}]-1,m_{I}\rangle\rightarrow\lvert[5s6s~{}^{1}S_{0}]0,m_{I}\rangle$
with a detuning large compared to $|\sqrt{\chi}\Omega_{\text{ps}}|$. In this
case, the state $\lvert[5s6s~{}^{1}S_{0}]0,m_{I}\rangle$ is barely populated
from $\lvert[5s5p~{}^{1}P_{1}]-1,m_{I}\rangle$ and population loss from it can
be avoided.

(iii) The transition $5s5p~{}^{1}P_{1}\rightarrow 5s15d~{}^{1}D_{2}$ is for
balancing the energies of
$\lvert[5s5p~{}^{1}P_{1}]-1,\uparrow(\downarrow)\rangle$. The detuning is
$\Delta_{\text{pd}}$ when addressing $|[5s15d~{}^{1}D_{2}]F=13/2,m_{F}\rangle$
and $\Delta_{\text{pd}}+E_{\text{hf}}$ when addressing
$|[5s15d~{}^{1}D_{2}]F=11/2,m_{F}\rangle$. The numerical example in Sec. IV
assumed $(\Omega_{\text{pd}},\Delta_{\text{pd}})/2\pi=(144.27,~{}-1700)$ MHz
where $E_{\text{hf}}/2\pi=1.3$ GHz. The laser is assumed $\sigma^{-}$
polarized, and polarization impurity can lead to dressing of the hyperfine
state $F=9/2$ which is an extra state not included in the discussion of Sec.
IV. The $F=9/2$ state is above the $F=11/2$ state by about $2\pi\times 1.1$
GHz shown in Appendix C, which means that it is unlikely to induce an extra AC
Stark shift to hamper the cooling when $\sqrt{\chi}\xi_{4}\Omega_{\text{pd}}$
is much smaller than these detunings, where $\xi_{4}$ is a factor smaller than
1 defined similar to those in Eq. (22). However, laser polarization impurity
can reduce the Rabi frequency from $\xi_{k}\Omega_{\text{pd}}$ to
$(1-\sqrt{\chi})\xi_{k}\Omega_{\text{pd}}$ for any of the desired dressings
where $\xi_{k}$ with $k=0,1,2$, and $3$ are shown above Eq. (22). The reduced
Rabi frequencies can result in unbalanced energies of
$\lvert[5s5p~{}^{1}P_{1}]-1,\uparrow(\downarrow)\rangle$, which will lead to
reduced cooling fidelity as discussed in Sec. V.1. Assuming $\chi=0.01$, the
Rabi frequencies for dressing the target hyperfine states of
$5s15d~{}^{1}D_{2}$ will decrease by $10\%$, and numerical simulation shows
that the cooling fidelity is $0.9981$ at or beyond $20~{}\mu$s. With a worse
polarization condition when $\chi=0.1$ which corresponds to a deduction of
Rabi frequencies by $32\%$, the final cooling fidelity is about $0.9878$.

##  VI Discussions

###  VI.1 Comparison with Ref. [19]

There are two differences between the present theory and that in Ref. [19].
First, the hyperfine-interaction-induced spin mixing is suppressed by large
Zeeman energy in Ref. [19], while here it is suppressed by AC Stark shift of
laser fields. Second, the theory of Ref. [19] depends on defining qubits with
$\pm m_{I}$, while the present theory depends on defining qubits with
$m_{I}=1-I,-I$, namely, the two lowest nuclear-spin states in the ground
state.

###  VI.2 Cooling nuclear spin qubits in other atoms

It is useful to discuss the application of the present theory with other
alkaline-earth elements like calcium and barium, and some alkaline-earth-like
transition-metal elements that have similar relevant level structures. The
theory hinges on suppression of the hyperfine-interaction-induced spin flip
via exciting the lowest ${}^{1}P_{1}$ state to a nearby state with a large
Rabi frequency $\Omega_{\text{ps}}$ which may not be available for all
elements.

####  VI.2.1 Ytterbium, calcium, and barium

Another widely studied candidate for nuclear-spin quantum memories with
neutral atoms is 171Yb. There is a relatively large hyperfine interaction in
$6s6p~{}^{1}P_{1}$ of 171Yb, with $A/2\pi=-213$ MHz [50] (the quadrupole
interaction is zero for $I$ is $1/2$ with 171Yb). To use the present theory
with 171Yb, a large AC Stark shift is required to suppress the hyperfine
interaction, so it is not practical to use the present theory for cooling
171Yb. For example, Eq. (13) shows that for 87Sr, a Rabi frequency
$\Omega_{\text{ps}}/2\pi$ of about 300 MHz can result in an overlap over 0.99
between the state used in the cooling and the correct state; for 171Yb, we
find that to reach a similar overlap over 0.99, a corresponding Rabi frequency
over $2\pi\times$2.15 GHz should be available, which may be challenging. For
173Yb that has $I=5/2$, the hyperfine splitting in the lowest ${}^{1}P_{1}$
state is stronger than that of 171Yb [51] and is characterized with
$(A,~{}Q)/2\pi\approx(60,600)$ MHz [50]. To realize an overlap over 0.99
between the state used in the cooling and the correct state as in Eq. (13), we
find that the strong hyperfine interaction in 173Yb requires
$\Omega_{\text{ps}}>2\pi\times$5.4 GHz which is unlikely to be realizable. For
this reason, we conclude that the present theory is applicable for AEL
isotopes where the lowest ${}^{1}P_{1}$ state has a small enough hyperfine
interaction.

