# An accordion superlattice for controlling atom separation in optical
potentials

Simon Wili  Tilman Esslinger  Konrad Viebahn<EMAIL_ADDRESS>Institute
for Quantum Electronics & Quantum Center, ETH Zurich, 8093 Zurich, Switzerland

###### Abstract

We propose a method for separating trapped atoms in optical lattices by large
distances. The key idea is the cyclic transfer of atoms between two lattices
of variable spacing, known as accordion lattices, each covering at least a
factor of two in lattice spacing. By coherently loading atoms between the two
superimposed potentials, we can reach, in principle, arbitrarily large atom
separations, while requiring only a relatively small numerical aperture.
Numerical simulations of our ‘accordion superlattice’ show that the atoms
remain localised to one lattice site throughout the separation process, even
for moderate lattice depths. In a proof-of-principle experiment we demonstrate
the optical fields required for the accordion superlattice using acousto-optic
deflectors. The method can be applied to neutral-atom quantum computing with
optical tweezers, as well as quantum simulation of low-entropy many-body
states. For instance, a unit-filling atomic Mott insulator can be coherently
expanded by a factor of ten in order to load an optical tweezer array with
very high filling. In turn, sorted tweezer arrays can be compressed to form
high-density states of ultracold atoms in optical lattices. The method can be
also be applied to biological systems where dynamical separation of particles
is required.

##  I Introduction

Optical lattices [1, 2] and optical tweezer arrays [3, 4, 5, 6, 7, 8, 9, 10]
recently emerged as prime platforms for quantum simulation and computation
with neutral atoms. On the one hand, optical lattices are generated by
interference of two or more light beams, resulting in potentials defined by
individual $k$-vectors. Consequently, the resulting periodic patterns are of
exceptional purity, often with negligible disorder, allowing strongly
correlated phases of matter to be realised with ultracold atoms [11, 12, 13,
14]. On the other hand, optical tweezers are usually projected via multiple
radio-frequency (RF) tones of acousto-optic deflectors or, alternatively, from
spatial light modulators. Tweezers allow the addressing of individual atoms,
including the sorting of atoms into defect-free arrays via external feedback
and control [3, 4, 5, 6, 7, 8]. When combined with Rydberg excitations,
optical tweezers have been used to create synthetic many-body states [15, 16,
17].

Recent years have shown tremendous advancements in both fields, tweezers and
lattices, towards the realisation of programmable and universal quantum logic
[18, 19, 20, 21]. The next frontier is combining both techniques and their
respective benefits – addressability of tweezers and disorder-free, narrowly
spaced potentials for lattices. Such a combination could revolutionize
neutral-atom quantum technologies and several studies have been promoting this
goal [22, 23]. However, the fundamental mismatch in interatomic spacing
between the two platforms has yet to be overcome. While the minimum spacing
between tweezers is usually on the order of a few micrometers, the spacing in
optical lattices is typically several hundred nanometers.

Here, we propose a method to separate atoms in optical lattices by, in
principle, arbitrary distances and we demonstrate the required optical fields
to do so. Thereby, the gap in atomic spacing between optical lattices and
optical tweezer arrays can be bridged.

##  II Theoretical background

Optical lattices are periodic or quasiperiodic interference patterns resulting
from several mutually coherent light beams. Due to the dipole force of light
on polarizable particles, such as atoms, molecules, or even macroscopic
objects, the interference pattern results in a potential landscape $U$ which
is proportional to the intensity of the light. Lattices can be easiest
understood by considering the interference of two plane waves,
$\vec{E}(\vec{r},t)_{j}=\vec{E}_{0}e^{i(\omega
t-\vec{k}_{j}\cdot\vec{r}+\varphi_{j})}$ of equal polarization and frequency
$\omega$. The other parameters are amplitudes $\vec{E}_{0}$, wavevectors
$\vec{k}_{j}$, and phases $\varphi_{j}$. Let us further assume that the
wavevectors $\vec{k}_{j}$ lie in the $xz$-plane and enclose an angle
$\theta/2$ with the $z$-axis, as shown in Figure 1. The resulting trapping
potential reads

