# Synthetic gauge field in two interacting ultracold atomic gases without an
optical lattice

J. Mumford  Department of Physics and Astronomy, McMaster University, 1280
Main St. W., Hamilton, ON, L8S 4M1, Canada

###### Abstract

A 2D Fock-state lattice (FSL) is constructed from the many-body states of two
interacting two-mode quantum gases. By periodically driving the interspecies
interactions and pulsing the tunneling between the two modes of each gas, a
synthetic gauge field is generated. We derive an effective Hamiltonian in the
short pulse limit which resembles the Harper-Hofstadter Hamiltonian where the
magnetic flux per plaquette is controlled by the ratio of the interaction
energy and the driving frequency. The quasispectrum of the Floquet operator of
the driving sequence shows the celebrated Hofstadter’s butterfly pattern as
well as the existence of edge states. From the calculation of the local Chern
marker, we establish that the FSL has non-trivial topology and by simulating
the dynamics of the edge states, show that they exhibit chirality. Finally,
the inclusion of intraspecies interactions creates an overall harmonic trap in
the lattice and introduces the nonlinear effect of macroscopic quantum self-
trapping which is shown to hinder movement along the edges of the lattice.
This work introduces a new avenue to explore synthetic gauge fields and the
link between condensed matter systems and quantum gases.

##  I Introduction

Ultracold atomic gases captured in optical lattices have become a versatile
testbed for simulating condensed matter systems [1]. One of the main methods
behind these simulations, called Floquet engineering, involves periodically
driving external fields of the gas and lattice to create synthetic gauge
fields [2, 3, 4, 6, 5, 7]. The driving can be finely tuned in order to
generate an effective Hamiltonian that has the desired properties of a static
one. This is especially useful when simulating the interactions between
electrons and applied magnetic fields in materials. Since ultracold gases are
kept in clean, highly controlled environments, the synthetic magnetic fields
produced in these systems are also highly controlled. Depending on the
specific driving schemes used, the parameters of the system can be tuned to
generate magnetic field strengths that are difficult to approach otherwise. An
early success in simulating a magnetic field involved the Coriolis force in a
rotating atomic gas [8, 9]. Later, synthetic staggered magnetic fields were
created with laser induced tunneling [10] and shaking of the lattice [11].
This lead to the creation of topological many-body phases in 2D optical
lattices through the experimental simulation of the Harper-Hofstadter (HH)
[12, 13] and Haldane [14] models.

An additional aid in the engineering of condensed matter systems is the
concept of synthetic dimensions [15, 16]. A synthetic dimension is a degree of
freedom, from either internal or external states of particles, which can
imitate real-space dimensions. Experimentalists have been able to create
synthetic dimensions using internal spin [18, 19, 17, 20], momentum [21, 22,
23], clock [26, 24, 25], harmonic oscillator [27], and rotational [28] states
of atoms and molecules. It is common practice to prepare a 1D lattice in real-
space and pair it with a synthetic dimension to create a 2D lattice. The
inclusion of a synthetic gauge field in such systems has lead to the
observation of chiral edge states [18, 19] and Hall drift, as well as the
construction of the local Chern marker [29].

Another avenue taken to generate topological effects is through photonic
systems [30, 31]. Usually waveguides and resonators are used as lattice sites
arranged in the desired geometry. In these systems, topological edge states
[32, 33], band structures [34] and transport properties [35] have been
observed. A relatively newer approach is to use a few photonic modes in the
form of optical cavities. The range of photon occupations of these modes form
a Fock-state basis which is used as a synthetic dimension and can be
generalized to show that $d$ cavities form a $d-1$ dimensional Fock-state
lattice (FSL) [36, 37]. There is some restriction to this approach, however,
as the geometry of the FSL is ingrained in the Hilbert space making it
difficult to manipulate. Nevertheless, theoretical treatments of various
lattice models with gauge fields have been proposed using FSLs [38].

Fock states also form the basis of few-mode matter-based systems such as a
two-component Bose-Einstein condensate (BEC) [39] or a BEC in a double well
potential, alternatively called a bosonic Josephson junction (BJJ) [40, 41].
There are three main ways in which these systems are periodically driven: the
chemical potential, the tunneling between modes, and the interactions between
particles of the gas. For instance, periodically driving the tilt between two
wells of a BJJ has been proposed to enhance tunneling [42] and periodically
driving the tunneling between two modes of a Bose-Hubbard (BH) dimer has been
used to explore the emergence of chaos in the system [43]. Recently,
periodically driven interactions have been proposed to control the quantum
collapse in a BJJ [44], to generate quantum phase transitions in the Lipkin-
Meshkov-Glick model [45] and to control the tunneling of ultracold atoms in
the BH dimer [46].

In this work, we show a 2D FSL lattice can be formed by the states of two
interacting two-mode quantum gases. By incorporating a periodic driving
sequence used on a real-space lattice to generate synthetic gauge fields [47],
we show the same can be done for the FSL. The common thread throughout this
paper is the adoption of real-space measures of topology and their conversion
to equivalent measures of the FSL. Consequently, we establish that the FSL is
HH-like with non-trivial topology giving rise to nonzero Chern numbers and
chiral edge states. Nonlinearities from the repulsive intraspecies
interactions of the gases play an important role in the static and dynamic
properties of the system. They act as an overall harmonic potential in the FSL
causing some merging of bulk bands and they manifest in the dynamics in the
form of self-trapping, hindering the movement around the edge of the lattice.

