# Flat-band-induced superconductivity in synthetic bilayer optical lattices

Tymoteusz Salamon  ICFO - Institut de Ciencies Fotoniques, The Barcelona
Institute of Science and Technology, Av. Carl Friedrich Gauss 3, 08860
Castelldefels (Barcelona), Spain  Bernhard Irsigler  ICFO - Institut de
Ciencies Fotoniques, The Barcelona Institute of Science and Technology, Av.
Carl Friedrich Gauss 3, 08860 Castelldefels (Barcelona), Spain  Debraj Rakshit
Harish-Chandra Research Institute, A CI of Homi Bhabha National Institute,
Chhatnag Road, Jhunsi, Allahabad 211 019, India  Maciej Lewenstein  ICFO -
Institut de Ciencies Fotoniques, The Barcelona Institute of Science and
Technology, Av. Carl Friedrich Gauss 3, 08860 Castelldefels (Barcelona), Spain
ICREA, Pg. Lluis Companys 23, Barcelona, Spain  Tobias Grass  ICFO - Institut
de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, Av.
Carl Friedrich Gauss 3, 08860 Castelldefels (Barcelona), Spain  Ravindra
Chhajlany  Faculty of Physics, Adam Mickiewicz University, 61614 Poznań,
Poland

###### Abstract

Stacking two layers of graphene with a relative twist angle gives rise to
moiré patterns, which can strongly modify electronic behavior and may lead to
unconventional superconductivity. A synthetic version of twisted bilayers can
be engineered with cold atoms in optical lattices. Here, the bilayer structure
is mimicked through coupling between atomic sublevels, and the twist is
achieved by a spatial modulation of this coupling. In the present paper, we
investigate the superconducting behavior of fermionic atoms in such a
synthetic twisted bilayer lattice. Attractive interactions between the atoms
are treated on the mean-field level, and the superconducting behavior is
analyzed via the self-consistently determined pairing gap. A strong
enhancement of the pairing gap is found, when a quasi-flat band structure
occurs at the Fermi surface, reflecting the prominent role played by the twist
on the superconductivity. The tunability of interactions allows for the
switching of superconducting correlations from intra (synthetic) layer to
inter (synthetic) layer. This includes also the intermediate scenario, in
which the competition between inter- and intra-layer coupling completely
destroys the superconducting behavior, resulting in re-entrant
superconductivity upon tuning of the interactions.

##  I Introduction

The Fermi-Hubbard model plays a central role in describing various aspects of
the many-body physics of condensed matter systems Hensgens  _et al._ (2017);
Esslinger (2010); Bloch  _et al._ (2008); Lewenstein  _et al._ (2007);
Truscott  _et al._ (2001); Chen  _et al._ (2021). In particular, along with
its variants, it is widely believed to encompass the basic ingredients
required to understand high-temperature superconductivity, e.g. in cuprates
Micnas  _et al._ (1990); Arovas  _et al._ (2022); Feiner  _et al._ (1996);
Berg  _et al._ (2008); Macridin  _et al._ (2005). Although Hubbard models are
effective simplified models of complex condensed matter systems, they can be
realized with high fidelity and control in various engineered systems such as
ultracold atoms in optical lattices. This, in turn, has lead to an exciting
branch of physics - quantum simulation of condensed matter phenomena
Lewenstein  _et al._ (2012); Bloch  _et al._ (2012); Grynberg and Robilliard
(2001); Hofstetter  _et al._ (2002). In recent years, one-, two- and three-
dimensional optical lattices have been generated paving the way to studies of
different quantum phases under various types of interactions Mazurenko  _et
al._ (2017); Schneider  _et al._ (2008). A distinguishing feature within the
cold atom set-up is the freedom of precise tuning of microscopic system
parameters, i.e. interactions and particle tunneling over wide ranges Chin
_et al._ (2010).

The control of material properties via band structure engineering has been a
long standing goal in condensed matter physics. A new frontier is twistronics,
where the relative rotational misalignment between layers in quasi two
dimensional (2D) systems leads to moiré patterns in real space. The moiré
patterns strongly influence the band structure and lead to enhanced collective
effects induced by interactions and topology Andrei  _et al._ (2021). In
graphene bilayers, tuning the twist angle to so-called magic values was
predicted to strongly quench the electronic kinetic energy leading to the
formation of quasi-flat bands Lopes dos Santos  _et al._ (2007); Bistritzer
and MacDonald (2011); McCann and Koshino (2013), where small interactions can
dominate the phenomenology. With the successful development of fabrication
methods for such devices, a series of experiments spectacularly unveiled
superconductivity and correlated insulators in these materials around the
magic angles Cao  _et al._ (2018a, b); Wang  _et al._ (2020). These results
have driven the new field of twistronics involving the study of various kinds
of Van der Waals stacked heterostructures beyond bilayer graphene Kennes  _et
al._ (2021). The unavoidable effect of twisting 2D materials is the
enlargement of the unit cell, usually by a few orders of magnitude compared to
the original unit cell of a single sheet of the material Andrei  _et al._
(2021); Kim  _et al._ (2017), into a so-called moiré supercell. This emergent
approximate crystal symmetry strongly complicates direct microscopic modelling
and non-approximating studies of correlations in these systems. As a result
direct quantum simulations of twistronics, in particular based on the
promising platform of ultra cold atoms trapped in optical lattices, that allow
for exquisite control of system parameters, offer an additional window to gain
fundamental understanding of moiré materials. Importantly, this approach
allows to study systems without certain practical difficulties associated with
materials such as the lack of control over the homogeneity of the twist angle
in samples and strain effects which lead to disorder. Moreover, apart from the
control of interactions, ultra cold atom systems allow for tuning of
interlayer coupling to strong values that can lead to enhanced correlation
effects even for comparatively small moiré supercells.

The interest in general 2D bilayer systems has led to the design of multiple
architectures and control schemes for bi-layer optical lattices Graß  _et al._
(2016); Kantian  _et al._ (2018); Gall  _et al._ (2021). Moiré systems can be
generated, on one hand, by effectively performing the direct analog of
material twisting in overlapping samples González-Tudela and Cirac (2019), or
spin dependent optical lattices Meng  _et al._ (2021); Luo and Zhang (2021). A
different versatile approach stems from the fact that physically the main
effect of twisting is the induction of incommensurate quasi-periodic
potentials and quasi-periodic interlayer tunnelings in layered systems Wu  _et
al._ (2018); Tarnopolsky  _et al._ (2019); Fu  _et al._ (2020); Chou  _et al._
(2020); Salamon  _et al._ (2020a); Lee and Pixley (2021); Yi  _et al._ (2022).
Such spatially modulated patterns can be directly imprinted on synthetic
bilayer systems Salamon  _et al._ (2020a, 2021), i.e. a single physical
optical lattice layer of atomic species with Raman coupled internal states
playing the role of the additional layer degree of freedom Boada  _et al._
(2012); Celi  _et al._ (2014a). This remarkably realizes moiré-type physics
without physical twisting.

In the present paper, we build on the idea originally presented in Salamon
_et al._ (2020a) which uses the concept of synthetic dimensions to engineer
twisted bilayers. For such scenario, we consider attractive on-site $s$-wave
interactions with full SU(4) symmetry and explore superconducting properties
in such synthetic bi-layers with a chosen size and shape of the supercell.
Near-flat bands with very small dispersion compared to its immediately
neighboring bands can be accessed with rather small unit cells in our
synthetic-dimension-based proposal, which allows us to adopt an multi-band
Hartree-Fock-Bogoliubov theory Goodman (1979a); Lewin and Paul (2014) for
probing superconductivity. The analysis is performed extensively for a wide
range of experimentally controllable parameters, such as interlayer coupling
and interaction strength. Our study seeks to understand and to characterize
the role played by a finite dispersion of the quasi-flat band on
superconductivity. In fact, the proposed setup allows to accurately control
the widths of the quasi-flat bands over a broad range Salamon  _et al._
(2020a, 2021).

The paper is arranged as follows: In Sec. II we present the lattice
Hamiltonian and discuss the band structure and interaction types appearing in
the model. In Sec. III the detailed description of Hartree-Fock-Bogoliubov
mean field decoupling is shown together with resulting Bogoliubov - de Gennes
Hamiltonian and self-consistent procedure. Sec. IV presents original results
of our study. First, we take into account the impact of band flattening,
caused by modulation of the inter-layer hopping, on the superconducting gap.
Then, we consider breaking the symmetry into SU(2) $\times$ SU(2) via
selectively tuning the interaction channels. We also tune one of interaction
types to be of negligible amplitude to facilitate the comparison of the
obtained results with standard bi-layer Fermi-Hubbard system. Finally,
conclusions are presented in Sec. V.