$U\propto I=|\vec{E}_{1}+\vec{E}_{2}|^{2}\propto\cos^{2}\left[\pi
x\left(\frac{\lambda}{2\sin\left(\frac{\theta}{2}\right)}\right)^{-1}+\Delta\varphi\right],$

where $\lambda$ is the wavelength of the light and
$\Delta\varphi\equiv\varphi_{1}-\varphi_{2}$. A one-dimensional standing wave
pattern emerges with periodicity

$a=\frac{\lambda}{2}\times\frac{1}{\sin(\theta/2)}\equiv\frac{\lambda}{2}\times\frac{1}{\text{NA}}~{}.$
(1)

While the periodicity can be tuned by varying the wavelength $\lambda$ or the
enclosed angle $\theta$, a change in $\Delta\varphi$ results in a translation
of the lattice. In general, a lattice with variable spacing is referred to as
as an (optical) accordion lattice [24, 25, 26, 27, 28]. The mathematical
description of the interference pattern gets more involved when assuming
realistic Gaussian beams instead of plane waves (Appendix A). Here we state
the approximate potential landscape in the $z=0$ plane,

$U\sim\exp\left[-2\left(\frac{\rho
z_{r}}{w_{0}z_{0}}\right)^{2}\right]\cos^{2}\left[\pi
x\left(\frac{\lambda}{2\sin\left(\frac{\theta}{2}\right)}\right)^{-1}+\Delta\varphi\right],$

where $\rho\equiv\sqrt{x^{2}+y^{2}}$, $z_{r}$ is the so-called Rayleigh range,
$w_{0}$ the beam waist, $z_{0}$ the distance of the beam waist to the lattice
where the last three quantities are assumed equal for both beams. Compared to
the plane-wave description, the optical lattice is now confined in space by a
Gaussian envelope, but the lattice spacing remains unchanged. The one-
dimensional case discussed here can easily be extended to two– or
three–dimensional lattices by adding pairs of beams with orthogonal
polarization.

Figure 1:  An optical lattice formed by interfering two coherent beams with a
stable phase relation. The lattice constant $a$ depends only on the optical
wavelength and the angle enclosed by the intersecting beams. While the
wavelength is typically fixed, the angle can be varied to create an optical
accordion. Figure 2: Transverse section through the center of a potential
formed by two Gaussian beams intersecting at an angle. We assume beams that
are red-detuned with respect to the relevant optical transitions, leading to
an attractive potential.

An important figure of merit in optical accordion lattices is the tuning range
of the lattice spacing and interparticle distance. Experimentally, the tuning
range is usually restricted by optical access, limiting the range of
interfering angles which can be accessed. If a single objective is used to
project both lattice beams, a large numerical aperture (NA, see Eq. 1) is
required. This brings about its own challenges due to optical aberrations [29]
and limited lattice diameters.

In general, the enclosed angle can be varied by mechanically moving parts,
such as translation stages [25, 27], rotating fiber tips [28], or
galvanometers [11]. Alternatively, one can rely on acousto-optic beam steering
[24, 26]. While the mechanical methods allow for a large range of angles,
acousto-optic steering offers high precision but the range of achievable
angles, hence lattice constants, is very limited.

In the following, we present a novel method that circumvents the challenge of
limited tuning ranges of acousto-optic steering. In addition, a scheme to
avoid optical aberrations and generate large-scale lattices is developed in
Section IV.2.

##  III Method for achieving arbitrary large interparticle separation

Let us assume an optical lattice of spacing $a_{0}$ uniformly filled with
atoms. We aim to stretch the lattice in space, for example to address
individual lattice sites with optical tweezers or in order to overcome the
diffraction-limited site-resolution of quantum gas microscopes. Now we
introduce two superimposed accordion lattices which each cover a factor of two
in lattice spacing, inspired by two-frequency Floquet engineering [30]. By
cyclicly transferring the atoms between the two lattices and expanding one
lattice at a time, we can separate the particles by, in principle, arbitrarily
large distances.