##  II Model

The system we will be investigating consists of two interacting bosonic
quantum gases. Each gas contains $N$ identical particles and each particle has
access to two states labeled 1 and 2. The states can be internal spin states
or external states such as the ground states in each well of a double well
potential. The basis we will be using is the Fock basis defined in terms of
half the particle number difference between states 1 and 2, $|n,m\rangle$,
where $n=\left(N_{1}-N_{2}\right)/2$ and $m=\left(M_{1}-M_{2}\right)/2$. In
the Schwinger representation, $n$ and $m$ are interpreted as $z$ \- components
of spin-$N/2$ particles, so $|n,m\rangle$ is an eigenstate of the operators
$\hat{J}_{z}=\frac{1}{2}\left(\hat{a}_{1}^{\dagger}\hat{a}_{1}-\hat{a}_{2}^{\dagger}\hat{a}_{2}\right)$
and
$\hat{S}_{z}=\frac{1}{2}\left(\hat{b}_{1}^{\dagger}\hat{b}_{1}-\hat{b}_{2}^{\dagger}\hat{b}_{2}\right)$
where $\hat{J}_{z}|n,m\rangle=n|n,m\rangle$ and
$\hat{S}_{z}|n,m\rangle=m|n,m\rangle$. The creation and annihilation operators
for the two states of the two gases follow the usual commutation relation
$[\hat{a}_{i},\hat{a}^{\dagger}_{j}]=[\hat{b}_{i},\hat{b}^{\dagger}_{j}]=\delta_{ij}$
where $i,j=1,2$. Here, we will interpret the Fock states as locations on a
$\left(N+1\right)\times\left(N+1\right)$ FSL with lattice spacing $a=1$ where
$n$ and $m$ are the coordinates along the $x$ and $y$ axis, respectively. The
origin located at $|0,0\rangle$ corresponds to an equal number of particles of
each gas occupying each of the two modes and $\hat{J}_{z}$ ($\hat{S}_{z}$)
becomes the displacement operator $\hat{x}$ ($\hat{y}$) on the FSL.

Our goal in this paper is to investigate the effects of a synthetic magnetic
field through the FSL. To generate the magnetic field, we are motivated by
past efforts which have generated one in real-space. For instance, in [47] it
was shown that a magnetic flux through a real-space lattice could be generated
from a combination of periodically driving a quadrupolar potential term
$V_{\mathrm{qp}}\sin(\omega t)\hat{x}\hat{y}$, where $\omega$ is the driving
frequency, and periodically flashing on tunneling in the $x$ and $y$
directions for a short period of time, $\tau\ll 1$. One period of the driving
sequence produces the Floquet operator

$\displaystyle\hat{U}^{\prime}\left(t=\frac{2\pi}{\omega}\right)=$
$\displaystyle\mathrm{e}$
${}^{-\mathrm{i}\tau\hat{T}_{y}/2}\mathrm{e}^{2\mathrm{i}V_{\mathrm{qp}}\hat{x}\hat{y}/\omega}\mathrm{e}^{-\mathrm{i}\tau\hat{T}_{x}}\mathrm{e}^{-2\mathrm{i}V_{\mathrm{qp}}\hat{x}\hat{y}/\omega}$
(1) $\displaystyle\times\mathrm{e}^{-\mathrm{i}\tau\hat{T}_{y}/2}$

where $\hat{T}_{x}$ and $\hat{T}_{y}$ are the kinetic energy operators in the
$x$ and $y$ directions, respectively. For $\omega\gg 1$, the evolution
operator can be written in terms of an effective Hamiltonian
$\hat{U}^{\prime}(\tau)=\mathrm{e}^{-\mathrm{i}\hat{H}^{\prime}_{\mathrm{eff}}\tau}$
where

$\displaystyle\hat{H}_{\mathrm{eff}}^{\prime}\approx\hat{H}_{\mathrm{HH}}=$
$\displaystyle-$ $\displaystyle J\sum_{n,m}|n,m\rangle\langle n,m-1|$ (2)
$\displaystyle+\mathrm{e}^{\mathrm{i}2\pi\alpha m}|n,m\rangle\langle
n-1,m|+\mathrm{h.c}\,.$

The effective Hamiltonian takes the form of the celebrated HH Hamiltonian
$\hat{H}_{\mathrm{HH}}$ (in the Landau gauge) which describes a uniform
magnetic field in the direction perpendicular to the plane of a 2D lattice.
The parameter $\alpha=V_{\mathrm{qp}}/\pi\omega=\Phi/\Phi_{0}$ is the ratio of
the magnetic flux through a unit cell $\Phi=Ba^{2}$ and the magnetic flux
quantum $\Phi_{0}=h/e$.

Switching the focus back to our model, according to our previous analogy
between $\hat{J}_{z}$ ($\hat{S}_{z}$) and $\hat{x}$ ($\hat{y}$), the
quadrupolar potential becomes an interaction term between the two gases
$V_{\mathrm{qp}}\hat{x}\hat{y}\to\kappa\hat{S}_{z}\hat{J}_{z}$ where $\kappa$
is the interaction energy. The kinetic energy operators in Eq. (1) are
responsible for moving a particle from one lattice site to an adjacent one.
The $x$-component spin operators accomplish the same thing on the FSL, so we
make the additional transformations
$\hat{T}_{x}\to-J\hat{J}_{x}=-\frac{J}{2}\left(\hat{a}_{1}^{\dagger}\hat{a}_{2}+\hat{a}^{\dagger}_{2}\hat{a}_{1}\right)$
and
$\hat{T}_{y}\to-J\hat{S}_{x}=-\frac{J}{2}\left(\hat{b}_{1}^{\dagger}\hat{b}_{2}+\hat{b}_{2}^{\dagger}\hat{b}_{1}\right)$
(keeping in mind that the subscripts on the kinetic energy operators represent
direction and the subscripts on the spin operators do not) where $J$ is the
tunneling energy and is the same for both gases. The driving sequence in Eq.
(1) becomes

$\hat{U}\left(t=\frac{2\pi}{\omega}\right)=\mathrm{e}^{2\mathrm{i}\kappa\hat{S}_{z}\hat{J}_{z}/\omega}\mathrm{e}^{\mathrm{i}\tau\hat{J}_{x}}\mathrm{e}^{-2\mathrm{i}\kappa\hat{S}_{z}\hat{J}_{z}/\omega}\mathrm{e}^{\mathrm{i}\tau\hat{S}_{x}}$
(3)

where, when comparing it to Eq. (1), we have made the small modification of
removing the left-hand factor of $\mathrm{e}^{\mathrm{i}\tau\hat{S}_{x}/2}$
and multiplying it to the right-hand side. Also, we have set the tunneling
energy to unity, $J=1$, so that from now on all other energies will be in
units of $J$. A single species version of $\hat{U}$ has been used to produce
the fractal spectrum of Hofstadter’s butterfly [48] and study the effects of
nonlinearities on it. This driving sequence is accomplished by periodically
modulating the interspecies interactions $\kappa\sin(\omega
t)\hat{S}_{z}\hat{J}_{z}$ and flashing on the hopping between the two single
particle states of one gas at time $\omega t=0$ for a duration of $\tau\ll 1$,
then flashing on the hopping of the other gas at time $\omega t=\pi$ for the
same duration. Altogether the driving results in the ’stirring’ of the FSL.
The Floquet operator can again be written in terms of an effective Hamiltonian
(Appendix A)