##  II The system

The considered cold-atom system consists of $N$ fermionic atoms loaded into an
optical square lattice with unit lattice constant. Atoms with large nucleus
manifold, e.g., 87Sr or 173Yb Cazalilla and Rey (2014), are prepared in four
internal states, which are then interpreted as two spin levels within two
synthetic layer levels. Accordingly, the four levels are denoted by two
quantum numbers, $\sigma=\uparrow,\downarrow$ for the (pseudo-)spin, and
${m=+,-}$ for the (pseudo-)layer. Throughout this study, the hopping amplitude
$t$ is set to unity, i.e. $t=1$, which fixes the units of the energies
reported in this work. The synthetic hopping amplitude between the pseudo-
layers, i.e., between $m=+$ and $m=-$, is denoted by $\Omega({\bf r})$. The
synthetic hopping is generated through a pair of Raman lasers, and the
position dependence of this coupling can be used to produce a synthetic twist,
which gives rise to the flattening of certain energy bands, see Salamon  _et
al._ (2020a). Accordingly, the kinetic part of the Hamiltonian reads

$H_{\rm kin}=H_{t}+H_{\Omega},$ (1)

where

$\begin{split}H_{t}=&-t\sum_{{\bf
r},m,\sigma}\left[a_{m,\sigma}^{\dagger}({\bf
r}+\mathbf{1}_{x})+a_{m,\sigma}^{\dagger}({\bf
r}+\mathbf{1}_{y})\right]a_{m,\sigma}({\bf r})\\\ &+\mathrm{h.c.}\end{split}$
(2)

denotes the hopping Hamiltonian between different sites in the optical
lattice, and

$H_{\Omega}=\sum_{{\bf r},m,m^{\prime},\sigma}\Omega({\bf
r})~{}a_{m,\sigma}^{\dagger}({\bf r})a_{m^{\prime},\sigma}({\bf
r})+\mathrm{h.c.}$ (3)

denotes the synthetic hopping Hamiltonian. Here,
$a_{m,\sigma}^{\dagger}(\bf{r})$ and $a_{m,\sigma}(\bf{r})$ represent the
fermionic creation and annihilation operators, respectively and ${\bf
r}=(x,y)$ denotes the position of a lattice site in the two-dimensional plane
of the optical square lattice, where $x$ and $y$ are integers. Moreover, we
define the unit vectors $\mathbf{1}_{x}=(1,0)$ and $\mathbf{1}_{y}=(0,1)$.
Note that in Eq. (3) we have chosen a spatially constant phase of the
coupling, but it is also straightforwardly possible to implement a position-
dependent phase term. Such a choice would allow for incorporating artificial
gauge fields into the synthetic dimension Celi  _et al._ (2014b); Price  _et
al._ (2017), and the effect of such artificial gauge fields in the context
twist-simulating optical lattices has been already discussed in Salamon  _et
al._ (2020a). The spatial modulation of the synthetic hopping strength is
given by

$\Omega({\textbf{r}})=\Omega_{0}\left\\{1-\alpha[1+\cos{(2\pi
x/l_{x})}\cos{(2\pi y/l_{y})}]\right\\}.$ (4)

The strength of the spatial modulation is controlled by the dimensionless
parameter $\alpha$ which imposes a twist on the hopping energies in the
lattice, i.e., a Moiré lattice. The two length scales, $l_{x}$ and $l_{y}$,
define the size of the unit cell of the Moiré lattice. Here, we focus on
$l_{x}=l_{y}=4$, which is the simplest case of the, so-called, first ”magic-
configuration”. This choice is, yet, large enough to make the band structure
sufficiently flat, see Ref. Salamon  _et al._ (2020a), but still keeps the
unit cell small enough for an efficient computational treatment. It is
important to note that, despite the original lattice being a square lattice,
the Moiré lattice has a graphene-like brick-wall geometry. Its unit cell
contains $8$ physical sites of the original lattice, as shown in Fig.
1(a).Among these eight sites, we distinguish four sets of sites
$\\{AA,AB,B,C\\}$: $AA$ and $AB$ denote the six black sites in Fig. 1(a) with
two or four ”black” nearest neighbours, respectively , and $B$ and $C$ denote
the red and green sites, respectively. According to Eq. (4), these three sets
of sites exhibit the following property:

$\Omega(\bf{r})=\begin{cases}\Omega_{0}(1-\alpha)&\text{ if }{\bf r}\in AA\lor
AB,\\\ \Omega_{0}&\text{ if }{\bf r}\in B,\\\ \Omega_{0}(1-2\alpha)&\text{ if
}{\bf r}\in C,\\\ \end{cases}$ (5)

that is, the coupling strength at sites B and C is shifted by
$\absolutevalue{\alpha}$ with respect to the coupling strength at sites of
type A, that is, type AA or AB.

Figure 1:  (a) Square lattice with unit cell of $8$ physical sites due to
synthetic coupling which is different on the black ($AA$ and $AB$ \- depending
on the amount of A-type neighbours), green ($B$), and red ($C$) sites. The
lattice is covered by unit cells in a brick wall arrangement, as indicated in
the inset. (b) Evolution of the average energy of the bands as a function of
the modulation parameter $\alpha$ at $\Omega_{0}=100$ (in units of $t$).
Quasi-flat bands have been plotted red while all the others - black. The
critical value $\alpha_{c}\approx 0.67$ is shown as a dashed line. (c)
Schematic representation of the three interaction types appearing in the
system. (d) Full spectrum of the system under periodic modulation
$\Omega_{0}=100$ and $\alpha=0.2$. (e) Zoomed plot of (d) highlighting the
negative six-fold subset of bands including the central quasi-flat bands. (f)
Spectrum of the system at $\Omega_{0}=100$ and $\alpha=0.7$, after the
flipping of the bands occurred (dashed line in panel (b)).

For our choice of $l_{x}$ and $l_{y}$, the spectrum of $H_{\rm kin}$ consists
of 16 energy bands. A symmetrical arrangement of the bands with respect to
$E=0$ reflects the particle-hole symmetry in the system. At sufficiently large
interlayer tunneling, e.g., $\Omega_{0}=100$, the spectrum is gapped, see Fig.
1(d) for $\alpha=0.2$ and (f) for $\alpha=0.7$. Among these 16 bands, we focus
on the subset of six bands at $E\approx\pm 80t$, i.e. we assume a Fermi energy
in the vicinity of this manifold. These two manifolds are plotted in black in
Fig. 1(d), and further analyzed in Fig. 1(e), where we zoom into one of the
six-fold manifold. Importantly, we observe that the two central bands of this
manifold are almost flat. Moreover, there are different degenercies, e.g. at
the $\Gamma$-point. In addition to these quasi-flat bands, the system also
exhibits isolated flat bands shown in blue and red in Fig. 1(d).

The impact of the twist parameter $\alpha$ on the structure of the whole
energy spectrum is shown in Fig. 1(b). For $\alpha<0.67$, the isolated bands
are located above and below the six-fold manifolds, whereas for $\alpha>0.67$
both isolated bands are above the sixfold manifold on the positive side of the
spectrum, and below the sixfold manifold on the negative side of the spectrum.

Within the sixfold manifold, the modulation of the interlayer tunneling
flattens the two central bands, as it mimics Moiré patterning. This implies
that, for $\alpha<0.67$, this quasi-flat band structure appears at the Fermi
surface, when the filling $\nu$ of the system is 1/4 or 3/4, i.e., the lowest
four or the lowest twelve bands are filled, see Fig. 1(d) and (e). For
$\alpha>0.67$, the quasi-flat bands are at the Fermi surface for $\nu=5/16$ or
$\nu=11/16$, i.e., the lowest five or eleven bands are filled, see Fig. 1(f).
In this way, the parameter $\alpha$ can be used to control the density of
states at the Fermi surface, which will later be shown to have a strong impact
on the superconducting behavior. We have defined the filling $\nu$ such that
$\nu=1$ corresponds to a lattice which is filled by 4 fermions per site. We
consider only the uniform value of filling (not distinguishing the filling at
each synthetic level). In the limiting case of $\alpha=1$, twelve bands become
fully degenerated at $E=0$, as can be seen from Fig. 1(b).