The present cooling scheme can work for 41Ca and 43Ca which have $I=7/2$ and a
level structure that is compatible with the cooling theory. The ground state
of calcium is $4s^{2}~{}^{1}S_{0}$ and the lowest ${}^{1}P_{1}$ state is
$4s4p~{}^{1}P_{1}$. The stable calcium isotope 43Ca has a relatively weak
hyperfine interaction with $(A,~{}Q)/2\pi=-(15.46,9.7)$ MHz [52]. For the
radionuclide odd calcium isotope that can be assumed stable in quantum optics
(its half-life is about $10^{5}$ years), 41Ca, the $4s4p~{}^{1}P_{1}$ state
has $(A,~{}Q)/2\pi=-(18.84,9.2)$ MHz [52]. The transition from
$4s4p~{}^{1}P_{1}$ to $4s5s~{}^{1}S_{0}$ has a wavelength 1034.66 nm [53]
which is close to that in the case of 171Yb as shown in Fig. 1, and this
transition has a rate $2.435\times 10^{7}s^{-1}$ [53] that is larger than the
corresponding value shown in Eq. (19) used in the numerical example studied in
this paper. This means that it should be easier to realize a large
$\Omega_{\text{ps}}$ for calcium, and it is possible to suppress the
hyperfine-interaction-induced spin mixing in calcium which is crucial for the
theory to work. We find that the minimal $\Omega_{\text{ps}}/2\pi$ to realize
an overlap over 0.99 between the state used in the cooling and the correct
state as in Eq. (13) shall be at least 490 and 580 MHz for 43Ca and 41Ca,
respectively. 41Ca and 43Ca were not well studied as 87Sr [54], but according
to the discussion in Appendix B, the numerical example shown in Figs. 1 can in
principle be realized with $\Omega_{\text{ps}}$ up to $2\pi\times 1$ GHz,
which means that the present theory can work with 41Ca and 43Ca since they
possess even larger transition rate in the infrared-laser transition for
suppressing nuclear spin mixing.

The two stable odd barium isotopes 135Ba and 137Ba have $I=3/2$ and the
electronic ground and optical clock states have similar configurations to
those of 171Yb [55, 56]. The spectra reported in Ref. [56] show that the
hyperfine interaction in $6s6p~{}^{1}P_{1}$ leads to an frequency separation
of 400.5 MHz and 457.2 MHz between the $F=5/2$ and $F=1/2$ levels for 135Ba
and 137Ba, respectively. Comparing to $5s5p~{}^{1}P_{1}$ in 87Sr, the
frequency separations of two nearby F levels in $6s6p~{}^{1}P_{1}$ of 135Ba
and 137Ba are roughly four times larger. We suppose that the value of
$\Omega_{\text{ps}}$ for achieving suppression of nuclear-spin mixing as in
Eq. (13) should be at least four times larger than that used in the example
shown in Sec. IV. However, the condition in Eq. (13) is for a cooling fidelity
over 0.999 with 87Sr shown in Table 1. for a rough estimate, we find that if
the values of $A$ and $Q$ used in Sec. IV are increased by four times while
other parameters are the same, the two numbers on the right sides of the two
lines of Eq. (13) become about $0.984$ and $0.999$, respectively, which
suggests that even with an infrared laser field of similar strength as used in
Sec. IV, a cooling fidelity of around 0.99 should be achievable with 135Ba and
137Ba.

####  VI.2.2 Zinc, cadmium, and mercury

There are some alkaline-earth-like transition-metal elements with nuclear
spins and low-lying states similar to the elements discussed above. For
example, the ground-clock transition has a wavelength 309, 332, and 266 nm for
zinc, cadmium, and mercury, respectively [57], and it is possible to achieve
high power UV laser fields for driving these transitions [58].