Figure 3: The accordion superlattice method. It consists of two accordion
lattices operating with lattice constants between $a_{0}$ and $2a_{0}$. The
atoms, molecules, or other polarizable particles, henceforth ‘atoms’, are
initially located in lattice 1 (left panel, blue lattice). This lattice is
expanded by a factor of two (top). Then the atoms are transferred into a
second lattice (green) with the same initial lattice constant $a_{0}$ (right).
Now the atoms occupy every second site. The second lattice can be expanded
before the atoms are loaded back into lattice 1 (bottom). Cyclic repetition of
those steps allows increasing the interatomic distance by an arbitrary factor
($>1$) utilizing two accordions that can be expanded by a factor of two. The
two colours represent lattices whose frequencies differ by a few Megahertz,
due to the diffraction of the AOD, but otherwise result from a single laser
source.

The scheme consists of four steps and is depicted in Figure 3, starting from
the left-most panel.

  1. 1.

Stretch the initial (blue) lattice until a lattice constant of $2a_{0}$ is
reached.

  2. 2.

Align a very shallow, second lattice with a lattice constant $a_{0}$ such that
all the potential minima of the first lattice coincide with minima of the
second lattice (green). Then ramp up the potential of the second lattice and
switch off the first lattice.

  3. 3.

Stretch the second lattice by a factor of two.

  4. 4.

Transfer the atoms back to the first lattice by aligning the lattices before
ramping up the first lattice while ramping down the second one.

Those four steps can be repeated and in each cycle the distance between
particles is increased by a factor of four. The expansion can be interrupted,
for instance at step 1 or step 3, to access lattice constants between
multiples of $a_{0}$ in the last cycle. Thus, the interparticle distance can
be increased by an arbitrary factor larger than one. Alternatively, the
protocol can be reversed, thereby shrinking the distance between the
particles. Compared to conventional accordion schemes which aim for
separations larger than a factor of two, the required numerical aperture of
this method is substantially reduced. This comes at the cost of aligning two
accordion lattices precisely onto one another, which can be achieved using the
precision offered by acousto-optic steering. Moreover, the two lattices which
differ by exactly a factor of two in spacing typically have a frequency
difference of several Megahertz, ensuring that interferences between both
lattices average out on atomic timescales. In the following, we present an
experimental setup to realise the optical fields for the accordion
superlattice scheme (Sec. IV) while experimental measurements are shown in
Section IV.2.

##  IV Setup and measurements

Figure 4: Schematic representation of the experimental setup. Two frequency
generators are connected in a primary-secondary configuration and provide a
first driving frequency for their acousto-optic modulator (AOD). An additional
generator provides a second driving frequency for both AODs. The two RF tones
in each AOD result in two separate beams which are illustrated by a solid and
a dashed line, respectively. The beams intersect due to two separate
microscope objectives (depicted by lenses) and form optical lattices in a
single plane. A camera in the lattice plane records images which are
transmitted to a computer and analysed in real-time. The computer provides
active feedback to the secondary generator.

An experimental test-setup for the accordion superlattice scheme is shown in
Figure 4. All lattices are derived from the same laser source, which is split
into two before each beam passes through its individual acousto-optic
modulator (AOD). One of the AODs is flipped with respect to the other one,
such that the deflections of the first orders occur in opposite directions.
Each AOD is driven with two RF tones, resulting in two first-order deflected
beams, illustrated by a solid and a dashed red line. The two sets of accordion
beams are projected onto a single plane via individual mirrors and objectives
to form optical lattices. A camera is placed in the lattice plane to record
the optical potential. The images acquired by the camera are analyzed by a
computer that is used to control three frequency generators and provide
feedback. Two frequency generators are arranged in a primary-secondary
configuration, each providing one driving signal for one AOD. The third
‘additional’ generator outputs a signal that is split and simultaneously used
to drive both AODs. The additional generator creates the first pair of
deflected beams that constitutes the first lattice. Tuning the driving
frequency, the angles of deflection change, resulting in a different lattice
constant. The pair of primary and secondary signal generators serves the same
purpose, outputting signals with identical frequency but controllable relative
phase to their respective AOD. This configuration of signal generators allows
independent tuning the position of one accordion lattice with respect to the
other.

In the following, we discuss two features of this setup which differ from
typical realisations of accordion lattices. First, due to acousto-optic
deflection, rays arriving at the objectives are not parallel but skew to the
optical axis. Second, the pair of beams constituting a lattice is not passing
through a single objective.