In order to investigate the superconductivity, we consider attractive
collisional Hubbard-type interactions between the atoms, i.e., local
interactions in the physical lattice. We assume that the internal state of the
atoms is not changed during the collision. In the most general form the
interaction Hamiltonian then reads

$H_{\rm
int}=-\sum_{\textbf{r},m,m^{\prime},\sigma,\sigma^{\prime}}U^{m,m^{\prime}}_{\sigma,\sigma^{\prime}}n_{m,\sigma}(\textbf{r})n_{m^{\prime},\sigma^{\prime}}(\textbf{r}),$
(6)

where $n_{m,\sigma}({\bf r})=a_{m,\sigma}^{\dagger}({\bf r})a_{m,\sigma}({\bf
r})$ is the density operator of a fermion in the $\\{m,\sigma\\}$ state.
$U^{m,m^{\prime}}_{\sigma,\sigma^{\prime}}$ denotes the (non-negative)
interaction strength between atoms in levels $\\{m,\sigma\\}$ and
$\\{m^{\prime},\sigma^{\prime}\\}$ and the negative sign indicates that we
consider these interactions to be attractive. In general,
$U^{m,m^{\prime}}_{\sigma,\sigma^{\prime}}$ describes 16 different kinds of
interactions. Out of these, the Pauli principle excludes all diagonal
interactions, i.e., $U^{m,m}_{\sigma,\sigma}=0$. We are left with possibly
twelve different non-zero interaction processes. Because of symmetry
arguments, these can be further grouped into three interaction types, as
illustrated in Fig. 1(c): The first type are the intralayer interactions
within a synthetic layer

$U_{1}\equiv
U^{+,+}_{\uparrow,\downarrow}=U^{-,-}_{\uparrow,\downarrow}=U^{+,+}_{\downarrow,\uparrow}=U^{-,-}_{\downarrow,\uparrow}.$
(7)

The second type groups the inter-layer interactions between particles of
opposite spin,

$U_{2}\equiv
U^{+,-}_{\uparrow,\downarrow}=U^{-,+}_{\uparrow,\downarrow}=U^{+,-}_{\downarrow,\uparrow}=U^{-,+}_{\downarrow,\uparrow}.$
(8)

The third groups contains the interlayer interactions of particles with equal
spin

$U_{3}\equiv
U^{+,-}_{\uparrow,\uparrow}=U^{-,+}_{\uparrow,\uparrow}=U^{+,-}_{\downarrow,\downarrow}=U^{-,+}_{\downarrow,\downarrow}.$
(9)

From the point of view of a realization with alkali-earth atoms, the case of
$U_{1}=U_{2}=U_{3}$ is the most natural/realistic one. Interactions with
nearly SU(N) symmetry are exhibited, for instance, between the internal states
obtained from the nuclear spin manifolds ($I=5/2$ and $I=9/2$, respectively)
for the fermionic isotopes ${}^{87}{\rm Sr}$ and ${}^{173}{\rm Yb}$, see Ref.
Cazalilla and Rey (2014). We also discuss the cases in which $U_{1}$ (or
$U_{2}$) become the dominant interactions, which is particularly relevant from
the point of view of bilayer interpretation.

##  III Mean-field decoupling

We apply a Hartree-Fock-Bogoliubov-de Gennes mean-field approach Koch (2016);
Goodman (1979b) to tackle the many-body Hamiltonian
$H_{\mathrm{kin}}+H_{\mathrm{int}}$ (see Eqs. (1) and (6)). As we consider
attractive interactions, we focus only on pairing fields in the mean-field
decomposition. Each on-site quadratic attractive term in Eq. (6) is thus
decoupled as:

$\begin{split}a_{m,\sigma}^{\dagger}a_{m,\sigma}a_{m^{\prime},\sigma^{\prime}}^{\dagger}a_{m^{\prime},\sigma^{\prime}}&\approx\langle
a_{m,\sigma}^{\dagger}a_{m^{\prime},\sigma^{\prime}}^{\dagger}\rangle
a_{m^{\prime},\sigma^{\prime}}a_{m,\sigma}\\\
&+a_{m,\sigma}^{\dagger}a_{m^{\prime},\sigma^{\prime}}^{\dagger}\langle
a_{m^{\prime},\sigma^{\prime}}a_{m,\sigma}\rangle\\\ &-\langle
a_{m,\sigma}^{\dagger}a_{m^{\prime},\sigma^{\prime}}^{\dagger}\rangle\langle
a_{m^{\prime},\sigma^{\prime}}a_{m,\sigma}\rangle,\end{split}$ (10)

where $\langle\cdot\rangle$ denotes the average. The last term is a constant
shift affecting the grand thermodynamic potential and is important for
obtaining the self consistent equations for the order parameters via
minimization of the thermodynamic potential or for assessing thermodynamic
stability of different solutions. We do not display this term in the
following. Combining Eqs. (6)-(10) and assuming symmetry between the layers
and spins, let us explicitly write down the pairing Hamiltonian:

$\displaystyle\begin{split}H_{\rm
P}&=\Delta_{1}(a_{+,\uparrow}^{\dagger}a_{+,\downarrow}^{\dagger}+a_{-,\uparrow}^{\dagger}a_{-,\downarrow}^{\dagger})\\\
&+\Delta_{2}(a_{+,\uparrow}^{\dagger}a_{-,\downarrow}^{\dagger}+a_{+,\downarrow}^{\dagger}a_{-,\uparrow}^{\dagger})\\\
&+\Delta_{3}(a_{+,\uparrow}^{\dagger}a_{-,\uparrow}^{\dagger}+a_{+,\downarrow}^{\dagger}a_{-,\downarrow}^{\dagger})+{\rm
h.c.}\end{split}$ (11)

Here, we have defined the following superconducting order parameters

$\displaystyle\Delta_{1}$ $\displaystyle\equiv U_{1}\langle
a_{+,\uparrow}a_{+,\downarrow}\rangle=U_{1}\langle
a_{-,\uparrow}a_{-,\downarrow}\rangle,$ (12) $\displaystyle\Delta_{2}$
$\displaystyle\equiv U_{2}\langle
a_{+,\uparrow}a_{-,\downarrow}\rangle=U_{2}\langle
a_{-,\uparrow}a_{+,\downarrow}\rangle,$ (13) $\displaystyle\Delta_{3}$
$\displaystyle\equiv U_{3}\langle
a_{+,\uparrow}a_{-,\uparrow}\rangle=U_{3}\langle
a_{+,\downarrow}a_{-,\downarrow}\rangle.$ (14)

The value of the order parameters can, in principle, vary within each unit
cell due to in-equivalence of the lattice sites and their surroundings caused
by spatial modulation of the synthetic coupling, described in Eq. (3). As has
been shown in Fig. 1(a) and defined in Eq. (5), one can differentiate the
sites in the unit cell into four types. While types $B$ and $C$ are taken into
account separately due to their different value of synthetic coupling
$\Omega({\bf r})$, the sites of A-type (black sites in Fig.1) are
distinguished based on geometric reasons and divided into ”bridge” (AA) and
”node” (AB) sites, depending on the amount of nearest A-type neighbours (two
for AA, and four for AB). A visual representation of this scheme is also shown
in Fig. 2, where yellow sites represent ”bridge” (AA) sites on panel (a) and
”node” (AB) on panel (b). Therefore, distinguishing between the four different
types of sites, we write the order parameter in the interaction channel
$i=1,2,3$ for site of type $I\in\\{AA,AB,B,C\\}$. For the eight sites of a
unit cell, we thus have three sets of order parameters defined below:

$\vec{\Delta_{i}}=(\Delta_{i}^{AB},\Delta_{i}^{AA},\Delta_{i}^{AB},\Delta_{i}^{AA},\Delta_{i}^{AA},\Delta_{i}^{B},\Delta_{i}^{AA},\Delta_{i}^{C}).$
(15)

It is also convenient to view the real-space fermionic operators
$a_{m,\sigma}({\bf r}_{j})$ as eight-component vectors, with each component
representing one site in the unit cell, and the $j$ representing the index of
unit cell within the Moiré lattice. Then, we Fourier transform the operators
to quasi-momentum space, via

$a_{m,\sigma}({\bf r}_{j})=\frac{1}{\sqrt{N_{s}}}\sum_{k}e^{-i{\bf k}{\bf
r}_{j}}a_{m,\sigma}({\bf k}),$ (16)

where $N_{s}$ is the number of unit cells in the lattice and
$c_{m,\sigma}({\bf k})$ is the eight-dimensional field operator of a fermion
with quasi-momentum ${\bf k}=(k_{x},k_{y})$. The real-space hopping $H_{t}$ is
diagonalized as $H_{t}=-t\sum_{m,\sigma}a_{m,\sigma}^{\dagger}({\bf
k})H^{m}_{t}({\bf k})a_{m,\sigma}({\bf k})$, with $H^{m}_{t}({\bf k})$ being a
diagonal matrix representing the eight bands per layer index $m\in\\{+,-\\}$.
Due to the symmetry between the synthetic layers, we have $H^{+}_{t}({\bf
k})=H^{-}_{t}({\bf k})$. The interlayer tunneling is also diagonal in $\bf k$,
but of the form $a_{+,\sigma}^{\dagger}({\bf k})H_{\Omega}({\bf
k})a_{-,\sigma}({\bf k})$.