For zinc, the stable odd isotope 67Zn has $I=5/2$, and its lowest
${}^{1}P_{1}$ state, $4s4p~{}^{1}P_{1}$, has both a fast decay rate with a
lifetime around $1.3$ ns [59, 60, 61] and a relatively weak hyperfine
interaction with $(A,Q)/2\pi\approx(17.7,20.0)$ MHz [62]. We note that Eq.
(13) in the case of 87Sr shows that a Rabi frequency $\Omega_{\text{pd}}/2\pi$
of about 300 MHz is quite useful; here, we find that to reach a wavefunction
overlap over 0.99 in an equation similar to Eq. (13) for the case of 67Zn, a
Rabi frequency about $2\pi\times$535 MHz is sufficient. Though we didn’t find
data for a strong transition between $4s4p~{}^{1}P_{1}$ and a higher state,
the data in Ref. [61] for the triplet states $4s4p^{3}P_{x}$ with $x=0,1,2$
indirectly suggest that it is possible to have a strong transition for the
singlet state as well. This suggests that the cooling theory can in principle
be applied with 67Zn due to the weak hyperfine interaction.

Both of the two stable odd cadmium isotopes 111Cd and 113Cd have a simple
nuclear spin state with $I=1/2$ and they are recognized as a useful candidate
for optical lattice clocks [63]. However, the hyperfine interaction in
$5s5p~{}^{1}P_{1}$ of 111Cd and 113Cd is strong [64], with $|A|/2\pi$ equal to
about 150 and 240 MHz, respectively [65], which is comparable to that of
$6s6p~{}^{1}P_{1}$ in 171Yb. We didn’t find data about dipole matrix elements
between $5s5p~{}^{1}P_{1}$ and a nearby state for suppressing the hyperfine-
interaction-induced spin mixing, but that the decay rate of $5s5p~{}^{1}P_{1}$
in 111Cd and 113Cd [63] is more than three times of that in 171Yb suggests
that the dipole matrix element between $5s5p~{}^{1}P_{1}$ and a nearby state
in Cd is likely to be much larger. So, it is possible to achieve a much larger
$\Omega_{\text{ps}}$ with reasonable laser powers for cooling 111Cd and 113Cd
and we think that it might be possible to use the present theory with cadmium.

Mercury is among the heaviest elements that were optically trapped for
precision physics [66, 67, 68]. The stable odd isotopes 199Hg and 201Hg have
$I=1/2$ and $3/2$, respectively. However, the hyperfine interaction in the
lowest ${}^{1}P_{1}$ state, $6s6p~{}^{1}P_{1}$, is so strong that the
frequency separation between the $F=1/2$ and $F=3/2~{}(5/2)$ states is about
$5$ GHz for 199Hg (201Hg) [69, 70] which suggests that it is challenging to
apply the present cooling theory for mercury. Nonetheless, the lifetime of the
$6s6p~{}^{1}P_{1}$ state in mercury is 1.31 ns [71] which is a quarter of that
of the $6s6p~{}^{1}P_{1}$ state in 171Yb. This suggests that the dipole matrix
element between $6s6p~{}^{1}P_{1}$ and a nearby state in 199Hg and 201Hg can
be much larger than that in 171Yb, and it is difficult to say that the present
cooling theory can’t be used with mercury.

The above discussions show that the theory shown with strontium as an example
in this paper can be used with zinc and calcium with a high cooling fidelity.
It may also be used with barium and cadmium but the cooling fidelity may not
be high unless strong laser fields are available for suppressing the hyperfine
interactions. We only studied AEL atoms in this paper and it is a question
whether the theory can be extended to quantum control over nuclear spins of
noble gas [72, 73, 74].

##  VII Conclusion

We present a theory to cool 87Sr atoms with resolved sideband excitation from
the ground state to the clock state quenched by two-photon excitation between
the clock state and the fast-decaying $5s5p~{}^{1}P_{1}$ state. The nuclear-
spin-changing process induced by the hyperfine interaction in
$5s5p~{}^{1}P_{1}$ is suppressed by using laser excitation between
$5s5p~{}^{1}P_{1}$ and nearby states. The suppression is achieved via the
$m_{J}$-dependent AC Stark shift that is large compared to the hyperfine
interaction. Numerical simulations with reasonable parameters indicate that a
cooling fidelity over $99.9\%$ can be easily achieved with 87Sr. The cooling
is not sensitive to fluctuation of intensities and frequencies of the lasers,
but depends on high polarization purity in the laser fields. The theory can be
used with some other alkaline-earth-like species like calcium, zinc, and
barium.

## ACKNOWLEDGMENTS

The author thanks T. A. B. Kennedy for valuable inputs during the initial
stage of this work, and thanks Yan Lu for helpful discussions. This work is
supported by the National Natural Science Foundation of China under Grants No.
12074300 and No. 11805146, the Innovation Program for Quantum Science and
Technology 2021ZD0302100, and the Fundamental Research Funds for the Central
Universities.

