# A Generalized Theory for Optical Cooling of a Trapped Atom with Spin

Saumitra S. Phatak  Department of Physics and Astronomy, Purdue University,
West Lafayette, Indiana, 47907, USA  Karl N. Blodgett  Department of
Chemistry, Purdue University, West Lafayette, Indiana, 47907, USA  David Peana
Department of Chemistry, Purdue University, West Lafayette, Indiana, 47907,
USA  Meng Raymond Chen  Department of Chemistry, Purdue University, West
Lafayette, Indiana, 47907, USA  Jonathan D. Hood  Department of Physics and
Astronomy, Purdue University, West Lafayette, Indiana, 47907, USA, Department
of Chemistry, Purdue University, West Lafayette, Indiana, 47907, USA
<EMAIL_ADDRESS>
###### Abstract

Cooling atoms to the ground-state of optical tweezers is becoming increasingly
important for high-fidelity imaging, cooling, and molecular assembly. While
extensive theoretical work has been conducted on cooling in free space, fewer
studies have focused on cooling in bound states. In this work, we present a
unified formalism for optical cooling mechanisms in neutral atom tweezers,
including resolved and unresolved sideband cooling with different trapping
potentials, polarization gradient cooling, gray molasses cooling,
$\Lambda$-enhanced gray molasses cooling, and Raman sideband cooling. We
perform simulations and demonstrate good agreement with a simplified spin
model. We derive and discuss the fundamental limits of each cooling mechanism
and propose new strategies for achieving ground-state cooling in optical
tweezers. Our findings provide valuable insights into optimizing cooling
schemes for neutral atoms in optical tweezers, paving the way for minimizing
thermal decoherence in Rydberg and molecular gates and improving efficiencies
of molecular assembly.

††preprint: APS/123-QED

##  I Introduction

Laser cooling of bound neutral atoms has recently gained renewed significance
in experiments involving tweezer arrays of atoms and molecules [1]. In systems
of Rydberg atoms [2] or molecules [3, 4, 5] with dipolar coupling, thermal
motion limits the coherence of interactions. Molecular assembly [6, 7] and
few-body physics [8, 9] studies require high-fidelity ground-state preparation
of multiple atoms.

A wide range of optical cooling techniques have been developed for trapped
atoms, illustrated in Fig. 1. Single-photon cooling schemes, such as sideband
cooling, have been used for atoms with narrow lines for high-fidelity imaging
and ground-state preparation [10, 11, 12]. Two-photon schemes, including
polarization gradient (PG) [13, 14, 15, 16], gray molasses (GM) and
$\Lambda$-enhanced gray molasses ($\Lambda$-GM) [17, 18, 19, 20, 21], and
Raman-sideband cooling (RSB) [22, 23, 24, 25, 26], have been successfully
applied for imaging and cooling of atoms without narrow lines. While extensive
theoretical work has been developed for sideband cooling [27, 28, 29], many of
the other cooling schemes for bound atoms are still interpreted through their
free-space pictures, for example in Sisyphus cooling where an atom moves
through a changing polarization [13]. However, this picture breaks down for a
tightly trapped atom in the Lamb-Dicke regime, where the atom wavefunction is
smaller than the wavelength.

In this work, we develop a generalized optical cooling model for bound atoms
with spin in a single or counter-propagating beam configuration with different
polarizations. We validate these models using master equation simulations that
include the ground and excited spins and the harmonic oscillator states. Our
analysis reveals a consistent picture across various cooling schemes, shown in
Fig. 1. Cooling occurs when $|n\rangle$ and $|n-1\rangle$ are Raman coupled
for different spin states and brought into resonance using a combination of
light shifts (PG, GM, $\Lambda$-GM) and two-photon detunings (RSB).

Sideband cooling has been extensively studied in previous works [30, 31], but
cooling of a bound atom with spin has been much more limited. Previous work on
cooling bound atoms with spin has primarily focused on specific cases, such as
$J\\!\\!=\\!\\!1/2$ to $J^{\prime}\\!\\!=\\!\\!3/2$ transitions in counter-
propagating orthogonal linear light [32] or linear light with standing waves
[33]. We develop a new formalism for arbitrary spin and polarization, with a
particular emphasis on the experimentally relevant $\sigma_{+}-\sigma_{-}$
configuration. Notably, while these previous theoretical studies found a
fundamental limit of $\langle n\rangle\approx 1$, we demonstrate that schemes
like GM and $\Lambda$-GM can be used for high ground-state preparation when
considering different polarization and spin configurations.

Figure 1:  (a) Laser cooling of a trapped atom with ground and excited state
spin $F$ and $F^{\prime}$ in a changing polarization from counter-propagating
beams. (b) Single photon cooling schemes such as resolved and unresolved
sideband cooling drive population from the ground-state to the excited state
with one reduced motional quanta $n-1$. In the Lamb-Dicke regime, the recoil
heating is smaller than the trapping frequency and the population mostly
returns to $n-1$, resulting in net cooling. (c) Two-photon cooling schemes
like polarization gradient (PG), gray molasses (GM), and $\Lambda$-enhanced
gray molasses ($\Lambda$-GM) drive $n\rightarrow n-1$ between two ground-state
spins with a two-photon Raman transition. (d) Temperature scale for cooling
schemes. A wide variety of cooling schemes have been used to trap, cool, and
image tightly trapped atoms.

A recent experimental result achieved a record-high imaging fidelity of 99.96%
for lithium using $\Lambda$-enhanced GM cooling with significant ground-state
preparation, despite lithium having the largest recoil heating [21]. This
raises the question of whether spin cooling techniques can be further
optimized for ground-state preparation. The unified model also allows us to
explore novel cooling schemes that transcend the conventions of any single
conventional cooling technique.

The paper is structured as follows. Section II develops the general master
equation for spin cooling and derive effective ground-state spin operators by
adiabatically eliminating the excited state. Section III applies this
formalism to sideband cooling, including cases with different excited state
trapping frequencies. Section IV develops a formalism for spin cooling and
applies it to PG, GM, $\Lambda$-GM, and RSB cooling. Each section compares
exact simulations to simplified models and contrasts various cooling
techniques, with the goal of creating a unified picture that will lead to more
coherent interactions and higher fidelity ground-state preparation.

##  II Theory: Laser cooling in a harmonic potential

###  II.1 Master equation for laser cooling

In this section, we develop a master equation for an atom confined in a tight
harmonic trap and interacting with a semi-classical electric field. In a semi-
classical approximation, we take the expectation of the electric field but
keep the atom position $\hat{x}$ as a quantum operator [34, 35]. The electric
field for a single laser frequency $\omega_{L}$ is

$\bm{E}(\hat{x},t)=E_{0}\,{\bm{\mathcal{\epsilon}}}(\hat{x})\,e^{-i\omega_{L}t}+h.c..$
(1)

$E_{0}$ is a real field amplitude and the last term is the hermitian
conjugate. The complex polarization and phase are contained within
$\bm{\epsilon}(\hat{x})$ and can describe a traveling wave or a standing wave
with various polarization configurations.

The atom has ground hyperfine spin states denoted as $|F_{g}m_{g}\rangle$ and
excited hyperfine spin states denoted as $|F_{e}m_{e}\rangle$. The ground and
excited state projection operators $P_{e}$ are

$\displaystyle P_{e}=\sum_{m_{e}}|F_{e}m_{e}\rangle\langle F_{e}m_{e}|$
$\displaystyle P_{g}=\sum_{m_{g}}|F_{g}m_{g}\rangle\langle F_{g}m_{g}|.$ (2)

For an optically trapped atom, the ground and excited states can generally
experience different potentials due to their differing polarizabilities. This
difference will be addressed later, but for now, we will assume that the
ground and excited states share the same harmonic trapping frequency $\nu$.
The total Hamiltonian for the bound atom interacting with the electric field
in the rotating frame of the laser is [34, 35]

$\displaystyle\hat{H}=\nu\hat{a}^{\dagger}\hat{a}+\left(\Delta-i\frac{\Gamma}{2}\right)P_{e}-\frac{\Omega}{2}\left(\bm{\mathcal{\epsilon}}(\hat{x})\cdot\bm{\hat{D}}^{\dagger}+h.c.\right).$
(3)

$\Delta=\omega_{A}-\omega_{L}$ is the detuning of the atom from the laser. The
Rabi frequency $\Omega$ is defined in terms of the reduced matrix element
times the electric field amplitude

$\Omega=\langle J_{e}||d||J_{g}\rangle E_{0}/\hbar.$ (4)

The atomic operator $\hat{\bm{D}}^{\dagger}$ is a raising operator for all the
hyperfine levels in the ground-state, and $\hat{\bm{D}}$ is the corresponding
lowering operator. They are defined in terms of the dipole operator
$\bm{\hat{d}}$ normalized by the reduced matrix element,

$\hat{\bm{D}}=\frac{P_{g}\bm{\hat{d}}P_{e}}{\langle
J_{e}||d||J_{g}\rangle},\quad\hat{\bm{D}}^{\dagger}=\frac{P_{e}\bm{\hat{d}}P_{g}}{\langle
J_{e}||d||J_{g}\rangle}.$ (5)

The Hamiltonian in Eq. 3 also contains a non-Hermitian imaginary decay rate
$\Gamma$ of the excited state due to spontaneous emission. In the Wigner-
Weisskopf model, the excited state decays due to spontaneous emission without
repopulating the ground-state. This leads to a reduction in the wavefunction
normalization. The master equation for the density matrix contains refeeding
terms with collapse operators $L_{i}$ that repopulate the ground-state and
preserve the density matrix normalization:

$\frac{d\hat{\rho}}{dt}=-i(\hat{H}\hat{\rho}-\hat{\rho}\hat{H}^{\dagger})+\sum_{i}\hat{L}_{i}\hat{\rho}\hat{L}^{\dagger}_{i}.$
(6)

The collapse operators contain both the lowering of the excited state as well
as the momentum recoil for emitting a photon:

$\hat{L}_{\hat{\bm{n}},q}=\sqrt{\Gamma}\,\hat{R}_{\bm{n},q}\hat{D}_{q}.$ (7)

The collapse operators have two effects when operating on the density matrix.
First, they lower the atom back to the ground-state. $\hat{D}_{q}$ is the
lowering operator in the spherical vector basis
$\hat{D}_{q}=\hat{\bm{e}}^{*}_{q}\cdot\hat{\bm{D}}$, where $\hat{\bm{e}}_{q}$
is a spherical basis vector. Second, the recoil operator $\hat{R}_{n,q}$
imparts a momentum kick due to the photon recoil after the emission of a
photon with polarization $q$ in $\bm{n}$ directions. In general, the recoil
operator is averaged over the angular emission profile for each polarization.
If we assume that the photon is emitted along either direction of the
$x$-axis, then the recoil operators are $R_{\pm,q}=e^{\pm ikx}$, and we have a
total of six collapse operators for the two emissions directions and three
polarizations

$\hat{L}_{\pm,q}=\sqrt{\Gamma}\,e^{\pm ik\hat{x}}\hat{D}_{q}.$ (8)

In 1D, the photon recoil operator $R_{\pm}$ = $e^{\pm ik\hat{x}}$ can be
understood as a momentum translation operator that induces a momentum kick due
to the recoil from an emitted photon, with $e^{\pm
ik\hat{x}}|\psi(p)\rangle=|\psi(p\pm\hbar k)\rangle$.

We can express the position operator in terms of the harmonic oscillator
creation and annihilation operators
$\hat{x}=x_{0}(\hat{a}+\hat{a}^{\dagger})$, where $x_{0}=\sqrt{\hbar/2m\nu}$
is the width of the ground-state wavefunction. The recoil operator becomes
$e^{ik\hat{x}}=e^{i\eta(\hat{a}+\hat{a}^{\dagger})}$, where the Lamb-Dicke
(LD) parameter $\eta=kx_{0}$ is the ratio of the ground-state wavefunction
width to the wavelength.

The square of the LD parameter also happens to be equal to the ratio of the
photon recoil energy $E_{\text{recoil}}=\hbar^{2}k^{2}/(2m)$ to the harmonic
energy $\hbar\nu$, or $\eta^{2}=E_{\text{recoil}}/(\hbar\nu)$. When $\eta\ll
1$, then the recoil from photon emission or absorption is unlikely to change
$n$, and the ground-state wavefunction is smaller than the cooling laser’s
wavelength. In this regime, we can also approximate the recoil operator as

$e^{ik\hat{x}}\approx 1+i\eta(\hat{a}+\hat{a}^{\dagger}).$ (9)

The result of a recoil is then
$e^{ik\hat{x}}|n\rangle\approx|n\rangle+\eta\sqrt{n}|n-1\rangle+\eta\sqrt{n+1}|n+1\rangle$.
The zeroth order term is from $n\rightarrow n$, and the first order LD term is
from $n\rightarrow n\pm 1$. The LD regime is defined as when
$\eta^{2}(2n+1)\ll 1$. One common misconception is that in the LD regime,
heating is suppressed. Although the probability of $n$ changing decreases, the
overall heating rate is still given by the scattering rate times the recoil
energy.