In order to present the full Hamiltonian, $H=H_{\rm kin}+H_{P}$, containing
all order parameters of interest in the quadratic form, we define the
following $8\times 8$-dimensional Nambu spinor:

$\Psi_{\textbf{k}}^{\dagger}=\begin{pmatrix}a_{-,\uparrow}^{\dagger}({\bf
k})\\\ a_{-,\downarrow}({\bf-k})\\\ a_{+,\uparrow}^{\dagger}({\bf k})\\\
a_{+,\downarrow}({\bf-k})\\\ a_{-,\uparrow}({\bf k})\\\
a_{-,\downarrow}^{\dagger}({\bf-k})\\\ a_{+,\uparrow}({\bf k})\\\
a_{+,\downarrow}^{\dagger}({\bf-k})\\\ \end{pmatrix}$ (17)

The mean field Hamiltonian in momentum space is of the Bogoliubov-de-Gennes
(BdG) form and is given by

$H(\textbf{k})=\Psi_{\textbf{k}}^{\dagger}H_{\rm
BdG}(\textbf{k})\Psi_{\textbf{k}}-\epsilon(\textbf{k}),$ (18)

where $\epsilon(\textbf{k})$ is a diagonal matrix which includes all constant
values coming from the decoupling. Note that $H_{\rm BdG}({\bf k})$ is a
$64\times 64$ matrix as each component of the Nambu spinor is an 8-vector
yielding eight bands. The structure of this matrix is constructed as follows:

$H_{\rm BdG}(\textbf{k})=\begin{pmatrix}H_{F}({\bf
k})&J_{2}\otimes\mathbf{1}\otimes{\rm diag}(\vec{\Delta_{3}})\\\
J_{2}\otimes\mathbf{1}\otimes{\rm diag}(\vec{\Delta_{3}})&H_{F}^{*}({\bf
k})\\\ \end{pmatrix},$ (19)

where $\mathbf{1}$ is the two-dimensional unity matrix, $J_{2}=[[0,1],[1,0]]$
is the first Pauli matrix, and $\otimes$ denotes the tensor product.

The matrix on the diagonal block has the following structure

$H_{F}(\textbf{k})=\begin{pmatrix}H^{m}({\bf k})&H_{R}({\bf k})\\\ H_{R}({\bf
k})&H^{m}({\bf k})\\\ \end{pmatrix},$ (20)

where the four $16\times 16$ blocks are defined as

$H^{m}(\textbf{k})=\begin{pmatrix}H_{t\uparrow}^{m}(\textbf{k})-\mu-\frac{n}{3}(\frac{U_{1}}{2}+\frac{U_{2}}{2}+\frac{U_{3}}{2})&{\rm
diag}(\vec{\Delta_{1}})\\\ {\rm
diag}(\vec{\Delta_{1}})&-H_{t\downarrow}^{m}(-\textbf{k})+\mu+\frac{n}{3}(\frac{U_{1}}{2}+\frac{U_{2}}{2}+\frac{U_{3}}{2})\\\
\end{pmatrix},$ (21)

and

$H_{R}(\textbf{k})=\begin{pmatrix}H_{\Omega}(\textbf{k})&{\rm
diag}(\vec{\Delta_{2}})\\\ {\rm
diag}(\vec{\Delta_{2}})&-H_{\Omega}(-\textbf{k})\end{pmatrix}.$ (22)

The quadratic matrix $H(\textbf{k})$ depends on the unknown superconducting
order parameters $\vec{\Delta_{i}}$, which we determine self-consistently by
diagonalizing $H(\textbf{k})$ using random initial guesses of
$\vec{\Delta_{i}}$, and subsequently updating the order parameters by the ones
obtained from the diagonalization until convergence is attained. We check that
this procedure leads to the same order parameters for different initial
guesses, or, if this is not the case, we choose the solution with the lowest
grand thermodynamic potential energy.

##  IV Results

In this section we present the original results of the pairing correlations in
the system within the framework described above. While we first focus on the
case of fully symmetric interactions that naturally arise in the context of
the experimental proposal presented in Salamon  _et al._ (2020a), namely
$U_{1}$=$U_{2}$=$U_{3}$ (see Fig. 1(c)), where all internal degrees of freedom
of each atom are coupled to the each others with the same strength, we also
study the effects of SU(4) symmetry breaking by considering the relative
alteration between interaction channels $U_{1}$ and $U_{2}$ in the subsequent
subsection. In this context, we set $U_{3}=0$, which is justified because any
pairing in the $U_{3}$ channel is strongly suppressed by the strong Raman
coupling, $\Omega$. This coupling (interlayer hopping) energetically penalizes
the state with two particles of equal pseudospin per site, as compared to the
single-particle states formed by the antisymmetric superposition of the states
with equal pseudospin and opposite pseudolayer degree of freedom.

A very interesting phase diagram is found in the regime of weak interactions:
superconductivity is exponentially suppressed in the symmetric case, i.e. near
$U_{1}=U_{2}$, but a significant non-zero SC gap can be again amplified if the
interactions are tuned to a sufficiently non-symmetric choice. We observe the
narrowing of the weakly-superconducting wedge shaped region in the phase
diagram with increasing interaction strength, as well as the coexistence of
both the inter and intra-layer superconducting order parameters. Techniques to
tune interactions, such as Feshbach resonance or magnetic/optical field
gradients Sonderhouse  _et al._ (2020a); Martone  _et al._ (2014), allow in
experiments for such a selective choice of a dominant interaction channel.
Increase of band flattening via the tunable parameter $\alpha$ leads to the
development of strong superconducting order for lower values of attractive
interactions, in particular, as compared to the standard ($\alpha=0$) bi-layer
Fermi-Hubbard model.

Numerical calculations were performed for a system of 2 $\times$ 256 sites
forming a $2\times L\times L$ bilayer square lattice with $L=16$. The low
temperature properties were studied by choosing the temperature to
$k_{B}T=0.02t$ (or inverse temperature $\beta\approx 3L$ for the finite size
system).

###  IV.1 Superconductivity in $SU(N)$ symmetric system

We set a strong interlayer (Raman) coupling $\Omega_{0}/t=100$ in order to
focus on effects in the quasi-flat band regime of our system. We investigate
the influence of two main parameters, namely the modulation amplitude $\alpha$
and interaction strength $U$ on the SC characteristics.

The parameter $\alpha$ controls the relative strength of the spatially
dependent part of the synthetic hopping $\Omega(\mathbf{r})$ in eq. (4). The
increase of $\alpha$ primarily results in flattening in the dispersion of the
set of quasiflat bands of our interest. Up to a value of $\alpha\approx 0.67$,
the flattened bands lie exactly at Fermi energy of the system for a filling
$\nu=1/4$ (that is, one fermion per physical site). As shown on Fig. 1(b), the
critical value of $\alpha\approx 0.67$ causes band flipping, which lifts the
Fermi energy of the quarter-filled system away from quasi-flat band (see Fig.
1(d) and (f)). We focus on the case with $\nu=1/4$ and $0\leq\alpha\leq 0.67$.
A large superconducting gap opens in the SU(4) symmetric system, for
interaction strengths, higher than a certain, $\alpha$-dependent cut-off
interaction amplitude, namely $U>U_{C}(\alpha)$. We note that $U_{C}(\alpha)$
decreases if one considers lower temperatures. Enhancing the density of states
at the Fermi energy by increasing the band flattening with increasing $\alpha$
is expected to lead to larger stability of the superconducting phase and
therefore lower threshold values $U_{C}(\alpha)$.Indeed, such behaviour is
markedly seen in Fig. 2(a,b). For an example value of $\alpha=0.6$, the
superconducting state appears above $U\sim 6.2t$. Similar effect has been
observed for the critical temperature of the system with respect to modulation
$\alpha$. Fig. 2(c) depicts the growth of the critical temperature with band-
flattening. We have limited the range of $\alpha$ in the figure such that it
covers only the scenario with Fermi energy matching the energy of quasi-flat
bands. Therefore, by tuning the modulation parameter $\alpha$ one can reach
superconducting state at lower interaction values as compared to the uniform
inter-layer hopping scheme. Moreover, resulting difference in critical
temperature between highly modulated system at $\alpha=0.6$ and un-modulated
one is of one order of magnitude. In other words, the superconductivity near
$U_{C}$ is truly induced by the synthetic twist.

