# Dissipative pure-optical trap for laser cooling and trapping of neutral
atoms

O.N. Prudnikov<EMAIL_ADDRESS>Institute of Laser Physics, 630090,
Novosibirsk, Russia  Novosibirsk State University, 630090, Novosibirsk, Russia
R. Ya. Ilenkov  Institute of Laser Physics, 630090, Novosibirsk, Russia
Novosibirsk State University, 630090, Novosibirsk, Russia  A. V. Taichenachev
Institute of Laser Physics, 630090, Novosibirsk, Russia  Novosibirsk State
University, 630090, Novosibirsk, Russia  V. I. Yudin  Institute of Laser
Physics, 630090, Novosibirsk, Russia  Novosibirsk State University, 630090,
Novosibirsk, Russia  Novosibirsk State Technical University, 630073,
Novosibirsk, Russia  S. N. Bagaev  Institute of Laser Physics, 630090,
Novosibirsk, Russia

###### Abstract

We show the possibility of implementing a deep dissipative optical lattice for
neutral atoms with a macroscopic period. The depth of the lattice can reach
magnitudes comparable to the depth of the magneto-optical traps (MOT), while
the presence of dissipative friction forces allows for trapping and cooling of
atoms. The area of localization of trapped atoms reaches sub-millimeter size,
and the number of atoms is comparable to the number trapped in MOT. As an
example, we study lithium atoms for which the macroscopic period of the
lattice $\Lambda=1.5$ cm. Such deep optical lattices with a macroscopic period
open up possibility for developing effective methods for cooling and trapping
neutral atoms without use of magnetic field as an alternative to MOT. This is
important for developing compact systems based on cold atoms.

Optical Lattices, Bichromatic Laser Cooling, Recoil Effects, Compact optical
trap

##  I Introduction

Magneto-optical traps (MOT) are the basic instrument for laser cooling and
trapping of neutral atoms Raab et al. (1987); Metcalf and Straten (1999). The
operation of MOT is based on a combination of dissipative forces of
spontaneous light pressure and a deep macroscopic potential (of the order of
several $K$) in a spatially non-uniform magnetic field. Cold atoms obtained in
MOT have a wide range of applications, including the development of quantum
sensors based on matter wave interference Berman (1997), in laser metrology
for developments of ultra-precise frequency standards (atomic clocks) Takamoto
et al. (2020); Lion et al. (2017); McGrew et al. (2018); Nicholson et al.
(2015) and others. However, in many cases fast turn-off of magnetic field used
in MOT and precise control over the residual magnetic field are required,
which may be technically difficult to realize. Therefore, the development of
alternative methods of primary laser cooling and trapping of neutral atoms
without using a magnetic field is an important direction in creating compact
and mobile ultra-precision devices based on cold atoms.

It is well known that optical lattices allow for atom trapping without a
magnetic field. Dissipative and non-dissipative optical lattices are
distinguished Metcalf and Straten (1999); Jessen and Deutsch (1996); Grynberg
and Robilliard (2001). The regime of dissipative lattices Metcalf and Straten
(1999); Grynberg and Robilliard (2001); Dalibard and Cohen-Tannoudji (1989);
Berg-Sørensen et al. (1993); Prudnikov et al. (2007a); Prudnikov and Arimondo
(2004a); Dalibard and Cohen-Tannoudji (1985a); Prudnikov et al. (2007b);
Prudnikov et al. (2011), which combines atom trapping and laser cooling, is
realized for small detunings, when the light field frequency is close to the
atom transition resonance with natural linewidth $\gamma$. However, since the
frequency is chosen close to the atomic transition and the intensities of
light waves required for deep cooling should be small, the depth of the
dissipative lattice is usually small and comparable to the temperature of
Doppler or sub-Doppler cooling. The absence of a deep macroscopic potential
does not allow the use of such lattices for efficient primary laser cooling
and trapping. Non-dissipative deep optical lattices are created by intense
light waves with a sufficiently large detuning. Their depth is determined by
the technical capabilities of laser systems and can reach hundreds of
microkelvin Metcalf and Straten (1999); Kulosa et al. (2015); Takamoto et al.
(2020). However, dissipative laser cooling mechanisms are negligible here,
which requires the use of MOT to load atoms into them.

The use of bichromatic waves opens up new possibilities to form a deep
dissipative potential of macroscopic scale. The first theoretical studies of
atomic kinetics in the presence of two monochromatic fields have been done in
Kazantsev and Krasnov (1987); Voitsekhovich et al. (1988), where the effect of
rectification of dipole force was demonstrated. In such fields the atom
acquires momentum $\Delta p=2\hbar k$ due to induced absorption of photons of
one of the counter-propagating waves and induced emission into the opposite
wave. Since the rate of induced processes is not directly related to the line
width $\gamma$, the force on the atom can significantly exceed the spontaneous
light pressure force from a single wave, which can be used for effective
control of atomic beams Grimm et al. (1990); Söding et al. (1997); Cashen and
Metcalf (2001); Liebisch et al. (2012).

Moreover, in bichromatic field, the force averaged over the wavelength is not
equal to zero, making it possible to create a deep optical potential with a
macroscopic period $\Lambda=\pi/\Delta k$, determined by the spatial beating
of the two frequency components ($\Delta k$ is difference in wave vectors). In
this case, the parameters of the light field can be chosen so that the
dissipative mechanisms of laser cooling lead to cooling and trapping of the
atoms in the area of minimum of macroscopic potential Prudnikov et al. (2013,
2017). However, because both field frequency are resonant with the same
optical transition, the difference in wave vectors is extremely small, and
$\Lambda$ is very large. For example, for a frequency difference of 10-100 MHz
is required for near-resonant interaction the macroscopic period is about 1-10
meters. In such fields, the curvature in the minimum of the macroscopic
potential ($\propto\Delta k/k$) is small and does not allow to reach a
distinct localization of atoms inside cell of centimeter size, since the
spatial phase difference of the fields at such scales is practically
unchanged.

In this work, to reduce the period of the macroscopic potential in a
bichromatic field and increase its curvature, we propose to use light waves,
which are resonant to different transitions between fine and hyperfine
components of atomic levels. As an example, we investigate lithium atoms, for
which the fine splitting of the ${}^{2}P_{3/2}$ and ${}^{2}P_{1/2}$ levels is
about 10 GHz, determining the macroscopic lattice period of $\Lambda=1.5$ cm.
In this case, each field component is near-resonant to the $D_{1}$ and $D_{2}$
lines, which leads to significant dipole forces and allows for the realization
of a deep optical potential with a macroscopic period even in low-intensity
field. The presence of dissipative effects in such an optical lattice leads to
laser cooling and trapping of neutral atoms with a sub-millimeter localization
area near the minimum of the macroscopic potential. The results of our
theoretical analysis show that the depth of the macroscopic potential is
comparable to the depth of the magneto-optical trap (approximately 1 K in
temperature units or more), and laser cooling temperatures can reach sub-
Doppler values. These studies open up the possibility of implementing deep
dissipative optical traps for neutral atoms as an alternative to MOT.

##  II Kinetics of lithium atoms in a resonant bichromatic field

In this section, we describe the kinetics of lithium atoms in a bichromatic
light field with frequencies $\omega_{1}$ and $\omega_{2}$

${\bf E}({\bf r},t)=\mbox{Re}\left\\{{\bf E}^{(1)}({\bf
r})e^{-i\omega_{1}t}+{\bf E}^{(2)}({\bf r})e^{-i\omega_{2}t}\right\\}\,,$ (1)

where vector amplitudes ${\bf E}^{(1)}$ and ${\bf E}^{(2)}$ are determined by
a superposition of running waves

${\bf E}^{(n)}({\bf r})=\sum_{m}{\bf E}^{(n)}_{m}e^{i{\bf
k}^{(n)}_{m}\cdot{\bf r}},\,\,\,n=1,2\,.$ (2)

The directions of the waves propagation are given by wave vectors ${\bf
k}^{(n)}_{m}$. The frequencies of the light field components $\omega_{1}$ and
$\omega_{2}$ are near the resonance lines $D_{2}$ and $D_{1}$ of lithium
atoms, respectively. The scheme of energy levels and optical transitions used
for laser cooling of 6Li atoms is shown in Fig. 1.