$\displaystyle\hat{H}_{\mathrm{eff}}\approx-\sum_{n,m}$ $\displaystyle
D_{+}(m)$ $\displaystyle|n,m+1\rangle\langle n,m|$ $\displaystyle+$
$\displaystyle D_{-}(m)|n,m-1\rangle\langle n,m|$ $\displaystyle+$
$\displaystyle e^{i2\pi\alpha m}D_{+}(n)|n+1,m\rangle\langle n,m|$
$\displaystyle+$ $\displaystyle e^{-i2\pi\alpha m}D_{-}(n)|n-1,m\rangle\langle
n,m|,$

where the new magnetic flux ratio $\alpha=\kappa/\pi\omega$ is controlled by
the interaction energy and the driving frequency. The major difference between
the real-space and Fock-space effective Hamiltonians is the presence of the
$D_{\pm}(x)=\frac{1}{2}\sqrt{\left(N/2\mp x\right)\left(N/2\pm x+1\right)}$
factors. This means that even without the magnetic field, the Fock lattice is
not translationally invariant. The factors have the shape of a semicircle as a
function of $x$, so for $N\gg 1$ and $x\ll N/2$, they are rather flat
$D_{\pm}(x)\approx N/4$, then as $x$ increases, they transition to a parabolic
shape with a hard wall edge at $x=\pm N/2$ because $D_{\pm}(\pm N/2)=0$.

The tunneling terms in the driving sequence can be implemented by pulsing
lasers at the specified times to induce tunneling between the two modes of
each gas. We assume that the interspecies interactions take place via s-wave
scattering, so the periodic interaction energy is controlled by alternating
between positive and negative scattering lengths corresponding to repulsive
and attractive interactions, respectively. This is achieved through Feshbach
resonance [49] between the two species and by applying an ac-magnetic field.
In the region of a Feshbach resonance, the s-wave scattering length takes the
form

$a_{s}(t)=a_{0}\left(1-\frac{\Delta}{B(t)-B_{\infty}}\right)$ (5)

where $a_{s}$ and $a_{0}$ are the modified and background s-wave scattering
lengths, respectively, $\Delta$ is the width of the resonance region,
$B_{\infty}$ is the magnetic field strength of the resonance point and $B(t)$
is the ac-magnetic field strength. If the ac-magnetic field is
$B(t)=(B_{\infty}+\Delta)+\delta B\sin(\omega t)$ and $\delta B\ll\Delta$,
then $a_{s}(t)\approx\frac{a_{0}\delta B}{\Delta}\sin(\omega t)$. Time varying
interaction energies were achieved experimentally not long after the creation
of the first BECs to study the effects of ramping through a Feshbach resonance
point [50, 51, 52]. Oscillating interaction energies have been used to study
the association of ultracold atoms of the same species [53] and of different
species [54, 55] and also to control the excitations in a BEC [56].
Experiments have also shown that periodically driving onsite interactions of
an optical lattice can lead to complete suppression of the tunneling [57].
Furthermore, paired time-dependent interactions and tunneling have been
proposed to generate density dependent momenta of ultracold atoms in an
optical lattice [58]. To avoid unwanted excitations, the strength of the
driving should be adiabatically ramped up. Heating effects from enhanced
inelastic collisions around the resonance point [59] are reduced naturally
because we shift the driving away from it by $\Delta$ with a small amplitude
of oscillation. It has also been shown that heating effects can be further
reduced with stronger confinement of the trapping potential [60].

##  III Results

###  III.0.1 Quasispectrum

Figure 1: The energy/quasienergy is plotted as a function of the magnetic flux
ratio $\alpha$ and the background shows a density plot of the probability for
a state of a given energy to occupy the edge of the lattice. (a) The energy is
calculated from the HH Hamiltonian in Eq. (2) with a lattice size of $11\times
11$. (b) The quasienergy is calculated from the Floquet operator in Eq. (3)
for two gases containing 10 particles each, so the Hilbert spaces of the real-
and Fock-spaces are equal. The pulse time is $\tau=0.01$.

In this section, we explore the hallmarks of magnetic fields in 2D lattices
starting with the comparison of the spectra of the $\hat{H}_{\mathrm{HH}}$
Hamiltonian and the Floquet operator in Eq. (3). In the HH model, the magnetic
field breaks translational symmetry, however, if the flux ratio is a rational
number, $\alpha=p/q$ where $p,q\in\mathbb{Z}$, then the translational symmetry
is restored if the unit cell increases by a factor of $q$. The increase in
size of the unit cell in real-space results in the reciprocal space being $q$
times smaller, so a single band will turn into $q$ bands. The resulting
pattern of the spectrum as a function of $\alpha$ is shown in Fig. 1(a) and is
called Hofstadter’s butterfly due to its shape. Usually the spectrum is
depicted as having a fractal structure and being without states flowing
through the gaps, however, for better visualization we have used a small
lattice size of $11\times 11$ which smooths out the fractals and applied an
edge in the numerics which produces edge states in the gaps. The latter point
is made clearer with the density plot in the background of the panel which is
a plot of the probability for an energy eigenstate to occupy the edge of the
lattice

$P_{i}=\sum_{n,m\in\mathrm{edge}}\left|\langle n,m|\psi_{i}\rangle\right|^{2}$
(6)

where $|\psi_{i}\rangle$ is the $i^{\mathrm{th}}$ energy eigenstate of
$\hat{H}_{\mathrm{HH}}$. The image shows that the edge states do indeed occur
in the gaps with the largest edge probability being $P_{\mathrm{max}}\approx
0.9$, so the states are quite localized.