Figure 2: (a) Expansion of superconducting gaps $\Delta^{AA}_{1}$ and
$\Delta^{AB}_{1}$ as a function of the $U=U_{1}=U_{2}=U_{3}\leq 16t$ under
different modulation values $\alpha$ resulting in specific cutoff value
$U_{c}$ ($\Delta_{2}$ has an identical behavior). (b) Critical temperature
dependence on the modulation strength $\alpha$. Inset plot represents the
energetic width $\delta_{F}$ of the quasi-flat bands as a function of the
modulation parameter $\alpha$.

###  IV.2 $SU(4)$ to $SU(2)\times SU(2)$ symmetry breaking

Figure 3: Superconducting order parameter $\Delta_{1}^{AA}$ (panel a and c)
and $\Delta_{2}^{AA}$ (panel b and d) as functions of relative amplitude of
$U_{1}$ and $U_{2}$ interactions at $\alpha=0.67$. Existence of SC phase in
this $SU(4)$ symmetric system is not present until $U=U_{1}=U_{2}$ reaches a
cut-off value $U_{c}$, at which the non-SC valley narrows to zero. The width
of the valley as well as the critical interaction strength depends on the
modulation parameter $\alpha$ (as shown on the Fig. 2). Panels (c) and (d)
represent the regime of small interaction strengths that are not covered on
the panels (a) and (b) due to limited colormap resolution. Sites of (AB) type
reveal qualitatively identical behavior, however, with much smaller pairing
amplitude (See Fig.4(b))

.

The SU(N=4) symmetry (spin and magnetic levels) of the system resulting from
the internal structure of the atoms forbid the free tuning of the strength of
each interaction type individually without an external fields applied. As
shown in Stellmer  _et al._ (2011); Sonderhouse  _et al._ (2020b), one can
tune the strength of interaction types by applying external state-dependent
force that effectively separates the $m_{F}$ manifold of the ground state. For
atoms with two valence electrons, such as 87Sr, this technique, also termed
as, optical Stern-Gerlach (OSG) has been already successfully applied
experimentally Stellmer  _et al._ (2011); Sonderhouse  _et al._ (2020a).
Following the scheme of the system proposed in Salamon  _et al._ (2020a, b),
we propose to use OSG to modify the energy gaps between specific $m_{F}$
states, thereby tuning the interaction strength of desired type. In this
paragraph, we study the effects on the SC properties due to a modification of
the interaction strength in such fashion. We note that the extreme scenario of
$U_{1}\gg U_{2}$ corresponds to standard spin-spin onsite only interactions
widely explored in Fermi-Hubbard model. Whereas the SU(N)-symmetric scenario
requires a threshold value of the coupling $U$ for the system to exhibit SC,
the symmetry-broken scenario allows us to observe SC pairing even for
$U_{1},U_{2}<U_{C}$ (with one of them possibly even being zero).
Interestingly, a phase where SC is dominated by $U_{1}$ is separated from a
$U_{2}$ dominated SC phase through an intermediate phase in which SC is
absent, leading to the interesting re-entrance phenomena, when one of the
interaction parameters is tuned.

Figure 4:  $\Delta_{AA,AB}$ (dashed) and $\Delta_{2}$ (solid) as functions of
$U_{2}$ (in units of $U_{1}$) for four different values of $U_{1}$ at
modulation strength $\alpha=0.67$. One can observe the shift of the SC gap
decay towards $U_{2}/U_{1}=1$ with the increase of interaction strength. For
$U_{1}=6t$, the decay of SC gap occurs at $U_{2}/U_{1}=1$ marking the $U_{C}$
for this particular value of $\alpha$.

The different phases are seen in Fig. 3 where the SC gaps $\Delta^{AA,AB}_{1}$
are plotted as a function of the interaction strengths $U_{1}$ and $U_{2}$, in
the interval $[1,6]$ in panels (a) and (b), and in the interval $[0,2]$ in
panels (c) and (d). Here we have chosen $\alpha=0.$, and the corresponding
$U_{C}$ from the SU(4) symmetric system is $U_{C}\approx 6$. Hence, the shown
regime is below $U_{C}$ everywhere, and accordingly, the system does not
exhibit SC along the line $U_{1}=U_{2}$. It is seen that this non-
superconducting regime, plotted in black, has a finite width, which narrows as
$U_{1}$ and $U_{2}$ approach towards $U_{C}$. The width and the rate of the
narrowing depends on the modulation parameter $\alpha$, as indicated in the
Fig. 4. Nevertheless, relatively small deviations from the symmetric
interaction are already sufficient to open a SC gap. This can be understood in
the following way: the SC pairing of each interaction type compete with each
other, but breaking the symmetry favors one interaction type with respect to
the other, and therefore facilitates the pairing in this channel.

Figure 5: Superconducting gap in the system of $U_{1}=2t$ and $U_{1}\gg
U_{2}$. (a) The order parameter $\Delta_{1}$ associated with interaction type
$U_{1}$ as a function of the inter-layer hopping strength $\Omega_{0}$ at
$\alpha=0.2$. We have set $U_{2}=0$ as its presence weakens the order
parameter $\Delta_{1}$, as shown on Fig. 4(b). The green line represents the
situation where the $\Omega(\textbf{r})$ is homogeneous, i.e. $\alpha=0$. The
separate plot for $\Delta_{C}$ has been omitted since it’s behaviour is
identical to $\Delta_{B}$, which has been plotted. (b) Dependence of the order
parameter $\Delta_{1}^{AA}$ on the modulation strength $\alpha$ for
$\Omega_{0}=100t$. We have omitted the plot of $\Delta_{1}^{AB}$ due to its
negligible amplitude. Sudden drop of pairing (marked by dashed gry line)
occurs around critical value of $\alpha\approx 2/3$ at which the Fermi level
does not reside at quasi-flat bands any more. Such situation has been shown on
Fig. 1(f).

Although one of the two different pairing channels becomes dominant, there is
still coexistence of the SC gaps corresponding to the two channels, $U_{1}$
and $U_{2}$, for most parameter choices (unless we are in the non-SC regime,
or one of the interaction parameters is zero). However, monotonicity of the
size of the gap with respect to the interaction strength is not obvious. We
present a closer insight to this phenomenon through the details presented in
Fig. 4. It depicts the dependence of the SC order parameters on $U_{2}$ for
four different values of $U_{1}$ at $\alpha=0.67$. The decay of
$\Delta^{AA}_{1}$ and $\Delta^{AA}_{2}$ as $U_{2}$ approaches $U_{1}=U_{2}$ is
the beginning of the zero-gap valley. With increasing value of $U_{2}$ the
corresponding gap $\Delta^{AA}_{2}$ opens and keeps increasing until reaching
its maximum value. In parallel, SC order parameter corresponding to $U_{1}$
constantly decreases.

We now consider the cases of $U_{1}\gg U_{2}$ which can be obtained
experimentally with help of OSG techniques. We would like to note that the
same results have been obtained for the opposite case, i.e. $U_{2}\gg
U_{1}$.This scenario allows us to directly compare our bi-layer model with
quasi-flat bands with a standard bi-layer Fermi-Hubbard model with on-site
spin-spin interactions. We choose this regime in order to obtain a maximum
possible value of the gap for a given amplitude of $U$. As it has been shown
on the Fig. 4, the widest gap appears for highly unequal interaction values,
i.e. $U_{1}\gg U_{2}$. We begin by investigating the dependence on the
synthetic hopping amplitude, $\Omega_{0}$. In its absence the system consists
of two uncoupled layers of square lattices. At finite $\Omega_{0}$, we can
flatten the bands through the spatial modulation provided by $\alpha$, or
realize the standard bi-layer model, i.e. $\alpha=0$. Panel (a) of Fig. 5
depicts both of these scenarios. In particular, we have plotted separately the
gap for A-type and (B,C) sites in $\alpha$-modulated case, as well as the mean
value of the gap averaged over all sites of the unit cell. The green line
represents the size of the gap for a standard bilayer model, that is, with
$\alpha=0$. Interestingly, with a small separation of the spectrum caused by
inter-layer hopping, SC gaps drop. However, after a full separation of the
bands into positive and negative branches, the system the size of the gap of
the quasi-flat band system starts to grow, in contrast to the SC order
parameter of the standard FH system. Panel (b) of Fig. 5 represent the
dependence of the order parameter $\Delta_{1}$ on the modulation parameter
$\alpha$. The plot depicts the situation where $\Omega_{0}/t=100$ and
therefore $\Delta^{B,C}_{1}$ can be neglected due to their vanishing values.
The amplitudes of $\Delta_{1}$ in panel (b) have been obtained for the fixed
filling $n=1/4$. Summarizing, modulation of the inter-layer hopping leads to
enhanced SC order parameter with respect to a non-modulated one at
sufficiently high $\Omega_{0}$. This effect is a result of band flattening and
therefore disappears once the modulation $\alpha$ crosses the critical value
or the Fermi energy does not match the energy of quasi-flat bands.