##  Appendix A Spin mixing by hyperfine interaction

Hyperfine interaction can mix nuclear spin states with $m_{I},m_{I}\pm
1,m_{I}\pm 2$. To understand this, we note that in Eq. (5),

$\displaystyle\hat{\mathbf{I}}\cdot\hat{\mathbf{J}}$ $\displaystyle=$
$\displaystyle\hat{I}_{x}\hat{J}_{x}+\hat{I}_{y}\hat{J}_{y}+\hat{I}_{z}\hat{J}_{z}$
(14) $\displaystyle=$
$\displaystyle\frac{1}{2}[(\hat{I}_{x}+i\hat{I}_{y})(\hat{J}_{x}-i\hat{J}_{y})+(\hat{I}_{x}-i\hat{I}_{y})(\hat{J}_{x}+i\hat{J}_{y})]$
$\displaystyle+\hat{I}_{z}\hat{J}_{z}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2}(\hat{I}_{+}\hat{J}_{-}+\hat{I}_{-}\hat{J}_{+})+\hat{I}_{z}\hat{J}_{z},$

where

$\displaystyle\hat{I}_{+}\hat{J}_{-}|m_{J},m_{I}\rangle=\sqrt{(I-m_{I})(I+m_{I}+1)}$
$\displaystyle~{}~{}\cdot\sqrt{(J+m_{J})(J-m_{J}+1)}|m_{J}-1,m_{I}+1\rangle$
$\displaystyle~{}~{}\equiv a(m_{J}m_{I})|m_{J}-1,m_{I}+1\rangle,$
$\displaystyle\hat{I}_{-}\hat{J}_{+}|m_{J},m_{I}\rangle=\sqrt{(I+m_{I})(I-m_{I}+1)}$
$\displaystyle~{}~{}\cdot\sqrt{(J-m_{J})(J+m_{J}+1)}|m_{J}+1,m_{I}-1\rangle$
$\displaystyle~{}~{}\equiv b(m_{J}m_{I})|m_{J}+1,m_{I}-1\rangle,$
$\displaystyle\hat{I}_{z}\hat{J}_{z}|m_{J},m_{I}\rangle=m_{J}m_{I}|m_{J},m_{I}\rangle.$

In the three equations above, $m_{J}\geq-J+1,m_{I}\leq I-1$ in the first
equation, and $m_{J}\leq J-1,m_{I}\geq-I+1$ in the second equation. The term
$(\hat{\mathbf{I}}\cdot\hat{\mathbf{J}})^{2}$ in Eq. (5) can be expanded as

$\displaystyle(\hat{\mathbf{I}}\cdot\hat{\mathbf{J}})^{2}$ $\displaystyle=$
$\displaystyle\hat{I}_{z}^{2}\hat{J}_{z}^{2}$
$\displaystyle+\frac{1}{2}[(\hat{I}_{+}\hat{J}_{-}+\hat{I}_{-}\hat{J}_{+})\hat{I}_{z}\hat{J}_{z}+\hat{I}_{z}\hat{J}_{z}(\hat{I}_{+}\hat{J}_{-}+\hat{I}_{-}\hat{J}_{+})]$
$\displaystyle+\frac{1}{4}(\hat{I}_{+}^{2}\hat{J}_{-}^{2}+\hat{I}_{-}^{2}\hat{J}_{+}^{2}+\hat{I}_{+}\hat{I}_{-}\hat{J}_{-}\hat{J}_{+}+\hat{I}_{-}\hat{I}_{+}\hat{J}_{+}\hat{J}_{-}),$

where

$\displaystyle\hat{I}_{+}\hat{I}_{-}\hat{J}_{-}\hat{J}_{+}$ $\displaystyle=$
$\displaystyle(I+m_{I})(I-m_{I}+1)$ (16)
$\displaystyle\cdot(J-m_{J})(J+m_{J}+1)$

when $m_{J}<J$ and $m_{I}>-I$, and

$\displaystyle\hat{I}_{-}\hat{I}_{+}\hat{J}_{+}\hat{J}_{-}$ $\displaystyle=$
$\displaystyle(I-m_{I})(I+m_{I}+1)$ (17)
$\displaystyle\cdot(J+m_{J})(J-m_{J}+1)$

when $m_{J}>-J$ and $m_{I}<I$.

##  Appendix B Rabi frequencies

In this appendix, we list the dipole matrix elements found in literature for
the atomic transitions involved in the model studied here, and discuss the
achievable Rabi frequencies for the transitions used in the cooling of the
nuclear spin qubits.