Figure 1: Level structure of 6Li atom.

It should be noted that the hyperfine splitting of the ${}^{2}P_{3/2}$ and
${}^{2}P_{1/2}$ states for lithium atoms is comparable to the natural
linewidth $\gamma/2\pi=5.9$ MHz. Moreover, the hyperfine components of the
${}^{2}P_{3/2}$ state are within the natural linewidth $\gamma$, which
requires the use of a model taking into account the interaction of light waves
with all hyperfine components. Further, to simplify notation, we introduce the
indices $e_{\alpha}$ ($\alpha=1..5$) for hyperfine components of the excited
states ${}^{2}P_{3/2}$ and ${}^{2}P_{1/2}$, and $g_{n}$ ($n=1,2$) for the
hyperfine components of the ground state ${}^{2}S_{1/2}$ (see Fig. 1). Note
that 7Li and 6Li isotopes have a similar level structure. Here we perform our
analysis for the 6Li atoms, that have lower values of total angular momentum.
However, the results can also be generalized for 7Li atoms as well.

The kinetics of atoms in light fields is described by the quantum kinetic
equation for the atomic density matrix

$\frac{\partial}{\partial
t}{\hat{\rho}}=-\frac{i}{\hbar}\left[{\hat{H}}_{kin}+{\hat{H}}_{int}+{\widehat{W}}(t),{\hat{\rho}}\right]+{\hat{\Gamma}}\left\\{{\hat{\rho}}\right\\}\,,$
(3)

where ${\hat{H}}_{kin}={\hat{p}}^{2}/2M$ is the kinetic part,
${\widehat{W}}(t)=-{\hat{\bf d}}\cdot{\bf E(t)}$ describes interaction with
the field in the dipole approximation (where ${\hat{\bf d}}$ is the dipole
moment operator), and ${\hat{H}}_{int}$ is the Hamiltonian of a free atom in
the rest frame

${\hat{H}}_{int}=\sum_{\alpha=1\dots 5}{\cal
E}_{e_{\alpha}}{\hat{P}}^{(e_{\alpha})}+\sum_{n=1,2}{\cal
E}_{g_{n}}{\hat{P}}^{(g_{n})}\,,$ (4)

where ${\cal E}_{e_{\alpha}}$ and ${\cal E}_{g_{n}}$ are energies of excited
states $|e_{\alpha}\rangle$ and ground states $|g_{n}\rangle$. Here
${\hat{P}}^{(e_{\alpha})}$ and ${\hat{P}}^{(g_{n})}$ are projection operators
onto the Zeeman components of the hyperfine level $|e_{\alpha}\rangle$ and
$|g_{n}\rangle$

$\displaystyle{\hat{P}}^{(e_{\alpha})}$ $\displaystyle=$
$\displaystyle\sum_{\mu_{e_{\alpha}}=-F_{e_{\alpha}}}^{F_{e_{\alpha}}}|F_{e_{\alpha}},\mu_{e_{\alpha}}\rangle\langle
F_{e_{\alpha}},\mu_{e_{\alpha}}|,\,\,\;\alpha=1\dots 5$
$\displaystyle{\hat{P}}^{(g_{n})}$ $\displaystyle=$
$\displaystyle\sum_{\mu_{g_{n}}=-F_{g_{n}}}^{F_{g_{n}}}|F_{g_{n}},\mu_{g_{n}}\rangle\langle
F_{g_{n}},\mu_{g_{n}}|,\,\,\;n=1,2.$ (5)

The operator ${\hat{\Gamma}}\left\\{{\hat{\rho}}\right\\}$ describes the
relaxation of an atom due to spontaneous emission.

To use the resonant approximation in a bichromatic field (1), let’s introduce
the operator

${\hat{T}}=\mbox{exp}\left[-i\,t\left(\sum_{\alpha=1,2,3}{\omega_{1}}\,{\hat{P}}^{(e_{\alpha})}+\sum_{\alpha=4,5}{\omega_{2}}\,{\hat{P}}^{(e_{\alpha})}\right)\right],$
(6)

where the summation is taken over all components of the levels ${}^{2}P_{3/2}$
and ${}^{2}P_{1/2}$. For the transformation (6), the evolution equation for
the transformed density matrix
${\hat{\tilde{\rho}}}={\hat{T}}^{{\dagger}}{\hat{\rho}}\,{\hat{T}}$ takes a
similar form to (3)

$\frac{\partial}{\partial
t}{\hat{\tilde{\rho}}}=-\frac{i}{\hbar}\left[{\hat{H}}_{kin}+{\widehat{\widetilde{H}}}_{int}+{\widehat{\widetilde{W}}},{\hat{\tilde{\rho}}}\right]+{\hat{\Gamma}}\left\\{{\hat{\tilde{\rho}}}\right\\}\,.$
(7)

However, the resonance approximation allows to eliminate the time dependence
in the interaction operator and divide it into the sum of contributions

${\widehat{\widetilde{W}}}=\hbar{\hat{V}}^{(1)}+\hbar{\hat{V}}^{(2)}+h.c.\,,$
(8)

determined by the interaction with two frequency components: ${\bf E}^{(1)}$
is close to the resonance with $D_{2}$ line, and ${\bf E}^{(2)}$ is close to
the resonance with $D_{1}$ line (Fig.1)

$\displaystyle{\hat{V}}^{(1)}$ $\displaystyle=$
$\displaystyle{\hat{V}}^{e_{1}g_{1}}+{\hat{V}}^{e_{2}g_{1}}+{\hat{V}}^{e_{3}g_{1}}$
$\displaystyle{\hat{V}}^{(2)}$ $\displaystyle=$
$\displaystyle{\hat{V}}^{e_{4}g_{2}}+{\hat{V}}^{e_{5}g_{2}}\,.$ (9)

Here, the operator blocks ${\hat{V}}^{e_{\alpha}g_{n}}$ are defined as

${\hat{V}}^{e_{\alpha}g_{n}}=-\left({\bf
E}^{(n)}\cdot{\bf\hat{D}}^{e_{\alpha}g_{n}}\right){\bar{d}}/\hbar\,,$ (10)

where ${\bar{d}}$ is the reduced matrix element of the dipole moment operator,
and ${\bf\hat{D}}^{e_{\alpha}g_{n}}$ are the matrix blocks of dipole moment
operator

${\bf{\hat{d}}}={\hat{\bf D}}\,{\bar{d}}+h.c.\,.$ (11)

Thus the the matrix blocks ${\bf\hat{D}}^{e_{\alpha}g_{n}}$ are

${\bf\hat{D}}^{e_{\alpha}g_{n}}={\hat{P}}^{(e_{\alpha})}\,{\hat{\bf
D}}\;{\hat{P}}^{(g_{n})}\,.$ (12)

The decomposition of dipole moment operator ${\bf\hat{D}}$ in circular basis

${\hat{\bf D}}=\sum_{q=0,\pm 1}{\hat{D}}_{q}\,{\bf e}^{q}\,,$ (13)

are determined by the corresponding matrix elements $\langle
I,J_{e_{\alpha}};F_{e_{\alpha}},\mu_{e_{\alpha}}|{\hat{D}}_{q}|I,J_{g_{n}};F_{g_{n}},\mu_{g_{n}}\rangle$
for atomic states with electronic angular momenta of the ground and excited
states: $J_{g_{n}}=1/2$ and $J_{e_{\alpha}}=1/2,\,3/2$ (Fig. 1). Here $I$ is
the nuclear spin ($I=1$ for 6Li atoms). According to the Wigner-Eckart theorem
Khersonskii et al. (1988), matrix elements are expressed through Clebsch-
Gordan coefficients and 6-j symbols

$\langle
I,J_{e_{\alpha}};F_{e_{\alpha}},\mu_{e_{\alpha}}|{\hat{D}}_{q}|I,J_{g_{n}};F_{g_{n}},\mu_{g_{n}}\rangle=C^{F_{e_{\alpha}},\mu_{e}}_{F_{g_{n}},\mu_{g_{n}};\,1,q}\,(-1)^{(J_{e_{\alpha}}+F_{g_{n}}+I+1)}\sqrt{(2F_{g_{n}}+1)(2J_{e_{\alpha}}+1)}\left\\{\begin{array}[]{ccc}J_{e_{\alpha}}&1&F_{g_{n}}\\\
J_{g_{n}}&I&J_{e_{\alpha}}\\\ \end{array}\right\\}.$ (14)

