# Use of vector polarizability to manipulate alkali-metal atoms

D. Cho111e-mail<EMAIL_ADDRESS>Department of Physics, Korea
University, Seoul 02841, Korea

###### Abstract

We review a few ideas and experiments that our laboratory at Korea University
has proposed and carried out to use vector polarizability $\beta$ to
manipulate alkali-metal atoms. $\beta$ comes from spin-orbit coupling, and it
produces an ac Stark shift that resembles a Zeeman shift. When a circularly
polarized laser field is properly detuned between the $D_{1}$ and $D_{2}$
transitions, an ac Stark shift of a ground-state atom takes the form of a pure
Zeeman shift. We call it the “analogous Zeeman effect”, and experimentally
demonstrated an optical Stern-Gerlach effect and an optical trap that behaves
exactly like a magnetic trap. By tuning polarization of a trapping beam, and
thereby controlling a shift proportional to $\beta$, we demonstrated
elimination of an inhomogeneous broadening of a ground hyperfine transition in
an optical trap. We call it “magic polarization”. We also showed significant
narrowing of a Raman sideband transition at a specific well depth. A Raman
sideband in an optical trap is broadened owing to anharmonicity of the trap
potential, and the broadening can be eliminated by a $\beta$-induced
differential ac Stark shift at what we call a “magic well depth”. Finally, we
proposed and experimentally demonstrated a cooling scheme that incorporated
the idea of velocity-selective coherent population trapping to Raman sideband
cooling to enhance cooling efficiency of the latter outside of the Lamb-Dicke
regime. We call it “motion-selective coherent population trapping”, and
$\beta$ is responsible for the selectivity. We include as Supplementary
Material a program file that calculates both scalar and vector
polarizabilities of a given alkali-metal atom when the wavelength of an
applied field is specified. It also calculates depth of a potential well and
photon-scattering rate of a trapped atom in a specific ground state when
power, minimum spot size, and polarization of a trap beam are given.

##  I INTRODUCTION

While a near-resonant laser field applies radiation pressure on an atom, an
off-resonant one can produce an energy shift by the ac Stark effect. The
former plays an important role in slowing and cooling atoms at an ambient
temperature, and the latter, coupled with an intensity gradient, produces an
electric dipole force to trap precooled atoms. When the laser field is far
detuned from allowed transitions, an optical dipole trap provides an almost
conservative potential well, which is ideal for precision spectroscopy or
quantum manipulation of trapped atoms. In addition, the ac Stark shift of an
alkali-metal atom in an elliptically polarized laser field depends on the
magnetic quantum number of the atom and degree of circularity of the field.
This dependence gives us a versatile tool to manipulate an alkali-metal atom
in an optical trap. In this paper, I will review a few ideas and experiments
that our group at Korea University has proposed and performed to use this
useful tool.

We first introduce spherical tensor formalism of the ac Stark shift to
quantify its polarization dependence in terms of a vector polarizability. The
theory section is followed by one on the analogous Zeeman effect analogous
Zeeman theory ; analogous Zeeman exp , where we show that when an applied
field is circularly polarized and detuned properly between the $D_{1}$ and
$D_{2}$ transitions, an ac Stark shift takes the form of a pure Zeeman shift.
The intensity gradient of such a field produces optical Stern-Gerlach effect
optical Stern-Gerlach  or an optical trap that behaves like a magnetic trap
analogous magnetic trap . Next two sections describe “magic” ideas; (i) A
shift proportional to vector polarizability can be used to eliminate an
inhomogeneous broadening of a ground hyperfine transition, which originates
from the difference in scalar polarizabilities of the two hyperfine levels.
The elimination is complete at a particular polarization, and we call it magic
polarization magic pol theory ; magic pol experiment . (ii) Stimulated Raman
transitions of a trapped atom that involve the change in vibrational quantum
number show inhomogeneous broadening from anharmonicity of an optical trap. It
can be eliminated by another broadening from the vector polarizability.
Anharmonicity-induced broadening is independent of well depth, and we use the
well depth as a tuning parameter to eliminate the broadening. We call it magic
well depth magic well depth . Finally, we describe a cooling method that we
call motion-selective coherent population trapping MSCPT theory ; MSCPT exp .
In this method, the ideas of Raman sideband cooling and velocity-selective
coherent population trapping are combined to complement each other, and vector
polarizability plays a pivotal role in producing motion-selectivity.

##  II Spherical Tensor formalism of ac Stark shift

When an atom in the state $|\Psi_{0}\rangle$ is irradiated by a laser field
with an oscillating electric field $\vec{E}(t)=\vec{\mathcal{E}}e^{-i\omega
t}+{\vec{\mathcal{E}}}^{*}e^{i\omega t}$, its ac Stark shift is, from second-
order perturbation theory,

$U_{\rm{AC}}(\Psi_{0})=\sum_{n\neq
0}\frac{\langle\Psi_{0}|\vec{d}\cdot\vec{\mathcal{E}}^{*}|\Psi_{n}\rangle\langle\Psi_{n}|\vec{d}\cdot\vec{\mathcal{E}}|\Psi_{0}\rangle}{\hbar[\,\omega-(\omega_{n}-\omega_{0})]}-\sum_{n\neq
0}\frac{\langle\Psi_{0}|\vec{d}\cdot\vec{\mathcal{E}}|\Psi_{n}\rangle\langle\Psi_{n}|\vec{d}\cdot\vec{\mathcal{E}}^{*}|\Psi_{0}\rangle}{\hbar[\,\omega+(\omega_{n}-\omega_{0})]},$
(1)

where $\omega_{0}$ and $\omega_{n}$ are the unperturbed energy eigenvalues of
$|\Psi_{0}\rangle$ and $|\Psi_{n}\rangle$, respectively, and $\vec{d}$ is the
electric dipole moment operator. When $|\Psi_{0}\rangle$ is the ground state,
the second term is off resonant and it is much smaller than the first in most
cases. $U_{\rm{AC}}(\Psi_{0})$ can be written as
$\langle\Psi_{0}|\Omega_{+}+\Omega_{-}|\Psi_{0}\rangle$, where

$\displaystyle\Omega_{+}$ $\displaystyle=$
$\displaystyle\sum_{i,\,j}d_{i}\mathcal{E}_{i}^{\ast}(\omega-H+\omega_{0})^{-1}d_{j}\mathcal{E}_{j},$
(2) $\displaystyle\Omega_{-}$ $\displaystyle=$
$\displaystyle\sum_{i,\,j}d_{i}\mathcal{E}_{i}(-\omega-H+\omega_{0})^{-1}d_{j}\mathcal{E}_{j}^{\ast},$
(3)

where $H$ is the unperturbed atomic Hamiltonian, and $\mathcal{E}_{i}$,
$\mathcal{E}_{j}$ and $d_{i}$, $d_{j}$ for $i,j=x,y,z$ are the Cartesian
components of $\vec{\mathcal{E}}$ and $\vec{d}$, respectively. The operators
can be rewritten in a spherical tensor form Happer , and explicitly for
$\Omega_{+}$,