The set of eigenvalues of Eq. (LABEL:eq:heff), are called quasienergies
because they come from the Floquet operator in Eq. (3) whose set of
eigenvalues is $\\{\lambda_{i}=\mathrm{e}^{-\mathrm{i}\epsilon_{i}\tau}\\}$
and so lose their uniqueness when $\epsilon_{i}\tau>2\pi$. However, when
$\tau$ is small enough, quasienergies play a similar role in periodically
driven systems as energies do in time independent Hamiltonians. We plot
$\frac{\mathrm{i}}{\tau}\mathrm{ln}(\\{\lambda_{i}\\})$ as a function of
$\alpha$ in Fig. 1(b) which shows all of the qualitative features of
Hofstadter’s butterfly have been replicated and the lack of translational
invariance of the bare FSL only has an effect on the finer details of the
quasispectrum. The edge states are again found in the gaps with a maximum
probability of $P_{\mathrm{max}}\approx 0.8$. The difference in the ranges
along the $y$ axis between both panels comes from the $D_{\pm}(x)$ factors
whose ranges are proportional to $N$.

Figure 2: Density plots of the probability for a site on the FSL to have a
given quasienergy and the LCM for different fictitious Fermi energies
($E_{F}$). (a) and (b) show $P_{\epsilon_{i}}(n)$ for $\alpha=1/3$ and
$\alpha=1/5$, respectively, where the colored horizontal lines represent
values of $E_{F}$. (c) and (d) show the LCM, $C(n)$, for the values of $E_{F}$
in (a) and (b), respectively. The density plots show bulk bands, gaps and edge
states and the number of them corresponds to the value $\alpha$. The LCM
agrees with the Chern number of the HH model with $C=\pm 1$ for $\alpha=1/3$
and $C=\pm 1,\pm 2$ for $\alpha=1/5$ for each band. The number of particles in
each gas is $N=30$ and the pulse time is $\tau=0.01$.

###  III.0.2 Local Chern Marker

We now turn our attention to the topological features of the model. For 2D
lattices, the Chern number is a quantity which can help identify topological
properties (trivial or non-trivial). The Chern number is usually calculated by
integrating the Berry curvature over the first Brillouin zone for all occupied
bands. This calculation requires the quasimomentum to be a well defined
quantity which in turn requires translational invariance. For non-homogeneous
systems, a real space version of the Chern number called the local Chern
marker (LCM) has been introduced [61] which defines it locally in a unit cell
of the lattice

$C=\frac{4\pi}{A_{c}}\mathrm{Im}\left(\mathrm{tr}_{\mathrm{cell}}\\{\hat{P}\hat{x}\hat{P}\hat{y}\\}\right)$
(7)

where $A_{c}$ is the area of the unit cell and $\hat{P}$ is the projection
operator onto the states of the occupied bands. It is expected that for large
enough system sizes the average of the LCM of the bulk bands matches the
quasimomentum version of the Chern number. We neglect the trace over the unit
cell because we will average the LCM over states of the bulk later on, so
$A_{c}=1$ and the LCM becomes completely localized

$C(n,m)=-4\pi\mathrm{i}\langle
n,m|\hat{P}\hat{S}_{z}\hat{P}\hat{J}_{z}|n,m\rangle\,.$ (8)

What remains is to determine which states $\hat{P}$ projects onto and for that
we need to know if the model contains any band-like structures. To this end,
we calculate the probability for a position on the lattice to have a given
quasienergy $\epsilon_{i}$ (calculated from the Floquet operator in Eq. (3)),
$P_{\epsilon_{i}}(n,m)=|\langle n,m|\epsilon_{i}\rangle|^{2}$, and plot an
$m=0$ slice of it, $P_{\epsilon_{i}}(n)=P_{\epsilon_{i}}(n,m=0)$, for two
values of the flux ratio $\alpha$ in Fig. 2 (a) and (b). Panel (a) shows three
distinct band-like structures for $\alpha=1/3$ and panel (b) shows five for
$\alpha=1/5$. Although these are not energy bands in the strictest sense, they
still follow the qualitative feature of the HH model that there are $q$ bands
when $\alpha=1/q$. Each band is separated by a gap with the exception of the
edges of the lattice at $n=\pm 15$ ($N=30$ for both gases) where the bands are
connected agreeing with the existence of edge states in the gaps in Fig. 1.

In normal real-space lattices there is a Fermi surface with an energy $E_{F}$
where states with energy below $E_{F}$ are occupied by electrons. It is these
states that belong to the projection operator in Eq. (7). Such a Fermi surface
does not exist for the FSL, however, we can imagine a fictitious one existing
in the gaps between bands to help facilitate the calculation of the LCM.
Different Fermi energies are shown as horizontal colored lines in panels (a)
and (b) of Fig. 2 and are placed at an arbitrary position within the gaps.
Therefore, we will take $\hat{P}$ as the projection operator of states with
energies below these lines. Panels (c) and (d) show $C(n)=C(n,0)$ for the
Fermi energies in (a) and (b), respectively. One can see that the LCM varies
significantly over the edges, but settles down to a constant value farther
inside the bulk. We take the average of the LCM in the region $|n|,|m|\leq 8$,
so that it is taken over a region comfortably inside the bulk. The average
LCMs for increasing Fermi energy are: $C_{\mathrm{ave}}=$ -0.996, 0.996 for
$\alpha=1/3$ and $C_{\mathrm{ave}}=$ -0.998, -1.934, 1.934, 0.998 for
$\alpha=1/5$. These values agree well with the Harper model Chern numbers of
$\pm 1$ for $\alpha=1/3$ and $\pm 1,\pm 2$ for $\alpha=1/5$ and satisfy the
condition that the sum of the Chern numbers for all of the bands is equal to
zero.

###  III.0.3 Intraspecies Interactions

Figure 3: The same images as in Fig. 2 except with intraspecies interactions
$\mu=0.02$ and the number of particles in each gas is now $N=40$ instead of
$N=30$ to make the gaps more visible. When $\mu\neq 0$ the intraspecies
interactions act as a harmonic trap on the FSL causing the parabolic shape of
the bands. The LCM shows that for weak interactions the bands retain their
topological features with values matching the Chern numbers of the HH model in
the bulk.