The non-Hamiltonian evolution of the system due to spontaneous emission of the
light field photons is described by
${\hat{\Gamma}}\left\\{{\hat{\tilde{\rho}}}\right\\}$ in the equation for the
density matrix (3) and (7). Taking into account the recoil effects, this
contribution has the form (see, for example, Prudnikov et al. (2007b);
Prudnikov et al. (2011))

${\hat{\Gamma}}\left\\{{\hat{\tilde{\rho}}}\right\\}=-\frac{\gamma}{2}\left({\hat{P}}^{(e)}{\hat{\tilde{\rho}}}+{\hat{\tilde{\rho}}}\,{\hat{P}}^{(e)}\right)+{\hat{\gamma}}\left\\{{\hat{\tilde{\rho}}}\right\\}\,,$
(15)

with

$\displaystyle{\hat{\gamma}}\left\\{{\hat{\tilde{\rho}}}\right\\}=\gamma\frac{3}{2}\times$
(16) $\displaystyle\left<\sum_{\xi=1,2}{\left({\hat{\bf D}}\cdot{\bf
e}_{\xi}({\bf k}_{\xi})\right)^{\dagger}}e^{-i{\bf
k}_{\xi}\cdot{\bf{\hat{r}}}}{\hat{\tilde{\rho}}}\,e^{i{\bf
k}_{\xi}\cdot{\bf{\hat{r}}}}\left({\hat{\bf D}}\cdot{\bf e}_{\xi}({\bf
k}_{\xi})\right)\right>_{\Omega_{\xi}}.$

Here the operator ${\hat{P}}^{(e)}=\sum_{\alpha}{{\hat{P}}^{(e_{\alpha})}}$ is
a projector onto excited states (the summation is taken over all states of the
${}^{2}P_{3/2}$ and ${}^{2}P_{1/2}$ levels),
$\left<\dots\right>_{\Omega_{\xi}}$ means averaging over the angles of
spontaneously emitted photons with two orthogonal polarizations ${\bf
e}_{\xi}$, ($\xi=1,2$), the wave vectors ${\bf k}_{\xi}$ are specifying the
direction of spontaneously emitted photon, and ${\bf{\hat{r}}}$ is position
operator.

The Hamiltonian of a free atom in the rest frame in the rotating basis (6)
takes the form:

${\widehat{\widetilde{H}}}_{int}=-\hbar\sum_{\alpha=1..5}{\Delta_{\alpha}}\,{\hat{P}}^{(e_{\alpha})}\,,$
(17)

where

$\displaystyle\Delta_{\alpha}$ $\displaystyle=$
$\displaystyle\omega_{1}-\omega_{e_{\alpha}g_{1}},\,\mbox{for}\,\,\alpha=1,2,3$
$\displaystyle\Delta_{\alpha}$ $\displaystyle=$
$\displaystyle\omega_{2}-\omega_{e_{\alpha}g_{2}},\,\mbox{for}\,\,\alpha=4,5\,,$
(18)

are detunings from resonance for light induced transitions between hyperfine
levels of $D_{2}$ and $D_{1}$ lines caused by two frequency components Fig. 1.
Here $\omega_{e_{\alpha}g_{n}}=({\cal E}_{e_{\alpha}}-{\cal E}_{g_{n}})/\hbar$
is the transition frequency between states $|e_{\alpha}\rangle$ and
$|g_{n}\rangle$.

Note, that the kinetics of lithium atoms is characterized by enough small
recoil parameter $\varepsilon_{R}=E_{R}/\hbar\gamma\simeq 0.01$
($E_{R}=\hbar^{2}k^{2}/2M$ is recoil energy), that makes it possible to use
the semiclassical approximation. Indeed, for such a small recoil parameter,
the semiclassical approximation gives a good agreement with quantum treatment
based on direct numerical solution of the density matrix equation (7) in
monochromatic light Kirpichnikova et al. (2022, 2020). Within the
semiclassical approximation, the equation for the density matrix (7) can be
reduced to the Fokker-Planck equation (see, for example Dalibard and Cohen-
Tannoudji (1985b); Javanainen (1991); Prudnikov et al. (1999a))

$\displaystyle\left(\frac{\partial}{\partial t}+\frac{{\bf
p}}{M}\cdot\nabla\right){\cal F}=$
$\displaystyle-\sum_{i}\frac{\partial}{\partial p_{i}}f_{i}({\bf r},{\bf
p}){\cal F}+\sum_{i,j}\frac{\partial^{2}}{\partial p_{i}\,\partial
p_{j}}D_{ij}({\bf r},{\bf p}){\cal F}\,,$ (19)

for the distribution function of atoms in the phase space ${\cal F}({\bf
r},{\bf p})=Tr\\{{\hat{\rho}}({\bf r},{\bf p})\\}$, where the trace is taken
over internal degrees of atom density matrix in the Wigner representation.
Here $\nabla$ is spatial gradient, $f_{i}$ are the Cartesian components of the
light force on atoms, and $D_{ij}$ are the Cartesian components of the
diffusion tensor in momentum space. The force and diffusion tensor can be
obtained in a process of reduction of the density matrix equation (3) to the
Fokker-Plank equation (II).

##  III Bichromatic optical lattice for lithium atoms

The light force on atom in a light field is determined by the spatial
gradients of the interaction operator (8)

$f_{i}({\bf r},{\bf
v})=-Tr\left\\{\bigg{(}\nabla_{i}\,{\widehat{\widetilde{W}}}({\bf
r})\bigg{)}\,{\hat{\sigma}}({\bf r},{\bf v})\right\\}\,,$ (20)

where ${\hat{\sigma}}({\bf r},{\bf v})$ is the stationary solution of the
equation (7) in zero order by recoil effects

$\displaystyle({\bf v}\cdot\nabla)\,{\hat{\sigma}}({\bf r},{\bf v})=$
$\displaystyle-\frac{i}{\hbar}\left[{\widehat{\widetilde{H}}}_{int}+{\widehat{\widetilde{W}}}({\bf
r}),{\hat{\sigma}}({\bf r},{\bf
v})\right]+{\hat{\Gamma}}^{(0)}\left\\{{\hat{\sigma}}({\bf r},{\bf
v})\right\\}\,,$ (21)

with the normalization condition $\mbox{Tr}\left\\{{\hat{\sigma}}\right\\}=1$.
The operator of spontaneous relaxation in zeroth order by recoil is

$\displaystyle{\hat{\Gamma}}^{(0)}\left\\{{\hat{\sigma}}\right\\}$
$\displaystyle=$
$\displaystyle-\frac{\gamma}{2}\left({\hat{P}}^{(e)}{\hat{\sigma}}+{\hat{\sigma}}\,{\hat{P}}^{(e)}\right)+{\hat{\gamma}}^{(0)}\left\\{{\hat{\sigma}}\right\\}$
$\displaystyle{\hat{\gamma}}^{(0)}\left\\{{\hat{\sigma}}\right\\}$
$\displaystyle=$ $\displaystyle\gamma\sum_{q=0,\pm
1}{\hat{D}}_{q}^{\dagger}\,{\hat{\sigma}}\,{\hat{D}}_{q}\,.$ (22)

Let us note, the expression (20) define general expression for the force is a
function of atom position ${\bf r}$ and velocity ${\bf v}$. The optical
lattice, the optical potential for atoms is determined by the force on
moveless atom $v=0$. The expression for the force can be obtained numerically
based on the solution of (III) for the density matrix ${\hat{\sigma}}$,
however, analytical solutions are of the greatest interest for analysis.

The force, by its nature, is divided into two components: the induced force,
associated with the transmission of momentum from light field to an atom in
the processes of stimulated absorption/emission of photons between different
spatial modes of the field, and the spontaneous light pressure force,
associated with the transfer of momentum to atoms in the cycle of induced
absorption and spontaneous emission of photons. Both components of the force
can lead to the formation of an optical potential, or the so-called optical
lattice Metcalf and Straten (1999); Jessen and Deutsch (1996); Grynberg and
Robilliard (2001); Bezverbnyi et al. (2005).