$\Omega_{+}=\sum_{L=0}^{2}(-1)^{L}\sum_{m=-L}^{L}(-1)^{m}{\cal{D}}_{m}^{(L)}{\mathcal{F}}_{-m}^{(L)},$
(4)

where ${\cal{D}}^{(L)}$ and ${\mathcal{F}}^{(L)}$ with $L$ = 0, 1, 2 are the
scalar, vector and the second-rank spherical tensor operators:

${\cal{D}}_{m}^{(L)}=(-1)^{m+1}\sqrt{2L+1}\sum_{\mu=-1}^{1}\left(\begin{array}[]{ccc}1&1&L\\\
\mu&m-\mu&-m\end{array}\right)d_{\mu}(\omega_{0}-H_{0}+\omega)^{-1}d_{m-\mu}.$
(5)

Here $d_{0}=z$ and $d_{\pm 1}=\mp(x\pm iy)/\sqrt{2}$.
${\mathcal{F}}_{m}^{(L)}$ are similarly defined in terms of
$\mathcal{E}_{i}^{\ast}$ and $\mathcal{E}_{j}$;

${\mathcal{F}}_{m}^{(L)}=(-1)^{m+1}\sqrt{2L+1}\sum_{\mu=-1}^{1}\left(\begin{array}[]{ccc}1&1&L\\\
\mu&m-\mu&-m\end{array}\right)\mathcal{E}_{\mu}^{\ast}\mathcal{E}_{m-\mu},$
(6)

where $\mathcal{E}_{0}=\mathcal{E}_{z}$ and $\mathcal{E}_{\pm
1}=\mp(\mathcal{E}_{x}\pm i\mathcal{E}_{y})/\sqrt{2}$. $\Omega_{-}$ can be
similarly reformulated.

While the scalar term in $U_{\rm{AC}}(\Psi_{0})$ is simply proportional to
$|\vec{\mathcal{E}}|^{2}$, the vector term is proportional to
$\langle\Psi_{0}|\vec{\sigma}|\Psi_{0}\rangle\cdot(\vec{\mathcal{E}}\times\vec{\mathcal{E}}^{\ast})$
leading to dependence of the ac Stark shift on magnetic quantum number of the
atom and polarization of the field. Here, ${\cal{D}}^{(1)}$ in the vector term
is replaced with the Pauli spin operator using the Wigner-Eckart theorem. The
second-rank term is nonvanishing only if the total angular momentum of the
state $\Psi_{0}$ is at least 1. Although the ground state of an alkali-metal
atom has a nuclear spin in addition to the electron spin 1/2 so that the total
angular momentum $F$ can be larger than 1, when the detuning of the field
$\vec{E}(t)$ from the $D$ transitions is much larger than the hyperfine
splitting, the second-rank term can be neglected. Under this approximation,
the ac Stark shift of the ground hyperfine state $|nS_{1/2},F,m_{F}\rangle$ of
an alkali-metal atom, where $m_{F}$ is the magnetic quantum number, can be
written as

$U_{\rm{AC}}(F,m_{F})=\alpha|\vec{\mathcal{E}}|^{2}-i\beta\langle
S_{1/2},F,m_{F}|\vec{\sigma}|S_{1/2},F,m_{F}\rangle\cdot(\vec{\mathcal{E}}\times\vec{\mathcal{E}}^{\ast}),$
(7)

where $\alpha$ and $\beta$ are the scalar and the vector polarizabilities,
respectively. Their expressions can be obtained by comparing Eq. (1) with Eq.
(7):

$\displaystyle\alpha$ $\displaystyle=$
$\displaystyle\frac{1}{6\hbar}\sum_{n^{\prime}}{|\langle
n^{\prime}P_{1/2}\,||\,d\,||\,nS_{1/2}\rangle|^{2}}\left[\frac{1}{\omega-(\omega_{n^{\prime}P_{1/2}}-\omega_{nS_{1/2}})}-\frac{1}{\omega+(\omega_{n^{\prime}P_{1/2}}-\omega_{nS_{1/2}})}\right]$
$\displaystyle+$ $\displaystyle\frac{1}{6\hbar}\sum_{n^{\prime}}{|\langle
n^{\prime}P_{3/2}\,||\,d\,||\,nS_{1/2}\rangle|^{2}}\left[\frac{1}{\omega-(\omega_{n^{\prime}P_{3/2}}-\omega_{nS_{1/2}})}-\frac{1}{\omega+(\omega_{n^{\prime}P_{3/2}}-\omega_{nS_{1/2}})}\right],$
$\displaystyle\beta$ $\displaystyle=$
$\displaystyle\frac{1}{6\hbar}\sum_{n^{\prime}}{|\langle
n^{\prime}P_{1/2}\,||\,d\,||\,nS_{1/2}\rangle|^{2}}\left[\frac{1}{\omega-(\omega_{n^{\prime}P_{1/2}}-\omega_{nS_{1/2}})}+\frac{1}{\omega+(\omega_{n^{\prime}P_{1/2}}-\omega_{nS_{1/2}})}\right]$
$\displaystyle-$ $\displaystyle\frac{1}{12\hbar}\sum_{n^{\prime}}{|\langle
n^{\prime}P_{3/2}\,||\,d\,||\,nS_{1/2}\rangle|^{2}}\left[\frac{1}{\omega-(\omega_{n^{\prime}P_{3/2}}-\omega_{nS_{1/2}})}+\frac{1}{\omega+(\omega_{n^{\prime}P_{3/2}}-\omega_{nS_{1/2}})}\right],$

where $\langle n^{\prime}P_{J}\,||\,d\,||\,nS_{1/2}\rangle$ for $J=1/2$ and
3/2 is the reduced matrix element of $\vec{d}$. The definition of $\langle
n^{\prime}P_{J}\,||\,d\,||\,nS_{1/2}\rangle$ we use is given in Appendix A.

The ac Stark shift in Eq. (7) can be written in a form which clearly shows its
dependence on $m_{F}$ and polarization of the laser beam:

$U_{\rm{AC}}(F,m_{F})=(\alpha-\eta\beta g_{F}m_{F})(c\mu_{0}I_{0}/2),$ (10)

where $I_{0}=2|\vec{\mathcal{E}}|^{2}/c\mu_{0}$ is the intensity of the laser
field $\vec{E}(t)=\vec{\mathcal{E}}e^{-i\omega
t}+{\vec{\mathcal{E}}}^{*}e^{i\omega t}$, $g_{F}$ is the Landé $g$ factor, and
$\eta$ is the degree of circularity:

$\eta=i\hat{z}\cdot(\hat{\epsilon}\times\hat{\epsilon}^{\ast}).$ (11)

Here $\hat{\epsilon}$ is a Jones vector for the field $\vec{\mathcal{E}}$ and
$\hat{z}$ is the quantization axis that defines $m_{F}$. $\eta$ is 0 for a
linearly polarized light, and $+1$ and $-1$ for a right and a left circularly
polarized light, respectively, when the light propagates along $\hat{z}$. We
note that the second term has the same form as a Zeeman shift,
$g_{F}m_{F}\mu_{B}B_{0}$, of an atom in the $|nS_{1/2},F,m_{F}\rangle$ state
inside a magnetic field $B_{0}\hat{z}$, and we define an effective magnetic
field,