##  V Conclusions

In this paper, we have used Bogoliubov-deGennes theory to study attractive
interactions of the synthetic bi-layer square lattice system. The studied
model goes beyond thoroughly explored bi-layer Hubbard models and tackles
correlated phases in quasi-flat band systems thanks to its modulated inter-
layer hopping. We have taken into account all possible density-density on-site
interaction channels.

First, we have focused on the natural case of equal interaction amplitudes and
observed strong dependence between the inter-layer modulation parameter
$\alpha$ and minimal interaction strength $U_{c}$ required to reach SC
pairing. Observed results confirm the followings: _(i) Flattening of the bands
in the vicinity of Fermi energy leads to opening of the SC gap at much lower
interaction amplitudes, when compared to uniform coupling systems, and (ii)
Band flattening causes the critical temperature to increase in these novel
synthetic-twist-induced superconducting latices with magic configurations._

Further results are obtained by altering relative interaction amplitudes in
the system: _(iii) The resulting phase diagram revealed the non-SC valley
along $U_{1}=U_{2}$. The width of this valley narrows with the growing
amplitude of interactions to finally completely vanish at $U_{c}$ specific for
each value of $\alpha$._ The nature of the valley and reason for its narrowing
is not clear, as well as the re-entry of SC phase at sufficient interaction
alteration. Similar behaviour has been observed in bi-layer Hubbard square
lattices with only one correlated layer Zujev  _et al._ (2014). There, the re-
entrance of the SC gap was a result of increasing inter-layer hopping. Here,
however, we observe similar effect as a function of inter-layer interactions
with inter-layer hopping being already high.

While the results discussed in this paper help to form a general understanding
of the effects of band flattening on superconductivity in the synthetically
twisted materials, further studies can be be pursued in future, particularly,
in context of topologically nontrivial bands with relatively weak dispersion,
which can be obtained via more involved, but experimentally viable, means
e.g., via imaginary next-to-nearest neighbor tunnelings driving the system
into a quantum anomalous Hall phase Salamon  _et al._ (2021). In such cases,
Wannier functions with algebraically decaying tails may originate from nonzero
Chern number Peotta and Törmä (2015). Novel understandings of the correlated
phenomenon could then be obtained via incorporation of new mechanisms, such as
correlated tunnelings in extended Hubbard bi-layer systems.

##  VI Acknowledgements

The authors would like to thank Utso Bhattacharya and Leticia Tarruell for all
the fruitful discussions and insightful comments. We acknowledge support from:
ERC AdG NOQIA; Agencia Estatal de Investigación (R&D project CEX2019-000910-S,
funded by MCIN/ AEI/10.13039/501100011033, Plan National FIDEUA
PID2019-106901GB-I00, FPI, QUANTERA MAQS PCI2019-111828-2, Proyectos de I+D+I
“Retos Colaboración” QUSPIN RTC2019-007196-7); Fundació Cellex; Fundació Mir-
Puig; Generalitat de Catalunya through the European Social Fund FEDER and
CERCA program (AGAUR Grant No. 2017 SGR 134, QuantumCAT U16-011424, co-funded
by ERDF Operational Program of Catalonia 2014-2020); EU Horizon 2020 FET-OPEN
OPTOlogic (Grant No 899794); National Science Centre, Poland (Symfonia Grant
No. 2016/20/W/ST4/00314); European Union’s Horizon 2020 research and
innovation programme under the Marie-Skłodowska-Curie grant agreement No
101029393 (STREDCH) and No 847648 (“La Caixa” Junior Leaders fellowships
ID100010434: LCF/BQ/PI19/11690013, LCF/BQ/PI20/11760031, LCF/BQ/PR20/11770012,
LCF/BQ/PR21/11840013). RWC acknowledges support from the Polish National
Science Centre (NCN) under the Maestro Grant No. DEC-2019/34/A/ST2/00081. DR
acknowledges support from Science and Engeenering Research Board (SERB),
Department of Science and Technology (DST), under the sanction No.
SRG/2021/002316-G.

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# Transport-enabled entangling gate for trapped ions

Holly N. Tinkey<EMAIL_ADDRESS>Craig R. Clark  Brian C. Sawyer
Kenton R. Brown  Georgia Tech Research Institute, Atlanta, GA 30332, USA

###### Abstract

We implement a two-qubit entangling Mølmer-Sørensen interaction by
transporting two co-trapped ${}^{40}\mathrm{Ca}^{+}$ ions through a
stationary, bichromatic optical beam within a surface-electrode Paul trap. We
describe a procedure for achieving a constant Doppler shift during the
transport which uses fine temporal adjustment of the moving confinement
potential. The fixed interaction duration of the ions transported through the
laser beam as well as the dynamically changing ac Stark shift require
alterations to the calibration procedures used for a stationary gate. We use
the interaction to produce Bell states with fidelities commensurate to those
of stationary gates performed in the same system. This result establishes the
feasibility of actively incorporating ion transport into quantum information
entangling operations.

Recent progress in the ability to control trapped-ion positions and velocities
offers opportunities to explore novel roles for active transport in quantum
logic operations. Transport operations are essential to most modern ion trap
experiments to enable loading, individual detection, and individual addressing
[1, 2]. Maturing trap design and electrical potential control hardware have
led to impressive feats of fast shuttling [3, 4], fast ion separation [5, 6],
optical phase control [7], junction transport [4, 8], and ion chain rotation
[9, 10].

An architecture that incorporates ion transport directly into quantum gates
was proposed as an approach to reduce optical power and timing precision
requirements within large trap arrays [11]. This architecture was partially
realized by de Clercq et al., who performed one-qubit operations in parallel
on two ${}^{9}\mathrm{Be}^{+}$ ions by transporting them through reflected,
co-propagating beams in separate trap regions [12]. This work did not
demonstrate the two-qubit entangling operations that would be necessary for a
universal gate set [13].

Here we demonstrate two-qubit entangling gates performed on trapped ions
during transport. In contrast to the hyperfine qubits envisioned in ref. [11],
we address an optical qubit transition between an electronic ground state and
a metastable excited state of 40Ca+ ions confined within a surface Paul trap,
where we can perform both one-qubit and two-qubit gates with a single, global
beam. We modify the temporal interpolation of the transport potential to
control the velocity of the ions across an 80 $\mu$m trap region, and we apply
a bichromatic field during this transit to produce a Mølmer-Sørensen (MS)
entangling interaction [14]. In order to compensate dynamically for the
varying ac Stark shift during the interaction, we leverage small changes in
the Doppler shift and thereby obtain fidelities tantamount to those of
stationary gates in the system.

In these experiments, two 40Ca+ ions are confined 58 $\mu$m above a surface-
electrode linear Paul trap [15]. A radio-frequency potential with 176 V
amplitude at 56.4 MHz applied to long electrodes on both sides of the trap
axis provides radial confinement. Arbitrary waveform generators (AWGs) deliver
potentials with a maximum amplitude of $\pm 12$ V and a 5 ns sampling rate to
42 segmented electrodes to control the strength and location of the axial
potential minimum [7]. The center-of-mass (COM) and breathing-motion (BM)
axial mode frequencies are $\omega_{\mathrm{COM}}/(2\pi)=1.41$ MHz and
$\omega_{\mathrm{BM}}/(2\pi)=2.45$ MHz [16]. We use the ground
$|S_{1/2},m_{j}=-1/2\rangle$ ($|S\rangle$) and metastable
$|D_{5/2},m_{j}=-1/2\rangle$ ($|D\rangle$) electronic states of the ions as
our qubit, and we coherently manipulate the populations in these states with a
narrow-linewidth 729 nm laser beam oriented at $45^{\circ}$ to the trap axis
and with a waist of 15 $\mu$m. We distinguish populations in the two-ion
bright state $P_{2}$ ($|SS\rangle$), the one-ion bright subspace $P_{1}$
(superpositions of $|SD\rangle$ and $|DS\rangle$), and the dark state $P_{0}$
($|DD\rangle$) using fluorescence detected by a single photomultiplier tube
while illuminating the ions with 397 nm and 866 nm light [17]. The trap is
housed in a room temperature vacuum chamber with windows for optical access,
and the chamber and optics are surrounded by a mu-metal enclosure.