For analysis of optical potential in bichromatic field, we can get analytical
expression for the force on slow atoms $kv<\gamma$ in the light fields of low
intensity

$S^{(n)}=\frac{|\Omega^{(n)}({\bf r})|^{2}}{\gamma^{2}+4\delta_{n}^{2}}\ll
1,\,\,\,n=1,2\,,$ (23)

where

$\Omega^{(n)}({\bf r})=-\frac{|{\bf E}^{(n)}({\bf r})|{\bar{d}}}{\hbar}\,,$
(24)

is the local Rabi frequency for $n$-th frequency component (2),

$\displaystyle\delta_{1}$ $\displaystyle=$
$\displaystyle\omega_{1}-\omega_{e_{3}g_{1}}$ $\displaystyle\delta_{2}$
$\displaystyle=$ $\displaystyle\omega_{2}-\omega_{e_{5}g_{2}}\,,$ (25)

are main detunings from the resonances, which are determined by the frequency
$\omega_{e_{3}g_{1}}$ of the transition
${}^{2}P_{3/2}(F=5/2)\to\,^{2}S_{1/2}(F=3/2)$ for component ${\bf E}^{(1)}$,
and the frequency $\omega_{e_{5}g_{2}}$ of the transition
${}^{2}P_{1/2}(F=1/2)\to\,^{2}S_{1/2}(F=1/2)$ for component ${\bf E}^{(2)}$ as
shown in Fig. 1.

In the limit (23), the populations of the excited states are small, and for
slow atoms $kv<\gamma$ the equation (III) can be reduced to the equation for
the density matrices of two ground states ${}^{2}S_{1/2}$ with angular momenta
$F=3/2$ and $F=1/2$

${\hat{\sigma}}^{(n)}={\hat{P}}^{(g_{n})}{\hat{\sigma}}\,{\hat{P}}^{(g_{n})},\,\,n=1,2\,.$
(26)

In this case, the equations for these matrices take the form

$\displaystyle({\bf v}\cdot\nabla)\,$ $\displaystyle{\hat{\sigma}}^{(1)}$
$\displaystyle\,=$ $\displaystyle-$ $\displaystyle
i\left[{\hat{H}}_{eff}^{(1)},{\hat{\sigma}}^{(1)}\right]+{\hat{P}}^{(g_{1})}\,{\hat{\gamma}}^{(0)}\left\\{{\hat{\sigma}}^{ee}\right\\}\,{\hat{P}}^{(g_{1})}\,,$
$\displaystyle({\bf v}\cdot\nabla)\,$ $\displaystyle{\hat{\sigma}}^{(2)}$
$\displaystyle\,=$ $\displaystyle-$ $\displaystyle
i\left[{\hat{H}}_{eff}^{(2)},{\hat{\sigma}}^{(2)}\right]+{\hat{P}}^{(g_{2})}\,{\hat{\gamma}}^{(0)}\left\\{{\hat{\sigma}}^{ee}\right\\}\,{\hat{P}}^{(g_{2})}\,.$

The Hamiltonian evolution here is described by two effective Hamiltonians

$\displaystyle{\hat{H}}_{eff}^{(1)}$ $\displaystyle=$
$\displaystyle\sum_{\alpha=1,2,3}\frac{\Delta_{\alpha}-i\gamma/2}{|\nu_{\alpha}|^{2}}\,\left({\hat{V}}^{e_{\alpha}g_{1}}\right)^{\dagger}{\hat{V}}^{e_{\alpha}g_{1}}\,,$
$\displaystyle{\hat{H}}_{eff}^{(2)}$ $\displaystyle=$
$\displaystyle\sum_{\alpha=4,5}\frac{\Delta_{\alpha}-i\gamma/2}{|\nu_{\alpha}|^{2}}\,\left({\hat{V}}^{e_{\alpha}g_{2}}\right)^{\dagger}{\hat{V}}^{e_{\alpha}g_{2}}\,.$
(28)

Here we use the notation $\nu_{\alpha}=\gamma/2-i\,\Delta_{\alpha}$. The
density matrix of excited states ${\hat{\sigma}}^{ee}$ in the zeroth order by
recoil for slow atoms is divided into a sum of two blocks

$\displaystyle{\hat{\sigma}}^{ee}$ $\displaystyle=$
$\displaystyle{\hat{\sigma}}^{ee}_{1}+{\hat{\sigma}}^{ee}_{2}$
$\displaystyle{\hat{\sigma}}^{ee}_{1}$ $\displaystyle=$
$\displaystyle\sum_{{\alpha},{\alpha^{\prime}}=1,2,3}\frac{1}{\nu_{\alpha}\,\nu_{\alpha^{\prime}}^{*}}\,{\hat{V}}^{e_{\alpha}g_{1}}\,{\hat{\sigma}}^{(1)}\,\left({\hat{V}}^{e_{\alpha^{\prime}}g_{1}}\right)^{\dagger}$
$\displaystyle{\hat{\sigma}}^{ee}_{2}$ $\displaystyle=$
$\displaystyle\sum_{{\alpha},{\alpha^{\prime}}=4,5}\frac{1}{\nu_{\alpha}\,\nu_{\alpha^{\prime}}^{*}}\,{\hat{V}}^{e_{\alpha}g_{2}}\,{\hat{\sigma}}^{(2)}\left({\hat{V}}^{e_{\alpha^{\prime}}g_{2}}\right)^{\dagger}\,.$
(29)

Note, that for sufficiently large detunings that exceed the hyperfine
splitting in excited states ${}^{2}P_{3/2}$ and ${}^{2}P_{1/2}$ and the
natural width $\gamma$, i.e.

$\displaystyle\delta_{1}$
$\displaystyle\,\gg(\omega_{e_{1}e_{2}},\,\omega_{e_{2}e_{3}},\,\gamma)$ (30)
$\displaystyle\delta_{2}$
$\displaystyle\,\gg(\omega_{e_{4}e_{5}},\,\gamma)\,,$

the following approximation can be used

$\displaystyle\Delta_{\alpha}$ $\displaystyle\simeq$
$\displaystyle\delta_{1}\,\,\,\mbox{for}\,\,{\alpha}=1,2,3$
$\displaystyle\Delta_{\alpha}$ $\displaystyle\simeq$
$\displaystyle\delta_{2}\,\,\,\mbox{for}\,\,{\alpha}=4,5\,.$ (31)

In this case, the effective Hamiltonians (III) are substantially simplified
and reduced to the shift operators for the hyperfine components of the ground
states

$\displaystyle{\hat{H}}_{eff}^{(n)}({\bf r})$ $\displaystyle=$
$\displaystyle\delta_{n}\,S^{(n)}({\bf r})\,\,{\hat{U}}^{(n)}({\bf
r}),\,\,\,n=1,2$ (32) $\displaystyle{\hat{U}}^{(1)}({\bf r})$ $\displaystyle=$
$\displaystyle\frac{2}{3}|{\bf A}_{1}({\bf
r})|^{2}\,{\hat{P}}^{(g_{1})}-\frac{i}{3}\bigg{(}\left[{\bf A}_{1}^{*}({\bf
r})\times{\bf A}_{1}({\bf r})\right]\cdot\frac{\hat{\bf
F}}{F_{g_{1}}}\bigg{)},$ $\displaystyle{\hat{U}}^{(2)}({\bf r})$
$\displaystyle=$ $\displaystyle\frac{1}{3}|{\bf A}_{2}({\bf
r})|^{2}\,{\hat{P}}^{(g_{2})}-\frac{i}{9}\bigg{(}\left[{\bf A}_{2}^{*}({\bf
r})\times{\bf A}_{2}({\bf r})\right]\cdot\frac{{\hat{\bf
F}}}{F_{g_{2}}}\bigg{)},$

are determined by the spatial configuration of local polarization vectors for
each frequency component: ${\bf A}_{1}({\bf r})={\bf E}^{(1)}({\bf r})/|{\bf
E}^{(1)}({\bf r})|$ and ${\bf A}_{2}({\bf r})={\bf E}^{(2)}({\bf r})/|{\bf
E}^{(2)}({\bf r})|$. Here ${\bf\hat{F}}$ is the total angular momentum
operator, and $\left[{\bf A}\times{\bf B}\right]$ denotes the cross product of
the vectors ${\bf A}$ and ${\bf B}$. The expressions (32) are written in the
invariant form and valid for an arbitrary nuclear spin (i.e., they are also
applicable to 7Li atoms). Note that the expression for ${\hat{U}}^{(1)}$
corresponds to optical lattice light shift in Hamann et al. (1998); Deutsch
and Jessen (1998) obtained for monochromatic light field far detuned to
optical resonance $D_{2}$ line of Cs atoms.