Figure 5: Decoupling the image plane (IP) from the focal plane (FP). Top:
conventional schemes relying on a focusing lens use parallel incoming rays to
create a lattice in the focal plane. Those lattices often have a small
diameter because a collimated incoming beam will have its beam waist in the FP
(Middle). Bottom: our setup uses diverging beams to form the lattice in an
image plane behind the FP. If the incoming beam is collimated, the beam
diameter in the IP and hence the lattice diameter can be large.

###  IV.1 Creating the lattice outside the focal plane

Typical accordion lattices are created by a pair of parallel beams [24, 25,
26, 27], approaching a lens parallel to its optical axis. The lens then
focuses the two beams in the focal plane where they interfere and form the
lattice. By varying the distance between parallel rays, the lattice constant
is tuned. Our protocol could, in principle, be realised by two sets of
parallel beams. Such a configuration is shown in Figure 5 (top panel). While
this method is conceptually simple and relatively easy to align, the fact that
the lattice is generated in the focal plane is often impractical. Firstly, the
diameter of the lattice is small due to the Gaussian beam profile (middle
panel), assuming collimated beams at the AOD. Secondly, the focussed light
suffers from aberrations which originate from dust or scratches on the
focussing lens [29]. Instead, we use pairs of large, collimated beams entering
the lens at an angle relative to the optical axis and the beams form the
lattice in a plane which is located behind the focal plane of the lens. Here,
the beam profile is larger and the lattice can contain a large number of
lattice sites. In addition, focussed aberrations are avoided.

As mentioned above, a typical challenge of optical accordions is that a small
minimal lattice constant requires a large NA of the focusing lens (Eq. 1). In
our configuration of skew incoming rays the image plane is further away from
the lens than its focal length. Hence, the requirement on the NA of the lens
for a given minimal lattice constant is more stringent compared to the
ordinary configuration. However, the focusing task can be split between two
lenses or objectives, reducing the requirements on NA, as described in the
following.

Figure 6: Schematic representation of a two-lens setup. The two lower beams
will produce the smallest possible lattice constant; the two upper beams will
produce the biggest one. The numerical apertures (NA, $sin(\alpha)$), in
combination with the chosen angles ($\theta_{1}$,$\theta_{2}$), determine the
range over which the lattice constant can be tuned. Given a minimal lattice
constant, using two lenses combined with the accordion superlattice method
reduces the requirement on the NA for arbitrary atom separation substantially.
The area around the vertical axis is kept free.

###  IV.2 Lowering the requirements on the NA

The accordion superlattice scheme presented in Figure 3 allows for arbitrarily
large interparticle separation while requiring only relatively little optical
access in the interval of angles $[\theta_{1},\theta_{2}]$. In practice, the
achievable interparticle separation will be limited by the number of available
lattice sites. When comparing the traditional single-accordion approach to the
novel accordion superlattice method, the maximal enclosed angle $\theta_{2}$
is the same and given by the minimal lattice constant. However, the minimal
angle $\tilde{\theta}_{1}$ is much smaller in the single-accordion method,
compared to the method employing the accordion superlattice. Here, the minimal
required angle $\theta_{1}$ corresponds to a lattice constant $a=2a_{0}$, even
for an arbitrary large interparticle separation. The angles between zero and
$\theta_{1}$ are not used, providing optical access for other applications.
Moreover, the accordion superlattice method allows to split the projecting
lens into two, as shown in Figure 6, similar to ref. [31]. This is
advantageous, as each of these lenses requires only a fraction
($<\frac{1}{2}$) of the NA that a single lens would need. The requirement is
initially lowered by a factor of two as two lenses are employed and then
further lowered by the fact that the range of angles $[0,\theta_{1}]$ does not
need to be covered in the accordion superlattice method. (A table of the
required NAs for a few exemplary configurations is provided in the Appendix
B.)

As a proof-of-principle, we use the setup described above to demonstrate the
accordion superlattice scheme in the following.