$\mathcal{B}_{\rm eff}=-\frac{\eta\beta c\mu_{0}I_{0}}{2\mu_{\rm B}},$ (12)

where $\mu_{\rm B}$ is the Bohr magneton. To understand the origin of this
Zeeman-like ac Stark shift, we write $U_{\rm{AC}}(F,m_{F})$ in Eq. (7) as an
expectation value of an electric dipole interaction Hamiltonian,
$-\vec{d}_{\rm ind}(\vec{\mathcal{E}})\cdot\vec{\mathcal{E}}^{\ast}$, where

$\vec{d}_{\rm
ind}(\vec{\mathcal{E}})=-\alpha\vec{\mathcal{E}}+i\beta(\vec{\sigma}\times\vec{\mathcal{E}})$
(13)

is an induced electric dipole moment. While the scalar term is from the usual
polarization by $\vec{\mathcal{E}}$ and parallel to the field, the vector term
is from perturbation by the spin-orbit coupling and is perpendicular to the
field. The spin-orbit coupling mediates the interaction between the magnetic
dipole moment of an electron with $\vec{E}(t)$, while their direct interaction
is forbidden by parity ($P$) and the time-reversal ($T$) symmetry. We point
out that the second term in Eq. (13) is odd under $P$ and even under $T$ like
an electric dipole moment, and the equation satisfies $P$ and $T$ symmetry.
According to this picture, $\beta$ depends on the strength of the spin-orbit
coupling. This can be shown by rewriting $\alpha$ and $\beta$ in Eqs. (II, II)
as

$\displaystyle\alpha$ $\displaystyle\approx$
$\displaystyle\frac{1}{6\hbar}\sum_{n^{\prime}}{|\langle
n^{\prime}P_{1/2}\,||\,d\,||\,nS_{1/2}\rangle|^{2}}\left[\frac{1}{\Delta_{1}(n^{\prime})}+\frac{2}{\Delta_{2}(n^{\prime})}\right],$
(14) $\displaystyle\beta$ $\displaystyle\approx$
$\displaystyle\frac{1}{6\hbar}\sum_{n^{\prime}}{|\langle
n^{\prime}P_{1/2}\,||\,d\,||\,nS_{1/2}\rangle|^{2}}\left[\frac{1}{\Delta_{1}(n^{\prime})}-\frac{1}{\Delta_{2}(n^{\prime})}\right],$
(15)

where
$\Delta_{1}(n^{\prime})=\omega-(\omega_{n^{\prime}P_{1/2}}-\omega_{nS_{1/2}})$
and
$\Delta_{2}(n^{\prime})=\omega-(\omega_{n^{\prime}P_{3/2}}-\omega_{nS_{1/2}})$.
Here we use $|\langle
n^{\prime}P_{3/2}\,||\,d\,||\,nS_{1/2}\rangle|^{2}\approx 2|\langle
n^{\prime}P_{1/2}\,||\,d\,||\,nS_{1/2}\rangle|^{2}$ and neglect the off-
resonant terms. Without the spin-orbit coupling, there is no fine structure
and $\Delta_{1}(n^{\prime})=\Delta_{2}(n^{\prime})$ and $\beta=0$. As a
consequence, heavy alkali-metal atoms with a strong spin-orbit coupling such
as Rb and Cs have larger $\beta$ than the lighter atoms.

In an optical trap, in addition to the ac Stark shift $U_{\rm AC}$ in Eq. (1),
which determines well depth, the rate of scattering $R_{\gamma}$ of the trap
beam by an atom, which determines heating and decoherence rate, is also an
important parameter. Using first-order time-dependent perturbation theory, the
scattering rate is given by

$R_{\gamma}(\Psi_{0})\approx\sum_{n\neq
0}\frac{\langle\Psi_{0}|\vec{d}\cdot\vec{\mathcal{E}}^{*}|\Psi_{n}\rangle\langle\Psi_{n}|\vec{d}\cdot\vec{\mathcal{E}}|\Psi_{0}\rangle}{\hbar^{2}\left\\{\,\omega-(\omega_{n}-\omega_{0})\right\\}^{2}}\Gamma(n),$
(16)

where $\Gamma(n)=1/\tau(n)$ with $\tau(n)$ being a lifetime of the upper state
$|\Psi_{n}\rangle$, and the off-resonant terms are neglected. If we further
assume that $\Gamma(n)$ is independent of $n$, $R_{\gamma}(\Psi_{0})$ can be
written as $\langle\Psi_{0}|\Theta|\Psi_{0}\rangle\Gamma$, where

$\Theta=\sum_{i,\,j}d_{i}\mathcal{E}_{i}^{\ast}(\omega-H+\omega_{0})^{-2}d_{j}\mathcal{E}_{j},$
(17)

which is smilar to $\Omega_{\pm}$ in Eqs. (2, 3). When $|\Psi_{0}\rangle$ is
the ground state of an alkali-metal atom and a trap beam is red detuned from
the $D$ transitions, only the contributions from the $D_{1}$ and $D_{2}$
transitions to $R_{\gamma}(\Psi_{0})$ are significant and we may neglect
couplings to the higher excited states. In addition, decay rates of the
$nP_{1/2}$ and $nP_{3/2}$ states are approximately the same. Under these
approximations, the transformation of $R_{\gamma}(\Psi_{0})$ in Eq. (16) to
$\langle\Psi_{0}|\Theta|\Psi_{0}\rangle\Gamma$ is justified. Following the
spherical tensor formalism that led to Eq. (10), the photon scattering rate of
the ground hyperfine state $|nS_{1/2},F,m_{F}\rangle$ of an alkali-metal atom
can be approximated as

$R_{\gamma}(F,m_{F})=(a-\eta bg_{F}m_{F})(c\mu_{0}I_{0}/2)\Gamma,$ (18)

where considering only the $D_{1}$ and $D_{2}$ couplings,

$\displaystyle a$ $\displaystyle=$
$\displaystyle\frac{1}{6\hbar^{2}}\left[\frac{|\langle
nP_{1/2}\,||\,d\,||\,nS_{1/2}\rangle|^{2}}{\\{\omega-(\omega_{n_{1/2}}-\omega_{nS_{1/2}})\\}^{2}}+\frac{|\langle
nP_{3/2}\,||\,d\,||\,nS_{1/2}\rangle|^{2}}{\\{\omega-(\omega_{nP_{3/2}}-\omega_{nS_{1/2}})\\}^{2}}\right],$
(19) $\displaystyle b$ $\displaystyle=$
$\displaystyle\frac{1}{12\hbar^{2}}\left[\frac{2|\langle
nP_{1/2}\,||\,d\,||\,nS_{1/2}\rangle|^{2}}{\\{\omega-(\omega_{nP_{1/2}}-\omega_{nS_{1/2}})\\}^{2}}-\frac{|\langle
nP_{3/2}\,||\,d\,||\,nS_{1/2}\rangle|^{2}}{\\{\omega-(\omega_{nP_{3/2}}-\omega_{nS_{1/2}})\\}^{2}}\right].$
(20)

In the Supplementary Material, we include a Microsoft Excel file that
calculates $\alpha,\beta$ and $a,b$ of alkali-metal atoms from lithium to
cesium when the wavelength of an applied laser beam is given. Because we
consider a far-detuned trap beam, effects of nuclear spin and isotope shift on
the polarizabilities and the scattering parameters are negligible. For a given
ground hyperfine state $|nS_{1/2},F,m_{F}\rangle$, the Excel file calculates
well depth $U_{\rm AC}(F,m_{F})$ and photon scattering rate
$R_{\gamma}(F,m_{F})$ of an optical trap according to Eq. (10) and Eq. (18),
respectively, when power, minimum spot size, and degree of circularity $\eta$
of the trap beam are given. Formulae and conversion factors used for the items
in the Excel file are explained in Appendix A.