More generally, the matrix element can be expressed in terms of the
generalized Laguerre polynomial as [36]

$\displaystyle\langle n^{\prime}|e^{ik\hat{x}}|n\rangle=$ (10) $\displaystyle
e^{-\eta^{2}/2}\left(\frac{n_{<}!}{(n_{<}+\Delta
n)!}\right)^{1/2}(i\eta)^{\Delta n}\,L_{n_{<}}^{\Delta n}(\eta^{2}).$

Here, $n_{<}$ is the least of $n$ and $n^{\prime}$, and $\Delta
n=|n^{\prime}-n|.$ This functional form becomes important when cooling outside
the LD regime. The $n\rightarrow n-1$ transition goes to zero for higher $n$,
creating dark motional states where the population gets stuck. Cooling can
still occur by addressing higher order sidebands, as shown experimentally in
Ref. [26].

The master equation here is simulated throughout this paper for different
cooling schemes using the steady-state solver in QuTip [37]. We include up to
15 harmonic states, which for the larger spin systems in this paper like $F=1$
to $F=2$ takes approximately one minute per simulation.

###  II.2 Effective master equation within ground-state subspace

Most optical cooling processes take place when there is low saturation,
meaning that only a small amount of the steady-state population is in the
excited state. The excited state can then be reasonably approximated from the
ground state, and can be adiabatically eliminated. In this section, we will
derive the effective ground-state master equation.

In situations where the excited state quickly reaches an equilibrium, the
excited state can be solved in terms of the ground-states and then eliminated
from the time evolution equations. The elimination results in an effective
Hamiltonian and collapse operator that gives the dynamics within the ground-
state. These effective operators can be derived by performing these operations
for a given Hamiltonian [38], but Ref. [39] provides a compact general
formalism that we use here.

This first form is valid even when the cooling light is close to resonance
such as in sideband cooling. The Hamiltonian and collapse operators within the
ground-state subspace are

$\displaystyle H_{\text{eff}}=\nu a^{\dagger}a$
$\displaystyle-\frac{1}{2}\left[\sum_{m_{g},m_{e}}\frac{P_{g}H|m_{e}\rangle\langle
m_{e}|H|m_{g}\rangle\langle
m_{g}|}{(\Delta_{m_{e},m_{g}}-i\Gamma/2)}+h.c.\right]$ (11) $\displaystyle
L_{\text{eff}}^{i}=L_{i}\sum_{m_{g},m_{e}}\frac{|m_{e}\rangle\langle
m_{e}|H|m_{g}\rangle\langle m_{g}|}{\Delta_{m_{e},m_{g}}-i\Gamma/2}.$ (12)

Here the laser detuning is
$\Delta_{m_{g},m_{e}}=\omega_{L}-(E_{m_{e}}-E_{m_{g}})$ appears in the
denominator. For the case of an atom with spin interacting with a single
frequency laser field, as in the Hamiltonian in Eq. 3, the effective ground-
state Hamiltonian is

$\displaystyle H_{\text{eff}}=\nu\hat{a}^{\dagger}\hat{a}-$ (13)
$\displaystyle\frac{\Omega^{2}}{8}\\!\left[\sum_{m_{g},m_{e}}\\!\\!\frac{(\bm{\epsilon}(\hat{x})\\!\cdot\\!\bm{D})|m_{e}\rangle\langle
m_{e}|(\bm{\epsilon}^{*}(\hat{x})\\!\cdot\\!\bm{D}^{\dagger})|m_{g}\rangle\langle
m_{g}|}{\Delta_{m_{g},m_{e}}-i\Gamma/2}+\text{h.c.}\right]$

and the collapse operator for the emission of a photon with polarization $q$
is:

$\displaystyle
L_{\text{eff}}^{q,\bm{n}}=\\!-\sqrt{\Gamma}\,\frac{\Omega}{2}\\!\\!\sum_{m_{g},m_{e}}\\!\\!\frac{(\mathcal{R}_{\bm{n}}\bm{e}_{q}^{*}\cdot\bm{D})|m_{e}\rangle\langle
m_{e}|(\bm{\epsilon}(\hat{x})\cdot\bm{D}^{\dagger})|m_{g}\rangle\langle
m_{g}|}{\Delta_{m_{g},m_{e}}-i\Gamma/2}.$ (14)

The effective Hamiltonian includes light shifts, two-photon Raman coupling,
and decay due to spontaneous emission. The collapse operators repopulate the
ground-state.

Two photon cooling schemes operate in the large detuning limit. In the large
detuning limit, the state dependence in the denominator detuning vanishes
$\Delta_{m_{g},m_{e}}\approx\Delta$. The sum over the excited states in the
numerator becomes the identity. The effective ground-state operators simplify
to:

$\displaystyle H_{\text{eff}}=\nu
a^{\dagger}a-\frac{1}{2\Delta}\left(P_{g}HP_{e}HP_{g}\right)$ (15)
$\displaystyle L_{\text{eff}}^{q}=L_{q}\frac{P_{e}HP_{g}}{\Delta-i\Gamma/2}.$
(16)

For the single frequency Hamiltonian in Eq. 3, it becomes

$\displaystyle
H_{\text{eff}}=\nu\hat{a}^{\dagger}\hat{a}-\frac{\Omega^{2}}{4\Delta}\left[(\bm{\epsilon}^{*}(\hat{x})\cdot\bm{D})(\bm{\epsilon}(\hat{x})\cdot\bm{D}^{\dagger})\right]$
(17)

and the ground-state collapse operator in the large detuning limit is

$\displaystyle
L_{\text{eff}}^{q,\bm{n}}=-\sqrt{\Gamma}\,\frac{\Omega}{2}\left[\frac{(\mathcal{R}_{\bm{n}}\hat{\bm{e}}_{q}^{*}\cdot\bm{D})(\bm{\epsilon}(\hat{x})\cdot\bm{D}^{\dagger})}{\Delta-i\Gamma/2}\right].$
(18)

The interaction part of the ground-state Hamiltonian describes two-photon
processes such as light shifts and Raman transitions. The right term in the
numerator $\bm{\epsilon}(\hat{x})\cdot\hat{\bm{D}}^{\dagger}$ excites the atom
due to absorption of a photon from the cooling light. The left term
$\bm{\epsilon}^{*}(\hat{x})\cdot\hat{\bm{D}}$ returns the atom to the ground-
state with the emission of a photon into the cooling light.

The ground-state collapse operator describes the redistribution of population
within the ground-state due to spontaneous emission. It has a similar form
compared to interaction Hamiltonian. The right term
$\bm{\epsilon}(\hat{x})\cdot\hat{\bm{D}}^{\dagger}$ excites the atom due to
the absorption of a photon from the cooling light. The left term
$\mathcal{R}_{\bm{n}}\bm{e}_{q}^{*}\cdot\hat{\bm{D}}$ returns the atom to the
ground-state by emitting a photon into free-space and gives the atom a
momentum kick through the recoil operator $\mathcal{R}_{\bm{n}}$. This
momentum kick is the key difference between the two operators.

After adiabatic elimination, the form of both the effective Hamiltonian and
collapse operators are in terms of the scalar products of the dipole operators
and field polarization,

$H\propto(\bm{A}^{*}\cdot\hat{\bm{D}})(\hat{\bm{D}}^{\dagger}\cdot\bm{B})=(\bm{A}^{*}\otimes\bm{B})\cdot(\hat{\bm{D}}\otimes\hat{\bm{D}}^{\dagger}).$
(19)

These operators act within the ground-state spin manifold just like the
hyperfine spin operators $\hat{F}_{i}$, $F_{\pm}$. If you observe, the right
hand side is actually a scalar product of two rank 2 matrices. Using the
Wigner-Eckart theorem and properties of spherical vectors, we can express
these in terms of spin operators. The spin Hamiltonian is [38]

$\displaystyle
H_{\text{eff}}=\nu\hat{a}^{\dagger}\hat{a}-\frac{\Omega^{2}}{4\Delta}\left[C^{(0)}|\bm{\epsilon}(\hat{x})|^{2}\right.$
(20)
$\displaystyle\left.+C^{(1)}i(\bm{\epsilon}^{*}(\hat{x})\times\bm{\epsilon}(\hat{x}))\right.$
$\displaystyle\left.+C^{(2)}\left(((\bm{\epsilon}^{*}(\hat{x})\\!\cdot\\!\hat{\bm{F}})(\bm{\epsilon}(\hat{x})\\!\cdot\\!\hat{\bm{F}})+c.c.)/2-\frac{1}{3}\hat{\bm{F}}^{2}|\bm{\epsilon}(\hat{x})|^{2}\right)\right]$

and the collapse operators which give the redistribution of population in the
ground-state due to spontaneous emission are

$\displaystyle
L_{\text{eff}}=-\sqrt{\Gamma}\frac{\Omega}{2\Delta}\left[C^{(0)}(\mathcal{R}_{\bm{n}}\hat{\bm{e}}_{q}^{*})\cdot\bm{\epsilon}(\hat{x})\right.$
(21)
$\displaystyle\left.+C^{(1)}i((\mathcal{R}_{\bm{n}}\hat{\bm{e}}_{q}^{*})\times\bm{\epsilon}(\hat{x}))\right.$
$\displaystyle\left.+\frac{C^{(2)}}{2}\left[((\mathcal{R}_{\bm{n}}\hat{\bm{e}}_{q}^{*})\\!\cdot\\!\hat{\bm{F}})(\bm{\epsilon}(\hat{x})\\!\cdot\\!\hat{\bm{F}})+c.c.\right)\right.$
$\displaystyle\left.-\frac{1}{3}\hat{\bm{F}}^{2}|(\mathcal{R}_{\bm{n}}\hat{\bm{e}}_{q}^{*})\cdot\bm{\epsilon}(\hat{x})|\right].$

The effective spin operators are valid for any ground and excited spin states.
The excited state only affects the $C^{(i)}$ coefficients.

The ground-state collapse operators in Eqs. 18 (far detuned limit) and 14
(near-resonant limit) describe the redistribution of the population within the
ground-state. With the effective ground-state Hamiltonian, a master equation
incorporating these operators describes the ground-state dynamics. The
Hamiltonian accounts for energy shifts and coherent transfer between states,
while the collapse operators account for incoherent redistribution due to
spontaneous emission.

Next, we will derive an approximate population rate equation by working in the
basis where $H_{\text{eff}}$ is diagonal. We define an interaction picture by
applying a unitary transformation $\tilde{A}=UAU^{\dagger}$ with
$U=e^{-iH_{\text{eff}}t}$. In the interaction picture, the time evolution of
the Hamiltonian is included within the density matrix and operators $U\rho
U^{\dagger}$ with unitary $U=e^{-iHt}$. In the interaction picture, the master
equation is:

$\dot{\tilde{\rho}}=\sum_{q}\left[\tilde{L}_{\text{eff},q}\,\tilde{\rho}\,\tilde{L}_{\text{eff},q}^{\dagger}-\frac{1}{2}\\{\tilde{L}_{\text{eff},q}^{{\dagger}}\tilde{L}_{\text{eff},q},\,\tilde{\rho}\\}\right].$
(22)

Finally, we derive rate equations for the populations by multiplying by
$\langle n|$ on the left and $|n\rangle$ on the right and defining the
populations $P(n)=\tilde{\rho}_{nn}=\rho_{nn}$. The resulting equation
contains coherence terms $\langle n|\rho|m\rangle$ between ground-states. If
we work in the basis where the ground-state Hamiltonian is diagonal, these
coherences are less important and can often be ignored. By setting all these
coherences to zero, we obtain a population rate equation:

$\frac{dP(n)}{dt}=-P(n)\sum_{m}\gamma_{n\rightarrow
m}+\sum_{m}\gamma_{m\rightarrow n}P(m),$ (23)

where the transfer rate between ground-states $n$ and $m$ is:

$\gamma_{n\rightarrow m}=\sum_{q}|\langle m|L_{\text{eff},q}|n\rangle|^{2}.$
(24)

Note that in this last equation, $L_{\text{eff}}$ can be taken out of the
interaction picture because as long as $n$ and $m$ are eigenstates of
$H_{\text{eff}}$, the time-dependence cancels upon taking the magnitude.
However, it is important to remember that the eigenstates must be those of
$H_{\text{eff}}$. The steady-state conditions can then be found by solving the
matrix equation for $\frac{dP(n)}{dt}=0$. In this paper, we derive analytic
solutions for different cooling schemes by picking an appropriate basis where
the coherences between eigenstates are least important and solving this rate
equation.