Figure 7: The top panel shows the RF tones required for the accordion
superlattice method. A ramp-up of a driving frequency results in an expansion
of an accordion lattice. Increasing the amplitude of the RF tone increases the
potential depth of the accordion. The initial and final frequencies have to be
tuned, such that the corresponding lattice constants have a ratio of exactly
$1:2$. Bottom: experimental demonstration of the accordion superlattice
method. Starting from an initial lattice constant of
$20.6\text{\,}\mathrm{\SIUnitSymbolMicro m}$, the lattice is expanded by a
factor of two within $3\text{\,}\mathrm{s}$ before the second lattice is
ramped up while the expanded lattice is ramped down (between
$3\text{\,}\mathrm{s}$ and $3.5\text{\,}\mathrm{s}$). The second lattice is
then expanded before another transition takes place after
$\approx$7\text{\,}\mathrm{s}$$. The discretisation in time is due to the
limited frame rate of the camera and does not reflect the expansion of the
true lattice potentials, which is entirely smooth.

###  IV.3 Experimental demonstration of the optical potential

The control protocol for the RF tones of the AODs is illustrated in the top
panel of Figure 7, realising one cycle of the hand-over process. The measured
optical potential is displayed in the lower panel of Figure 7. Each column of
pixels, corresponding to one snap-shot, is a line-sum of the camera image. To
account for the dependence of the AODs’ efficiencies on the driving frequency,
the measured intensities for each point in time have been normalized. Length
and time scales in this measurement were chosen for illustrative purposes,
dominated by the pixel size ($3.75\text{\,}\mathrm{\SIUnitSymbolMicro m}$) and
the limited framerate (10 fps) of the camera.

Figure 8: Simulated transfer fidelity (probability of finding the atom at the
intended site after the expansion) of the accordion superlattice method. The
probability is shown for a particle at site $N=25$. The interatomic spacing is
increased by a factor of eight, i.e. two transitions between the lattices are
simulated. The transition offset is the two lattices’ misalignment during the
transition process (assumed to be identical for both transitions). On the
left-hand side, the simulated initial potential depth at the center of the
lattice is 40 recoil energies; on the right-hand side, it is $70$ recoil
energies.

##  V Numerical simulation of the atomic separation

To highlight the capabilities of the accordion superlattice scheme, we
numerically simulate the transport of atoms for an expansion of the lattice by
a factor of eight, that is, one and a half cycles of the hand-over scheme
(Fig. 3). A possible application of this process is the loading of a linear
tweezer array with unit fidelity at $4.2\text{\,}\mathrm{\SIUnitSymbolMicro
m}$ tweezer spacing starting, for example, from a defect-free atomic Mott
insulator. The beams constituting the lattice are assumed to be Gaussian with
a beam diameter of $500\text{\,}\mathrm{\SIUnitSymbolMicro m}$ in the lattice
plane, starting with an initial lattice constant of $532\text{\,}\mathrm{nm}$.
The particles in the periphery of the lattice are accelerated much stronger
than the inner atoms. Moreover, the lattice depth in the outer regions is
reduced due to the Gaussian profile. Assuming a unit-filled Mott insulator
across 51 lattice sites, the outermost atoms ($N=\pm 25$) are the hardest to
translate. While an atom adjacent to the trap centre only has to move by eight
sites, the outermost atom effectively has to move by $8\times 25=150$ sites,
corresponding to the extreme scenario. Therefore, we simulate the trajectory
of one (the outermost) particle, initialised as a maximally localized Wannier
state of a $\mathrm{{}^{87}Rb}$ atom, by numerically solving the Schrödinger
equation in one dimension. We assume a lattice depth of 40 or 70 recoil
energies at the central site. The intended trajectory of the particle follows
a Gaussian error function, interpolating between the initial and the final
point in space. We take the total transport time as a variable parameter. In
general, sudden acceleration or deceleration should be avoided to reduce the
risk of the particle leaving its intended site. Therefore, the transition
between lattices is realised without stopping the particles. When the first
lattice reaches a lattice constant of $1.8a_{0}$, its lattice constant is
further increased to $2.2a_{0}$ while its amplitude is simultaneously ramped
down. Meanwhile, the amplitude of the second lattice is ramped up, while it is
already expanding. The alignment of the two lattices during the transition is
crucial. Therefore we simulated a realistic misalignment of the two lattices
during the transition, described by the ‘transition offset’. A transition
offset of $\pi$ rad corresponds to an offset of one full lattice site. The
misalignment is assumed to be of constant, identical absolute value but of
opposite sign for both transitions. We call the probability of finding the
particle within the intended site after the expansion ‘transfer fidelity’. The
results of the simulation are summarized in Figure 8. We clearly see that it
is possible to transfer an atom from the initial to its final intended
position with high fidelity. However, it is essential that the alignment of
the two lattices during transition is precise, while there is only a weak
dependence on the ramp time in the investigated parameter space. For a one-
dimensional lattice, we expect our simulation to be a lower bound of the
individual transfer fidelity in an array of 51 atoms. This makes the accordion
superlattice scheme a promising candidate for loading tweezer arrays as large
as $51\times 51=2601$ with high fidelity.