In the considered limit (23), the force on atoms can be represented as sum of
two parts

${\bf
f}=\hbar\,\mbox{Tr}\left\\{{\bf{\hat{f}}}_{1}\,{\hat{\sigma}}^{(1)}\right\\}+\hbar\,\mbox{Tr}\left\\{{\bf{\hat{f}}}_{2}\,{\hat{\sigma}}^{(2)}\right\\}\,,$
(33)

where the force vector operators ${\bf{\hat{f}}}_{n}$ are determined by the
spatial gradients

${\bf{\hat{f}}}_{n}=-\nabla{\hat{H}}_{eff}^{(n)}\,,\,\,n=1,2\,.$ (34)

It should be noted that a distinctive feature of the considered bichromatic
configuration, compared to optical lattices formed by monochromatic fields, is
the presence of an additional shift ${\hat{U}}^{(2)}$ with a spatial
dependence that may be different from ${\hat{U}}^{(1)}$. Spatially non-uniform
optical pumping of the ground state levels $|g_{1}\rangle$ and
$|g_{2}\rangle$, defined by equation (III) under the motion in two potentials,
leads to the rectification of the dipole force (33) on the wavelength scale.
In result, the optical potential with a macroscopic period can be formed. To
demonstrate this, we consider several examples bellow.

###  III.1 One-dimensional optical lattices created by a bichromatic field

Let us consider the general form of a one-dimensional bichromatic field
configuration created by counter-propagating waves of equal scalar amplitude
along the $z$ axis

${\bf E}^{(n)}(z)=E^{(n)}_{0}\left({\bf A}^{(n)}_{+}e^{ik^{(n)}z}+{\bf
A}^{(n)}_{-}e^{-ik^{(n)}z}\right),\,\,\,n=1,2\,.$ (35)

Here $E^{(n)}_{0}$ are scalar amplitudes, and ${\bf A}^{(n)}_{\pm}$ are unit
polarization vectors of the counter-propagating light waves of different
frequency components ($n=1,2$), which can be elliptical in general case. If we
denote $k^{(2)}=k$, and $\Delta k=k^{(1)}-k^{(2)}>0$, then the field (35) can
be rewritten as:

$\displaystyle{\bf E}^{(1)}(z)$ $\displaystyle=$ $\displaystyle
E^{(1)}_{0}\left({\bf A}^{(1)}_{+}e^{ikz+i\Delta\phi}+{\bf
A}^{(1)}_{-}e^{-ikz-i\Delta\phi}\right)\,,$ $\displaystyle{\bf E}^{(2)}(z)$
$\displaystyle=$ $\displaystyle E^{(2)}_{0}\left({\bf A}^{(2)}_{+}e^{ikz}+{\bf
A}^{(2)}_{-}e^{-ikz}\right).$ (36)

Since $\Delta k=k^{(1)}-k^{(2)}\ll k$, the relative spatial phase

$\Delta\phi=\Delta k\,z\,,$ (37)

can be considered constant over the wavelength scale $\lambda=2\pi/k$. The
spatial polarization configurations of different frequency components are
determined by the mutual spatial orientation of the polarization vectors ${\bf
A}^{(n)}_{+}$ and ${\bf A}^{(n)}_{-}$. Bellow we consider several
configurations.

####  III.1.1 Double $lin\,||\,lin$ field configuration

Figure 2: Double $lin\,||\,lin$ field configuration

For a bichromatic field created by standing waves with identical linear
polarizations ${\bf A}^{(1)}_{\pm}={\bf A}^{(2)}_{\pm}={\bf A}={\bf A}^{*}$
(the so-called double $lin\,||\,lin$ configuration, see Fig.2), the effective
Hamiltonians (32) in the limit of large detunings (30) are reduced to the
scalar light shifts $u_{1}=2\delta_{1}S_{1}|{\bf A}|^{2}/3$ and
$u_{2}=\delta_{2}S_{2}|{\bf A}|^{2}/3$ with a spatial dependence represented
by standing waves of each frequency components

$\displaystyle{\hat{H}}_{eff}^{(n)}$ $\displaystyle=$ $\displaystyle
u_{n}\,{\hat{P}}^{(g_{n})},\,\mbox{for}\,\,\;n=1,2$ $\displaystyle u_{1}$
$\displaystyle=$
$\displaystyle\frac{8}{3}\,\delta_{1}\,s_{1}\,\cos^{2}\left(kz+\Delta\phi\right)$
$\displaystyle u_{2}$ $\displaystyle=$
$\displaystyle\frac{4}{3}\,\delta_{2}\,s_{2}\,\cos^{2}\left(kz\right)\,.$ (38)

Here $s_{1}$ and $s_{2}$ are the saturation parameters per one running wave,
$s_{n}=|\Omega_{0}^{(n)}|^{2}/(4\delta_{n}^{2}+\gamma^{2})$, where
$\Omega_{0}^{(n)}=-E_{0}^{(n)}{\bar{d}}/\hbar$ are the corresponding Rabi
frequencies. The optical potential of the light field is determined by the
force (20), (33) on atom at rest. The solution of Eq. (III) for $v=0$ in this
field leads to an isotropic distribution on each of the ground levels
$|g_{1}\rangle$ and $|g_{2}\rangle$

$\displaystyle{\hat{\sigma}}^{(1)}$ $\displaystyle=$
$\displaystyle\frac{\cos^{2}(kz)\,s_{2}}{2s_{1}\cos^{2}(kz+\Delta\phi)+4s_{2}\cos^{2}(kz)}\,{\hat{P}}^{(g_{1})}\,,$
$\displaystyle{\hat{\sigma}}^{(2)}$ $\displaystyle=$
$\displaystyle\frac{\cos^{2}(kz+\Delta\phi)\,s_{1}}{2s_{1}\cos^{2}(kz+\Delta\phi)+4s_{2}\cos^{2}(kz)}\,{\hat{P}}^{(g_{2})}\,,$
(39)

since $\left[{\bf A}_{n}^{*}\times{\bf A}_{n}\right]=0$ in (32) for this field
configuration Fig.2. Thus, the force (33) resulting to formation of an optical
potential is divided into two contributions from each of the frequency
components

$\displaystyle f$ $\displaystyle=$ $\displaystyle f_{1}+f_{2}\,,$
$\displaystyle f_{1}$ $\displaystyle=$ $\displaystyle\hbar
k\frac{16}{3}\frac{\delta_{1}s_{1}s_{2}\,\sin(2kz+2\Delta\phi)\cos^{2}(kz)}{s_{1}\cos^{2}(kz+\Delta\phi)+2s_{2}\cos^{2}(kz)}\,,$
$\displaystyle f_{2}$ $\displaystyle=$ $\displaystyle\hbar
k\frac{4}{3}\frac{\delta_{2}s_{1}s_{2}\,\sin(2kz)\cos^{2}(kz+\Delta\phi)}{s_{1}\cos^{2}(kz+\Delta\phi)+2s_{2}\cos^{2}(kz)}\,.$
(40)

Figure 3: Macroscopic optical potential $U(z)$ in units of $\hbar\gamma$ of a
bichromatic lattice formed by field of double $lin\,||\,lin$ configuration.
The solid line represents the solution based on the analytical expressions
obtained in the approximation of small saturation and large detunings
(III.1.1), while the dotted line represents the optical lattice potential
obtained by direct numerical solution for the force (20), i.e. outside the
indicated approximations. The detunings are $\delta_{1}=-2\gamma$ and
$\delta_{2}=-3\gamma$, the saturation parameters are $s_{1}=s_{2}=0.1$ (i.e.
Rabi per one wave $\Omega_{0}^{(1)}\simeq 1.3\gamma$, $\Omega_{0}^{(2)}\simeq
1.9\gamma$). The depth of macroscopic optical potential for considered
parameters reaches $\Delta U=1720\,\hbar\gamma$ that corresponds $\simeq 0.5$
K in temperature units.