Intraspecies interactions are introduced with the term
$U\left(\hat{S}_{z}^{2}+\hat{J}_{z}^{2}\right)$. For simplicity we take the
interaction energy $U$ to be the same for both gases and it is controlled by
the s-wave scattering length of the particles. Only repulsive interactions
$U>0$ will be considered, so their inclusion results in a harmonic trap in the
FSL. The evolution operator in Eq. (3) gains the factor
$\mathrm{e}^{-\mathrm{i}\frac{\mu}{N}\left(\hat{S}_{z}^{2}+\hat{J}_{z}^{2}\right)}$
at the first and third positions (left to right) where $\mu=UN\pi/\omega$.
Harmonic traps have been studied for real-space lattices [62] because they are
needed as an overall trapping potential in experimental setups involving
ultracold atomic gases. It was found that harmonic confinement creates a soft-
wall boundary for the lattice which makes edge states harder to identify due
to smearing. However, bulk bands were still visible, as well as gaps between
them for large enough lattices and weak enough confinements, so topological
features still persisted. We plot $P_{\epsilon_{i}}(n)$ with $\mu\neq 0$ for
the two values of $\alpha$ in Fig. 3 (a) and (b) and find that the soft wall
of the harmonic trap does indeed smear out the bands near the edges, however,
they still connect bands together due to the hard wall at the very edge. The
interactions also decrease the gap size in the longitudinal direction. This
suggests that there is a critical interaction strength for a finite sized
system where the bands get so smeared out that the gaps disappear. Even so,
for weak enough interactions the bands and gaps are still distinguishable
which is what we see for $\mu=0.02$.

In Fig. 3 (c) and (d) we once again plot the LCM, $C(n)$, for the fictitious
Fermi energies in (a) and (b), respectively, however, we only include the
lower energy LCMs since we have established that the upper ones are the same,
but with a flipped sign. With interactions we find that the average LCM values
are $C_{\mathrm{ave}}=-0.987$ for $\alpha=1/3$ and
$C_{\mathrm{ave}}=-0.996,-1.918$ for $\alpha=1/5$. These values again agree
with the HH model Chern numbers signaling that although the interactions
change the shape of the bands, their topological properties remain intact.
Away from the center of the lattice there are some qualitative differences
compared to the case with no interactions. In (c) the low Fermi energy LCM
(red, smooth) quickly goes to zero because, due to the curvature of the bands,
$E_{F}$ enters a region where there are no states as seen in (a). This
crossing occurs at $n=\pm 12$. The higher Fermi energy LCM in (d) (orange,
dot-dashed) varies significantly away from the center because $E_{F}$ crosses
the second bulk band and enters the lower gap at $n=\pm 10$, so the LCM
briefly picks up the lower band LCM value of $-1$. Near the edges of the
lattice, the LCM goes to zero again because there are no more states below
$E_{F}$ in this area.

###  III.0.4 Transport Properties

In condensed matter systems the Chern number is usually measured through the
Hall conductance [63], however, it is difficult to use the same technique for
neutral particles in an ultracold atomic gas. One proposed method uses hybrid
time-of-flight (HTOF) images [64] which involves in situ measurements of a gas
cloud’s density in one direction of the lattice and measurements of the free
expansion of the gas in the other direction. The HTOF images are quantified in
the density of particles with a given position and transverse quasimomentum,
$\rho(k_{x},y)$. This quantity takes advantage of the fact that the Chern
number is related to adiabatic transport properties of the lattice and that a
2D lattice can also be thought of as a 1D charge pump where $k_{x}$ plays the
role of the pumping parameter. When $k_{x}$ is cycled over the Brillouin zone,
$k_{x}\to k_{x}+2\pi$, the particles on the lattice are displaced if there is
nontrivial topology present. The displacement is related to the Chern number
by $\delta y_{\mathrm{COM}}=Cq$ where $\delta y_{\mathrm{COM}}$ is the center-
of-mass displacement, $C$ is the Chern number and $q$ is the length of the
magnetic unit cell in the $y$ direction when $\alpha=1/q$.

Figure 4: The particle density function, $\rho(\theta_{x},m)$, is plotted as a
function of the relative phase of one gas and the number difference of the
other gas. Panels (a) and (b) show $\rho(\theta_{x},m)$ for the lowest bands
for $\alpha=1/3$ and $\alpha=1/5$, respectively. A peak of the density is
displaced by -3 in (a) and -5 in (b) agreeing with the charge pump result of
$\delta m=Cq$ in real-space lattices. The other parameter values match those
in Fig. 3.

In the current system, the relative phases between states 1 and 2 of each BEC,
$\theta_{x}$ and $\theta_{y}$, are the conjugate variables to the number
differences $n$ and $m$, respectively, so they are analogous to the
quasimomenta in real-space. In the Schwinger representation, the relative
phase is also the azimuthal angle on the surface of a spin-$N/2$ Bloch sphere.
The particle density takes the form

$\rho(\theta_{x},m)=\frac{1}{\mathcal{N}}\sum_{i=1}^{\mathcal{N}}|\Phi_{i}(\theta_{x},m)|^{2}$
(9)

where the sum is over the $\mathcal{N}$ states below the Fermi energy and

$\Phi_{i}(\theta_{x},m)=\left(N+1\right)^{-1/2}\sum_{n=-N/2}^{N/2}\mathrm{e}^{\mathrm{i}\theta_{x}n}\psi_{i}(n,m)$
(10)

is the Fourier transform of the $\mathrm{i}^{\mathrm{th}}$ eigenstate of the
Floquet operator. The relative phase also takes discrete values,
$\theta_{x}=\frac{2\pi r}{N+1}$ where $r$ is an integer in the range $-N/2\leq
r\leq N/2$. Similarly to the real-space transport, we expect there to be a
displacement of $\delta m=Cq$ on the FSL when $\theta_{x}$ changes by $2\pi$.

We will consider the lowest Fermi energy (red) in Fig. 3 for both values of
$\alpha$ which means $C=-1$. In one cycle of $\theta_{x}$ the expected
displacement is $\delta m=-3$ for $\alpha=1/3$ and $\delta m=-5$ for
$\alpha=1/5$. Figure 4 shows $\rho(\theta_{x},m)$ and we find agreement with
the predicted result. Following a peak of $\rho(\theta_{x},m)$ (red) as
$\theta_{x}$ increases by $2\pi$ results in the peak being displaced by
$\delta m=-3$ for $\alpha=1/3$ in (a) and by $\delta m=-5$ for $\alpha=1/5$ in
(b). A helpful guide to identify the displacement in the figure is to note
that $m$ is discrete, so the images are broken up into horizontal strips. The
number of strips going from one peak to another is three in (a) and five in
(b). The downward slope is attributed to the Chern number being negative,
however, a Chern number of $+1$ can also be checked for the lowest band by
flipping the sign of the interspecies interactions $\kappa$ in Eq. (3). The
intraspecies interactions do not have an effect on the transport properties of
the FSL except to prevent any displacement near the sides of the lattice where
the sites are energetically inaccessible (white-space in the images).