##  VI Conclusion and Outlook

We have introduced a novel technique for tuning the distance between optically
trapped particles. The method offers advantages in terms of precision, dynamic
range and experimental requirements compared to known methods. In particular,
it allows arbitrarily large particle separation, limited only by optical
power, while using a small numerical aperture. Furthermore, we have
demonstrated the feasibility of the optical potential in a proof-of-principle
experiment and assessed the transfer fidelity with a numerical simulation.
Thus, the method can bridge the spacing gap between optical tweezers and
optical lattices, allowing the transfer of atoms from lattices to tweezers and
vice versa. The setup can straightforwardly be extended to higher dimensions
by applying the same concept to the orthogonal directions. In addition,
mutliple RF tones can be added to superimpose more than two lattices.

The tuneable accordion lattice setup provides multiple technological
applications. First, the amplitudes and phases of both lattices with spacings
$a_{0}$ and $2a_{0}$ are fully tuneable, realising a dynamic optical
superlattice. Therefore, it can be used to prepare charge-density waves [32],
realise singlet-triplet oscillations [33, 34], and study topological pumping
[35, 36, 37]. Second, the tuneable lattice spacing can be used to perform
Bragg spectroscopy [38, 39, 40, 41] and light-assisted tunnelling [42, 43]
over a wide range of $k$-vectors. The frequency offset in the kilohertz-range
between two lattice beams renders the diffraction angle almost unchanged,
given typical centre frequencies of AODs around
$80\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$. Thus, both energy and momentum of
the induced transitions are essentially free-tuneable. Third, all lattice
spacings between $a_{0}$ and $2a_{0}$ are readily accessible, including
incommensurate frequency ratios leading to quasi-periodic [44] or quasi-
crystalline [45] lattice structures. In general, multiple RF tones can be used
to compose complex optical potentials in Fourier space. Last, the separation
of atoms in small-spacing optical lattices represents a pathway towards
single-site detection and addressability beyond the diffraction limit, apart
from matter-wave expansion in a harmonic trap [46]. Applications outside
quantum computing and quantum simulation are also conceivable. For instance,
the accordion superlattice method could be used to sort or assemble biological
samples on different lengthscales [47].

## Acknowledgments

We acknowledge funding by the Swiss National Science Foundation (Grant Numbers
182650 and NCCR-QSIT) and European Research Council advanced grant TransQ
(Grant No. 742579). K.V. acknowledges financial support from the ETH
fellowship. S.W., T.E., and K.V. are authors of the patent appl. EP22188223.6
(method and setup for changing an inter-particle distance).

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##  Appendix A Corrections to the lattice spacing from the interference of
Gaussian beams

Here, we provide the mathematical derivation of the potential landscape
corresponding to an optical lattice formed by two interfering Gaussian beams.
Each of the Gaussian beams is conveniently described in cylindrical
coordinates; we denote the radial coordinate of beam $j\in{1,2}$ with
$\rho_{j}$, and the axial coordinate with $z_{j}$. The electric field reads