To compare different cooling schemes we need to set some parameter as a
metric. The energy or temperature of a single atom confined in a 1D harmonic
potential is given by $E=\hbar\nu(\langle n\rangle+1/2)$, where $\nu$ is the
trapping frequency, and $\langle n\rangle$ is the average motional quantum
number. However, it is important to note that the energy of a single atom in
an optical tweezer can be manipulated by adiabatically varying the optical
trap power [14], where both the trapping frequency and the energy scale with
the square root of the optical trap power. As a result, the motional quantum
number, $\langle n\rangle$, becomes a more relevant quantity to consider, as
it remains conserved during adiabatic changes in the trap depth. Therefore, we
choose to use $\langle n\rangle$ as the cooling metric for comparing all the
techniques in the following discussions.

##  III Sideband cooling

Figure 2:  (a) Resolved ($\Gamma<\nu$) and unresolved ($\Gamma>\nu$) sideband
cooling in the Lamb-Dicke regime. (b) The cooling laser drives the three blue
transitions. Spontaneous emission (red) most likely does not change $n$ in the
LD regime. (c-d) Simulation of $\langle n\rangle$ in the unresolved and
resolved regime for $\eta=0.05$. (e) The population versus trapping frequency.
(f) Cooling in a standing wave can lower the energy by positioning the atom at
the node, thereby decreasing scattering.

For an atom confined in a harmonic trap within the Lamb-Dicke (LD) regime,
Doppler cooling operates differently than in free space, as illustrated in
Fig. 2(a). In this scenario, the cooling laser drives transitions from the
motional state $n$ to $n-1$, while spontaneous emission predominantly returns
the atom to the $n-1$ state. This process leads to net cooling because, in the
LD regime, the recoil energy from photon emission is smaller than the energy
spacing between trap levels ($\hbar\nu$). Doppler cooling of trapped atoms can
be classified into two regimes based on the relationship between the trapping
frequency $\nu$ and the atomic transition linewidth $\Gamma$:

  * •

Resolved sideband cooling: when $\nu>\Gamma$

  * •

Unresolved sideband cooling: when $\nu<\Gamma$

These regimes exhibit distinct cooling dynamics and efficiencies, which we
will explore in subsequent sections. Sideband cooling was extensively studied
theoretically in the 1980’s by Refs. [36, 40, 30, 31]. It has been
demonstrated experimentally in ions [29], neutral atoms [41, 42], and
nanomechanical oscillators [28]. Here we present these results in our
formalism. We then look at the case of different trapping frequencies and
derive an analytic expression for the steady-state temperature.

Doppler cooling for a bound atom in the LD regime only requires a single
traveling wave $\bm{\epsilon}(\hat{x})=e^{-ik\hat{x}}$. For a two-level
system, the dipole operators are $\hat{D}=\hat{\sigma}=|g\rangle\langle e|$.
The Hamiltonian is

$H=\nu\hat{a}^{\dagger}\hat{a}+\Delta|e\rangle\langle
e|-\frac{\Omega}{2}\left(e^{-ik\hat{x}}\hat{D}^{\dagger}+h.c.\right),$ (25)

and the collapse operators for emission into the positive and negative
$x$-direction is

${L}_{\scriptscriptstyle\pm}=\sqrt{\Gamma}\,e^{\pm ikx}\hat{D}.$ (26)

The resonant and spontaneous emission processes are shown in Fig. 2(b). To
estimate the final population distribution, we can use the effective ground-
state collapse operator in Eq. 14 and the population transfer rates from Eq.
24. From these equations, the transfer rate between ground-states $n_{1}$ and
$n_{2}$ in the ground-state through the excited harmonic states $n_{e}$ is

$\gamma_{n_{1}\rightarrow
n_{2}}=\Gamma\,\,\Omega^{2}\sum_{\pm}\left|\sum_{n_{e}}\frac{\langle
n_{2}|e^{\pm ik\hat{x}}|n_{e}\rangle\langle
n_{e}|e^{ik\hat{x}}|n_{1}\rangle}{\Delta+\nu(n_{1}-n_{e})-i\Gamma/2}\right|^{2}.$
(27)

The sum inside the magnitude is over the excited state $n_{e}$ and pathways
through each excited motional state can interfere. The sum on the outside is
over the emission direction. Emission into each direction results in a
different phase, and the average over all directions averages out the
coherences between the different pathways. As a result, we can take the sum
over the excited states outside of the sum and omit the average over
directions,

$\gamma_{n_{1}\rightarrow
n_{2}}=\Gamma\,\,\Omega^{2}\sum_{n_{e}}\left|\frac{\langle
n_{2}|e^{ik\hat{x}}|n_{e}\rangle\langle
n_{e}|e^{ik\hat{x}}|n_{1}\rangle}{\Delta+\nu(n_{1}-n_{e})-i\Gamma/2}\right|^{2}.$
(28)

This rate describes two processes. The first term in the numerator $|\langle
n_{2}|e^{ik\hat{x}}|n_{e}\rangle|^{2}$ describes the process of recoil due to
spontaneous emission. The rest of the expression gives the rate of absorption
and recoil from photons from the cooling light. Together, these two terms give
the total rate of photon absorption and emission into motional states.

In the LD regime, the allowed motional state transition rates are

$\displaystyle\gamma_{n\rightarrow
n+1}=\eta^{2}(n+1)\,A_{\scriptscriptstyle+}$ (29)
$\displaystyle\gamma_{n\rightarrow n-1}=\eta^{2}(n)\,A_{\scriptscriptstyle-}$
(30)

where $A_{\scriptscriptstyle+}$ and $A_{\scriptscriptstyle-}$ are

$\displaystyle A_{\scriptscriptstyle+}=R(\Delta-\nu)+\alpha R(\Delta)$ (31)
$\displaystyle A_{\scriptscriptstyle-}=R(\Delta+\nu)+\alpha R(\Delta).$ (32)

The function $R(\Delta)$ is the low-saturation two-level system scattering
rate

$R(\Delta)=\Gamma\,\frac{\Omega^{2}/4}{\Delta^{2}+(\Gamma/2)^{2}}.$ (33)

The constant $\alpha$ is related to the averaging over emission directions.
Assuming emission into 1D gives $\alpha=1$, but averaging over emission into
all directions leads to $\alpha=1/3$ [40].

In the LD regime, transfer only occurs between neighboring states, in which
case the rate equation from Eq. 23 is

$\displaystyle\dot{P}(n)=\eta^{2}\,A_{\scriptscriptstyle-}\,(n+1)\,P(n+1)+\eta^{2}A_{\scriptscriptstyle+}\,n\,P(n-1)$
$\displaystyle-\eta^{2}\,A_{\scriptscriptstyle+}\,(n+1)\,P(n)-\eta^{2}A_{\scriptscriptstyle-}\,n\,P(n).$
(34)

The cooling laser (Fig. 2(b), blue) drives the $n+1$ and $n-1$ sidebands with
effective Rabi frequencies $\Omega\eta\sqrt{n+1}$ and $\Omega\eta\sqrt{n}$.
Decay (Fig. 2(b), red) of excited state to sideband occurs with probability
$\Gamma\eta^{2}n$.

Because the population only transfers between neighboring $n$ in the LD
regime, we can calculate the steady-state population by assuming no flow
between two states: $P(n+1)\,\gamma_{n+1\rightarrow
n}=P(n)\,\gamma_{n\rightarrow n+1}$. Solving this equation leads to the
steady-state condition

$\frac{P(n+1)}{P(n)}=\frac{A_{\scriptscriptstyle+}}{A_{\scriptscriptstyle-}}.$
(35)

We can set this equal to the Boltzmann factor to get the steady-state
temperature:

$\frac{P(n+1)}{P(n)}=\exp{(-\hbar\nu/k_{B}T)}.$

This represents a geometric distribution. Defining this ratio as $s$, the
normalized solution is $P(n)=(1-s)s^{n}$, and the expectation and variance of
$n$ is $\langle n\rangle=\frac{s}{1-s}$ and $\langle(\Delta
n)^{2}\rangle=\frac{s}{(1-s)^{2}}$.

Multiplying by $n$ and summing over $n$ gives the time rate equation for
$\langle n\rangle$:

$\displaystyle\frac{d\langle
n\rangle}{dt}=\eta^{2}\left[A_{\scriptscriptstyle+}(\langle
n\rangle+1)-A_{\scriptscriptstyle-}\langle n\rangle\right]$ (36)

The cooling rate is
$W_{A}=\eta^{2}(A_{\scriptscriptstyle-}-A_{\scriptscriptstyle+})$, which is
balanced by the constant heating from spontaneous emission
$\eta^{2}A_{\scriptscriptstyle+}$. The steady-state solution is

$\displaystyle\langle
n\rangle=\frac{A_{\scriptscriptstyle+}}{A_{\scriptscriptstyle-}-A_{\scriptscriptstyle+}}=\frac{R(\Delta-\nu)+\alpha
R(\Delta)}{R(\Delta+\nu)-R(\Delta-\nu)}.$ (37)

In the resolved-sideband regime ($\Gamma\ll\nu$), the lowest temperature
occurs at the cooling sideband $\Delta=-\nu$, at which the steady-state is

$\langle
n\rangle=\frac{1}{4}\left(\frac{\Gamma}{\nu}\right)^{2}\left(\alpha+\frac{1}{4}\right).$
(38)

In the unresolved-sideband regime ($\Gamma\gg\nu$), the steady state is [36,
30]

$\langle
n\rangle=\frac{(1+\alpha)}{8}\left(\frac{\Gamma}{2\Delta}+\frac{2\Delta}{\Gamma}\right)-\frac{1}{2}.$
(39)

In Fig. 2 we simulate the exact solution using the master equation from Eq. 3.
Fig. 2(c-d) are in the unresolved and resolved sideband regimes for
$\eta=0.05$ and in the low power limit $\Omega=0.05/\Gamma$. In the resolved
regime in (d), best cooling occurs at $\Delta=-\nu$ and agrees closely with
Eq. 38. The unresolved regime in (e) is a zoomed plot of the lower left corner
of (d). Theoretically, the lowest temperature occurs for $\Delta=-\Gamma/2$,
although the exact solution shows that it is broad and not much better than
the resolved criteria $\Delta=-\Gamma$.

In plot Fig. 2(e), we plot steady-state population versus the trapping
frequency for various LD parameters $\eta$. The population converges for
$\eta<0.1$.

The cooling performance of sideband cooling can be improved by adding a second
counter-propagating beam with the same polarization to form a standing wave
and placing the atom at the node of the field [43]. Although the atom now sees
zero intensity at the equilibrium position, the sidebands are still driven.
This can be seen by expanding the atom-field interaction as a series to first
order in terms of the LD parameter and field gradient

$H_{I}=-\frac{\Omega}{2}\left[\left(\bm{\epsilon}(0)+\hat{x}\frac{d\bm{\epsilon}(0)}{d\hat{x}}\right)\cdot\bm{D}^{\dagger}+h.c.\right].$
(40)

The gradient can be either a phase or amplitude gradient, both of which drive
the sideband. Intensity at the position of the atom does not lead to cooling
and, in fact, only leads to extra heating. If we place the atom in a standing
wave with linear polarization, the field is
$\bm{\epsilon}(\hat{x})=\sin(k\hat{x})$. Fig. 2(f) simulates the steady-state
population versus atom position with $\nu$ = $-\Delta$. Theoretically, this
gets rid of the process in Fig. 2(b) of driving $n$ to $n$, followed by
spontaneous emission heating to the sideband. The steady-state solution is the
same except with $\alpha=0$. Cooling can be further enhanced by placing the
atom in a cavity that can enhance the red sideband [44, 45].

###  III.1 Excited state with different trapping frequency

The Doppler cooling section assumed that the electronic ground and excited
states have the same trapping frequency. But typically, the ground and excited
states of neutral atoms have different polarizabilities except at “magic
wavelengths.” The different potentials lead to heating due to the abruptly
changing dipole force during absorption or emission, shown in the schematics
in Fig. 3. In the extreme case, the excited state is anti-trapped resulting in
loss of the atom if the excited lifetime is long compared to $1/\nu$.

Because magic wavelengths are not often convenient for optical trapping,
neutral atoms more often use two-photon cooling schemes, such as polarization
gradient cooling and Raman sideband cooling. However, alkaline-earth atoms
contain narrow linewidth transitions that have been used for imaging and
cooling [46, 10], even ground-state cooling in Ref. [47]. Alkali atoms also
contain narrow lines from forbidden transitions like the quadrupole transition
of the S to D transition [48] or (n)S to (n+1)P transitions [49].