As was mentioned above, due to the different spatial dependencies of the light
shifts $u_{1}$, $u_{2}$ and the spatially inhomogeneous optical pumping of
$|g_{1}\rangle$ and $|g_{2}\rangle$ states, the effect of force rectification
on the wavelength scales appears. This rectification effect is similar in
nature to those described for a two-level atom in a bichromatic field
Kazantsev and Krasnov (1987); Grimm et al. (1990); Prudnikov et al. (2013,
2017), as well as in monochromatic fields formed by counter-propagating waves
with elliptical polarizations Prudnikov et al. (2001). Thus, the averaged over
the wavelength force

$\displaystyle{\bar{f}}(\Delta\phi)=\frac{1}{\lambda}\int_{0}^{\lambda}f(z)\,dz={\bar{f}}_{1}(\Delta\phi)+{\bar{f}}_{2}(\Delta\phi)\,,$
(41)

in the general case, is not equal to zero, and its magnitude and sign are
determined by the relative spatial phase $\Delta\phi$

$\displaystyle{\bar{f}}_{1}$ $\displaystyle=$ $\displaystyle\hbar
k\frac{16}{3}\frac{\delta_{1}s_{1}\,s_{2}\,\,\sin(2\Delta\phi)}{\left(8s_{1}s_{2}\cos^{2}(\Delta\phi)+(2s_{2}-s_{1})^{2}\right)^{2}}\left[|\sin(\Delta\phi)|\sqrt{2s_{1}s_{2}}\left((2s_{2}+3s_{1})(2s_{2}-s_{1})-8s_{2}s_{1}\cos^{2}(\Delta\phi)\right)\right.$
$\displaystyle+\left.16s_{2}^{2}s_{1}\cos^{2}(\Delta\phi)-s_{1}(6s_{2}+s_{1})(2s_{2}-s_{1})\right]\,,$
$\displaystyle{\bar{f}}_{2}$ $\displaystyle=$ $\displaystyle\hbar
k\frac{4}{3}\frac{\delta_{2}\,s_{1}\,s_{2}\,\,\sin(2\Delta\phi)}{\left(8s_{1}s_{2}\cos^{2}(\Delta\phi)+(2s_{2}-s_{1})^{2}\right)^{2}}\left[|\sin(\Delta\phi)|\sqrt{2s_{1}s_{2}}\left((6s_{2}+s_{1})(2s_{2}-s_{1})+8s_{2}s_{1}\cos^{2}(\Delta\phi)\right)\right.$
(42)
$\displaystyle-\left.8s_{2}s_{1}^{2}\,\cos^{2}(\Delta\phi)-2\,s_{2}\,(2s_{2}+3s_{1})(2s_{2}-s_{1})\right]\,.$

As the result, on a distance exceeding the wavelength, the rectified force
components ${\bar{f}}_{1}$ and ${\bar{f}}_{2}$ form a deep macroscopic
potential

$\displaystyle U(z)$ $\displaystyle=$
$\displaystyle-\int{\bar{f}}_{1}\big{(}\Delta\phi(z)\big{)}\,dz-\int{\bar{f}}_{2}\big{(}\Delta\phi(z)\big{)}\,dz$
(43) $\displaystyle=$ $\displaystyle U_{1}(z)+U_{2}(z)$

with a period corresponding to change of the relative spatial phase
$\Delta\phi$ from 0 to $\pi$, which for the lithium atoms in one-dimension
field (35) is $\Lambda\simeq 1.5$ cm (see Fig. 3).

Figure 4: The functions $\mu_{1}(s_{2}/s_{1})$ and $\mu_{2}(s_{2}/s_{1})$,
which determine the depth of macroscopic potentials $U_{1}$ and $U_{2}$ in the
field of double $lin\,||\,lin$ configuration.

Figure 5: The force on lithium atoms in bichromatic field as function of
velocity in the region of the global minimum of the optical potential. The
field parameters correspond to Fig.3.

Figure 6: The Wigner function of the atomic phase distribution ${\cal F}(z,p)$
(a) and the momentum distribution of cold atoms (b) in the macroscopic
potential of the bichromatic field of double $lin\,||\,lin$ configuration. The
field parameters correspond to Fig.3.

In this case, the positions of the global maxima and minima of the potentials
$U_{n}$ (43) correspond to position where $\Delta\phi=0,\pm\pi/2,\dots$. The
depth of each term in (43) is determined by

$\Delta U_{n}=\frac{1}{\Delta
k}\int_{0}^{\pi/2}{\bar{f}}_{n}\left(\Delta\phi\right)\,d\Delta\phi,\,\,n=1,2\,,$
(44)

that results the following expressions for potentials depth

$\displaystyle\Delta U_{1}$ $\displaystyle=$
$\displaystyle\hbar\delta_{1}\frac{k}{\Delta k}s_{1}\mu_{1}(s_{2}/s1)\,,$
$\displaystyle\Delta U_{2}$ $\displaystyle=$
$\displaystyle\hbar\delta_{2}\frac{k}{\Delta k}s_{1}\mu_{2}(s_{2}/s1)\,.$ (45)

The $\mu_{1}$ and $\mu_{2}$ here are dimensionless functions of the ratio
$a=s_{2}/s_{1}$ (see Fig.4)

$\displaystyle\mu_{1}(a)$ $\displaystyle=$
$\displaystyle-\frac{8}{3}\,a\left[\ln{\left(\frac{2\,a+1}{(\sqrt{2a}+1)^{2}}\right)}+\frac{2}{\sqrt{2a}+1}\right]$
$\displaystyle\mu_{2}(a)$ $\displaystyle=$
$\displaystyle\frac{1}{3}\left[\ln{\left(\frac{2\,a+1}{(\sqrt{2a}+1)^{2}}\right)}+\frac{\sqrt{8a}}{\sqrt{2a}+1}\right]\,,$
(46)

which allow to estimate the depth of the macroscopic potential (43) for
various light field parameters.

The relatively large depth of the macroscopic potential, which significantly
exceeds the magnitude of the light shifts of each frequency component, is
provided by the multiplier $k/\Delta k$ (for lithium atoms $k/\Delta k\simeq
4.4\times 10^{4}$). As the result, for parameters of Fig. 3, the depth of the
macroscopic potential reaches $\simeq 0.5\,K$ in temperature units. However,
for the localization of trapped atoms, the curvature at the points of minimum
macroscopic potential plays a significant role. In the case of bichromatic
field it is proportional to $\Delta k/k$, that allows to localize the trapped
Li atoms at sub-millimeter scales (see bellow).

For the bichromatic field with detunings $\delta_{1}$, $\delta_{2}$ near the
resonant lines $D_{2}$ and $D_{1}$, the presence of dissipative Doppler force
potentially allows trapping and cooling of atoms directly from the room
temperature vapor. The Doppler mechanisms of laser cooling are result of
imbalance of spontaneous light pressure forces from counter-propagating waves
on moving atoms Metcalf and Straten (1999) and acts on atoms over the all
points of macroscopic potential. The Fig. 5 shows dependence of the force on
lithium atoms as function of atomic velocity. In particular, the force $f(v)$
for considered detunings $\delta_{1}=-2\gamma$ and $\delta_{2}=-3\gamma$ takes
on maximum values for the velocity groups of atoms near resonance lines, near
$|v|\simeq 3\gamma/k$. As can be seen here the force has a sufficiently wide
velocity range, comparable to the dissipative force in a standard MOT. The
result for the force here is obtained on a base of eq. (20), i.e. beyond the
limit of the low fields intensity (23) and slow atoms $kv<\gamma$.

The Fig. 6 shows steady-state Wigner function ${\cal F}(z,p)$ and the momentum
distribution of atoms in the trap, obtained by solving the Fokker-Planck
equation (II) with taking into account the nonlinear dependence of the force
(see Fig.5) and diffusion coefficients on atoms velocity. For considered
parameters, the momentum distribution of atoms is well approximated by a
Gaussian distribution with temperature $T\simeq 1.1\,\hbar\gamma/k_{B}\simeq
300\,\mu K$ and corresponds to temperatures near the Doppler limit of laser
cooling. The size of cold atom cloud in a macroscopic potential can be defined
as

$\beta_{z}=\sqrt{\int z^{2}{\cal F}(z,p)\,dzdp}\,\,,$ (47)

where integration by coordinate is taken over the macroscopic period. For the
parameters of Fig. 6 the size of the atomic cloud is $\beta_{z}\simeq 320\,\mu
m$.