The spontaneous emission from $(5s5p)^{1}P_{1}$ to $(5s^{2})^{1}S_{0}$ has a
decay rate $\Gamma=2.0\times 10^{8}~{}s^{-1}$ [34, 35], which is related with
the dipole-transition matrix element in the context of the Weisskopf-Wigner
approximation [see Eq. (11.33) of Ref. [39]]

$\displaystyle\Gamma_{\text{p}}$ $\displaystyle=$
$\displaystyle\frac{\omega_{0}^{3}}{9\pi\epsilon_{0}\hbar
c^{3}}|\langle[5s^{2}]^{1}S_{0}||\mathbf{d}||[5s5p]^{1}P_{1}\rangle|^{2},$
(18)

where $\mathbf{d}$ is the atomic dipole operator, $\epsilon_{0}$ is the free-
space dielectric permittivity, $c$ is the light speed in vacuum, $\hbar$ is
the Planck constant, and $\omega_{0}/2\pi\approx 6.51\times 10^{14}$ Hz is the
transition frequency, which lead to
$|\langle[5s5p]^{1}P_{1}||\mathbf{d}||[5s^{2}]^{1}S_{0}\rangle|=5.38ea_{0}$.
This estimate should have overestimated the value of
$\langle[5s5p]^{1}P_{1}||\mathbf{d}||[5s^{2}]^{1}S_{0}\rangle$ because the
decay rate of $(5s5p)^{1}P_{1}$ is not only from the coupling between it and
the ground state, but also from the coupling between it and $(5s4d)^{1}D_{2}$.
Indeed, a value of about $5.25ea_{0}$ was suggested in Refs. [75, 37]. The
above analyses show that Eq. (11.33) of Ref. [39] is useful for the estimation
of dipole matrix elements.

The transition from $(5s5p)^{1}P_{1}$ to $(5s6s)^{1}S_{0}$ is with a
wavelength of 1124.232 nm [44]. As in Eq. (18), we have

$\displaystyle\Gamma_{\text{s}}$ $\displaystyle=$
$\displaystyle\frac{\omega_{0}^{3}}{\pi\epsilon_{0}\hbar
c^{3}}|\langle[5s5p]^{1}P_{1}||\mathbf{d}||[5s6s]^{1}S_{0}\rangle|^{2}.$ (19)

With $\Gamma_{\text{s}}=1.86\times 10^{7}$s-1 [45], we estimate
$|\langle[5s5p]^{1}P_{1}||\mathbf{d}||[5s6s]^{1}S_{0}\rangle|=2.09ea_{0}$.
With this value and $\pi$-polarized laser for the transition, a Rabi frequency
$\Omega_{\text{ps}}=\mathcal{E}|\langle[5s6s]^{1}S_{0}||\mathbf{d}||[5s5p]^{1}P_{1}\rangle|=2\pi\times
300$ MHz would require an electric field $\mathcal{E}=1.12\times 10^{4}V/m$,
which corresponds to a beam intensity $16.7$ W$/$cm2, or a laser power of
$0.21$ mW if the radius of the laser spot at the atom is $20~{}\mu$m. This
estimate shows that in principle a GHz-scale $\Omega_{\text{ps}}$ is
realizable with a laser power over 2 mW.

We did not find data about the transition probability from $(5s15d)^{1}D_{2}$
to $(5s5p)^{1}P_{1}$ in literature. One can estimate the dipole matrix element
by using Coulomb wave functions as done in Refs. [76, 77]. By the angular
momentum coupling rules one can find
$\langle[5s5p]^{1}P_{1}||\mathbf{d}||[5s15d]^{1}D_{2}\rangle=\langle
5p||\mathbf{d}||15d\rangle$, where [76]

$\displaystyle|\langle 5p||\mathbf{d}||15d\rangle|\approx\sqrt{2}\int
rP_{\text{5p}}(r)P_{\text{15d}}(r)dr,$ (20)

which is about $0.092ea_{0}$ by using the effective principal quantum numbers
for the 5p and 15d states suggested in Ref. [46]. With this estimate, A Rabi
frequency $2\pi\times 144.27$ MHz would require an electric field
$\mathcal{E}=1.23\times 10^{5}V/m$, which corresponds to a laser power of
$25.1$ mW if the radius of the laser spot at the atom is $20~{}\mu$m (this
should be experimentally feasible for the UV laser, for a power about 30 mW
with a 316.6 nm UV laser was achieved for exciting Rydberg states of 88Sr in
Ref. [12]). Note that the method via Eq. (20) can be not as accurate in the
two-electron atoms as in the alkali-metal atoms such as rubidium or cesium.
Let us examine if this estimate is acceptable when we would like to argue that
the Rabi frequency for the 424 nm laser field can be around $2\pi\times 140$
MHz as in this paper. Note that the highest $(5snd)^{1}D_{2}$ state with
transition probability to $(5s5p)^{1}P_{1}$ studied is with $n=9$ [45]. The
dipole matrix element $|\langle 5p||\mathbf{d}||nd\rangle|$ extracted by using
Eq. (11.33) of Ref. [39] via the data from Ref. [45] is $0.21ea_{0}$ for
$n=9$, while the method as in Eq. (20) leads to $0.094ea_{0}$ for $n=9$ by
using the effective principal quantum numbers suggested in Ref. [46]. This
means that the estimate by Eq. (20) is likely to be smaller than the actual
value, which further means that the above estimate about the required value of
$\Omega_{\text{pd}}$ is within experimental feasibility.