While sideband cooling with different trapping frequencies has been
investigated through simulations [50], in this section we derive a novel
analytical formula for capturing the effects of different trapping
frequencies, which also demonstrates additional opportunities for cooling on
higher order sidebands.

We now derive the equations for an atom with a ground-state frequency $\nu$
and an excited state frequency $\nu_{e}$. The excited state harmonic
eigenstates are stretched or compressed relative to the ground-state
eigenstates. The squeezing operator from quantum optics can therefore be
utilized to relate the wavefunctions in the two potentials: [51, 52, 53]

$|m_{e}\rangle=\hat{S}^{\dagger}(r)|n_{e}\rangle,$ (41)

where $|n_{e}\rangle$ is the ground-state basis $|m_{e}\rangle$ is the excited
state basis. The squeezing operator $\hat{S}(r)$ is

$\hat{S}(r)=\exp{\left(\frac{r}{2}(\hat{a}^{2}-\hat{a}^{{\dagger}2})\right)}.$
(42)

The squeezing parameter $r$ is related to the trapping frequencies by

$r=\frac{1}{2}\ln\left(\frac{\nu_{e}}{\nu}\right).$ (43)

The excited state annihilation operator is related to the ground-state
annihilation operator by

$\hat{a}_{e}=\hat{S}^{\dagger}(r)\,\hat{a}\,\hat{S}(r).$ (44)

The harmonic Hamiltonian is

$\displaystyle H_{\text{ h}}$
$\displaystyle=\nu(\hat{a}^{\dagger}\hat{a})P_{g}+\nu_{e}(\hat{a}_{e}^{\dagger}\hat{a}_{e})P_{e}$
$\displaystyle=\nu(\hat{a}^{\dagger}\hat{a})P_{g}+\nu_{e}\,\,\hat{S}^{\dagger}(r)(\hat{a}^{\dagger}\hat{a})\hat{S}(r)\,\,P_{e}.$
(45)

Figure 3:  Sideband cooling with different trapping frequencies. (a) Two
regimes, where the excited state is less trapped (left) or more trapped
(right) result in extra heating. This heating comes from coupling many
different motional states in the quantum picture, or a fluctuating dipole
force in the classical picture. For different trapping frequencies, the $n$ to
$n-1$ transition can be resonant for only one specific $n$. For $\nu_{e}<\nu$,
higher $n$ gradually shift more blue towards heating. For $\nu_{e}>\nu$,
higher $n$ gradually shifts more red, leading to cooling of higher order
sidebands. (b) Population for a different excited state trapping frequency
$\nu_{e}$, plotted for different $\nu/\Gamma$ and $\eta=0.1$ and
$\Delta=-\nu$. The magic condition is $\nu_{e}/\nu=1$. (c) Population versus
detuning for different excited to ground trapping frequency ratios
$\nu_{e}/\nu$. A minimum occurs at $\Delta=-\nu$, but a second minimum occurs
at $\Delta=-2\nu$ when $\nu_{e}\neq\nu$. Plot is for $\eta=0.1$,
$\nu/\Gamma=2$. Eq. 59 is plotted in a light dashed line. Different trapping
frequencies allow the coupling of higher order sidebands, providing new
opportunities for cooling.

The transfer rates between the ground-states are similar to Eq. 27, where now
the squeezing operators are used to modify the excited harmonic states:

$\displaystyle\gamma_{n_{1}\rightarrow n_{2}}=\Gamma\,\,\Omega^{2}$ (46)
$\displaystyle\times\sum_{\scriptscriptstyle\pm}\left|\sum_{n_{e}}\frac{\langle
n_{2}|e^{\pm ik\hat{x}}\hat{S}^{\dagger}(r)|n_{e}\rangle\langle
n_{e}|\hat{S}(r)e^{ik\hat{x}}|n_{1}\rangle}{(\Delta+\nu
n_{1}-\nu_{e}n_{e})-i\Gamma/2}\right|^{2}.$

The sum over the excited states cannot generally be taken out of the magnitude
now because each term does not necessarily have the directional emission
terms. The transition matrix elements now contain the squeezing operator. To
first order in $r$ and $\eta$, the matrix elements are

$\displaystyle\langle n_{e}|\hat{S}(r)e^{ik\hat{x}}|n_{1}\rangle$
$\displaystyle=\langle
n_{e}|\left(1+i\eta(\hat{a}+\hat{a}^{\dagger})+\frac{r}{2}(\hat{a}^{2}-{(\hat{a}^{\dagger})}^{2})\right)|n_{1}\rangle.$
(47)

The first order LD terms couple $n\rightarrow n\pm 1$. In contrast, the first-
order squeezing terms couple $n\rightarrow n\pm 2$, preserving the symmetry of
the wavefunction.

The resonance condition for the light absorption is now determined by the
detuning $\Delta+\nu n_{1}-\nu_{e}n_{e}$, which we can re-express in terms of
the difference trapping frequency can be rewritten as $\Delta+\nu
n_{1}-\nu_{e}n_{e}=(\Delta+n_{e}\Delta\nu)+\nu(\nu_{1}-\nu_{e})$, where the
trapping frequency difference is $\Delta\nu=\nu_{e}-\nu$. For increasing $n$,
the detuning between $n_{1}=n_{e}$ changes by $\nu n$. We define the resonance
condition for the $n$’th harmonic state as

$\Delta_{n}=\Delta+n\,\Delta\nu.$ (48)

The $n\rightarrow n\pm 1$ transition are similar to before

$\displaystyle\gamma_{n\rightarrow
n+1}=\eta^{2}\,(n+1)\,A_{\scriptscriptstyle+}$ (49)
$\displaystyle\gamma_{n\rightarrow n-1}=\eta^{2}\,n\,A_{\scriptscriptstyle-}.$
(50)

The rates are a function of the $n$ dependent detunings

$\displaystyle A_{\scriptscriptstyle+}=R({\Delta}_{n}-\nu)+\alpha
R({\Delta}_{n})$ (51) $\displaystyle
A_{\scriptscriptstyle-}=R({\Delta}_{n}+\nu)+\alpha R({\Delta}_{n}).$ (52)

In this context, the rates add incoherently due to the randomized phase
acquired during emission in multiple directions. As illustrated in Fig. 3(a),
the $n$-dependent detuning results in the first-order cooling sideband being
resonant only for a specific vibrational quantum number $n$. For a less
tightly trapped excited state (left panel), the cooling light shifts towards
heating sidebands for higher $n$ values. Conversely, lower $n$ values
gradually shift towards cooling of higher-order sidebands. This effect leads
to instability when $\nu_{e}/\nu<1$, where higher-energy populations heat out
of the trap. When $\nu_{e}/\nu>1$, a type of “cap” emerges, with heating
occurring below a certain $n$ value and cooling above it. Consequently, this
cap effectively traps population below it.

The squeezing operator now also allows $n\rightarrow n\pm 2$ transitions to
first order in $r$:

$\displaystyle\gamma_{n\rightarrow
n+2}=r^{2}\,(n+2)(n+1)\,\,B_{\scriptscriptstyle+}$ (53)
$\displaystyle\gamma_{n\rightarrow
n-2}=r^{2}\,n(n-1)\,\,B_{\scriptscriptstyle-}.$ (54)

The new rates $B$ are a sum over different excited states, but the
coherence/interference now matters because the coupling does not involve
spontaneous emission to first order,

$\displaystyle
B_{\scriptscriptstyle-}=\left|{T}({\Delta}_{n})-{T}({\Delta}_{n}+2\nu)\right|^{2}$
(55) $\displaystyle
B_{\scriptscriptstyle+}=\left|{T}({\Delta}_{n})-{T}({\Delta}_{n}-2\nu)\right|^{2}.$
(56)

The complex scattering rates are

$T(\Delta)=\sqrt{\Gamma}\frac{\Omega/2}{\Delta-i\Gamma/2}.$ (57)

and are related to the $R$ rates by $|T(\Delta)|^{2}=R(\Delta)$.
Interestingly, the destructive interference between the two excited pathways
in the $n\rightarrow n\pm 2$ transition results in a transfer rate that
cancels for large detuning, leading to much lower heating rates at large
detuning.

Next we estimate the final temperature as a function of $\Delta\nu$. We
construct a rate equation for $\langle n\rangle$ by multiplying the rate
equation in Eq. 23 by $n$ and summing over all $n$. This leads to the rate
equation

$\displaystyle\frac{d\langle
n\rangle}{dt}=\eta^{2}\left[A_{\scriptscriptstyle+}(\langle
n\rangle+1)-A_{\scriptscriptstyle-}\langle n\rangle\right]$
$\displaystyle+2r^{2}\left[B_{\scriptscriptstyle+}\langle(n+2)(n+1)\rangle-
B_{\scriptscriptstyle-}\langle n(n-1)\rangle\right].$ (58)

Although we would need a separate expression for $\langle n^{2}\rangle$ to
solve this exactly, we can find an approximate solution by assuming the
population follows a geometric series with a Boltzmann factor, as was the case
in the Doppler section. Then we can use the property $\langle
n^{2}\rangle=2\langle n\rangle^{2}+\langle n\rangle$ for geometric
distributions and solve the equation numerically to estimate the steady-state
$\langle n\rangle$, which gives the equation

$0=\eta^{2}\left[\langle
n\rangle(A_{\scriptscriptstyle+}-A_{\scriptscriptstyle-})+A_{\scriptscriptstyle+}\right]\\\
+4r^{2}\left[\langle
n\rangle^{2}(B_{\scriptscriptstyle+}-B_{\scriptscriptstyle-})+B_{\scriptscriptstyle+}(2\langle
n\rangle+1)\right].$ (59)

$B_{\pm}$ and $A_{\pm}$ are also $n$ dependent through the detuning
$\Delta_{n}$. This solution agrees well for small $r$ and $\eta$, with the
main assumption that the distribution follows a geometric series. This will
break down for high populations as the cooling and heating become $n$
dependent.

Cooling is a balance between the sideband and squeezing terms. Cooling due to
sidebands is $W_{A}\approx\eta^{2}(A_{-}-A_{+})$, while squeezing cooling is
$W_{B}\approx 2r^{2}(B_{-}-B_{+})$, with the squeezing dependence being
quadratic. Additionally, there is heating from both that goes as
$\eta^{2}A_{+}$ and $2r^{2}B_{+}$, respectively. Interestingly, the squeezing
results in a coupling between different motional states and can be used for
cooling, as shown below.

Fig. 3(b) is a master equation simulation of steady-state population versus
the excited to ground-state trapping frequency ratio $\nu_{e}/\nu$. The magic
condition is $\nu_{e}/\nu=1$. Cooling performance drops on both sides, with
much more dramatic for a weaker excited state ($\nu_{e}/\nu<1$). Fig. 3(c) is
a detuning scan for $\eta=0.1$, $\nu=2\Gamma$, and various $\nu_{e}/\nu$.
There is a resonance at $\Delta=-\nu$ due to sideband cooling, but also a
second resonance at $\Delta=-2\nu$ due to the squeezing coupling.

For cooling the atom in a ‘non-magic’ trap, a technique known as “Sisyphus
Cooling”, has been recently demonstrated in tweezers [54, 55]. In situations
where the polarizability of the excited state is greater (or less) than that
of the ground-state, the atom experiences a deeper (or shallower) trap upon
excitation by the cooling laser. This difference enables the implementation of
‘attractive’ (or ‘repulsive’) Sisyphus cooling. In the repulsive Sisyphus
cooling method, there is a limit known as the Sisyphus cap to how much the
atom can be cooled. However, in the attractive Sisyphus regime, significantly
lower temperatures can be achieved within the trap. Ref. [50] proposes that
sweeping the cooling laser’s frequency adiabatically could achieve
temperatures well below the Sisyphus cap.

##  IV Sub-doppler cooling with spin

Figure 4:  Spin cooling. (a) An atom with spin in a harmonic trap with
changing polarization. (b) Lin$\perp$lin and $\sigma_{+}-\sigma_{-}$
polarization. (c-d) Free-space (top) and quantum picture (bottom) for PG
cooling, which uses $F\rightarrow F+1$ and red detuning. Population is
optically pumped into the lower energy state. (e-f) Free-space (top) and
quantum picture (bottom) for GM cooling, which uses $F^{\prime}\leq F$ and
blue-detuning to create dark states, which act as the lower energy state into
which the population is pumped.

Doppler cooling alone is often insufficient for achieving ground-state cooling
of neutral atoms in optical traps due to two main issues. First, the trapping
frequencies in optical traps are typically less than 100 kHz, while the line
widths of the D1 and D2 transitions are around 5 MHz, placing them far into
the unresolved regime. Second, the ground and excited states have different
optical polarizabilities, which can lead to additional heating and atom loss
due to far-off-resonant trapping light.