Like $k_{x}$, $\theta_{x}$ can also be measured from time-of-flight expansion
through the interference fringes of absorption images [41]. This means that in
order to see the transport effects one must be able to release one gas from
the trap for the $\theta_{x}$ measurement and keep the other gas trapped for
the $n$ measurement. One of the major difficulties in performing these
measurements is the requirement that the gases be separate, however, in
general, they will end up mixing. One way to aid in the measurements is if the
two modes of each gas are spatially separated like the two wells of a BJJ
where the tunneling is through a central potential barrier. On the FSL,
quadrants two and four represent immiscible states of the BJJ, so measurements
taken in these regions will have some separation between the gases.

Staying with the BJJ system, the edge state chirality can be measured by
preparing the initial state as a spin coherent state in the $x$-direction and
a Fock state at $m=-N/2$ in the $y$-direction. This results in a Gaussian
state in the $x$-direction centered at $(n,m)=(0,-N/2)$ on the lattice. Such a
state can be prepared by having relatively strong tunneling between the two
wells of the $x$-direction gas and loading all of the particles of the
$y$-direction gas into one well (well 2 in this case) before the driving
starts. This initial state is shown at the bottom edge of panel (a) in Fig. 5.
Panel (b) shows the wavefunction has moved to the right along the edge after
50 steps of $\hat{U}$ for parameter values $\alpha=1/6$ and $\mu=0.02$. The
chirality can be flipped by flipping the sign of $\kappa$ ($\alpha$), as
previously mentioned. Equivalently, due to the symmetry of the quasispectrum
in Fig. 1, the chirality can be flipped by setting $\alpha=5/6$ which is shown
in panel (c) where the wavefunction has now moved to the left. The bottom
right and left corners of the Fock lattice represent complete separation and
complete overlap of the two gases in the BJJ, so the chirality can be inferred
by the miscibility of the final state. Tunable miscibility experiments have
been conducted by controlling the Feshbach resonances of both the interspecies
and intraspecies interactions of quantum gases [65, 66].

Figure 5: Initial and final probability distributions in the FSL for different
parameter values. (a) An initial Gaussian distribution (spin coherent state)
along the horizontal centered at $(n,m)=(0,-N/2)$. (b) After 50 steps of the
Floquet operator in Eq. (3) the Gaussian has moved along the edge to the right
while maintaining its shape for $\alpha=1/6$ and $\mu=0.02$. (c) The same as
(b) except the flux ratio has been changed to $\alpha=5/6$ which reverses the
chirality of the Gaussian, so it has moved along the edge to the left after 50
steps. (d) An initial Fock distribution located in the bottom left corner at
$(n,m)=(-N/2,-N/2)$. (e) After 500 steps of the Floquet operator the
distribution remains in its initial vicinity for $\alpha=1/6$ and $\mu=0.02$
which is a qualitative sign of MQST. (f) The interaction energy is reduced to
$\mu=0.005$ ceasing MQST and after 500 steps the distribution has spread along
the left and bottom edges. The images were generated using $N=40$ for both
gases and with a pulse time of $\tau=0.01$.

A distinguishing feature of the single species BJJ is a dynamical process
called macroscopic quantum self-trapping (MQST). This process comes from the
intraspecies interactions and leads to $n$ having a nonzero long time average.
In extreme cases when the interactions are large enough, the number difference
does not evolve much at all. This is a rather counterintuitive result because
one would think that for larger $\mu$ (more repulsive) the gas would tend
toward a state that is maximally spread between the two wells, not prefer one
well over the other. Clarification of MQST can be gained by considering the
classical rigid pendulum. If given a large enough kinetic energy, the pendulum
will swing over the top and make full loops as it evolves. For a very large
kinetic energy, the pendulum swings around rapidly maintaining a near constant
angular momentum. This happens because there is no angular displacement state
that makes the gravitational potential energy comparable to the kinetic
energy, so the kinetic energy and therefore the angular momentum are locked.
In the BJJ, the intraspecies interaction energy and the boson number
difference $n$ ($m$) are analogous to the kinetic energy and the angular
momentum of the pendulum, respectively. Therefore, small oscillations of the
BEC moving back and forth between the two wells is similar to small
oscillations of the pendulum. Likewise, the pendulum having enough kinetic
energy to make full loops to maintain a nonzero angular momentum is the same
as the BEC having enough intraspecies interaction energy to maintain a nonzero
boson number difference and some of the BEC ends up trapped in one well over
the other.

Figure 6: The long-time average of
$\frac{1}{2}\langle\hat{S}_{z}+\hat{J}_{z}\rangle(t)$ (denoted by the overline
in the image) for different values of the ratio between the intraspecies
interaction energy and the pulse time, $\mu/\tau$. The brackets
$\langle...\rangle$ denote the expectation value with respect to the Fock
state $|-N/2,-N/2\rangle$ shown in Fig. 5 (d). For small values of the ratio
the state spreads along the edges of the lattice and the long-time average is
zero. Around the critical ratio of $\left(\mu/\tau\right)_{\mathrm{c}}=1$ MQST
begins to take effect stopping the wavefunction from spreading, so the long-
time average is close to its initial value of $-N/2$. The parameter values are
$N=40$ for both gases, $\tau=0.01$ and the time average was taken over 10000
steps of $\hat{U}$.