##  III Analogous Zeeman effect

In a typical optical trap, a trap beam is far red detuned from the $D_{1}$
transition as shown in Fig. 1(a). Here, we consider a case of tuning the beam
between the $D_{1}$ and $D_{2}$ transitions as in Fig. 1(b). In this case,
$\Delta_{1}(n)$ and $\Delta_{2}(n)$ have opposite signs, and we note from Eqs.
(14, 15) that $\alpha$ tends to be suppressed while $\beta$ tends to be
enhanced. Especially when

$\Delta_{2}(n)=-2\Delta_{1}(n),$ (21)

neglecting couplings other than $D_{1}$ and $D_{2}$, $\alpha$ vanishes and the
ac Stark shift becomes

$U_{\rm{AC}}(F,m_{F})=-\eta\beta g_{F}m_{F}(c\mu_{0}I_{0}/2).$ (22)

It is identical to a Zeeman shift of the $|nS_{1/2},F,m_{F}\rangle$ state of
an alkali-metal atom with an effective magnetic field $\mathcal{B}_{\rm eff}$
given in Eq. (12). We call it the analogous Zeeman effect analogous Zeeman
theory , which makes a laser beam play the role of a magnetic field with the
field strength proportional to its intensity. This effect gives a versatile
tool to manipulate alkali-metal atoms. However, it is applicable to only heavy
alkali-metal atoms such as Rb and Cs with large fine structure. For the
lighter atoms, the small fine structure limits detuning. Because $R_{\gamma}$
in Eq. (16) is inversely proportional to the detuning squared, excessive
scattering of the trap beam becomes a problem. In Table I, for each alkali-
metal atom, the wavelength $\lambda_{0}$ that satisfies the condition of Eq.
(21) and $\beta$ at $\lambda_{0}$ are given. $\mathcal{B}_{\rm eff}$ and
$R_{\gamma}$ for a state with $g_{F}m_{F}=1$ are also tabulated when the laser
intensity $I_{0}$ is $1.4\times 10^{9}$ W/m2, corresponding to laser power of
0.5 W and minimum spot size of 15 $\mu$m, and with right-circular polarization
so that $\eta=1$.

Figure 1: (a) A trap beam can be either red detuned from the $D_{1}$ transition or (b) tuned between the $D_{1}$ and $D_{2}$ transitions. Table 1:  Parameters for analogous Zeeman effect. $\lambda_{0}$ is the wavelength that satisfies Eq. (21), $\beta$ is the vector polarizability at $\lambda_{0}$. $\mathcal{B}_{\rm eff}$ and $R_{\gamma}$ are effective magnetic field and photon scattering rate, respectively, for a state with $g_{F}m_{F}=1$ when the laser intensity is $1.4\times 10^{9}$ W/m2 with right-circular polarization.  atom |  $\lambda_{0}$ (nm) |  $\beta$ (atomic units) |  $\mathcal{B}_{\rm eff}$ (G) |  $R_{\gamma}$ (s-1)
---|---|---|---|---
Li | 670.8 | $5.4\times 10^{6}$ | $2.5\times 10^{4}$ | $2.0\times 10^{8}$
Na | 589.4 | $1.2\times 10^{5}$ | $560$ | $1.4\times 10^{5}$
K | 768.8 | $4.8\times 10^{4}$ | $230$ | $1.0\times 10^{4}$
Rb | 789.8 | $1.2\times 10^{4}$ | $58$ | $6.5\times 10^{2}$
Cs | 879.8 | $6.0\times 10^{3}$ | $28$ | $1.2\times 10^{2}$

We experimentally confirmed the Zeeman-like ac Stark shift of Eq. (22) by
performing stimulated Raman spectroscopy analogous Zeeman exp  on a slow Rb
beam leaked from a magneto-optical trap LVIS . The intensity gradient of an
elliptically polarized beam with a detuning of Eq. (21) exerts an
$m_{F}$-dependent force on an alkali-metal atom in the same way as a magnetic
field gradient does. The strength of the force is proportional to the degree
of circularity $\eta$ in Eq. (11). We demonstrated that a spin-polarized Rb
atom traversing an intensity gradient on one side of a Gaussian beam profile
showed $\eta$-dependent deflection optical Stern-Gerlach . The experimental
apparatus is shown in Fig. 2, and the results are shown in Fig. 3; for the
right and left circularly polarized beam of $\eta=+1$ and $-1$, respectively,
atoms are deflected in the opposite directions. We call it optical the Stern-
Gerlach effect.

Figure 2: Apparatus for the optical Stern-Gerlach experiment. It consists of a
low-velocity intense source of atoms, a spin-polarization region (A), an
interaction region (C), and a probe region (B). In the C region, a vertically
propagating Gaussian beam is tightly focused. Figure 3: Hot-wire scan results.
Triangles are when the Stern-Gerlach beam is blocked. Circles and squares are
when the Stern-Gerlach beam is right and left circularly polarized,
respectively.

While the term was used to describe the difference in radiation pressure
experienced by two states of a fictitious spin associated with a two-level
atom OSG old theory ; OSG old exp , our experiment is a more direct analogy to
the original Stern-Gerlach experiment. We also used the idea to create an
optical trap that behaves like a magnetic trap and observed 98% spin
polarization of the trapped atoms analogous magnetic trap . This idea has been
applied to many interesting experiments. As an example, in a quantum walk
experiment in a one dimensional optical lattice, the optical Stern-Gerlach
force by a circularly polarized lattice beam plays the role of a coin toss on
an atom in a superposition of opposite $g_{F}m_{F}$ states quantum walk .
Quantum gate operation by entangling two atoms in a three dimensional optical
lattice was realized by merging nearby sites by controlling lattice
polarization Porto .