To overcome these limitations, sub-Doppler cooling schemes such as
polarization gradient (PG) [13], gray molasses (GM) and $\Lambda$-enhanced GM
[17, 18], electromagnetically induced transparency (EIT) cooling [56], and
Raman sideband (RSB) [27] employ two-photon transitions between ground-states,
as illustrated in Fig. 4(a). By detuning the cooling laser far from the
excited states, these techniques mitigate the issues associated with the
unresolved regime and differential polarizabilities. Furthermore, the ground
hyperfine states in neutral atoms generally experience the same light shifts,
resulting in similar trapping frequencies for different spin states.

Two-photon cooling schemes rely on coupling two spin states with different
motional quantum numbers, $n$, and energies. The quantum model for PG and GM
cooling is depicted in Fig. 4(d,f). In the first stage, a two-photon coherent
transition (green) changes the spin and motional states, where the motional
state decreases from $n$ to $n-1$. In the second stage, spontaneous emission
(red) from the cooling light or a separate beam then re-pumps the atom back to
the original spin state, allowing the cooling cycle to repeat. As long as the
trapping frequency energy, $\hbar\nu$ is larger than the recoil energy,
$E_{\text{recoil}}$, net cooling can be achieved, as the average heating from
spontaneous emission is limited to one photon recoil.

In this section, we present a unified spin cooling formalism that encompasses
PG, GM, EIT, $\Lambda$-GM, and RSB cooling. We show that all these schemes
exhibit similar forms of the effective Hamiltonian and collapse operators in
the LD regime and rely on the same principles. PG and GM cooling use a single
counter-propagating laser frequency to create light shifts similar to the
trapping frequency, as well as the Raman coupling and optical pumping. EIT and
$\Lambda$-GM cooling incorporate a second coherent laser to address other spin
states and exploit the dark states of a three-level $\Lambda$ system to create
darker reservoir states. RSB cooling separates all three processes, using two
lasers to couple different ground spin states and a third frequency and beam
for optical pumping that uses angular momentum selection rules to keep the
cooling state dark.

###  IV.1 Polarization Gradient Cooling and Gray Molasses

PG cooling was described theoretically by Cohen-Tannoudji in the late 1980’s
[13] and resulted in sub-doppler temperatures in the first magneto-optical
traps [57]. Two different PG models were presented in free space for the case
of lin$\perp$lin and
$\sigma_{\scriptscriptstyle+}\\!-\\!\sigma_{\scriptscriptstyle-}$
polarizations, shown in Fig. 4(b). The lin$\perp$lin configuration comes from
counter-propagating orthogonal linear polarization beams, resulting in a
polarization that changes from linear to circular. This model works on a
$F=1/2$ to $F^{\prime}=3/2$ atom, and the free-space picture is shown in Fig.
4(c). The circular polarization results in a vector shift $\Delta E\propto
m_{J}$ that modulates every wavelength. PG cooling works by having an optical
pumping that pumps the atom to the lower potential. As the atom travels, it
goes up the potential hill, loses energy, and then is pumped back to the
bottom. The cycle repeats until the atom reaches the recoil temperature
$E_{\text{recoil}}$, which is typically a few $\mu$K. This style of cooling is
also called Sisyphus cooling, named after the Greek mythological figure who
was condemned to eternally push a boulder up a hill.

The $\sigma_{\scriptscriptstyle+}\\!-\\!\sigma_{\scriptscriptstyle-}$ model is
applicable when counter-propagating laser beams with opposite circular
polarizations are used. As shown in Fig. 4(b), the resulting polarization is
linear at every point in space but rotates along the propagation direction
with a period equal to the wavelength. The
$\sigma_{\scriptscriptstyle+}\\!-\\!\sigma_{\scriptscriptstyle-}$ model relies
on the tensor shift and therefore requires a ground-state spin of at least
$F=1$. This version is more common because it is also the polarization
configuration used by a magneto-optical trap (MOT).

Both forms of PG cooling work by using red-detuned light with a $J\rightarrow
J+1$ transition, which is the same configuration used by a MOT for its stretch
state cycling condition.

Gray Molasses (GM) is a closely related cooling scheme that operates with
blue-detuned light with a $F\rightarrow F$ or $F\rightarrow F-1$ transition.
GM has achieved even colder temperatures than PG cooling [58, 17, 59] by
storing cold atoms in dark states that do not interact with the cooling light
due to polarization and angular momentum selection rules. For example in Fig.
4(e), certain ground-states are not coupled to the excited states. GM works
with blue detuned light so that the bright states are higher in energy. Atoms
that fall into these dark states are effectively shelved, reducing their
interaction with the cooling light. The residual velocity couples the atom
back into a bright state, where they can undergo Sisyphus cooling again.

However, these free-space models break down for a trapped atom in the Lamb-
Dicke (LD) regime, where the atom’s wavefunction is smaller than the cooling
wavelength. While PG and GM have been implemented for atoms in optical
lattices, tweezers [21, 60, 61], and ions [62], they have not been described
theoretically. The most related works have been Wineland et al. in Ref. [33],
which studied a bound atom trapped in a linearly polarized standing wave, and
Cirac et al. in Ref. [32] which investigated a $J=1/2\rightarrow
J^{\prime}=3/2$ atom in the lin$\perp$lin configuration. Both of these studies
found a lower limit for the mean vibrational quantum number, $\langle
n\rangle\approx 1$. Here we develop a generalized spin cooling model that
applies to all spins and polarizations and shows that certain configurations
can lead to ground-state cooling.

###  IV.2 Spin cooling theory

First we look at the case of two counter-propagating beams with opposite-
handed circular polarized light
($\sigma_{\scriptscriptstyle+}\\!-\\!\sigma_{\scriptscriptstyle-}$ ). The
complex polarization is

$\displaystyle\bm{\epsilon}_{\sigma_{\scriptscriptstyle+}\\!-\\!\sigma_{\scriptscriptstyle-}}(\hat{z})=\frac{i}{\sqrt{2}}\left(e^{ik\hat{z}}\hat{\bm{e}}_{1}+e^{-ik\hat{z}}\hat{\bm{e}}_{-1}\right)$
$\displaystyle=\sin{(k\hat{z})}\,\hat{\bm{x}}+\cos{(k\hat{z})}\,\hat{\bm{y}}$
$\displaystyle\approx\bm{\hat{y}}+(k\hat{z})\,\bm{\hat{x}}$ (60)

where $\hat{\bm{e}}_{1,0,-1}$ are the spherical unit vectors
$\hat{\bm{e}}_{\pm 1}=\mp\frac{1}{\sqrt{2}}(\hat{\bm{e}}_{x}\pm
i\hat{\bm{e}}_{y})$ and $\hat{\bm{e}}_{0}=\hat{\bm{e}}_{z}$, and also The
field is Taylor expanded in the last line. A fixed atom sees only linear
light.. The intensity is constant everywhere. The polarization is linear but
rotates every wavelength, as shown in Fig. 4(b). As it oscillates back and
forth in the trap, it sees a small rotation of the field. For convenience, we
choose the quantization axis $\hat{\bm{z}}$ to be aligned with linear
polarization at equilibrium. The cooling is however independent of the atom
position because the light is linear everywhere.

The Hamiltonian and collapse operator in the LD regime are

$\displaystyle
H_{\sigma_{\scriptscriptstyle+}-\sigma_{\scriptscriptstyle-}}=\nu\hat{a}^{\dagger}\hat{a}+(\Delta-i\Gamma/2)P_{e}$
$\displaystyle+\frac{\Omega}{\sqrt{2}}\left[\hat{D}^{\dagger}_{z}-i\eta\,(\hat{a}+\hat{a}^{\dagger})\hat{D}_{x}^{\dagger}+h.c.\right]$
(61) $\displaystyle\hat{L}_{q,\pm}=\sqrt{\Gamma}\,(1\pm ik\hat{x})\hat{D}_{q}$
(62)

The Cartesian dipole operators are defined as
$D_{i}=\hat{\bm{i}}\cdot\hat{\bm{D}}$.

The PG cooling processes are illustrated in Fig. 4(d) for a $F=1$ atom. The
position-independent interaction $\hat{D}_{z}$ leads to a tensor shift and
optical pumping. For the case of $F=1$ ground-state, the population is pumped
(red) to the $m=0$ state. The position-dependent term in the interaction
$\eta(\hat{a}+\hat{a}^{\dagger})\hat{D}_{x}$ drives an orthogonal polarization
transition $m\rightarrow m\pm 1$ while also changing the motional state. This
drive is reduced by the LD parameter and produces heating due to spontaneous
emission.

The last ingredient for cooling is a two-photon coherent Raman transition
(green). One branch of this transition comes from the position-independent
term, while the other branch comes from the position-dependent term. As shown
in the illustration, together these terms produce a coherent transition from
$|m=0,n\rangle$ to $|m=\pm 1,n-1\rangle$. As drawn, the resonance for this
two-photon condition is satisfied when the differential light shift between
the $m=0$ and $m=\pm 1$ states are equal to the trapping frequency. This
process then repeats, cooling the atom.

These processes can be seen more clearly by looking at the effective ground-
state operators from Eq. 20. The effective ground-state Hamiltonian is

$\displaystyle
H^{\text{eff}}_{\sigma_{\scriptscriptstyle+}-\sigma_{\scriptscriptstyle-}}=\nu\hat{a}^{\dagger}\hat{a}-$
(63)
$\displaystyle\frac{\Omega^{2}}{4\Delta}\left[\hat{D}_{z}\hat{D}_{z}^{\dagger}+k\hat{x}(\hat{D}_{z}\hat{D}_{x}^{\dagger}+\hat{D}_{x}\hat{D}_{z}^{\dagger})\right].$

Converting it to spin operators using Eq. 19 gives an effective spin model

$\displaystyle
H^{\text{eff}}_{\sigma_{\scriptscriptstyle+}-\sigma_{\scriptscriptstyle-}}=\nu\hat{a}^{\dagger}\hat{a}+\frac{\Omega^{2}}{\Delta}\left[C^{(0)}+C^{(2)}(\hat{F}_{z}^{2}-\hat{F}^{2}/3)\right]$
$\displaystyle+k\hat{x}\,\,\frac{\Omega^{2}}{\Delta}\,C^{(2)}\,(\hat{F}_{x}\hat{F}_{z}+\hat{F}_{z}\hat{F}_{x}).$
(64)

The collapse operators are

$\displaystyle
L_{q,\pm}^{\text{eff}}=\sqrt{\Gamma}\frac{\Omega/2}{\Delta-i\Gamma/2}\left[\hat{D}_{q}\,(\hat{D}_{z}^{\dagger}+ik\hat{x}(\hat{D}_{z}^{\dagger}+\hat{D}_{x}^{\dagger})))\right].$
(65)

A spin version of this operator can also be found using Eq. 21.

This spin Hamiltonian is valid for all ground and excited spin combinations.
While the ground-state spin determines the spin operators $F_{i}$, the excited
state spin only affects the $C^{(i)}$ coefficients. The Hamiltonian and
collapse operator both have a position-independent and position-dependent
term.

This expansion can be performed for other polarizations as well to get a
similar form. We can also look at the lin$\perp$lin case, where the
polarization changes from circular to linear depending on the atom position,

$\bm{\epsilon}_{\text{lin}\perp\text{lin}}(\hat{x},\phi)=\frac{1}{2}(e^{ik\hat{x}}\hat{\bm{x}}+e^{i\phi}e^{-ik\hat{x}}\hat{\bm{y}}).$
(66)

For circular polarization, the interaction is

$H^{I}_{\text{lin}\perp\text{lin}}(\phi=\pi/2)=\frac{\Omega}{2}\left[\hat{D}_{1}^{\dagger}+\eta(\hat{a}+\hat{a}^{\dagger})\hat{D}_{-1}^{\dagger}+h.c.\right],$
(67)

and for linear polarization,

$H^{I}_{\text{lin}\perp\text{lin}}(\phi=0)=\frac{\Omega}{2}\left[\hat{D}_{z}^{\dagger}+\eta(\hat{a}+\hat{a}^{\dagger})\hat{D}_{x}^{\dagger}+h.c.\right].$
(68)

Before performing the full master equation simulations, our next goal is to
develop a simple analytical model for a system with a Hamiltonian and collapse
operator in these forms.