To show MQST in the current system we change the initial state to the Fock
state $|-N/2,-N/2\rangle$ which corresponds to both gases being loaded into
the same well before the driving starts. This state can be seen in the bottom
left corner of Fig. 5 (d). Panel (e) shows the wavefunction after 500 steps of
$\hat{U}$ for parameter values of $\alpha=1/6$ and $\mu=0.02$. We can see that
MQST has prevented the wavefunction from leaving the corner (confirmed for all
intermediate steps) resulting in a localized state on the FSL. By reducing
$\mu$ we reduce the amount of energy stored in the intraspcecies interactions
and the wavefunction will be able to explore more of the lattice. This is what
we see in panel (f) where we keep the same $\alpha$, but now $\mu=0.005$. The
localization can also be explained through the examination of the high energy
eigenstates of $\hat{U}$ as was done for the HH model in a harmonic trap [67].
It was found that as $\mu$ (strength of the harmonic trap) increases, the
higher energy states form a near four-fold degenerate group and the initial
state $|-N/2,-N/2\rangle$ can be approximately constructed from a
superposition of these states. The result is that the initial state is close
to being an eigenstate of $\hat{U}$, so it does not evolve much.

To find the critical value of $\mu$ where the onset of MQST occurs we derive
the meanfield Hamiltonian of $\hat{H}_{\mathrm{eff}}$ with interactions giving
(Appendix B)

$\displaystyle H_{\mathrm{MF}}=\frac{\mu}{\tau}$
$\displaystyle\left(x^{2}+y^{2}\right)-\left[\sqrt{1-y^{2}}\cos(\theta_{y})\right.$
$\displaystyle+\left.\sqrt{1-x^{2}}\cos(\theta_{x}+\pi\alpha Ny)\right]$ (11)

where $x=2n/N$ and $y=2m/N$. If we imagine both $x$ and $y$ as angular
momentum variables ($\theta_{x}$ and $\theta_{y}$ as angular displacement
variables), then $H_{\mathrm{MF}}$ resembles the Hamiltonian of a system of
coupled pendula, albeit a strange one due to the angular momentum dependent
potential energy. Nevertheless, we can use the pendulum analogy discussed
earlier to say that MQST should take effect when the coupled pendula in Eq.
(11) have enough energy to make a full loop, or in other words, when the total
energy is greater than the maximum potential energy. The initial state is
located at $(n,m)=(-N/2,-N/2)$, or $(x,y)=(-1,-1)$, so its energy is
$2\mu/\tau$. The maximum potential energy occurs at
$(x,y,\theta_{x},\theta_{y})=(0,0,\pi,\pi)$ and is equal to 2, so MQST happens
when $2\mu/\tau\geq 2$ giving a critical value of

$\left(\frac{\mu}{\tau}\right)_{\mathrm{c}}=1\,.$ (12)

We stress that the critical value depends on the initial state as different
initial states have different total energies. In fact, the dependence of the
critical value on $\mu$ and $\alpha$ can be quite complicated in some systems
as seen in the periodically driven BEC in a quadruple well [68]. Figure 6
shows $\frac{1}{2}\overline{\langle\hat{S}_{z}+\hat{J}_{z}\rangle(t)}$, which
is the the long-time average (denoted by the line) of the expectation value of
the sum of the number difference operators, for different values of the ratio
$\mu/\tau$. The expectation is taken with respect to the Fock state
$|-N/2,-N/2\rangle$ shown in the bottom left corner of Fig. 5 (d). The image
shows that for small values of $\mu/\tau$ the long-time average is zero
because the state spreads along the edges of the entire lattice as it evolves.
Around the critical ratio of unity the long-time average jumps signifying the
onset of MQST and asymptotically approaches $-N/2$. This analysis has
highlighted the key difference between the Fock-space and real-space lattices
which is that the sites on the FSL are fundamentally many-body states and the
real space sites are not. Thus, many-body phenomena like MQST can appear on
the FSL and in this case disrupt edge states.

##  IV Conclusion

In this work we showed that two interacting bosonic quantum gases can be
periodically driven to simulate a particle on a 2D lattice in the presence of
a magnetic field. The quasispectrum of the driving sequence Floquet operator
reproduces the celebrated Hofstadter’s butterfly pattern found in the HH
model. Through the calculation of the LCM we found that the driven gases
retain the topological features of the HH model with the Chern numbers
agreeing between the two systems. A method to measure the topological
properties of the driven gases is proposed involving a combination of
measuring the relative phase of one gas and the number difference of the other
gas. This method is akin to hybrid time-of-flight images and measures the
transport properties of the lattice associated with the Chern number. Finally,
we showed that when intraspecies interactions are included, the nonlinear
effect of MQST can hinder the movement of edge states.

The implementation of the driving sequence raises some interesting
possibilities for the types of synthetic lattices one can build. For instance,
instead of coupling two many-particle two-mode gases together, only the
addition of a single two-mode particle to one $N$ particle gas is enough for a
second dimension. The FSL then takes the shape of a ladder with the rungs made
of the two-mode single particle and the legs made of the two-mode $N$ particle
gas. Also, a single three-mode gas has a hexagonal shaped Hilbert space with a
triangular hard wall, so this may open up possibilities to explore other
important systems like the Haldane model. Further study of simulating lattices
with ultracold atomic gases will provide opportunities to explore the relation
between seemingly disparate phenomena in the condensed matter, optical and
atomic fields.

###### Acknowledgements.

Acknowledgements J. Mumford thanks J. Larson, W. Kirkby, and D. H. J. O’Dell
for helpful comments in the writing of this manuscript.

##  Appendix A  Derivation of $\hat{H}_{\mathrm{eff}}$

To derive Eq. (LABEL:eq:heff) we start with the unitary Floquet operator in
Eq. (3) and using the fact that
$\mathrm{e}^{\mathrm{i}\phi\hat{J}_{z}}f\left(\hat{J}_{x}\right)\mathrm{e}^{-\mathrm{i}\phi\hat{J}_{z}}=f\left(\hat{J}_{x}\cos(\phi)-\hat{J}_{y}\sin(\phi)\right)$
where $f(x)$ is a general function of $x$ gives

$\hat{U}=\mathrm{e}^{\mathrm{i}\tau\left[\hat{J}_{x}\cos\left(2\pi\alpha\hat{S}_{z}\right)-\hat{J}_{y}\sin\left(2\pi\alpha\hat{S}_{z}\right)\right]}\mathrm{e}^{\mathrm{i}\tau\hat{S}_{x}}$
(13)

where the substitution $\alpha=\kappa/\omega\pi$ has been made. Using the
Baker-Campbell-Hausdorff formula