$\displaystyle E_{j}(\rho_{j},z_{j},t)$ $\displaystyle=$
$\displaystyle\hat{\varepsilon}A_{0}\frac{w_{0}}{w(z_{j})}e^{-(\rho_{j}/w(z_{j}))^{2}}e^{ik\rho_{j}^{2}/2R(z_{j})}\times
e^{i(k_{j}z_{j}-\eta(z_{j})+\varphi_{j})}$
$\displaystyle\text{with}~{}~{}z_{r}$ $\displaystyle=$ $\displaystyle\frac{\pi
w_{0}^{2}}{\lambda}$ $\displaystyle w(z)$ $\displaystyle=$ $\displaystyle
w_{0}\sqrt{1+\left(\frac{z}{z_{r}}\right)^{2}}$ $\displaystyle R(z)$
$\displaystyle=$ $\displaystyle z(1+(z_{r}/z)^{2})$ $\displaystyle\eta(z)$
$\displaystyle=$ $\displaystyle\tan^{-1}(z/z_{r}),$

where $\hat{\varepsilon}$ is called polarization vector, $A_{0}$ denotes the
amplitude, $k_{j}$ the wavewector, and $\varphi_{j}$ the phase offset. These
quantities are also used to describe plane waves. In addition, we have the
following new terms: the beam radius $w$, its single-beam waist $w_{0}$, the
Rayleigh range $z_{r}$, the radius of curvature $R$, and the Gouy phase
$\eta$. A longitudinal section of such a beam is depicted in Figure 9.

Figure 9: Transverse section of a Gaussian beam propagating along an axis
$z_{j}$, described by cylindrical coordinates. The minimal radius $w_{0}$ is
called (single) beam waist. The radius $w$ increases by a factor of $\sqrt{2}$
within one Rayleigh range $z_{r}$. Far away from the beam waist, $w$ increases
approximately linearly as a function of the distance to the beam waist

If the Gaussian beam has a large beam waist ($w_{0}\gg\lambda$), then its
Rayleigh range is much longer than its wavelength ($z_{r}\gg\lambda$), i.e.
the intensity depends only weakly on $z_{j}$ in the beam waist’s vicinity.
Moreover, we have $\eta\approx 0$ and $R\rightarrow\infty$. Consequently, we
essentially have plane waves with a Gaussian envelope in radial direction.
However, the situation becomes more complicated when we let the beams
interfere away from their beam waists where their wavefronts are curved. This
is the case for the setup described in Section IV. For the sake of simplicity,
we will restrict ourselves to the case that was implemented experimentally.
Therefore, we assume that the beam waists of both beams lie an equal distance
of $z_{0}$ in front of the lattice. Let us further assume that the centers of
both beams cross at the origin of a Cartesian coordinate system $x=y=z=0$, and
$z_{1}=z_{2}=z_{0}>0$ in the cylindrical coordinate systems of both beams.
Additionally, we assume that $z_{r}\ll z_{0}$ and $x,y,\rho\ll z_{0}$. Those
conditions can be described as having the waist far away from the lattice
compared to the Rayleigh range and the lattice’s diameter being small compared
to its distance to the beam waist. The lattice is created in the $xy$-plane,
where the following relations hold:

$\begin{split}z_{1,2}(x,y)&=z_{1,2}(x)=z_{0}\pm\sin\left(\frac{\theta}{2}\right)x\\\
\rho_{1,2}(x,y)&=\sqrt{\left[x\cos\left(\frac{\theta}{2}\right)\right]^{2}+y^{2}}\\\
w(z)&\approx w_{0}\frac{z}{z_{r}}\\\ R(z)&\approx z\\\ \eta(z)&\approx\rm
const.\end{split}$ (2)