Figure 7: Double $lin$ $\perp$ $lin$ field configuration

####  III.1.2 Double $lin$ $\perp$ $lin$ field configuration

To achieve laser cooling below the Doppler limit, the light fields with
polarization gradients are required. In our case, such a field can be formed
with counter-propagating waves with different polarizations Prudnikov et al.
(1999a); Prudnikov et al. (2001); Prudnikov et al. (1999b). For example, the
well-known $lin$ $\perp$ $lin$ field configuration, formed by waves with
orthogonal linear polarizations, and $\sigma_{+}-\sigma_{-}$ configuration,
formed by waves with opposite circular polarizations Dalibard and Cohen-
Tannoudji (1989). However, the $\sigma_{+}-\sigma_{-}$ field does not lead to
spatial modulation of light shifts according to (32) in the limit of large
detunings (30), and rectification effect is absent.

In this section, we consider the double $lin$ $\perp$ $lin$ configuration (see
Fig. 7), where ${\bf A}^{(1)}_{+}={\bf A}^{(2)}_{+}={\bf e}_{x}$ is unit
vector along $x$-axis, and ${\bf A}^{(1)}_{-}={\bf A}^{(2)}_{-}={\bf e}_{y}$
is unit vector along $y$-axis

${\bf E}^{(n)}(z)=E^{(n)}_{0}\left({\bf e}_{x}e^{ik^{(n)}z}+{\bf
e}_{y}e^{-ik^{(n)}z}\right),\,n=1,2\,.$ (48)

For this field, as well as for the double $lin||lin$ configuration, a deep
macroscopic potential can be realized. The effective Hamiltonians (32) take
the forms

$\displaystyle{\hat{H}}_{eff}^{(1)}$ $\displaystyle=$ $\displaystyle
2\,\delta_{1}s_{1}\left(\frac{2}{3}{\hat{P}}^{(g_{1})}-\frac{1}{3}\sin(2kz+\Delta\phi)\,\frac{{\hat{F}}}{F_{g_{1}}}\right)\,,$
$\displaystyle{\hat{H}}_{eff}^{(2)}$ $\displaystyle=$ $\displaystyle
2\,\delta_{2}s_{2}\left(\frac{1}{3}{\hat{P}}^{(g_{2})}-\frac{1}{9}\sin(2kz)\,\frac{{\hat{F}}}{F_{g_{2}}}\right)\,.$
(49)

The force on an atom in this field can also be divided into the sum of two
parts

$\displaystyle f_{1}$ $\displaystyle=$ $\displaystyle-\hbar
k\frac{8}{3}\frac{\delta_{1}s_{1}s_{2}}{Q}\cos(2kz+2\Delta\phi)\left[\sin(2kz+2\Delta\phi)\sin(2kz)+5\right]\left[2\sin(2kz+2\Delta\phi)+\sin(2kz)\right]\,,$
$\displaystyle f_{2}$ $\displaystyle=$ $\displaystyle-\hbar
k\frac{4}{3}\frac{\delta_{2}s_{1}s_{2}}{Q}\cos^{2}(2kz+2\Delta\phi)\cos(kz)\left[2\sin(2kz+2\Delta\phi)+\sin(2kz)\right]\,,$
(50) $\displaystyle Q$ $\displaystyle=$ $\displaystyle
s_{1}9\cos^{2}(2kz+2\Delta\phi)+s_{2}\left[20-\cos(4kz)-\cos(4kz+4\Delta\phi)+14\sin(2kz)\sin(2kz+\Delta\phi)\right]\,.$

As in the case above, the rectified force in double $lin$ $\perp$ $lin$ field
configuration creates a macroscopic potential, that takes zero values at
points where the relative phase of the fields is
$\Delta\phi=0,\pm\pi/2,\pm\pi\dots$. The macroscopic potential is determined
by the average force over the wavelength (43). For the double $lin$ $\perp$
$lin$ configuration, the analytical expression for the depths of the
macroscopic potentials $\Delta U_{n}$ [see Eq. (44)] are quite complicated.
However, they can also be represented in the form (III.1.1), where the
dependencies of the dimensionless functions $\mu_{1}$ and $\mu_{2}$ on the
saturation parameter ratio $s_{2}/s_{1}$ are presented in Fig. 8.

Figure 8: The functions $\mu_{1}(s_{2}/s_{1})$ and $\mu_{2}(s_{2}/s_{1})$,
which determine the depth of macroscopic potentials $U_{1}$ and $U_{2}$ in the
field of double $lin$ $\perp$ $lin$ configuration.

Figure 9: Macroscopic optical potential $U(z)$ in units of $\hbar\gamma$ of a
bichromatic lattice formed by field of double $lin$ $\perp$ $lin$
configuration. The detunings are $\delta_{1}=-1.5\gamma$ and
$\delta_{2}=-\gamma$, the saturation parameters are $s_{1}=0.1$, $s_{2}=0.3$.
The depth of macroscopic optical potential for considered parameters reaches
$\Delta U=890\,\hbar\gamma$ that corresponds $\simeq 0.25$ K in temperature
units.

Figure 10: The Wigner function of the atomic phase distribution ${\cal
F}(z,p)$ (a) and momentum distribution of cold atoms (b) in the macroscopic
potential of the bichromatic field of double $lin$ $\perp$ $lin$
configuration. The waves parameters correspond to Fig.9.

The macroscopic potential for the double field $lin$ $\perp$ $lin$ is
presented in Fig. 9. The main feature of the double $lin$ $\perp$ $lin$
configuration is the presence of polarization gradient cooling mechanisms,
which lead to the possibility of sub-Doppler cooling Dalibard and Cohen-
Tannoudji (1989). Indeed, for low field intensity we observe narrow
distributions of trapped atoms in the phase space, as shown in Fig. 10. Note
that in the conditions of sub-Doppler cooling, the distribution of atoms in
the momentum space is essentially non-equilibrium and generally cannot be
described in terms of temperature. However, as shown in Kalganova et al.
(2017), the momentum distribution of atoms is well approximated by a two-
temperature Gaussian functions. The momentum distribution of trapped atoms
presented in Fig. 10(b) is approximated by a two-temperature Gaussian function
with temperature $T_{C}\simeq 0.035\,\hbar\gamma/k_{B}\simeq 10\mu K$ for
“cold” fraction of atoms $N_{C}\simeq 20\%$, and temperature $T_{H}\simeq
0.5\,\hbar\gamma/k_{B}\simeq 140\,\mu K$ for “hot” fraction of atoms
$N_{H}\simeq 80\%$. For the parameters of Fig. 9 the size of trapped atoms
cloud (47) in the macroscopic potential is $\beta_{z}\simeq 45\,\mu m$. Thus,
the sub-Doppler cooling mechanisms allow to significantly decrease the trapped
atoms cloud size in comparison with double $lin\,||\,lin$ configuration.

###  III.2 Multidimensional optical lattices created by a bichromatic field

Figure 11: Two-dimensional configuration of bichromatic optical lattice with
phase independent topology.

In general, by combining bichromatic waves, as well as for monochromatic
fields, one can create sufficiently complex spatial configurations of
multidimensional optical lattices Jessen and Deutsch (1996). The spatial
topology of light shifts (32) for each frequency component $\omega_{1}$ and
$\omega_{2}$ depends not only on the chosen geometry of the waves and their
polarizations, but may also depend on the relative phases of the light waves
forming the field of each frequency component separately. The phase dependence
of the topology of optical lattices poses certain challenges for their
experimental implementation and requires some efforts to control and stabilize
the phase of the light waves forming the field Hemmerich and Hänsch (1993).
Nevertheless, among all the diversity, it is possible to distinguish phase-
independent configurations of optical lattices. Thus, in the case where the
number of waves $M$ forming the field for each of the frequency components
exceeds the dimension of the space $N$ by 1, $M=N+1$, then the light shifts
formed by each frequency component (32) have a phase-independent topology
Jessen and Deutsch (1996). We also note that the processes of optical pumping
between the ground-state levels $|g_{1}\rangle$ and $|g_{2}\rangle$ are
determined by quadratic combinations of the amplitudes of the light waves of
each frequency component individually (III) to (III), and are not dependent on
the relative temporal phases between different frequency components of the
bichromatic field. Thus, for the field configuration leading to the phase-
independent topology of monochromatic lattices, the bichromatic field
configuration will also lead to a phase-independent topology for a bichromatic
lattice with a macroscopic period for lithium atoms.