##  IV Magic polarization

Motivation of the early research on laser cooling and trapping of atoms in the
1980s was to increase interrogation time to improve precision in spectroscopy.
However, an optically trapped atom suffers a state-dependent ac Stark shift,
which leads to inhomogeneous broadening of a transition, and most of the
benefit of long interrogation time is lost. For an optical transition, such as
the $D$ transitions of an alkali-metal atom, the idea of magic wavelength can
eliminate the broadening when there is a proper three-level configuration
magic wavelength . For a ground hyperfine transition of an alkali-metal atom,
the inhomogeneous broadening originates from different detuning for the $F$
and $F+1$ states

Figure 4: Inhomogeneous broadening of a ground hyperfine transition of an
alkali-metal atom.

owing to the ground hyperfine splitting $\delta_{\rm hfs}$ as shown in Fig. 4.
Atoms in the $F$ and $F+1$ states have different ac Stark shifts, and
consequently their well depths $U_{1}$ and $U_{2}$ and vibration frequencies
$\nu_{1}$ and $\nu_{2}$, respectively, are different. This implies that
transition frequency between the vibrational states $|\chi_{1}(n)\rangle$ and
$|\chi_{2}(n)\rangle$ of $U_{1}$ and $U_{2}$, respectively, depends on the
vibrational quantum number $n$, and even a transition between
$|\chi_{1}(n_{1})\rangle$ and $|\chi_{2}(n_{2})\rangle$ with $n_{1}\neq n_{2}$
is possible.

Because the ground hyperfine transition plays an important role in precision
measurement such as an atomic clock and search for symmetry violation, there
were many discussions Meschede-inhomo  and experimental efforts to overcome
this problem: for example, use of an extra laser beam to compensate the
differential shift two-beam1 ; two-beam2 , spin-echo or dynamic decoupling to
undo Stark-induced dephasing spin echo , and a blue-detuned optical bottle to
minimize atom-light interaction blue bottle . However, these approaches are
designed only to reduce the broadening and dephasing. In 2007, we proposed an
idea that could eliminate the differential shift completely by having
$U_{1}=U_{2}$ for a transition of interest magic pol theory . The idea relies
on the $\beta$-induced ac Stark shift in Eq. (10) again; for a transition from
the $|nS_{1/2},F,m_{F}\rangle$ to the $|nS_{1/2},F+1,m_{F}^{\prime}\rangle$
state, the differential shift vanishes when the degree of circularity
satisfies

$\eta=\frac{\Delta\alpha}{\beta g_{F}(m_{F}+m_{F}^{\prime})},$ (23)

where $\Delta\alpha=\alpha_{F+1}-\alpha_{F}$ with $\alpha_{F+1}$ and
$\alpha_{F}$ being the scalar polarizabilities of the $F+1$ and $F$ state,
respectively, and $\beta$ is an average value of those for the $F+1$ and $F$
states. We call it the magic polarization. The same idea was published in 2008
by a theory group Flambaum , who proposed to tune the angle $\theta$ between a
quantization axis defined by a magnetic field and trap-beam propagation
direction to eliminate the differential ac Stark shift by noting that the
effective degree of circularity is given by

$\eta_{\rm eff}=\eta\cos\theta.$ (24)

Although both proposals in Refs. magic pol theory  and Flambaum  considered a
hyperfine transition of an optically trapped Cs atom, it is not the best atom
to apply the idea because its large $\beta$ puts very stringent limit on
polarization control. Optically trapped atoms at temperature $T$ have an
average vibrational quantum number $\langle n\rangle=k_{B}T/\hbar\nu$ in a
one-dimensional model, where $k_{B}$ is the Boltzmann constant. When a
polarization error results in $\delta\eta$ from the magic condition in Eq.
(23), frequency of a $\Delta n=0$ transition for $n=\langle n\rangle$ is
shifted by $\delta\\!f$ from that for $n=0$ with $\delta\eta$ and $\delta\\!f$
being related by

$\delta\eta=\frac{\alpha}{\beta
g_{F}(m_{F}+m_{F}^{\prime})}\cdot\frac{h}{k_{B}T}\delta\\!f.$ (25)

Table 2: Tolerance on polarization control for most abundant isotopes of alkali-metal atoms. $\lambda_{\rm trap}$ is the wavelength when a trap beam is 100 THz red detuned from the $D_{1}$ transition. $f_{\rm hf}$ is the ground hyperfine splitting. $\alpha$ and $\beta$ at $\lambda_{\rm trap}$ are given in atomic units. $\eta_{\rm magic}$ satisfies the magic-polarization condition in Eq. (23) when $g_{F}(m_{F}+m_{F}^{\prime})=1$, and $\delta\eta_{\rm max}$ is tolerance on $\eta$ when $\delta\\!f=1$ Hz and atomic temperature is 100 $\mu$K.  atom |  $\lambda_{\rm trap}$ (nm) |  $f_{\rm hf}$ (GHz) |  $\alpha$ (a.u.) |  $\beta$ (a.u.) |  $\eta_{\rm magic}$ |  $\delta\eta_{\rm max}$
---|---|---|---|---|---|---
7Li | 864 | $0.803$ | $407$ | 0.016 | $0.18$ |  $1.2\times 10^{-2}$
23Na | 734 | $1.772$ | $453$ | $0.66$ | $1.1\times 10^{-2}$ |  $3.3\times 10^{-4}$
39K | 1036 | $0.462$ | $629$ | $3.3$ | $7.5\times 10^{-4}$ |  $9.2\times 10^{-5}$
87Rb | 1082 | $6.835$ | $652$ | 13 | $2.8\times 10^{-3}$ |  $2.4\times 10^{-5}$
133Cs | 1275 | $9.193$ | $714$ | 32 | $1.5\times 10^{-3}$ |  $1.1\times 10^{-5}$

Any imperfection in polarization-control optics such as a linear polarizer or
a retardation plate and residual birefringence of a lens or a window can
introduce $\delta\eta$. We note that for a given $\delta\\!f$, tolerance
$\delta\eta_{\rm max}$ on polarization control is inversely proportional to
$\beta$, and it becomes small for heavy atoms. In Table II, $\delta\eta_{\rm
max}$ for $\delta\\!f=1$ Hz is given when a trap beam is 100 THz red detuned
from the $D_{1}$ transition, $T=100$ $\mu$K, and
$g_{F}(m_{F}+m_{F}^{\prime})=1$. For experimental demonstration of magic
polarization, we used the $|2S_{1/2},F=1,m_{F}=1\rangle$ to
$|2S_{1/2},F=2,m_{F}=2\rangle$ transition of 7Li atoms in a 1064-nm optical
trap magic pol experiment . Fig. 5(a) and (b) show full width at half maximum
(FWHM) of Rabi lineshape and coherence time $\tau_{c}$ from Ramsey
spectroscopy, respectively, versus $\eta$.

Figure 5: (a) Full width at half maximum when rf field is applied for 200 ms
and (b) coherence time of the $|2S_{1/2},F=1,m_{F}=1\rangle$ to
$|2S_{1/2},F=2,m_{F}=2\rangle$ transition of 7Li atoms in a 1064-nm optical
trap versus degree of circularity $\eta$.

We observed minimum linewidth and maximum $\tau_{c}$ at $\eta=0.413$ while
$\eta_{\rm magic}$ calculated from the published data Safronova  is 0.39. When
we applied 803-MHz field for 2 s, FWHM was as small as $0.59\pm 0.02$ Hz,
while the Fourier-limited value was 0.4 Hz. The maximum $\tau_{c}$ in Fig.
5(b) was $820\pm 60$ ms, largely limited by magnetic field noise.