##  Appendix C Hamiltonian matrix for numerical simulation

The theory depends on different AC Stark shifts for different $m_{J}$ states.
To numerically investigate them, we detail the Hamiltonian for the states. The
state $|[5s5p~{}^{1}P_{1}]-1,\uparrow\rangle$ is optically excited from the
state $|[5s5p~{}^{3}P_{0}]0,\uparrow\rangle$ via an intermediate state with an
effective Rabi frequency $\Omega_{\text{eff}}$, but is further coupled by
hyperfine interaction to $|[5s5p~{}^{1}P_{1}]0,\downarrow\rangle$. The state
$|[5s5p~{}^{1}P_{1}]-1,\downarrow\rangle$ is optically excited from the state
$|[5s5p~{}^{3}P_{0}]0,\downarrow\rangle$, and is not coupled with other
$|[5s5p~{}^{1}P_{1}],m_{J},m_{I}\rangle$ states because
$|[5s5p~{}^{1}P_{1}]-1,\downarrow\rangle$ has the maximal $m_{J}+m_{I}$. To
suppress the hyperfine induced spin mixing, namely, the coupling between
$|[5s5p~{}^{1}P_{1}]-1,\uparrow\rangle$ and
$|[5s5p~{}^{1}P_{1}]0,\downarrow\rangle$, a strong $\pi$ polarized laser field
is used to couple $|[5s5p~{}^{1}P_{1}]0,\downarrow\rangle$ and
$|[5s6s~{}^{1}S_{0}]0,\downarrow\rangle$ with a Rabi frequency
$\Omega_{\text{ps}}$ (for brevity, we assume all laser Rabi frequencies real
in this paper). There is a differential energy shift between
$[5s5p~{}^{1}P_{1}]-1,\downarrow\rangle$ and
$|[5s5p~{}^{1}P_{1}]-1,\uparrow\rangle$. To effectively remove it so as to
remove frequency resolution in the spontaneous emission, a highly detuned
laser field can couple the $5s5p~{}^{1}P_{1}$ state with
$|[5s15d~{}^{1}D_{2}]F,m_{F}\rangle$ via a 424.2399 nm [44] laser. The
hyperfine interaction constants are $(A,Q)/2\pi=(-194,-75)$ MHz for
$5s15d~{}^{1}D_{2}$ as determined experimentally [33], from which we find that
the energies ($/\hbar$) are about $2\pi\times(-1765,-463,604,1453,2100)$ for
$F=(13/2,11/2,9/2,7/2,5/2)$. When we use left-hand polarized laser field to
couple
$[5s5p~{}^{1}P_{1}]0,\downarrow\rangle,|[5s5p~{}^{1}P_{1}]-1,\downarrow\rangle,|[5s5p~{}^{1}P_{1}]-1,\uparrow\rangle$
with $5s15d~{}^{1}D_{2}$, only the states $[5s15d~{}^{1}D_{2}]F,m_{F}\rangle$
with $F=13/2,11/2$ are coupled. We label the Rabi frequency for coupling
$|[5s15d~{}^{1}D_{2}]F=13/2,m_{F}=-13/2\rangle$ and
$|[5s5p~{}^{1}P_{1}]-1,\downarrow\rangle$ by $\Omega_{\text{pd}}$, then the
Rabi frequencies for
$\\{|[5s15d~{}^{1}D_{2}]F=13/2,m_{F}=-11/2\rangle\leftrightarrow|[5s5p~{}^{1}P_{1}]0,\downarrow\rangle,|[5s15d~{}^{1}D_{2}]F=13/2,m_{F}=-11/2\rangle\leftrightarrow|[5s5p~{}^{1}P_{1}]-1,\uparrow\rangle,|[5s15d~{}^{1}D_{2}]F=11/2,m_{F}=-11/2\rangle\leftrightarrow|[5s5p~{}^{1}P_{1}]0,\downarrow\rangle,|[5s15d~{}^{1}D_{2}]F=11/2,m_{F}=-11/2\rangle\leftrightarrow|[5s5p~{}^{1}P_{1}]-1,\uparrow\rangle\\}$
are $\\{\xi_{0},\xi_{1},\xi_{2},\xi_{3}\\}\Omega_{\text{pd}}$, where
$\xi_{j}$, $j=0-3$ are angular momentum factors. With a $\sigma^{-}$-polarized
laser, we have

$\displaystyle\langle[5s15d~{}^{1}D_{2}]F,m_{F}|\mathbf{d}|[5s5p~{}^{1}P_{1}]m_{J},m_{I}\rangle$
$\displaystyle\propto$
$\displaystyle\sum_{m_{J}^{\prime}}C_{m_{J}(-1)m_{J}^{\prime}}^{11J^{\prime}}C_{m_{J}^{\prime}m_{I}m_{F}}^{J^{\prime}IF}$
(21)

where $J^{\prime}=2$ and
$m_{J}^{\prime}\in\\{-J^{\prime},-J^{\prime}+1,\cdots,J^{\prime}\\}$ are the
total electron angular momentum and its $\mathbf{z}$-projection of the
$5s15d~{}^{1}D_{2}$ state, from which we find