Figure 12: (a) Macroscopic $2$D optical potential formed by a bichromatic
field of light waves with linear polarization along the $z$ axis. The
parameters of the waves correspond to the parameters of Figs. 3. (b)
Macroscopic $2$D optical potential generated by a bichromatic field of
linearly polarized light waves oriented at an angle $\varphi=\pi/4$ to the $z$
axis. The parameters of the waves correspond to the parameters of Fig. 9.

An example of a two-dimensional (2D) configuration of fields forming a phase-
independent bichromatic optical lattice is presented in Fig. 11, where the
angles between the wave vectors of the running waves are 120∘. Figure 12 shows
the spatial dependencies of the optical potentials formed by three pairs of
waves in a symmetric configuration of Fig. 11. There are two cases
represented: the waves with linear polarization along the $z$ axis [see Fig.
12(a)], and the waves with linear polarizations oriented at an angle
$\varphi=\pi/4$ to the $z$ axis [see Fig. 12(b)]. In the second case, the
field, in addition to spatial non-uniform intensity, also has spatial non-
uniformity of the local polarization parameters (ellipticity and orientation)
of the fields ${\bf E}^{(1)}({\bf r})$ and ${\bf E}^{(2)}({\bf r})$. As is
well known, the spatial non-uniformity of polarization leads to sub-Doppler
mechanisms of laser cooling Berg-Sørensen et al. (1993); Prudnikov et al.
(2007a). The periods of macroscopic potential formed by three pairs of light
waves in Fig. 11 along the $x$ and $y$ axes are

$\displaystyle\Lambda_{x}$ $\displaystyle=$
$\displaystyle\frac{\lambda}{3}\frac{k}{\Delta k}\simeq 1.9\,\,\mbox{cm}\,,$
$\displaystyle\Lambda_{y}$ $\displaystyle=$
$\displaystyle\frac{\lambda}{\sqrt{3}}\frac{k}{\Delta k}\simeq
3.4\,\,\mbox{cm}\,.$ (51)

Note that the depths of the bichromatic potentials Fig. 12(a) and (b) in order
of magnitude correspond to the potential depths for the one-dimensional
configurations Fig. 3 and Fig. 9 respectively.

Figure 13: The phase-space trajectory of an atom entering to the macroscopic
trap formed by double $lin\perp lin$ configuration of bichromatic field. The
red (lower) line corresponds to atom trajectory in the trap with parameters of
Fig. 9 ($\delta_{1}=-1.5\gamma$, $\delta_{2}=-\gamma$, $s_{1}=0.1$,
$s_{2}=0.3$). The initial velocity of atoms on the trap boundary
$v=7\,\gamma/k\simeq 25$ m/s. The green (upper) line corresponds to the atom’s
trajectory in the trap with $\delta_{1}=-10\gamma$, $\delta_{2}=-2.5\gamma$,
$s_{1}=0.05$, $s_{2}=0.5$. The initial velocity of atoms on the trap boundary
$v=17\,\gamma/k\simeq 60$ m/s. The blue arrows define the direction of the
trajectory evolution.

An alternative way to avoid phase dependence in the topology of the
multidimensional optical trap formed by a combination of bichromatic standing
waves is to use the waves along different directions with slightly shifted
frequencies. For the frequency shift of a few MHz the mutual phase gets enough
fast oscillation and is much faster then the rate of the kinetic evolution
processes. In this case the atom’s motion will be determined by the phase-
averaged dissipative force that also forms 2D, or three-dimensional (3D)
optical traps with macroscopic period.

The presented deep macroscopic potentials can be used as an alternative to
magneto-optical traps for the cooling and simultaneous trapping of lithium
atoms. The number of atoms trapped from vapors can be estimated using the
relation presented in Monroe et al. (1990); Gibble et al. (1992)

$N_{c}=\frac{L^{2}}{\sigma_{c}}\left(\frac{v_{c}}{u}\right)^{4}\,,$ (52)

where $v_{c}$ is the capture velocity, $u=(2k_{B}T/m)^{1/2}$ is the most
probable speed, $L$ is the size of the trap determined by the size of the
laser beam, and $\sigma_{c}$ is the cross section for an atom to eject atom
from a trap. The capture velocity can be found from the analysis of
trajectories of atoms coming to the trap with different velocities, separating
those that lead to the capturing of atoms. This analysis can be done on the
basis of Langevin equations (see, for example, Javanainen (1992); Prudnikov
and Arimondo (2004b)) describing the atom trajectory under the light force and
associated stochastic diffusion. Here we used numerical expressions for the
force and diffusion as functions of velocity and the position of the atom.
Figure 13 demonstrates the trajectory in the phase space of the atom captured
in the macroscopic potential formed by bichromatic field of double $lin\perp
lin$ configuration. The direct numerical analysis results in $v_{c}\simeq
10\,k/\gamma\simeq 35$ m/s for the traps with parameters of Fig. 9, which
leads to estimation $N_{c}\simeq 10^{7}$ for the number of trapped atoms. We
additionally note that the number of trapped atoms $N_{c}$ is determined not
only by the capture velocity, $v_{c}$, which can be increased by adjusting the
parameters of the bichromatic field, but also by the most probable velocity at
which atoms enter the trap, $u$ (52). For lithium atoms loaded from vapors
$u\simeq 900$ m/s (at temperature T=300K) is sufficiently large compared to
heavy elements ($190$ m/s for Cs atoms and $240$ m/s for Rb atoms at
temperature T = 300 K). Therefore, in many works, the methods of pre-cooling
and slowing down atoms are used to increase the number of lithium atoms in MOT
Schünemann et al. (1998); Hulet et al. (2020); Hernandez-Rajkov et al. (2020).
A similar technique can also be used to increase the number of atoms loaded
into a bichromatic trap.

##  IV Conclusion

In this paper, we present an idea of a deep pure-optical trap for neutral
atoms based on a dissipative bichromatic optical lattice. For lithium atoms,
as an example, a low-intensity bichromatic field (about 5 mW/cm2) allows for
the creation of a deep optical potential with a macroscopic period of
centimeter scale, which provides laser cooling and trapping of atoms. The
analysis, which was carried out within the one-atom approximation, showed that
the temperature of laser cooling can reach values comparable to and below the
Doppler limit (is approximately $\simeq 140\,\mu K$ for lithium atoms). At the
same time, the localization of trapped atoms reaches sub-millimeter sizes, and
the number of trapped atoms is comparable to the number trapped in MOT.

It should also be noted that the presented idea of deep dissipative lattice
with a macroscopic period in a bichromatic field can be generalized to other
neutral atoms. For example, for Rb and Cs atoms transitions between hyperfine
components of $D_{2}$ line can be used as resonant transitions for two
frequency components of bichromatic lattice. In this case, the period of the
macroscopic potential will be determined mainly by the splitting between the
hyperfine components of the ground state ${}^{2}S_{1/2}$ and gives:
$\Lambda\simeq 4.9$ cm for 85Rb, $\Lambda\simeq 2.2$ cm for 87Rb, and
$\Lambda\simeq 1.6$ cm for 133Cs. Also, for odd isotopes of mercury atoms
199Hg and 201Hg, the intercombination transition ${}^{1}S_{0}\to\,^{3}P_{1}$
($\lambda=253.7$ nm, $\gamma/2\pi=1.3$ MHz) can be used, where hyperfine
splitting of the upper level ${}^{3}P_{1}$ exceeds 20 GHz that allows to
create a bichromatic lattice with macroscopic period of $\Lambda\simeq 0.7$
cm.

Such deep pure-optical macroscopic potential can be used for cooling and
trapping of atoms in compact devices with low power consumption.

We thank Ennio Arimondo for fruitful discussion and valuable comments.

The research was supported by the Russian Science Foundation (Project No
23-22-00198), https://rscf.ru/project/23-22-00198/

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