Figure 5:  PG and GM simulations. (a-b) Clebsch-Gordon coefficients for PG
($F=1$ to $F^{\prime}=1$) and GM ($F=1$ to $F^{\prime}=2$). For PG in linear
light, the population pumps to the $m=0$ state, which is also the brightest
state. PG cooling requires red-detuned light $(\Delta<0)$. For GM in linear
light, the population pumps to $m=0$ which is dark. GM cooling requires blue-
detuned light $(\Delta>0)$. (c-d) Simulations of $\langle n\rangle$ versus
cooling power for PG and GM. Optimal cooling occurs on a line (light-dashed)
where the differential light shift equals the trapping frequency. PG is
limited to $\langle n\rangle\approx 1$, while GM cools close to the ground-
state due to the dark state. Simulations use $\eta=0.05$ and 8 harmonic
states. (e) Optimal cooling for PG and GM versus the Lamb-Dicke parameter
$\eta$. Optimal cooling occurs when the differential light shift is equal to
the trapping frequency and in the limit of large detuning. The simple model
from Eq. 73 is plotted in a lighter color. For PG, $\Delta=-20\Gamma$ and for
GM $\Delta=20\Gamma.$ The optical cooling powers were adjusted to make the
Raman transition resonant.

We derive an approximate temperature for a simplified model that captures both
regimes. We use two spin states $|\\!\\!\downarrow,n\rangle$ and
$|\\!\\!\uparrow,n\rangle$ and an effective Hamiltonian and collapse operators

$\displaystyle
H=\nu\hat{a}^{\dagger}\hat{a}+V_{0}(\hat{F}_{z}+1/2)+\Omega_{R}k\hat{x}\,\hat{F}_{x}$
$\displaystyle
L_{1}=\sqrt{\gamma_{\scriptscriptstyle+}}\,\,(\hat{F}_{+}+ik\hat{x}\hat{F}_{x})$
$\displaystyle
L_{-1}=\sqrt{\gamma_{\scriptscriptstyle-}}\,\,(\hat{F}_{-}+ik\hat{x}\hat{F}_{x}).$
(69)

$V_{0}$ is a spin-dependent light shift. Position-independent collapse parts
with pumping rates $\gamma_{+}$ and $\gamma_{-}$ are the optical pumping
between the spin states. The $\Omega_{R}k\hat{x}\hat{F}_{x}$ term produces
Raman transitions between $|\uparrow,n\rangle$ and $|\downarrow,n-1\rangle$.
$\Omega_{R}=\Omega^{2}/\Delta C^{(2)}$. The optimal light shift for cooling is
to make these two states the same frequency by setting $V_{0}=\nu$. The
$|\downarrow,n\rangle$ couples to $|\uparrow,n-1\rangle$, but off-resonantly
so we can ignore it.

To find an approximate expression for the steady-state temperature, we first
use the master equation to determine the steady-state coherences between the
Raman-coupled states:

$\displaystyle\langle\uparrow,n|\rho|\downarrow,n\\!-\\!1\rangle=\
\frac{i\sqrt{n}\,\eta\,\Omega_{R}}{\Delta-i(\gamma_{\scriptscriptstyle-}\\!+\\!\gamma_{\scriptscriptstyle+})/2}\,\left[P_{\scriptscriptstyle\uparrow}(n)-P_{\scriptscriptstyle\downarrow}(n\\!-\\!1)\right]$
(70)

By substituting this coherence into the density matrix equations and ignoring
off-resonance coherences, we obtain the transfer rate $\Gamma_{R}$ between
$|\uparrow,n\rangle$ to $\downarrow,n-1\rangle$:

$\Gamma_{R}=\frac{4\eta^{2}\Omega_{R}^{2}}{(\gamma_{\scriptscriptstyle+}+\gamma_{\scriptscriptstyle-})}\,\,\frac{1}{1+4\Delta_{r}^{2}/(\gamma_{\scriptscriptstyle+}+\gamma_{\scriptscriptstyle-})^{2}}.$
(71)

$\Delta_{r}$ is the relative detuning between the two states. Optimal cooling
occurs when the light shifts bring these states into resonance, i.e.,
$\Delta_{r}=0$. Although there are off-resonant transitions, such as
$|n,\downarrow\rangle\rightarrow|n-1,\uparrow\rangle$, we ignore them in this
simple model. By neglecting all off-resonant coherences, we arrive at the
population rate equations:

$\displaystyle\frac{dP_{\scriptscriptstyle\downarrow}(n)}{dt}=\Gamma_{R}\,\,(n\\!+\\!1)\left[-P_{\scriptscriptstyle\downarrow}(n)\\!+\\!P_{\scriptscriptstyle\uparrow}(n\\!+\\!1)\right]$
(72)
$\displaystyle+\gamma_{\scriptscriptstyle-}P_{\scriptscriptstyle\uparrow}(n)-\gamma_{\scriptscriptstyle+}P_{\scriptscriptstyle\downarrow}(n)$
$\displaystyle+\eta^{2}\gamma_{\scriptscriptstyle
h}\left[(n+1)P_{\scriptscriptstyle\uparrow}(n\\!+\\!1)+nP_{\scriptscriptstyle\uparrow}(n\\!-\\!1)-(2n+1)P_{\scriptscriptstyle\downarrow}(n)\right]$
$\displaystyle\frac{dP_{\scriptscriptstyle\uparrow}(n)}{dt}=n\Gamma_{R}\,\left[P_{\scriptscriptstyle\downarrow}(n-1)-P_{\scriptscriptstyle\uparrow}(n)\right]$
$\displaystyle-\gamma_{\scriptscriptstyle-}P_{\scriptscriptstyle\uparrow}(n)+\gamma_{\scriptscriptstyle+}P_{\scriptscriptstyle\downarrow}(n)$
$\displaystyle+\eta^{2}\gamma_{\scriptscriptstyle
h}\left[(n+1)P_{\scriptscriptstyle\downarrow}(n\\!+\\!1)+nP_{\scriptscriptstyle\downarrow}(n\\!-\\!1)-(2n+1)P_{\scriptscriptstyle\uparrow}(n)\right]$

Here $\gamma_{h}=\gamma_{+}+\gamma_{-}$ is a scattering rate related to
spontaneous emission heating. Next, we calculate the steady state solution for
$\langle n\rangle$. The derivation is as follows. First, we sum the two
equations in Eq. 72, yielding a single equation. Next, we multiply the
distribution equations by $n$ and sum from $n=0$ to $\infty$, resulting in two
equations containing higher-order moments $\langle n^{2}\rangle_{\downarrow}$
and $\langle n^{2}\rangle_{\uparrow}$.

In these four expressions, we have six unknowns: $\langle
n^{2}\rangle_{\uparrow}$, $\langle n^{2}\rangle_{\downarrow}$, $\langle
n\rangle_{\uparrow}$, $\langle n\rangle_{\downarrow}$, and the fraction of
population in either up or down $P_{\uparrow}=\sum_{n=0}P_{\uparrow}(n)$,
$P_{\downarrow}=\sum_{n=0}P_{\downarrow}(n)$. We make these equations solvable
by simplifying the higher-order moments by assuming that both
$P_{\uparrow}(n)$ and $P_{\downarrow}(n)$ follow a geometric distribution.

We can express $\langle n^{2}\rangle_{\uparrow}$ as $\langle
n^{2}\rangle_{\uparrow}=2\langle n\rangle_{\uparrow}/P_{\uparrow}+\langle
n\rangle_{\uparrow}$, where $P_{\uparrow}$ and $P_{\downarrow}$ appear because
they are not a normalized distribution. Then we solve the system of equations
for $\langle n\rangle_{\uparrow}+\langle n\rangle_{\downarrow}=\langle
n\rangle$.

We then take the limit of large detuning to obtain a simpler analytical
expression compared to the complete solution. In the large detuning limit, the
cooling rate
$\Gamma_{R}=4\Omega_{R}^{2}\eta^{2}/(\gamma_{\scriptscriptstyle+}+\gamma_{\scriptscriptstyle-})$
dominates. The cooling rate $\Gamma_{R}$ is independent of detuning because
both $\Omega_{R}^{2}$ and the scattering rate scale as $1/\Delta^{2}$.
Consequently, for large detuning, $\Gamma_{R}$ becomes the dominant rate, and
the population quickly distributes between the two Raman-coupled states. The
asymmetry of the optical pumping then determines the cooling efficiency and
the final temperature. In this limit of large detuning, the Raman coupling
rate is faster than any scattering rate, and the population simplifies to

$\langle n\rangle=\frac{s}{(1-s)},\quad
s=\frac{4\eta^{2}\gamma_{h}+\gamma_{\scriptscriptstyle+}}{\eta^{2}\gamma_{h}+\gamma_{\scriptscriptstyle-}}.$
(73)

We note that a similar expression can also be derived from the no-flow
condition solving for $s=P(n+1)/P(n)$ without having to take the expectation
values, although other approximations have to be made.

This analytical model is the final result of this model and provides intuition
for the cooling performance of PG and GM. It agrees very well with the exact
spin 1/2 solution for $\eta<0.1$. For other spin models, it also serves as a
fitting function. The ratio $\gamma_{+}/\gamma_{-}$ is then an effective
asymmetry of the optical pumping to the lower energy state. This asymmetry is
close to unity for PG cooling, and much smaller for cooling with dark state,
i.e. GM and $\Lambda$-GM. The ratio $\gamma_{h}/\gamma_{+}$ is an effective
scattering rate of the lower energy state. Together with the Lamb-Dicke
parameter $\eta$, these three parameters determine the lowest achievable
population.

In Fig. 5, we simulate the full master equation with ground and excited states
for PG and GM for $F=1$ ground-state in
$\sigma_{\scriptscriptstyle+}\\!-\\!\sigma_{\scriptscriptstyle-}$ light,
meaning that the light is linear at the atom equilibrium position. The PG case
is $F=1$ to $F^{\prime}=1$, and the Clebsch-Gordan (CG) coefficients are shown
in (a). In the
$\sigma_{\scriptscriptstyle+}\\!-\\!\sigma_{\scriptscriptstyle-}$
configuration, the $m=0$ state has the largest CG and, consequently, the
largest Stark shift. In linear light, however, the population still pumps to
the $m=0$ state because of strong diagonal CG coefficients from
$|J^{\prime}=2,m=\pm 1\rangle$. Cooling requires that the population is pumped
to the lowest energy state. Therefore, cooling requires the red-detuned
configuration where the $m=0$, which is the most stark shifted state, is also
the lowest energy, as shown in Fig. 4(d).

In the PG simulation in Fig. 5(b), cooling is optimum along the line where the
Raman transition is resonant. The optical power of the cooling light is such
that the energy difference between $m=0$ and $m=\pm 1$ equals the trapping
frequency. The lowest population converges to $\langle n\rangle=0.9$ in the
limit of large detuning $\Delta$ and small LD parameter $\eta$. The optimal
value is plotted in red in Fig. 5(e). The light red line is the simple model
from Eq. 73 with pumping asymmetry $\gamma_{-}/\gamma_{+}=0.45$, and heating
scattering $\gamma_{h}/\gamma_{+}=27$. There is not a large asymmetry in the
optical pumping, and
$\gamma_{\scriptscriptstyle+}\sim\gamma_{\scriptscriptstyle-}$, which are both
larger than $\eta^{2}\gamma_{s}$ in the LD regime. In this case, the final
population is
$s\approx\gamma_{\scriptscriptstyle+}/\gamma_{\scriptscriptstyle-}$. In this
situation, the cooling reaches a fundamental limit for large detuning and
small $\eta$.

Next we look at GM for $F=1$ to $F^{\prime}=1$. The CG coefficients are in
Fig. 5(c). One optical selection rule is that the $|F,m=0\rangle$ to
$|F^{\prime},m=0\rangle$ is forbidden, due to the photon needing to add one
unit of angular momentum. Because of this dark state, all the population pumps
to $m=0$. But in contrast to PG, this state is now dark. For that reason GM
requires blue-detuned light ($\Delta>0$) to make this state the lowest energy.
The simulation in Fig. 5(d) again shows that optimal cooling occurs when the
light shift equals the trapping frequency. The population however drops close
to ground-state. The optimal cooling is plotted in Fig. 5(e), along with the
simple model with effective parameters $\gamma_{-}/\gamma_{+}=7.5\times
10^{-4}$ and heating scattering $\gamma_{h}/\gamma_{+}=1.5$. For this large
optical pumping asymmetry where the reservoir state is dark, such as found in
GM and EIT cooling, $\gamma_{\scriptscriptstyle-}=0$, and
$s\approx\frac{\gamma_{\scriptscriptstyle+}}{\eta^{2}\gamma_{h}}$. In this
dark cooling limit, the final population is instead limited by $\eta$.