Figure 1 illustrates the trap geometry and experimental sequences. The ions
first undergo Doppler cooling, sideband cooling, and state preparation at
position ‘C’ and are transported 40 $\mu$m to the left (position ‘L’) in 25
$\mu$s. Then they experience the transport of primary interest for this work,
an 80 $\mu$m portion of the trap (‘L’ to ‘R’) at 0.5 m/s. For our beam
geometry, this motion Doppler shifts the 729 nm beam by 0.5 MHz. The ions then
are returned to ‘C’ in 10 $\mu$s for additional 729 nm pulses and final state
detection. For interactions that take place during the entire
‘L’$\rightarrow$‘R’ transport duration, the 729 nm beam is switched on 2
$\mu$s before the transport starts and off 2 $\mu$s after the transport has
finished [Fig. 1(b)]. For detailed in-flight spectroscopy experiments, we
divide the transport into eight 20 $\mu$s portions and probe these segments
with shorter 729 nm pulses [Fig. 1(c)].

The MS entangling interaction requires two optical tones applied
simultaneously with frequencies detuned near red and blue motional sidebands
[18]. We create this bichromatic field by passing the 729 nm beam through an
acousto-optic modulator (AOM) with two rf tones applied; both diffracted
output beams are coupled into a single optical fiber to deliver co-propagating
laser fields to the trapped ions [19, 20]). For an interaction duration
$\tau$, the optical tone frequencies and powers are chosen such that the gate
transforms ions initialized in the ground state $|SS\rangle$ into a maximally
entangled Bell state $(|SS\rangle-i|DD\rangle)/\sqrt{2}$. To reliably
reproduce this Bell state, we impose a fixed phase upon the spin-dependent
force created by the bichromatic field at the beginning of the transport. We
quantify the fidelity $F$ of the state produced in this way using a
combination of two experimental results: (1) the populations $P_{0}$ and
$P_{2}$ after the gate and (2) the amplitude $A$ of a parity signal
constructed by applying a global ${\pi/2}$ pulse with varying phase after the
gate [21].

Figure 1:  (a) Diagram of the trap and 729 nm beam intensity profile along the
trap axis with rf electrodes in dark gold parallel to the trap axis and
segmented electrodes above and below them. Ions begin in the center of a 60
$\mu$m wide electrode at position ‘C’. ‘L’ and ‘R’ denote the leftmost and
rightmost portions of the transport region. (b) Experimental sequence for full
transport measurements. Green blocks indicate optical operations (cooling,
state preparation) before and after transport, red blocks indicate 729 nm
pulses, and blue blocks indicate transport operations between indicated
positions. (c) Segmented measurement sequence in which the 729 nm beam is
pulsed only during a 20 $\mu$s segment of the full transport duration. Red
pulses in parentheses indicate 729 nm pulses used in parity analysis and red-
sideband spectroscopy measurements.

We determine several initial sets of dc electrode voltages to confine the ions
statically at 2 $\mu$m intervals along the trap axis by creating a three-
dimensional discretized model of the ideal trap and solving for potentials
that produce harmonic confinement [22]. For simple shuttling operations,
linear interpolation between these harmonic potentials is sufficient to
transport ions from one position to another. We use this preliminary sequence
of voltages, called a waveform, as a starting point for transport across the
80 $\mu$m experimental region. Unfortunately, the confining fields produced in
practice can differ from the ideal solutions due to fabrication imperfections,
stray fields, and filter distortions, so we calibrate modifications to the
waveform to correct for variations in axial confinement strength and transport
velocity.

Because the entangling operation relies on interactions with an axial motional
mode, producing constant axial confinement during transport simplifies the
gate implementation. For this we take measurements of the COM mode frequency
of an ion in a stationary potential every 5 $\mu$m along the transport region:
we apply an rf excitation to a nearby electrode and measure the detected
fluorescence as the excitation frequency is varied. These measurements reveal
deviations in the harmonic potential strength (as high as 4.6% variation in
$\omega_{\mathrm{COM}}$), which we correct with a multiplicative scaling of
the electrode voltages.

Measurements of the Doppler-shifted qubit transition frequency after this
procedure reveal undesired deviations in the transport velocity. We expect the
frequency of the 729 nm beam to be shifted by 500 kHz for ions moving at 0.5
m/s, but initial spectroscopy of the qubit resonance realized by shuttling an
ion over the entire region from ‘L’ to ‘R’ reveals a multi-peaked feature
[black points in Fig. 2(a)]. This indicates that the Doppler shift is changing
during transport. We instead probe the in-flight Doppler shift in smaller 20
$\mu$s segments [see Fig. 1(c)], which produces single-peaked spectra
representing the local Doppler shifts plotted in Fig. 2(b). To achieve a
constant Doppler shift during transport, we modify the ion velocity in each
segment of the waveform with an appropriate change in sampling density. When
this correction is applied to each segment, the full-transit spectrum exhibits
a narrower, single peak [red points in Fig. 2(a)], confirming that the
corrected waveform greatly reduces Doppler shift variation. We refer to this
corrected waveform as the ‘constant-velocity waveform’ below.

Figure 2:  Black circles (red triangles) indicate data taken before (after)
waveform correction. (a) Frequency spectrum of the qubit transition; points
represent the mean of two-ion fluorescence counts over 500 experiments, and
error bars represent the standard error of the mean. (b) Doppler shift
measured in each 10 $\mu$m segment of the transport region as determined from
fits to spectral peaks; error bars (fit errors) are smaller than the size of
the points.

In the absence of Stark shifts, the blue and red BM motional sideband
frequencies are given by $\omega_{b}=\omega_{c}+\omega_{\mathrm{BM}}$ and
$\omega_{r}=\omega_{c}-\omega_{\mathrm{BM}}$, respectively, where $\omega_{c}$
is the qubit carrier frequency and $\omega_{\mathrm{BM}}$ is the frequency of
the breathing mode. When the ions are illuminated with the two MS interaction
tones, the sideband frequencies experience an intensity-dependent light-shift
of $\Delta_{S}$, assumed to be equal for both sidebands. To implement the
entangling interaction, we apply the two MS tones at detunings (from the bare
carrier) of $\delta_{b}=\omega_{\mathrm{BM}}+\delta_{m}+\delta_{g}$ for the
blue tone and $\delta_{r}=-\omega_{\mathrm{BM}}-\delta_{m}+\delta_{g}$ for the
red tone. Here, $\delta_{m}$ is the usual ‘mode detuning’, and $\delta_{g}$ is
a ‘global detuning’ that can be used to compensate for the light shift.

We consider the interaction of co-propagating laser fields of equal intensity
and polarization at frequency detunings $\delta_{b}$ and $\delta_{r}$ with the
electric quadrupole transition of the ions [23]. After taking both Lamb-Dicke
and rotating wave approximations and neglecting off-resonant terms, we write
the MS Hamiltonian in the (Stark-shifted) spin and motion interaction frame
as:

$\displaystyle
H_{\mathrm{MS}}=\frac{\hslash\Omega\eta_{\mathrm{BM}}}{2}(a^{\dagger}e^{-i(\delta_{m}+\delta_{g}-\Delta_{S})t}+ae^{i(\delta_{m}-\delta_{g}+\Delta_{S})t})$
$\displaystyle\times~{}(\sigma^{+}_{1}-\sigma^{+}_{2})+h.c.$

where $\Omega$ is the carrier Rabi frequency, $\eta_{BM}\sim 0.042$ is the BM
Lamb-Dicke parameter, $a^{\dagger}$ and $a$ are creation and annihilation
operators for BM vibrations, and $\sigma^{+}$ and $\sigma^{-}$ are raising and
lowering operators for the internal electronic states of the indicated ion. We
model the dynamics of the interaction by solving the time-dependent
Schrodinger equation for $\tau=160$ $\mu$s, and we allow the parameters of
$H_{\mathrm{MS}}$ to vary during the interaction to analyze the effects of a
changing Rabi frequency and Stark shift in cases where the ions are
transported across the 729 nm beam.