In this regime, we find

$\displaystyle I(\rho,z_{j}=z_{0})$ $\displaystyle=$ $\displaystyle
I(x,y)=|E_{1}(x,y)+E_{2}(x,y)|^{2}$ (3) $\displaystyle\approx$
$\displaystyle\underbrace{|A_{0}|^{2}\left(\frac{z_{r}}{z_{0}}\right)^{2}}_{:=I_{0}/4}e^{-2\left(\frac{\rho
z_{r}}{w_{0}z_{0}}\right)^{2}}\cdot|e^{ik\left(\frac{\rho^{2}}{2z_{1}}+z_{1}\right)}+e^{ik\left(\frac{\rho^{2}}{2z_{2}}+z_{2}\right)}|^{2}$
$\displaystyle=$ $\displaystyle I_{0}/4\cdot e^{-2\left(\frac{\rho
z_{r}}{w_{0}z_{0}}\right)^{2}}\left(2+2\cos\left[k\\{\frac{\rho^{2}}{2}(\frac{1}{z_{1}}-\frac{1}{z_{2}})+z_{1}-z_{2}\\}\right]\right)$
$\displaystyle=$ $\displaystyle I_{0}e^{-2\left(\frac{\rho
z_{r}}{w_{0}z_{0}}\right)^{2}}\cos^{2}\left[\frac{k}{2}\\{\frac{\rho^{2}}{2}(\frac{1}{z_{1}}-\frac{1}{z_{2}})+z_{1}-z_{2}\\}\right]$
$\displaystyle=$ $\displaystyle I_{0}e^{-2\left(\frac{\rho
z_{r}}{w_{0}z_{0}}\right)^{2}}\cos^{2}\left[\frac{k}{2}\\{\frac{-\rho^{2}\sin(\theta/2)x}{z_{0}^{2}-\sin^{2}(\theta/2)x^{2}}+2\sin(\theta/2)x\\}\right]$
$\displaystyle=$ $\displaystyle I_{0}e^{-2\left(\frac{\rho
z_{r}}{w_{0}z_{0}}\right)^{2}}\cos^{2}\left[\frac{\pi}{\underbrace{\left(\frac{\lambda}{2\sin(\theta/2)}\right)}}_{=a}x\left(1-\underbrace{\frac{1}{2}\frac{\rho^{2}}{z^{2}_{0}-\sin^{2}(\theta/2)x^{2}}}_{\rm
correction\>term}\right)\right].$

Note that the approximation we made in the second line neglects minor effects
regarding the contrast, but the value of the lattice constant is exact. In the
last line, we find the familiar form of a $\cos^{2}$ potential, up to a
correction term. The correction term makes the lattice constant larger towards
the edge of the lattice. Its effect is tolerable, as we show in the following.
During the transition, at a position $x,y$, the two lattices have a phase
offset due to the correction term given by

$\displaystyle|\delta\varphi|$ $\displaystyle=$
$\displaystyle\left|\frac{\pi}{\left(\frac{\lambda}{2\sin(\theta/2)}\right)}x\frac{1}{2}\rho^{2}\left(\frac{1}{z_{0}^{2}-\sin^{2}(\theta/2)x^{2}}-\frac{1}{2z_{0}^{2}-\frac{1}{2}\sin^{2}(\theta/2)x^{2}}\right)\right|$
(4) $\displaystyle\approx$
$\displaystyle\left|\frac{\pi}{4\left(\frac{\lambda}{2\sin(\theta/2)}\right)}\frac{x\rho^{2}}{z_{0}^{2}}\right|$

where $\theta$ denotes the larger of the two enclosed angles. For realistic
values of $x/a=100$ and $\rho/z_{0}=0.035$ during the last lattice transition
of the outermost atom of interest, we find $|\delta\varphi|<0.1$, which
according to our simulation is well within the range of tolerance. To further
mitigate this effect, a compensation term linear in $x$ (note that
$\rho^{2}=x^{2}+y^{2}$) can be introduced by choosing the lattice constant of
the second lattice slightly larger than two times the lattice constant of the
first lattice. While this results in an increase of said offset for the atoms
close to the center of the lattice, the maximal phase offset experienced by
the atoms that are affected the most can be reduced by a factor larger than
two.

In conclusion, the plane-wave lattice with a Gaussian envelope is a good model
for the lattice created at the beam waists of two Gaussian beams. Far away
from the beam waists, this still holds true approximately if the diameter of
the lattice is small compared to its distance to the beam waist.

##  Appendix B Exemplary NA requirements

$a_{0}/\lambda$ | naive NA | accordion superlattice NAs
---|---|---
$0.5$ | $1$ | $0.5$
$0.75$ | $0.67$ | $0.19$
$1$ | $0.5$ | $0.135$
$1.5$ | $0.33$ | $0.086$
$2$ | $0.25$ | $0.064$
Table 1: The table shows the required NAs for an increase in interparticle
distance by an arbitrarily large factor, starting from an initial spacing
$a_{0}$. The middle column states the required NA when relying on a single
lattice projected by a single objective. The right column contains the
required NAs when using the accordion superlattice method, relying on two
objectives.