In Fig. 6, we also investigate the case of $J=1/2$ to $J^{\prime}=3/2$ for
lin$\perp$lin, which was studied in Ref. [32]. There is no cooling for
$\sigma_{\scriptscriptstyle+}\\!-\\!\sigma_{\scriptscriptstyle-}$, which
requires at least three ground spins due to the tensor shift. This
configuration is not relevant for neutral atoms, but it is for ions. Because
the polarization in lin$\perp$lin changes with position, we plot the cooling
as a function of atom position, with $\phi=0$ linear and $\phi=\pi/2$
circular. Similar to the other PG model, the population converges to $\langle
n\rangle\approx 1$. Interestingly though, this occurs for when the
polarization is linear, not circular. This is because circular polarization
cooling also requires three spins because the Raman transition supplies two
angular momenta. The cooling still occurs for when the light shift is equal to
the trapping frequency [32]. However, we do find that for larger spin states,
circular polarization is the best cooling, in agreement with our model.

Figure 6:  $J=1/2$ to $J^{\prime}=3/2$ PG cooling in lin$\perp$lin light. (a)
Classical picture of cooling with spins in polarization gradient of
lin$\perp$lin light, inspired from Ref. [32]. (b) Cooling for different atom
positions, which corresponds to different polarizations. Best cooling occurs
at linear light, and give $\langle n\rangle\approx 1$. (c) Clebsch-Gordon
coefficients for $J=1/2$ to $J^{\prime}=3/2$.

###  IV.3  $\Lambda$-GM and EIT Cooling

Figure 7:  (a) EIT and $\Lambda$-enhanced GM decrease temperature by
establishing more dark states to cool into. (b) EIT cooling uses two lasers
with Rabi frequencies $\Omega_{1}$ and $\Omega_{2}$ to drive a two-photon
transition from $|B,n\rangle$ to $|D,n-1\rangle$. (c) Simulations of
population versus the two-photon detuning. Best cooling occurs at the two-
photon resonance when $\Delta_{1}=\Delta_{2}$. A heating resonance occurs to
the right. Parameters are $\eta=0.05$, $\Delta=10\Gamma$, and
$\Omega_{1}=\Omega_{2}$. The cooling power is set so that the light shift
equals the trapping frequency. The inset shows that as the cooling power is
lowered, the resonance shifts over. When the light shift becomes small
compared to the trapping frequency, the two-photon detuning must compensate to
drive $n\rightarrow n-1$. (d) Comparison of GM and $\Lambda$-GM. Light dashed
is simple model from Eq. 73 with parameters $\gamma_{h}/\gamma_{+}=2.0$ and
$\gamma_{-}/\gamma_{+}=0.006$ for $\Lambda$-GM, and
$\gamma_{s}/\gamma_{+}=5.0$ and $\gamma_{+}/\gamma_{-}=0.11$ for GM.

The simple models reveal a crucial insight: lower populations are achieved
when pumping to darker states. Gray molasses (GM) cooling generates dark
states through angular momentum selection rules. Electromagnetically induced
transparency (EIT) cooling and $\Lambda$-type gray molasses ($\Lambda$-GM)
create additional dark states by incorporating multiple ground-states
addressed with a second laser frequency in an EIT configuration, as
illustrated in Fig. 7(a). These dark states arise from the interference of two
lasers coupling the ground-states to a common excited state.

Historically, the term “EIT cooling” has been primarily associated with
trapped ions [63, 64], where the two ground-states are typically different
Zeeman sublevels. In contrast, “$\Lambda$-GM” is commonly used in the context
of neutral atoms [20, 61], where the two ground-states belong to different
hyperfine manifolds of the ground-state.

Cooling to the motional ground-state using EIT has been theoretically
predicted and achieved experimentally in $Ca^{+}$ ion using Zeeman sublevels
of the D1 line [56, 63]. $\Lambda$-enhanced GM cooling on the other hand is
proven effective not only for bound atoms [21] but also for cooling and
imaging molecules in optical traps [65].

EIT addresses two ground-states $|g_{1}\rangle$ and $|g_{2}\rangle$ with two
different Rabi frequencies $\Omega_{1}$ and $\Omega_{2}$. These two states are
typically from different hyperfine ground-states. Each ground-state is
addressed by a laser frequency to a single excited state with corresponding
detunings $\Delta_{1}$ and $\Delta_{2}$. The Hamiltonian is written with
$g_{1}$ in the rotating frame of laser one and $g_{2}$ in the rotating frame
of laser two,

$H_{0}=\nu a^{\dagger}a+\Delta_{1}|g_{1}\rangle\langle
g_{1}|+\Delta_{2}|g_{2}\rangle\langle g_{2}|.$ (74)

We have two dipole operators, $\hat{D}_{1}=|g_{1}\rangle\langle e|$ and
$\hat{D}_{2}=|g_{2}\rangle\langle e|$. The interaction Hamiltonian is

$H_{I}=\frac{\Omega_{1}}{2}(\hat{D}_{1}^{\dagger}e^{ik_{1}\hat{x}}+{h.c.})+\frac{\Omega_{2}}{2}(\hat{D}_{2}^{\dagger}e^{ik_{2}\hat{x}}+{h.c.}).$
(75)

The two collapse operators represent the spontaneous emission of the single
excited state to both ground-states,

$L_{1,\pm}=\sqrt{\Gamma}\,e^{\pm ik_{1}\hat{x}}\hat{D}_{1},\quad
L_{2,\pm}=\sqrt{\Gamma}\,e^{\pm ik_{2}\hat{x}}\hat{D}_{2}.$ (76)

Next, we perform a unitary transformation of the two ground-states into two
superpositions so that only one of them is coupled to the two lasers, and the
other is the EIT dark state, shown in Fig. 7(b). The dark state is dark due to
the destructive interference of the population in the excited state from the
two ground-states. The basis transformation is

$\displaystyle|g_{\scriptscriptstyle
B}\rangle=\frac{1}{\Omega_{\text{rms}}}\left(\Omega_{1}\,|g_{1}\rangle+\Omega_{2}\,|g_{2}\rangle\right)$
$\displaystyle|g_{\scriptscriptstyle
D}\rangle=\frac{1}{\Omega_{\text{rms}}}\left(-\Omega_{2}\,|g_{1}\rangle+\Omega_{1}\,|g_{2}\rangle\right)$
(77) $\displaystyle|e\rangle=|e\rangle$

where $\Omega_{\text{rms}}=\sqrt{\Omega_{1}^{2}+\Omega_{2}^{2}}$.

Under this transformation, the bare Hamiltonian is

$\displaystyle H_{0}=\nu a^{\dagger}a+\Delta_{\scriptscriptstyle
B}|g_{\scriptscriptstyle B}\rangle\langle g_{\scriptscriptstyle
B}|+\Delta_{\scriptscriptstyle D}|g_{\scriptscriptstyle D}\rangle\langle
g_{\scriptscriptstyle D}|$ $\displaystyle+\Omega_{\scriptscriptstyle
BD}\bigl{(}|g_{\scriptscriptstyle B}\rangle\langle g_{\scriptscriptstyle
D}|+|g_{\scriptscriptstyle D}\rangle\langle g_{\scriptscriptstyle
B}|\bigr{)},$ (78)

with the energies of the bright and dark states as

$\displaystyle\Delta_{\scriptscriptstyle
B}=\left(\Omega_{1}^{2}\,\,\Delta_{1}+\Omega_{2}^{2}\,\,\Delta_{2}\right)/\Omega_{\text{rms}}^{2}$
$\displaystyle\Delta_{\scriptscriptstyle
D}=\left(\Omega_{1}^{2}\,\,\Delta_{2}+\Omega_{1}^{2}\,\,\Delta_{2}\right)/\Omega_{\text{rms}}^{2},$
(79)

and a coupling between the dark and bright states

$\Omega_{\scriptscriptstyle
BD}=(\Delta_{2}-\Delta_{1})\frac{\Omega_{1}\Omega_{2}}{\Omega_{1}^{2}+\Omega_{2}^{2}}.$
(80)

The transformed atom-field interaction is

$\displaystyle
H_{I}=\frac{1}{2\Omega_{\text{rms}}}\left[D_{B}^{\dagger}\,\left(\Omega_{1}^{2}\,e^{ik_{1}\hat{x}}+\Omega_{2}^{2}\,e^{ik_{2}\hat{x}}\right)\right.$
$\displaystyle\left.+D_{D}^{\dagger}\Omega_{1}\Omega_{2}\,\left(e^{ik_{1}\hat{x}}-e^{ik_{2}\hat{x}}\right)+h.c.\right].$
(81)

In the case where the two lasers are two-photon resonant, the Rabi frequencies
are equal $\Omega_{1}=\Omega_{2}$, and the two lasers are counter-propagating
$k_{1}=-k_{2}$, the expressions takes a simple form

$\displaystyle H_{0}=\nu a^{\dagger}a+\Delta\left(|g_{\scriptscriptstyle
B}\rangle\langle g_{\scriptscriptstyle B}|+|g_{\scriptscriptstyle
D}\rangle\langle g_{\scriptscriptstyle D}|\right)$ (82) $\displaystyle
H_{I}=\Omega\left(\cos(k\hat{x})D_{B}^{\dagger}+\sin(k\hat{x})D_{D}^{\dagger}\right)$
(83)
$\displaystyle\approx\Omega\left(D_{B}^{\dagger}+k\hat{x}\,\,D_{D}^{\dagger}\right).$
(84)

In the LD regime, only the bright state is coupled to the excited state. To
the zeroth order, the dark state is not coupled. But the dark state can be
driven directly only after changing the motional state. This Hamiltonian now
looks similar to the spin cooling case.

An interesting case is if, instead, the lasers are co-propagating with
$k_{1}=k_{2}$, then $H_{I}=\Omega D_{B}^{\dagger}e^{ikx}$, and the dark state
is dark even to $n$ changing transitions. Co-propagating light cannot change
the momentum, prohibiting cooling.

Next, we can adiabatically eliminate the excited state using Eq. 15 and 16 to
get the effective ground-state dynamics. In the original basis, the effective
ground-state Hamiltonian is

$\displaystyle
H_{\text{eff}}=\nu\hat{a}^{\dagger}\hat{a}(\Delta_{1}-\frac{\Omega_{1}^{2}}{\Delta})|g_{1}\rangle\langle
g_{1}|+(\Delta_{2}-\frac{\Omega_{2}^{2}}{\Delta})|g_{2}\rangle\langle g_{2}|$
$\displaystyle-\frac{\Omega_{1}\Omega_{2}}{\Delta}\left(|g_{1}\rangle\langle
g_{2}|e^{i(k_{2}-k_{1})\hat{x}}+h.c.\right).$ (85)

In the bright and dark basis with $\Omega_{1}=\Omega_{2}$ and $k_{2}=-k_{1}$,
the effective ground-state Hamiltonian is

$\displaystyle H_{\text{eff}}=\nu
a^{\dagger}a+\frac{\Omega_{1}^{2}+\Omega_{2}^{2}}{2\Delta}|g_{\scriptscriptstyle
B}\rangle\langle g_{\scriptscriptstyle B}|$
$\displaystyle+\left(ik\hat{x}\frac{\Omega_{1}\Omega_{2}}{\Delta}|g_{\scriptscriptstyle
B}\rangle\langle g_{\scriptscriptstyle D}|+h.c.\right)$ (86)

and the collapse operators are

$\displaystyle
L_{\text{eff},\pm,i}=\frac{\sqrt{\Gamma}\,\Omega_{\text{rms}}}{\Delta-i\Gamma/2}\,\hat{D}_{i}\left(\,\hat{D}_{B}\pm
ik\hat{x}(\,\hat{D}_{B}+\frac{2\Omega_{1}\Omega_{2}}{\Omega^{2}_{\text{rms}}}\,\hat{D}_{D})\right).$
(87)

We can now see that the Hamiltonian and collapse operator take a similar form
as before. The position-independent Hamiltonian results in a light shift for
only the bright state. Therefore, just like GM, EIT also works in the blue-
detuned regime so that the dark state is lower energy. The position-dependent
Hamiltonian results in coupling between neighboring motional states. The
collapse operator also has a position-independent term, which optically pumps
from the bright state to the dark state, and a position-dependent term from
recoil heating. So if you are at two photon resonance condition
($\Lambda$-enhancement) in GM, the formalism and the effectiveness of
$\Lambda$-GM is identical to EIT.

In Fig. 7(c), we plot the cooling performance as a function of detuning. In
the main plot, the light shift is equal to the trapping frequency, so the
optimal cooling occurs when $\Delta_{1}=\Delta_{2}$. The inset shows the
cooling performance as the cooling power is decreased. The relative detuning
can be adjusted to bring the system back into resonance and recover the
cooling efficiency. Fig. 7(d) compares the optimal cooling performance for the
$F=1/2,3/2$ to $F^{\prime}=3/2$ system, with and without the
$\Lambda$-enhancement (i.e., with and without the $F=1/2$ state). The
inclusion of the $F=1/2$ state in the $\Lambda$-configuration significantly
improves the cooling performance. The simulation results are also fitted to
the theoretical model to extract the effective cooling parameters.