We briefly describe typical calibration procedures and results for an MS
entangling gate on ions in a stationary potential in order to highlight the
differences between this simpler situation and the interaction we later
implement on transported ions. For a constant-intensity interaction acting on
ions in a stationary well, we calibrate the Stark-shifted sideband frequencies
such that $\delta_{g}=\Delta_{S}$ and balance their relative Rabi frequencies
using a Rabi experiment for each tone with the other tone applied
simultaneously at $\delta_{m}/(2\pi)=25$ kHz. Then, for a fixed bichromatic
interaction duration, we analyze the ion state populations $P_{0}$, $P_{1}$,
and $P_{2}$ as a function of $\delta_{m}$ to determine that value which
minimizes $P_{1}$, and we adjust the beam powers at this optimum detuning to
balance $P_{0}$ and $P_{2}$. For $\tau=160$ $\mu$s, we choose
$\delta_{m}/(2\pi)=12.5$ kHz (implementing two loops in motional phase space),
and achieve a gate fidelity $F=97.0(4)$%. This fidelity is limited largely by
the coherence of our laser: we measure a 1.4(3)% contrast loss in a single-ion
Ramsey experiment with 160 $\mu$s delay, suggesting a decoherence-induced two-
ion process error of 2.8(6)%.

With the Stark shifts compensated ($\delta_{g}=\Delta_{S}$), the interaction
Hamiltonian reduces to a form symmetric in $\delta_{m}$
($H_{\mathrm{MS}}\propto\cos(\delta_{m}t)$). This results in a symmetry of the
state population curves about $\delta_{m}=0$, visible in Fig. 3(a). The
constant-intensity square pulse of 729 nm light produces minima in $P_{1}$ at
mode detunings $\delta_{m}=n/\tau$ for integers $n$, so that only these values
are optimal. Revivals in $P_{1}$ between these optimal values can be
diminished through a smooth ramp up and down of the beam intensity, often
achieved via manipulation of the AOM rf drive power [24], thereby reducing
this constraint on detunings.

Figure 3:  State populations $P_{0}$ (red), $P_{1}$ (blue), and $P_{2}$
(green) as a function of mode detuning $\delta_{m}$ for $\tau=160$ $\mu$s.
Points represent experimental data (error bars give the 68% confidence
interval assuming binomial statistics). Solid lines represent fits to
simulations with our model Hamiltonian $H_{\mathrm{MS}}$ and with $\Omega$ and
$\Delta_{S}$ as free parameters; in the transport gate case these are given a
Gaussian time envelope. Vertical dashed lines indicate the detunings
corresponding to the reported gate fidelities. (a) Constant intensity
interaction with ions in a stationary potential; $F=97.0(4)\%$. (b)
Interaction on ions transported with modulated velocity through the beam;
$F=97.1(4)\%$.

When we transport ions through the stationary 729 nm Gaussian beam, they
experience a natural intensity ramp without AOM rf drive modulation, but this
smoothly varying optical intensity also creates a time-dependent Stark shift
$\Delta_{S}(t)$ that cannot be completely compensated through a constant
offset in drive frequencies. The uncompensated Stark shift breaks the mode-
detuning symmetry of the interaction, complicating the dynamics and making it
difficult to create the desired Bell state. We implement two distinct Stark
shift compensation methods for these ‘transport gates’ performed on
transported ions; the first ‘static’ method involves the application of a
constant $\delta_{g}$ that roughly counteracts an average effective
$\Delta_{S}$. The second ‘dynamic’ method leverages in-flight adjustments to
the velocity of the moving potential to smoothly counteract $\Delta_{S}(t)$
with a varying Doppler shift. We model these effects in both cases using a
Gaussian envelope on $\Omega(t)$ and $\Delta_{S}(t)$ that takes into account
the beam waist, the moving potential, and the ion spacing.

All calibrations for the transport gate are performed with the beams turned on
during the full transport duration [Fig. 1(b)]. We begin by calibrating the
Doppler-shifted, bare sideband frequencies ($\omega_{b}$ and $\omega_{r}$)
using a reduced optical power with the constant-velocity waveform; the lowered
intensity avoids over-driving the transitions and minimizes the Stark shift.
We then perform the MS sequence at full power and analyze the resulting
populations to determine a value for the detuning $\delta_{m}$ that minimizes
$P_{1}$. We first attempt the static compensation method, for which we scan
the global detuning $\delta_{g}$ to further minimize $P_{1}$ and then optimize
the beam powers to balance $P_{0}$ and $P_{2}$. We perform the MS gate at
$\delta_{m}/(2\pi)=14.2$ kHz and $\delta_{g}/(2\pi)=4.4$ kHz, and we measure a
gate fidelity $F=96.6(4)$%.

Because the transported ions spend less time in the most intense beam portion
compared to ions in a stationary well, the overall intensity of the optical
tones necessary to produce the entangled Bell state is larger: a three-fold
increase in optical intensity between the stationary and transport
interactions shown in Fig. 3. To quantify the light shift variation during the
transport gate, we measure the Stark-shifted sideband frequencies during 20
$\mu s$ segments of the transport at the gate intensity. As described above
for the stationary potential well, each sideband is calibrated with the other
tone applied simultaneously at $\delta_{m}/(2\pi)=25$ kHz. We measure a 5 kHz
difference between sideband frequencies at the edge of the beam and those in
the most intense, center portion - a quantity consistent with the value of
$\delta_{g}$ used to optimize gate performance.

Rather than applying a constant frequency correction for the light shift, our
control over ion velocity during transport allows us to implement a dynamic
Stark shift correction. We create a ‘modified-velocity waveform’ by making
further small adjustments to the sampling density in each segment of the
waveform. The adjustments produce an additional time-varying Doppler shift
$\delta_{D}(t)$ on the laser tones that counteracts the intensity-dependent
Stark shift ($\delta_{g}=\delta_{D}(t)=\Delta_{S}(t)$). Using the bare,
Doppler-shifted sideband frequencies calibrated with the constant-velocity
waveform, we implement the MS interaction on ions transported with the
modified-velocity waveform. We choose a value of $\delta_{m}$ that minimizes
$P_{1}$ [Fig. 3(b)], and we optimize the global beam power to balance $P_{0}$
and $P_{2}$. For $\delta_{m}/(2\pi)=-15$ kHz, we measure a gate fidelity of
$F=97.1(4)$%. The interaction model fit to the dynamic correction transport
gate data determines an uncompensated Stark shift of 150 Hz responsible for
the asymmetry of the population curves at small $\delta_{m}$ values. The use
of fine adjustments to the moving potential’s velocity allows us largely to
remove the symmetry-breaking light shift effects while maintaining a smooth
optical intensity ramp and therefore allows a wider range of detunings and
optical intensities for the gate.

We note that implementation of the transport gate relies heavily on the
hardware chosen to produce and deliver the dc electrode voltages. The
transport gates require a well defined phase for the optical force during
transport (relative to the instant when shuttling begins) to ensure that the
same motional phase-space trajectory is achieved during every experimental
repetition. The subdivision of the transport waveform into eight 20 $\mu$s
segments does not approach the limits of the AWG sampling rate (5 ns) nor the
response time of the electrode filters (2 $\mu$s). Shorter waveform segments
would provide finer velocity control for correcting Doppler shift variations
and for dynamically compensating Stark shifts. Through observations of mode
detuning asymmetry, we can leverage the gate itself to calibrate small
corrections to the waveform interpolation which optimally compensate the
variable Stark shifts. Despite the coarse discretization of the transport
waveform used here, we obtain gate fidelities equalling those of stationary
gates in our system and demonstrate the feasibility of a transport-based
universal quantum gate set.

Ongoing advances in trapped-ion experimental control provide opportunities to
explore new quantum logic architectures and techniques. Fine spatial and
temporal manipulation of trap electric fields yields freedom in the confining
potential position, velocity, and strength that can be exercised to modify
qubit interactions in ways that can reduce constraints on experimental
requirements and can improve gate performance. The transport gate technique
demonstrated here could be extended to longer ion strings with different
motional modes and ion species with appropriate changes in beam geometry. In
particular, the scheme is amenable to $\sigma^{z}\sigma^{z}$ interactions such
as the optical-transition dipole force gate [25, 26] where the Stark shift
effects could be eliminated entirely with an echo pulse.

###### Acknowledgements.

Research was sponsored by the Army Research Office and was accomplished under
Grant Number W911NF-18-1-0166. The views and conclusions contained in this
document are those of the authors and should not be interpreted as
representing official policies, either expressed or implied, of the Army
Research Office or the U.S. Government. The U.S. Government is authorized to
reproduce and distribute reprints for Government purposes notwithstanding any
copyright notation herein.

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