$\mathrm{e}^{\mathrm{i}\tau\hat{A}}\mathrm{e}^{\mathrm{i}\tau\hat{B}}=\mathrm{e}^{\mathrm{i}\tau\left(\hat{A}+\hat{B}\right)+\frac{\left(\mathrm{i}\tau\right)^{2}}{2}\left[\hat{A},\hat{B}\right]+...}$
(14)

and taking $\tau\ll 1$, so that terms of $\mathcal{O}\left(\tau^{2}\right)$
and higher can be neglected gives

$\hat{U}\approx\mathrm{e}^{\mathrm{i}\tau\left[\hat{J}_{x}\cos\left(2\pi\alpha\hat{S}_{z}\right)-\hat{J}_{y}\sin\left(2\pi\alpha\hat{S}_{z}\right)+\hat{S}_{x}\right]}\,.$
(15)

Writing the Floquet operator as
$\hat{U}=\mathrm{e}^{-\mathrm{i}\hat{H}_{\mathrm{eff}}\tau}$, we identify the
effective Hamiltonian as

$\hat{H}_{\mathrm{eff}}=-\left[\hat{J}_{x}\cos\left(2\pi\alpha\hat{S}_{z}\right)-\hat{J}_{y}\sin\left(2\pi\alpha\hat{S}_{z}\right)+\hat{S}_{x}\right]\,.$
(16)

In terms of the spin raising and lowering operators, the Hamiltonian becomes

$\hat{H}_{\mathrm{eff}}=-\frac{1}{2}\left[\hat{J}_{+}\mathrm{e}^{\mathrm{i}2\pi\alpha\hat{S}_{z}}+\hat{J}_{-}\mathrm{e}^{-\mathrm{i}2\pi\alpha\hat{S}_{z}}+\hat{S}_{+}+\hat{S}_{-}\right]$
(17)

and expressing the spin operators explicitly in terms of the $\hat{J}_{z}$
($\hat{S}_{z}$) eigenstates produces the effective Hamiltonian in Eq.
(LABEL:eq:heff).

##  Appendix B  Derivation of $H_{\mathrm{MF}}$

We start the derivation of $H_{\mathrm{MF}}$ by adding intraspecies
interactions to the effective Hamiltonian. With intraspecies interactions the
unitary operator in Eq. (13) becomes

$\displaystyle\hat{U}=$
$\displaystyle\mathrm{e}^{-\mathrm{i}\frac{\mu}{N}\left(\hat{S}_{z}^{2}+\hat{J}_{z}^{2}\right)}\mathrm{e}^{\mathrm{i}\tau\left[\hat{J}_{x}\cos\left(2\pi\alpha\hat{S}_{z}\right)-\hat{J}_{y}\sin\left(2\pi\alpha\hat{S}_{z}\right)\right]}$
(18)
$\displaystyle\times\mathrm{e}^{-\mathrm{i}\frac{\mu}{N}\left(\hat{S}_{z}^{2}+\hat{J}_{z}^{2}\right)}\mathrm{e}^{\mathrm{i}\tau\hat{S}_{x}}\,.$

In this paper we are using small values of $\mu$ of $\mathcal{O}(\tau)$, so we
can neglect terms of $\mathcal{O}(\mu\tau)$ in the Baker-Campbell-Hausdorff
formula and simply combine all of the exponentials into one

$\hat{U}=\mathrm{e}^{\mathrm{i}\tau\left\\{-\frac{2\mu}{N\tau}\left(\hat{S}_{z}^{2}+\hat{J}_{z}^{2}\right)+\hat{J}_{x}\cos\left(2\pi\alpha\hat{S}_{z}\right)-\hat{J}_{y}\sin\left(2\pi\alpha\hat{S}_{z}\right)+\hat{S}_{x}\right\\}}\,,$
(19)

so the effective Hamiltonian with interactions is

$\displaystyle\hat{H}_{\mathrm{eff}}=\frac{2\mu}{N\tau}$
$\displaystyle\left(\hat{S}_{z}^{2}+\hat{J}_{z}^{2}\right)-\hat{J}_{x}\cos\left(2\pi\alpha\hat{S}_{z}\right)$
(20)
$\displaystyle+\hat{J}_{y}\sin\left(2\pi\alpha\hat{S}_{z}\right)-\hat{S}_{x}\,.$

The mean-field effective Hamiltonian is derived by replacing the creation
(annihilation) operators with complex numbers

$\hat{a}_{1,2}\to\sqrt{N_{1,2}}\mathrm{e}^{\mathrm{i}\theta_{1,2}},\hskip
15.0pt\hat{b}_{1,2}\to\sqrt{M_{1,2}}\mathrm{e}^{\mathrm{i}\phi_{1,2}}$ (21)

which transforms the spin operators into

$\displaystyle\hat{J}_{z}\to\frac{1}{2}\left(N_{1}-N_{2}\right)=n,\,\hat{S}_{z}\to\frac{1}{2}\left(M_{1}-M_{2}\right)=m$
$\displaystyle\hat{J}_{x}\to\sqrt{N^{2}/4-n^{2}}\cos\left(\theta_{x}\right),\,\hat{S}_{x}\to\sqrt{N^{2}/4-m^{2}}\cos\left(\theta_{y}\right)$
$\displaystyle\hat{J}_{y}\to\sqrt{N^{2}/4-n^{2}}\sin\left(\theta_{x}\right),\,\hat{S}_{y}\to\sqrt{N^{2}/4-m^{2}}\sin\left(\theta_{y}\right)$

where $\theta_{x}=\theta_{1}-\theta_{2}$ and $\theta_{y}=\phi_{1}-\phi_{2}$.
We take the mean-field Hamiltonian to be
$H_{\mathrm{MF}}=\lim_{N\to\infty}\hat{H}_{\mathrm{eff}}/(N/2)$, so it becomes

$\displaystyle H_{\mathrm{MF}}=$
$\displaystyle\frac{\mu}{\tau}\left(x^{2}+y^{2}\right)-\sqrt{1-x^{2}}\cos\left(\theta_{x}+\pi\alpha
Ny\right)$ (23) $\displaystyle-\sqrt{1-y^{2}}\cos\left(\theta_{y}\right)$

where $x=2n/N$ and $y=2m/N$ and we have used the trigonometric identity
$\cos\left(a+b\right)=\cos(a)\cos(b)-\sin(a)\sin(b)$.

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