The effects of EIT on the cooling performance can now be clearly seen. By
creating darker reservoir states, EIT decreases the effective asymmetry
between the pumping rates, $\gamma_{-}/\gamma_{+}$. As a result, the
fundamental limit for EIT cooling is similar to that of GM cooling, but
reduced by the factor by which the dark states are darker. The cooling
efficiency still depends on the inverse square of the Lamb-Dicke parameter,
$1/\eta^{2}$. In Fig. 7(d), we plot the cooling performance for the
$F=1/2,3/2$ to $F^{\prime}=3/2$ system, comparing the cases with and without
the $F=1/2$ state. The inclusion of the $F=1/2$ state in the
$\Lambda$-configuration leads to a significant improvement in the cooling
efficiency, demonstrating the power of EIT in enhancing the performance of
sub-Doppler cooling techniques.

###  IV.4 Raman-sideband cooling

Figure 8:  Raman-sideband cooling. The first step is a Raman transition
between spin states and decreasing the motional state by 1. The second step is
spontaneous emission back to original spin, which keeps atom in $n-1$ in the
Lamb-Dicke regime. The cycle repeats until the population is in the ground-
state.

Raman sideband (RSB) cooling has achieved the highest ground-state populations
in neutral atoms [66, 22, 67], ions [27], optical tweezers [24, 25, 26], and
molecules [68, 69]. The working principle of RSB cooling is illustrated in
Fig. 8(a).

In RSB cooling, two coherent, far-detuned lasers drive the transition from
$|g_{1},n\rangle$ to $|g_{2},n-1\rangle$, where $|g_{1}\rangle$ and
$|g_{2}\rangle$ are two ground-states, and $n$ represents the motional quantum
number. The light shifts and scattering rates from these cooling lasers are
typically negligible due to the large detuning. A third laser, tuned closer to
resonance, performs optical pumping and exploits selection rules to ensure
that the dark state remains dark.

The effective Raman coupling strength between the two ground-states is given
by $\Omega_{1}\Omega_{2}/\Delta$, where $\Omega_{1}$ and $\Omega_{2}$ are the
Rabi frequencies of the two cooling lasers, and $\Delta$ is the detuning from
the excited state. The effective ground-state Hamiltonian for RSB cooling is:

$\displaystyle\langle
g_{1},n_{1}|H_{\text{eff}}|g_{2},n_{2}\rangle=-\frac{\Omega_{1}\Omega_{2}}{\Delta}\langle
n_{1}|e^{i(k_{1}-k_{2})\hat{x}}|n_{2}\rangle,$ (88)

where $\hat{x}$ is the position operator, and $k_{1}$ and $k_{2}$ are the wave
vectors of the two cooling lasers. By carefully tuning the laser frequencies
and polarizations, this Raman coupling can be engineered to drive transitions
that reduce the motional quantum number, leading to cooling.

In 3D, the Raman coupling term becomes
$e^{i(\bm{k}_{1}-\bm{k}_{2})\cdot\hat{\bm{r}}}\approx
1+i\hat{x}\,(\bm{k}_{1}-\bm{k}_{2})\cdot\hat{\bm{x}}$, which is related to the
projection of the difference wave-vector onto the cooling axis. Multiple laser
beams are typically used so that pairs of beams have projections on all three
axes, enabling 3D cooling.

RSB cooling improves upon EIT and $\Lambda$-GM cooling techniques, which
introduced dark states in an electromagnetically induced transparency (EIT)
configuration. While EIT and $\Lambda$-GM achieved lower temperatures, they
were limited by spontaneous emission from the dark state, parameterized by the
ratio $\eta^{2}\gamma_{h}/\gamma_{+}$, where $\eta$ is the Lamb-Dicke
parameter, $\gamma_{h}$ is the heating rate, and $\gamma_{+}$ is the pumping
rate.

RSB cooling addresses this limitation by increasing the detuning of the
cooling lasers until the collapse operator $L$ becomes negligible, effectively
eliminating spontaneous emission and repumping from the cooling lasers. The
additional repump beam, optimized using angular momentum selection rules,
ensures that the state $|g_{1}\rangle$ remains dark to it, making the dark
state immune to position-dependent spontaneous emission.

An effective spin model for RSB cooling can be created, where the light shift
from the cooling light is ignored, and the two-photon detuning is adjusted to
bring the $|g_{1},n\rangle$ to $|g_{2},n-1\rangle$ transition into resonance:

$\displaystyle
H=\nu\hat{a}^{\dagger}\hat{a}+(\Delta_{1}-\Delta_{2})|g_{2}\rangle\langle
g_{2}|+\Omega_{R}k\hat{x}\,\hat{F}_{x}$ $\displaystyle
L_{+}=\sqrt{\gamma_{\scriptscriptstyle+}}\,\,\hat{F}_{+}(1+ik\hat{x}).$ (89)

While RSB cooling offers significant advantages, its fundamental limitations
are often technical rather than theoretical. The coherence time between the
two ground-states is typically limited to a few milliseconds due to factors
such as magnetic field fluctuations or laser decoherence. For efficient
cooling, the Raman Rabi frequency $\Omega_{R}$ should exceed 10 kHz. However,
the Rabi frequency for the $n$ to $n$ transition is larger than that of the
$n$ to $n-1$ transition by a factor of $1/\eta$, resulting in both transitions
being driven simultaneously. Further analysis is needed to determine the
cooling limit in the presence of these technical constraints.

While RSB cooling offers a pathway to overcome the limitations of EIT and
$\Lambda$-GM cooling schemes, its implementation is more complex
experimentally. However, it potentially enables more efficient ground-state
cooling in a wider range of atomic and molecular systems.

##  V Analysis

The formalism in this paper reveals that all two-photon cooling schemes share
a fundamental similarity: they involve a Raman transition between neighboring
motional states with different spin states. Close-to-resonance cooling schemes
like PG and GM utilize light shifts to bring the states into resonance, while
further detuned cooling methods, such as Raman sideband (RSB) cooling, use the
two-photon detuning of two frequencies and two far detuned ground-states.
Similarly, techniques like $\Lambda$-GM cool by establishing dark states
states through destructive interference of two frequencies and two far detuned
ground-states.

PG cooling is fundamentally limited to the optical pumping asymmetry, which
for $F$ to $F+1$ is $\langle n\rangle\approx 1$. GM and $\Lambda$-GM lower the
population to $\langle n\rangle\approx\eta^{2}$. But, spontaneous emission in
these schemes still drives the population out of the dark state through either
spontaneous emission from the first-order LD parameter or in the effective
ground-state formalism, the position dependent component of the collapse
operator. The dark $n=0$ state is thus not completely dark, as it still can be
driven through the $n=1$ excited state. However, when using the Raman light in
an optical pump, this is unpreventable, as the light requires a position-
dependent term to produce the Raman coupling between motional states.

RSB cooling represents the ultimate limit of cooling, detuning so far from the
excited state that the spontaneous emission of the Raman coupling light is
negligible. The re-pumping is achieved through a separate spontaneous emission
beam optimized to make the dark state truly dark.

In practical laboratory settings, PG, GM, and $\Lambda$-GM are more convenient
due to their operational simplicity and similarity to the MOT configuration.
But RSB is still employed for high ground-state preparation despite its
significant experimental overhead. This paper demonstrates that Grey molasses
cooling combined with the $\Lambda$-enhancement can achieve significant
ground-state populations. This raises the question of how schemes resembling
MOT configurations can be adapted for near-ground-state cooling. Likely, this
adaptation involves adding an additional beam specialized for optical pumping.
For example adding an additional circularly polarized beam close-to-resonance
beam for optical pumping (not counter-propagating). This should, from this
model, provide the optical pumping without driving sidebands of the dark
state. Then in the limit of large detuning for the GM beams, it would
effectively create a $L_{+}=\sqrt{\gamma_{+}}e^{ik\hat{x}}\hat{F}_{+}$ from
the RSB model. This will be the investigation of future experimental and
theoretical work.

##  VI Conclusion

This study presents a comprehensive theoretical framework for major cooling
mechanisms in neutral atom tweezers, uncovering shared principles across
diverse techniques. Our approach combines detailed full-level structure
simulations with a simplified spin model, offering novel insights for
optimizing cooling schemes. By extending previous research to encompass
arbitrary spins and polarizations, we show that gray molasses cooling may
potentially reach the ground-state, challenging conventional limits and
opening up new avenues for exploration.

Future research will investigate polarization gradients in three-dimensional
beam configurations and examine deviations from the idealized one-dimensional
cases presented here. We also plan to conduct simulations for specific alkali
atoms, including all relevant ground and excited states, to assess the impact
of multiple excited states and cross-coupling between lasers addressing
different hyperfine ground-states. These simulations will help establish
fundamental cooling limits for each atomic species for various techniques.
Future research will explore innovative cooling approaches that blur the
boundaries between established techniques.

Rapid, robust, and efficient ground-state optical cooling will increase the
coherence and fidelity of experiments throughout the ultracold field. Reducing
thermal motion in Rydberg atoms will pave the way for higher-fidelity quantum
gates. Rapid optical cooling techniques could also reduce the long times
associated with thermal evaporation for producing degenerate gases.

## Acknowledgements

This work was supported by the NSF Career Award (Award No. 0543784). We
express our gratitude to Yichao Yu for valuable early discussions regarding
the relationship between spin cooling and Raman-sideband cooling.

*

## Appendix A

### Dipole spin operator

For the atom spin, we will use the hyperfine spin states
$\bm{F}=\bm{J}+\bm{I}$. Note that setting $I=0$ recovers the $J$ formalism for
comparison. The excited states are $|F_{e},m_{e}\rangle$ and ground states are
$|F_{g},m_{g}\rangle$. The dipole operator in Eq. 5 is in this spin basis and
using the Wigner-Eckart theorem is

$\hat{D}_{q}=\sum_{m_{g}m_{e}}O^{J_{g}J_{e}}_{IF_{g}F_{e}}\,\,\langle
F_{g}m_{g}|F_{e}m_{e};1q\rangle\,\,|F_{g}m_{g}\rangle\langle F_{e}m_{e}|.$
(90)

The oscillator strength coefficients are given by

$O^{J_{g}J_{e}}_{IF_{g}F_{e}}=(-1)^{F_{e}\\!+\\!J_{g}+\\!I\\!+\\!1}\sqrt{(2F_{e}\\!+\\!1)(2\\!J_{g}\\!+\\!1)}\begin{Bmatrix}J_{e}&J_{g}&1\\\
F_{g}&F_{e}&I\end{Bmatrix},$ (91)

which are on the order of unity and give the strength of a $F_{g}$ to $F_{e}$
transition relative to the $J_{g}$ to $J_{e}$ transition with
$\sum_{F_{e}}|O^{J_{g}J_{e}}_{IF_{g}F_{e}}|^{2}=1$

### Spin tensor coefficients

The coefficients are determined by the Wigner-Eckart reduced matrix in the
ground and excited spins. For the case of $J,F$ in the ground and
$J^{\prime},F^{\prime}$ in the excited, the coefficients are [38]:

$\displaystyle
C^{(0)}=(-1)^{3F-F^{\prime}}\sqrt{\frac{1}{3}}\frac{2F^{\prime}+1}{\sqrt{2F+1}}$
(92) $\displaystyle\times\begin{pmatrix}F&1&F^{\prime}\\\
1&F&0\end{pmatrix}\left|\begin{Bmatrix}1&F&F^{\prime}\\\
0&1/2&F\end{Bmatrix}\right|^{2},$ $\displaystyle
C^{(1)}=(-1)^{3F-F^{\prime}}\sqrt{\frac{3}{2}}\frac{2F^{\prime}+1}{\sqrt{F(F+1)(2F+1)}}$
(93) $\displaystyle\times\begin{pmatrix}F&1&F^{\prime}\\\
1&F&1\end{pmatrix}\left|\begin{Bmatrix}1&F&F^{\prime}\\\
0&1/2&F\end{Bmatrix}\right|^{2},$ $\displaystyle
C^{(2)}=(-1)^{3F-F^{\prime}}\sqrt{\frac{30(2F^{\prime}+1)}{F(F+1)(2F+1)(2F-1)(2F+3)}}$
(94) $\displaystyle\times\begin{pmatrix}F&1&F^{\prime}\\\
2&F&0\end{pmatrix}\left|\begin{Bmatrix}1&F&F^{\prime}\\\
0&1/2&F\end{Bmatrix}\right|^{2}.$

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