The narrow linewidth allowed us to address individual Li atoms in a 1D optical
lattice with a site-specific resolution, and the long coherence time allowed
us to coherently manipulate individual atoms independently of neighboring ones
PRL 2019 . Figure 6 shows Rabi signal from single atoms at three adjacent
sites versus rf frequency while a magnetic field gradient along the lattice
introduces differential Zeeman shift $\Delta f_{\rm Zeeman}$ of 180 Hz between
neighboring sites. Resolving power in the frequency domain, defined as $\Delta
f_{\rm Zeeman}$ divided by FWHM of each Rabi lineshape, is 2.3. It is three
times that in the spatial domain of 0.8 obtained by a state-of-the-art
quantum-gas microscope on Li atoms in an optical lattice Choi KAIST . Here,
the resolving power in the spatial domain is defined as half the lattice
wavelength divided by FWHM of a point spread function of an imaging optics.

Figure 6: Rabi line shapes for single atoms residing at the $q_{-1}$ (red
circle), $q_{0}$ (black square), and $q_{+1}$ sites (blue triangle). Magnetic
field gradient is 1.6 G/cm. Solid curves are theory predictions for a $\pi$
pulse driven by a 20-ms Blackman window.

For a clock transition between $m_{F}=0$ states, where the first-order Zeeman
shift vanishes and the idea of magic polarization is not applicable,
Derevianko proposed to use the second-order shift to eliminate the
inhomogeneous broadening Derevianko . When $\vec{B}=B_{0}\hat{z}$ is applied,
the second-order Zeeman shift of a clock transition at $f_{0}$ is

$\Delta f_{Z}^{(2)}\approx
2f_{0}\left(\frac{\mu_{B}B_{0}}{hf_{0}}\right)^{2}.$ (26)

In addition, inside a right circularly polarized trap beam, the total shift
$\Delta f_{0}$ of the clock transition is given by

$h\Delta
f_{0}=\Delta\alpha|\vec{\mathcal{E}}|^{2}+\frac{2{\mu_{B}}^{2}}{hf_{0}}\left(B_{0}-\frac{\beta}{\mu_{B}}|\vec{\mathcal{E}}|^{2}\right)^{2},$
(27)

where $\Delta\alpha=\alpha_{F+1}-\alpha_{F}$ and $B_{0}$ in Eq. (26) is
replaced by $B_{0}+\mathcal{B}_{\rm eff}$. Inhomogeneous broadening owing to
$\Delta\alpha$ is eliminated when

$B_{0}=\frac{\Delta\alpha}{\beta}\cdot\frac{hf_{0}}{4\mu_{B}},$ (28)

which may be called a magic B field. Effect from $|\mathcal{B}_{\rm eff}|^{2}$
is negligible for an alkali-metal atom owing to the smallness of
$|\mu_{B}\mathcal{B}_{\rm eff}/hf_{0}|$ in a typical optical trap.

##  V Magic well depth

One of the most effective ways to cool trapped atoms is Raman sideband cooling
(RSC) RSC Wineland . We consider RSC applied to alkali-metal atoms trapped in
a 1D optical lattice formed by a standing wave of a Gaussian beam with a
minimum spot size $w_{0}$ and wavelength $\lambda_{\rm OL}$. Its potential
well is

$U(\vec{r}\,)=U_{0}e^{-2(x^{2}+y^{2})/w_{0}^{2}}\cos^{2}kz,$ (29)

where $U_{0}=U_{\rm{AC}}(F,m_{F})$ in Eq. (10) and $k=2\pi/\lambda_{\rm OL}$.
We assume a large Rayleigh range and therefore a constant spot size. In a
harmonic approximation, the state of a lattice-bound atom is described by
$|\psi_{j}(\mathbf{n})\rangle=|\phi_{j}\rangle\otimes|\chi(\mathbf{n})\rangle$,
where $|\phi_{j}\rangle$ is one of the ground hyperfine states and
$|\chi(\mathbf{n})\rangle$ with vibrational quantum numbers
$\mathbf{n}=(n_{x},n_{y},n_{z})$ represents a 3D harmonic oscillator.
Longitudinal and transverse vibration frequencies are
$\nu_{z}=k\sqrt{2|U_{0}|/m}$ and $\nu_{\bot}=(2/w_{0})\sqrt{|U_{0}|/m}$,
respectively, with $m$ being the atomic mass. For RSC along the $z$ axis, we
use a Raman transition
$|\psi_{1}(\mathbf{n})\rangle\rightarrow|\psi_{2}(\mathbf{n}^{\prime})\rangle$
with $|\phi_{1}\rangle=|nS_{1/2},F,m_{F}\rangle$,
$|\phi_{2}\rangle=|nS_{1/2},F+1,m_{F}^{\prime}\rangle$, and
$n_{x}=n_{x}^{\prime}$, $n_{y}=n_{y}^{\prime}$. The transition frequency
$\omega_{R}(\mathbf{n},\mathbf{n^{\prime}})$ is $\omega_{12}+\Delta
n_{z}\nu_{z}$, where $\omega_{12}$ is for the
$|\phi_{1}\rangle\rightarrow|\phi_{2}\rangle$ transition of a free atom and
$\Delta n_{z}=n_{z}^{\prime}-n_{z}$.

In practice, the Raman transition suffers an inhomogeneous broadening from
anharmonicity of the potential well as well as from the usual differential ac
Stark shift between $|\phi_{1}\rangle$ and $|\phi_{2}\rangle$. For
anharmonicity, we consider the quartic terms from $U(\vec{r}\,)$:

$W_{q}=U_{0}\left(\frac{1}{3}k^{4}z^{4}+2\,\frac{k^{2}\rho^{2}z^{2}}{w_{0}^{2}}+2\frac{\rho^{4}}{w_{0}^{4}}\right),$
(30)

where $\rho^{2}=x^{2}+y^{2}$. While the carrier frequency is not shifted by
$W_{q}$ in the first-order perturbation, the sidebands for $\Delta n_{z}=\pm
1$ suffer a shift by $\Delta\omega_{q}(\mathbf{n},\mathbf{n^{\prime}})$:

$\Delta\omega_{q}(\mathbf{n},\mathbf{n^{\prime}})=-\Delta
n_{z}\left\\{\Delta\omega_{q}^{(1)}\left(n_{z}+\frac{1}{2}\right)+\Delta\omega_{q}^{(2)}(n_{\bot}+1)\right\\},$
(31)

where $n_{\bot}=n_{x}+n_{y}$, $\Delta\omega_{q}^{(1)}=\hbar k^{2}/2m,$ and
$\Delta\omega_{q}^{(2)}=k\hbar/\sqrt{2}mw_{0}$. We neglect an offset term
which is independent of $\mathbf{n}$ and $\mathbf{n}^{\prime}$. In addition,
from the dependence of $U_{0}$ on $F$ and $m_{F}$ in Eq. (10),
$\omega_{R}(\mathbf{n},\mathbf{n^{\prime}})$ is further shifted by
$\Delta\omega_{\beta}(\mathbf{n},\mathbf{n^{\prime}})$:

$\Delta\omega_{\beta}(\mathbf{n},\mathbf{n^{\prime}})=\Delta\omega_{\beta}^{(1)}\left(n_{z}+\frac{1}{2}\right)+\Delta\omega_{\beta}^{(2)}(n_{\bot}+1),$
(32)

where $\Delta\omega_{\beta}^{(1)}=\\{\eta\beta
g_{F}(m_{F}+m_{F}^{\prime})/2\alpha\\}\omega_{z}$ and
$\Delta\omega_{\beta}^{(2)}=\\{\eta\beta
g_{F}(m_{F}+m_{F}^{\prime})/2\alpha\\}\omega_{\bot}$. Here we assume
$|\Delta\alpha|\ll|\eta\beta g_{F}(m_{F}+m_{F}^{\prime})|$, which is valid for
a heavy alkali-metal atom in a circularly polarized trap. From Eqs. (31) and
(32), we note that the total shift
$\Delta\omega_{q}(\mathbf{n},\mathbf{n^{\prime}})+\Delta\omega_{\beta}(\mathbf{n},\mathbf{n^{\prime}})$
vanishes if $\Delta\omega_{\beta}^{(1)}=\Delta n_{z}\Delta\omega_{q}^{(1)}$
and $\Delta\omega_{\beta}^{(2)}=\Delta n_{z}\Delta\omega_{q}^{(2)}$. Because
the ratio $\Delta\omega_{q}^{(1)}/\Delta\omega_{q}^{(2)}$ is the same as
$\Delta\omega_{\beta}^{(1)}/\Delta\omega_{\beta}^{(2)}$, the two conditions
are simultaneously satisfied when

$|U_{0}|=\left[\frac{\alpha}{\beta
g_{F}(m_{F}+m_{F}^{\prime})}\right]^{2}\\!\\!\\!\cdot\frac{\hbar^{2}k^{2}}{2m}$
(33)

and $\eta=\Delta n_{z}$. We may call $|U_{0}|$ a magic well depth. By using
either a right or a left circularly polarized beam, the total shift can be
eliminated for either blue or red sideband, respectively. If a trap beam is
elliptically polarized with $|\eta|<1$, the magic $|U_{0}|$ increases by
$1/\eta^{2}$, giving flexibility in designing an experiment.

Figure 7: Stimulated Raman spectrum at the magic well depth. Red sideband of
the $|5S_{1/2},F=1,m_{{}_{F}}=-1\rangle$ $\rightarrow$
$|5S_{1/2},F=2,m_{{}_{F}}=-1\rangle$ transition (solid line with blue circles)
is narrowed by cancellation of broadening from anharmonicity and vector
polarizability. Spectrum of the clock transition (dashed line with red
squares), which suffers broadening from only the anharmonicity, is shown for
comparison. Each spectrum is normalized to its carrier signal.

We have experimentally demonstrated the idea of magic well depth by
eliminating inhomogeneous broadening of a Raman sideband transition using 87Rb
atoms trapped in a 1D optical lattice magic well depth . Figure 7 shows the
lineshape of simulated Raman transitions: blue circles are for the transition
of interest, $|5S_{1/2},F=1,m_{F}=-1\rangle$ $\rightarrow$
$|5S_{1/2},F=2,m_{F}^{\prime}=-1\rangle$, and red squares are for the clock
transition, whose broadening is only from the anharmonicity. We use a left
circularly polarized lattice beam at $\lambda_{\rm OL}=980$ nm and the magic
well depth of $|U_{0}|/k_{B}=140$ $\mu$K, determined by $\lambda_{\rm OL}$ and
the transition of interest. Focusing on the red sideband of both transitions,
we observe the line-narrowing effect of magic well depth. The carrier of the
clock transition is the narrowest because it is free from both
$\Delta\omega_{q}(\mathbf{n},\mathbf{n^{\prime}})$ and
$\Delta\omega_{\beta}(\mathbf{n},\mathbf{n^{\prime}})$.

##  VI motion-selective coherent population trapping

RSC puts trapped atoms to the motional ground state with high probability only
when the Lamb-Dicke parameter $\eta_{\rm LD}=\sqrt{\mathcal{E}_{R}/\hbar\nu}$
is much smaller than 1. Here, $\mathcal{E}_{R}$ is the recoil energy
accompanying the emission of a photon and $\nu$ is the vibrational frequency
of a trapped atom. While the condition is well satisfied along tightly
confined directions in an optical lattice or an optical tweezer, it is
difficult to satisfy the condition in a typical optical trap. To improve the
effectiveness of RSC outside the Lamb-Dicke regime, we proposed to incorporate
the phenomenon of coherent population trapping (CPT) to RSC MSCPT theory .
Figure 8 shows the inverted $\mathsf{Y}$ configuration for the cooling scheme.
A usual RSC (shown in black) consists of a red-detuned Raman transition
$|\phi_{1}\rangle\rightarrow$ $|\phi_{3}\rangle$, which we call $p$
transition, and the $D$ transition
$|\phi_{3}\rangle\rightarrow|\phi_{4}\rangle$ for optical pumping. To
incorporate CPT, we add $q$ Raman transition
$|\phi_{2}\rangle\rightarrow|\phi_{3}\rangle$ (shown in red) to form a
$\Lambda$ configuration. To make the CPT resonance condition dependent on the
motional quantum number $n$, we use a circularly polarized trap beam so that
$\nu_{1}\neq\nu_{2}$ owing to the vector polarizability. Here, $\nu_{1}$ and
$\nu_{2}$ are the vibration frequencies of the motional states,
$|\chi_{1}\rangle$ and $|\chi_{2}\rangle$, for the potential wells of
$|\phi_{1}\rangle$ and $|\phi_{2}\rangle$, respectively. Their difference is

$\Delta\nu_{12}=\frac{\beta}{4\alpha}\,\nu_{0},$ (34)

where $\nu_{0}$ is the vibration frequency in a linearly polarized trap. When
the $p$ and $q$ pairs of Raman beams are tuned to the $\Lambda$ resonance
between the $|\phi_{1},\chi_{1}(0)\rangle$ and $|\phi_{2},\chi_{2}(0)\rangle$
states, the motional ground states form a CPT dark state:

$|\Psi(\rm
CPT)\rangle=\frac{|\phi_{1},\chi_{1}(0)\rangle-|\phi_{2},\chi_{2}(0)\rangle}{\sqrt{2}},$
(35)

where atoms are accumulated.

Figure 8: Level schemes for RSC and inverted $\mathsf{Y}$ configuration of
MSCPT experiment. RSC scheme is shown in black and $q$ transition of MSCPT is
shown in red. Specific levels and transitions in 87Rb are assigned in the
inset.