$\displaystyle\\{\xi_{0},\xi_{1},\xi_{2},\xi_{3}\\}$ $\displaystyle=$
$\displaystyle\\{\sqrt{2/13},~{}3/\sqrt{13},~{}3/\sqrt{26},~{}-2/\sqrt{13}\\}.$
(22)

The state $5s6s~{}^{1}S_{0}$ can be populated via the excitation of
$|[5s5p~{}^{1}P_{1}]0,\downarrow\rangle$ in the cooling scheme, and
$5s6s~{}^{1}S_{0}$ decays to the state $5s5p~{}^{1}P_{1}$ at a rate
$18.6\times 10^{6}$ s-1 [45], which means that when excited from
$|[5s5p~{}^{1}P_{1}]0,\downarrow\rangle$, the state
$|[5s6s~{}^{1}S_{0}]0,\downarrow\rangle$ can decay to
$|[5s5p~{}^{1}P_{1}]0,\downarrow\rangle$,
$|[5s5p~{}^{1}P_{1}]1,\downarrow\rangle$, or
$|[5s5p~{}^{1}P_{1}]-1,\downarrow\rangle$ via emission of $\pi$, $\sigma^{-}$,
or $\sigma^{+}$ polarized photons. However, due to that there is a large AC
Stark shift for $|[5s5p~{}^{1}P_{1}]0,\downarrow\rangle$, it is barely
populated, leading to negligible population in $5s6s~{}^{1}S_{0}$. As a
result, the population in $|[5s5p~{}^{1}P_{1}]1,\downarrow\rangle$ is
negligible. For this reason, we do not consider the laser excitation of
$|[5s5p~{}^{1}P_{1}]1,\downarrow\rangle$ when we analyze the AC Stark shift
for suppressing the hyperfine interaction. However, we include this state for
it is involved in the decay of the $5s6s~{}^{1}S_{0}$ state.

In the basis of

$\displaystyle\\{|[5s15d~{}^{1}D_{2}]F=13/2,m_{F}=-13/2\rangle,|[5s15d~{}^{1}D_{2}]F=13/2,m_{F}=-11/2\rangle,~{}|[5s15d~{}^{1}D_{2}]F=11/2,m_{F}=-11/2\rangle,$
$\displaystyle|[5s6s~{}^{1}S_{0}]0,\downarrow\rangle,~{}|[5s5p~{}^{1}P_{1}]0,\downarrow\rangle,~{}|[5s5p~{}^{1}P_{1}]-1,\uparrow\rangle~{}|[5s5p~{}^{1}P_{1}]-1,\downarrow\rangle,|[5s5p~{}^{3}P_{0}]0,\uparrow\rangle,~{}|[5s5p~{}^{3}P_{0}]0,\downarrow\rangle,$
$\displaystyle~{}|[5s^{2}~{}^{1}S_{0}]0,\uparrow\rangle,~{}|[5s^{2}~{}^{1}S_{0}]0,\downarrow\rangle,~{}|\mathscr{A}\rangle,~{}|[5s5p~{}^{1}P_{1}]1,\downarrow\rangle\\},$
(23)

the Hamiltonian consists of the atom-laser interaction $\hat{H}_{\text{a-l}}$,

$\displaystyle\hat{H}_{\text{a-l}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\begin{array}[]{ccccccccccccc}2(\Delta_{\text{pd}}+\Delta)&0&0&0&0&0&\Omega_{\text{pd}}&0&0&0&0&0&0\\\
0&2(\Delta_{\text{pd}}+\Delta)&0&0&\xi_{0}\Omega_{\text{pd}}&\xi_{1}\Omega_{\text{pd}}&0&0&0&0&0&0&0\\\
0&0&2(\Delta_{\text{pd}}+E_{\text{hf}}+\Delta)&0&\xi_{2}\Omega_{\text{pd}}&\xi_{3}\Omega_{\text{pd}}&0&0&0&0&0&0&0\\\
0&0&0&2\Delta&\Omega_{\text{ps}}&0&0&0&0&0&0&0&0\\\
0&\xi_{0}\Omega_{\text{pd}}&\xi_{2}\Omega_{\text{pd}}&\Omega_{\text{ps}}&2\Delta&0&0&0&0&0&0&0&0\\\
0&\xi_{1}\Omega_{\text{pd}}&\xi_{3}\Omega_{\text{pd}}&0&0&2\Delta&0&\Omega_{\text{eff}}&0&0&0&0&0\\\
\Omega_{\text{pd}}&0&0&0&0&0&2\Delta&0&\Omega_{\text{eff}}&0&0&0&0\\\
0&0&0&0&0&\Omega_{\text{eff}}&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&\Omega_{\text{eff}}&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0&0&0&0&0&0\end{array}\right),$ (37)