A pair of $|\phi_{1},\chi_{1}(n)\rangle$ and $|\phi_{2},\chi_{2}(n)\rangle$
states are detuned from the CPT resonance by $n\Delta\nu_{12}$, and a pair
with the larger $n$ is brighter. We call this method motion-selective coherent
population trapping (MSCPT), and it is a trapped-atom analogue to the
velocity-selective CPT developed for subrecoil cooling of free He atoms VSCPT
1988 . Heavy alkali-metal atoms with large $\beta$ are favored for MSCPT. For
a 87Rb atom, if the trap wavelength is 850 nm and $\eta_{LD}=1$,
$\Delta\nu_{12}/2\pi=75$ Hz, while FWHM of the CPT resonance in our radio
frequency (rf) experiment was 150 Hz M1 CPT . In the inset of Fig. 8, specific
levels in 87Rb are assigned: $|\phi_{1}\rangle=|5S_{1/2},F=2,m_{F}=2\rangle$,
$|\phi_{2}\rangle=|2,1\rangle$, $|\phi_{3}\rangle=|1,1\rangle$, and
$|\phi_{4}\rangle=|5P_{1/2},2,2\rangle$. We note that the inverted
$\mathsf{Y}$ configuration is closed owing to the angular momentum selection
rule and, as a consequence, recoil heating during optical pumping is
minimized.

Recently, we demonstrated the MSCPT cooling scheme in an experiment using 87Rb
in a 1D optical lattice MSCPT exp . The lattice parameters were $w_{0}=50$
$\mu$m, $\lambda_{\rm OL}=980$ nm, and $U_{0}/k_{B}=125$ $\mu$K, resulting in
$\eta_{\rm LD}=2.3$ and $\Delta\nu_{12}/2\pi=5$ Hz along the transverse
direction. Although the parameters were not optimal for MSCPT because we used
an apparatus originally built for a different experiment, we observed
significant improvement in cooling efficiency compared with RSC when the
detuning $\delta_{\rm CPT}(0)$ of the $p$ and $q$ Raman beams from the CPT
resonance of the $n=0$ states was close to 0. In Fig. 9, temperature of the
atoms in a steady state versus $\delta_{\rm CPT}(0)$ is shown for MSCPT (blue
circles) and RSC (black triangles). To demonstrate the full benefit of the
MSCPT scheme, we are upgrading the apparatus so that $\eta_{\rm LD}\simeq 1$
and $\Delta\nu_{12}\geq 75$ Hz by using $w_{0}=16$ $\mu$m, $\lambda_{\rm
OL}=850$ nm. Finally, we note that a diatomic polar molecule in an optical
trap polar molecule  provides an excellent opportunity to apply MSCPT because
its Stark shift depends strongly on the rotational quantum number. Considering
a MgF molecule in a 532-nm optical trap as an example, the fractional
difference between vibration frequencies of a pair of states in the same ro-
vibrational level can be as large as 12%. This may be compared with
$\Delta\nu_{12}/\nu_{0}$ of less than 2% for 87Rb in the new design.

Figure 9: Temperature versus detuning $\delta_{\rm CPT}(0)$ from the CPT
resonance for the motional ground state. Blue circles represent results from
MSCPT cooling scheme, and black triangles represent results when the $p$ and
$q$ Raman beams are alternately turned on at 1 kHz.

##  VII conclusion

Over the last two and a half decades, our laboratory at Korea University has
proposed several ideas using vector polarizability to manipulate alkali-metal
atoms and carried out experiments to demonstrate those ideas. In this paper,
we reviewed the ideas and experimental results. Currently, we are working on
an experiment to probe the cooling limit of a scheme based on motion-selective
coherent population trapping and study strategies to apply the scheme to
diatomic polar molecules in an optical trap.

## ACKNOWLEDGMENTS

This work was supported by the National Research Foundation of Korea (Grant
No. NRF-2022R1F1A1075131).

##  Appendix A Numerical calculation of scalar and vector polarizabilities

In the Excel file of the Supplementary Material, to obtain numerical values of
$\alpha$ and $\beta$, we use the definitions in Eq. (II) and Eq. (II),
respectively, including the off-resonant terms. Those terms are not negligible
when a trap beam is far detuned. When the principal quantum number of the
ground state of an alkali-metal atom is $n$, we include couplings to only the
$nP_{J^{\prime}}$ and $(n+1)P_{J^{\prime}}$ states with $J^{\prime}=1/2,3/2$.
To obtain numerical values of $a$ and $b$, we use Eq. (19) and Eq.(20),
respectively, and restrict the sums to $n^{\prime}=n$. Because we consider a
far detuned trap beam, either ground or excited state hyperfine splitting is
not important, and the effect of an isotope shift is negligible too.

There are two definitions for a reduced matrix element commonly used in the
literature. We follow that of Racah Racah , according to which the Wigner-
Eckart theorem is written as Edmonds

$\langle
n^{\prime},J^{\prime},m_{J}^{\prime}|d_{q}|n,J,m_{J}\rangle=(-1)^{J^{\prime}-m^{\prime}}\left(\begin{array}[]{ccc}J^{\prime}&1&J\\\
-m_{J}^{\prime}&q&m_{J}\end{array}\right)\langle
n^{\prime}J^{\prime}||d||nJ\rangle,$ (36)

and the reduced matrix element for an electric dipole moment between the
$|n^{\prime}P_{J^{\prime}}\rangle$ and $|nS_{1/2}\rangle$ states of an alkali-
metal atom is related to the lifetime $\tau(n^{\prime}P_{J^{\prime}})$ by

$|\langle
n^{\prime}P_{J^{\prime}}||d||nS_{1/2}\rangle|^{2}=\frac{1}{\tau(n^{\prime}P_{J^{\prime}})}\frac{3\pi\epsilon_{0}\hbar
c^{3}}{\omega_{0}^{3}}(2J^{\prime}+1),$ (37)

where $\omega_{0}$ is the $|nS_{1/2}\rangle$ $\rightarrow$
$|n^{\prime}P_{J^{\prime}}\rangle$ transition frequency. For spectroscopic
data, we use NIST Atomic Spectra Database Lines Data NIST , where the Einstein
coefficient $A(n^{\prime}P_{J^{\prime}})=1/\tau(n^{\prime}P_{J^{\prime}})$ of
the upper state $|n^{\prime}P_{J^{\prime}}\rangle$ and the wavelength
$\lambda_{0}=2\pi c/\omega_{0}$ are tabulated.

In the Excel file, for a given wavelength $\lambda$ of a trap beam,
$\alpha,\beta$ and $a,b$ are calculated in atomic units. For a given power
$P$, minimum spot size $w_{0}$ ($e^{-2}$ intensity radius), and degree of
circularity $\eta$ of the trap beam, the well depth $U_{\rm AC}(F,m_{F})$ for
the ground hyperfine state $|nS_{1/2},F,m_{F}\rangle$ according to Eq. (10) is
given (in mK) by

$U_{\rm{AC}}(F,m_{F})=(\alpha-\eta\beta g_{F}m_{F})(c\mu_{0}I_{0}/2).$ (38)

The photon scattering rate $R_{\gamma}(F,m_{F})$ according to Eq. (18) is
given (in s-1) by

$R_{\gamma}(F,m_{F})=(a-\eta bg_{F}m_{F})(c\mu_{0}I_{0}/2)\Gamma,$ (39)

where we use $\Gamma=A(nP_{1/2})$ of the $D_{1}$ transition.

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