the Zeeman shift
$\hat{H}_{\text{Zee}}=g_{J}\mu_{\text{B}}J_{z}B-g_{I}\mu_{\text{n}}I_{z}B$ (we
ignore the Zeeman shift for the three $5s15d~{}^{1}D_{2}$ states and
$|\mathscr{A}\rangle$), and the hyperfine interaction $\hat{h}$ which couples
$|[5s5p~{}^{1}P_{1}]-1,\uparrow\rangle$ and
$|[5s5p~{}^{1}P_{1}]0,\downarrow\rangle$, where $E_{\text{hf}}=2\pi\times 1.3$
GHz as shown above Eq. (21); here the detuning is defined as the dipole
transition frequency deducted by the laser frequency. The matrix element of
the hyperfine interaction is

$\displaystyle\langle
m_{J}^{\prime},m_{I}^{\prime}|\hat{h}|m_{J},m_{I}\rangle$
$\displaystyle=\langle
m_{J}^{\prime},m_{I}^{\prime}|A\hat{\mathbf{I}}\cdot\hat{\mathbf{J}}+Q\frac{3(\hat{\mathbf{I}}\cdot\hat{\mathbf{J}})^{2}+1.5\hat{\mathbf{I}}\cdot\hat{\mathbf{J}}-IJ(I+1)(J+1)}{2IJ(2I-1)(2J-1)}|m_{J},m_{I}\rangle$
$\displaystyle=\delta_{m_{J}m_{J}^{\prime}}\delta_{m_{I}m_{I}^{\prime}}\bigg{[}Am_{I}m_{J}+Q\frac{3m_{I}^{2}m_{J}^{2}+1.5m_{I}m_{J}-IJ(I+1)(J+1)}{2IJ(2I-1)(2J-1)}$
$\displaystyle+Q\frac{3(I+m_{I})(I-m_{I}+1)(J-m_{J})(J+m_{J}+1)\Theta(J-m_{J})\Theta(m_{I}+I)}{8IJ(2I-1)(2J-1)}$
$\displaystyle+Q\frac{3(I-m_{I})(I+m_{I}+1)(J+m_{J})(J-m_{J}+1)\Theta(J+m_{J})\Theta(I-m_{I})}{8IJ(2I-1)(2J-1)}\bigg{]}$
$\displaystyle+\delta_{m_{J}^{\prime}(m_{J}-1)}\delta_{m_{I}^{\prime}(m_{I}+1)}\bigg{[}\frac{1}{2}Aa(m_{I}m_{J})+Q\frac{1.5(m_{I}m_{J}+(m_{I}+1)(m_{J}-1))a(m_{I}m_{J})+0.75a(m_{I}m_{J})}{2IJ(2I-1)(2J-1)}\bigg{]}$
$\displaystyle+\delta_{m_{J}^{\prime}(m_{J}+1)}\delta_{m_{I}^{\prime}(m_{I}-1)}\bigg{[}\frac{1}{2}Ab(m_{I}m_{J})+Q\frac{1.5(m_{I}m_{J}+(m_{I}-1)(m_{J}+1))b(m_{I}m_{J})+0.75b(m_{I}m_{J})}{2IJ(2I-1)(2J-1)}\bigg{]}$
$\displaystyle+\delta_{m_{J}^{\prime}(m_{J}-2)}\delta_{m_{I}^{\prime}(m_{I}+2)}\bigg{[}Q\frac{3a(m_{I}m_{J})a\big{(}(m_{I}+1)(m_{J}-1)\big{)}}{8IJ(2I-1)(2J-1)}\bigg{]}$
$\displaystyle+\delta_{m_{J}^{\prime}(m_{J}+2)}\delta_{m_{I}^{\prime}(m_{I}-2)}\bigg{[}Q\frac{3b(m_{I}m_{J})b\big{(}(m_{I}-1)(m_{J}+1)\big{)}}{8IJ(2I-1)(2J-1)}\bigg{]},$
(38)

where $\Theta(x)=1$ if $x>0$ and $\Theta(x)=0$ if $x\leq 0$, and

$\displaystyle a(m_{J}m_{I})$ $\displaystyle=$
$\displaystyle\sqrt{(I-m_{I})(I+m_{I}+1)}\sqrt{(J+m_{J})(J-m_{J}+1)},$
$\displaystyle b(m_{J}m_{I})$ $\displaystyle=$
$\displaystyle\sqrt{(I+m_{I})(I-m_{I}+1)}\sqrt{(J-m_{J})(J+m_{J}+1)}.$

The hyperfine constants are $(A,Q)/2\pi=(-3.4,~{}39)$ MHz [40] for
$(5s5p)^{1}P_{1}$.

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