# Synthesizing arbitrary dispersion relations in a modulated tilted optical
lattice

Jean Claude Garreau  Univ. Lille, CNRS, UMR 8523 - PhLAM - Laboratoire de
Physique des Lasers Atomes et Molécules, F-59000 Lille, France  Véronique
Zehnlé  Univ. Lille, CNRS, UMR 8523 - PhLAM - Laboratoire de Physique des
Lasers Atomes et Molécules, F-59000 Lille, France

(August 27, 2024)

###### Abstract

Dispersion relations are fundamental characteristics of the dynamics of
quantum and wave systems. In this work we introduce a simple technique to
generate arbitrary dispersion relations in a modulated tilted lattice. The
technique is illustrated by important examples: the Dirac, Bogoliubov and
Landau dispersion relations (the latter exhibiting the roton and the maxon).
We show that adding a slow chirp to the lattice modulation allows one to
reconstruct the dispersion relation from dynamical quantities. Finally, we
generalize the technique to higher dimensions, and generate graphene-like
Dirac points and flat bands in two dimensions.

##  I Introduction

Dispersion relations, connecting the energy to the momentum,
$E=E(\boldsymbol{p})$, of a quantum particle, or the frequency to the wave
number $\omega=\omega(\boldsymbol{k})$ of a wave, are a fundamental concept in
many domains of physics. For example, relativistic particles are characterized
by the Einstein’s dispersion relation $E^{2}=p^{2}c^{2}+m^{2}c^{4}$,
crystalline solids by their bands $E=E(\boldsymbol{q})$ (with $\boldsymbol{q}$
the quasimomentum) and superfluidity by Landau’s dispersion relation
presenting exotic features like the _maxon_ and the _roton_ [1, 2, 3].
Dispersion relations provide a great deal of information on the physics of a
system.

Recent developments both in condensed matter and ultracold-atom systems have
generated the concept of “quantum simulator”. Devising, for instance, a system
displaying a given dispersion relation provides information on several aspects
of the physics of other systems exhibiting the same dispersion relation.
Ultracold atoms in optical lattices or trapped ions have proved to be one of
the most clean and flexible systems in physics, both theoretically and
experimentally. Optical lattices can mimic an almost perfect lattice (no
phonons, controllable decoherence, etc.) and make it easy to create low-
dimensional systems. Moreover, being formed by the interference of laser
beams, they can be “engineered” in many ways, e.g one can create a wealth of
different lattices. A few, non-exaustive, examples are Kagome [4], Lieb [5, 6,
7], quasiperiodic [8] and disordered lattices [9]. They can also be easily
modulated in time, producing, notably, oscillating [10] or accelerated
lattices [11, 12]. These properties make such systems an outstanding platform
to the realization of analog quantum simulators. In many cases, one can
construct a given Hamiltonian from its building blocks, e.g. Bose- and Fermi-
Hubbard’s [13, 14, 15], Anderson’s [9, 8, 16, 17] and Dirac’s [18, 19, 20, 21,
22] Hamiltonians, bringing new light into many aspects of their physics,
notably the Mott transition [14], the Anderson localization and transition
[17, 23, 24, 25], Bloch oscillations and Wannier-Stark systems [11, 12], or
the Klein tunneling [26, 27].

In the present work we present a simple one dimensional system able to
“synthesize” almost _any_ desired dispersion relation, that we illustrate with
important examples. An original technique allowing the direct detection of
these dispersion relations is also proposed. In the last part, we generalize
the synthesis of dispersion relations to higher dimensions, illustrated by the
generation of graphene-like Dirac cones and Lieb lattices (displaying a flat
band) in 2D.

##  II Synthesizing dispersion relations in one-dimensional Wannier-Stark
optical lattices

Our present model is based on the framework introduced in Ref. [21] and
further developed in [22]. Ultracold atoms are placed in the interference
pattern of counter-propagating laser beams which generates a sinusoidal
potential, proportional to the atom-laser coupling, acting on on the center-
of-mass degree of freedom of the atoms [28]. The atomic cloud density is
assumed low enough that atomic interactions are negligible. We consider here a
one-dimensional Wannier-Stark system consisting of a sinusoidal optical
lattice to which a constant force $F$ is applied:

$H=\frac{p^{2}}{2m}-V_{0}\cos\left(\frac{2\pi x}{a}\right)+Fx.$

This Hamiltonian can obtained by placing ultracold atoms on an accelerated
standing laser wave [11, 12, 29] 111The force can be generated by applying a
linear chirp to one of the beams forming the standing wave, so that the nodes
of the resulting standing wave are uniformly accelerated. In the (non-
inertial) reference frame where the standing wave is at rest, the atoms feel a
constant inertial force. of wavelength $k_{L}$, hence $a=\pi/k_{L}$. The
properties of such a system are well known [31, 32, 33, 34, 35, 36, 37]. In
short, this system is invariant under discrete spatial translations
corresponding to an integer multiple $n$ of the lattice step $a$ provided the
energy is also shifted of $nFa=n\hbar\omega_{B}$ where $\omega_{B}\equiv
Fa/\hbar$ is the so-called _Bloch frequency_. In the following, $n$ denotes
the site index corresponding to potential minima localized at the position
$x=na$. The symmetry of the system implies that the eigenfunctions (called
Wannier-Stark states) are invariant under a translation of an integer number
of lattice steps, i.e $\varphi_{n}^{(\ell)}(x)=\varphi_{0}^{(\ell)}(x-na)$,
with the corresponding eigenenergies
$E_{n}^{(\ell)}=E_{0}^{(\ell)}+n\hbar\omega_{B}$, thus forming (depending on
parameters $V_{0}$ and $F$) different ladders (labeled by $\ell$) of levels
separated by a step $\hbar\omega_{B}$ and characterized by a ground energy
shift $E_{0}^{(\ell)}$. In what follows we will mainly consider potential
parameters allowing two ladders $(\ell=g,e)$ of localized “ground” and
“excited” eigenstates, that is, there are two states localized at each site
$n$, $\varphi_{n}^{(g)}(x)$ and $\varphi_{n}^{(e)}(x)$, separated by an energy
shift $\Delta=E_{0}^{(e)}-E_{0}^{(g)}$. Other eigenstates of $H$, localized or
belonging to the continuum, are supposed to be irrelevant for the system’s
dynamics, as explained below.

A convenient set of dimensionless units is obtained by measuring space in
units of $a$, energy in units of the so-called atom’s _recoil energy_
$E_{R}\equiv\hbar^{2}k_{L}^{2}/2m$$=\hbar\omega_{R}$ and time in units of
$\omega_{R}^{-1}$ ($m$ is the atom’s mass). This leads to the dimensionless
Hamiltonian

$H_{0}=\frac{p^{2}}{2m^{*}}-V_{0}\cos\left(2\pi x\right)+\omega_{B}x$ (1)

where $m^{*}=\pi^{2}/2$, $\omega_{B}$ is measured in $\omega_{R}$ units and
Planck’s constant is $\hbar=1$.

Controllable dynamics can be induced in such systems by external modulations
of the parameters $V_{0}$ (or $F$) at frequencies close to resonances, i.e
close to $\Delta$ and to multiples of $\omega_{B}$. We thus add to $H_{0}$ a
time-dependent potential

$H_{1}(t)=f(t)V(x)$ (2)

where $V(x)=\cos(2\pi x)$. Using this flexibility, the feasibility of models
reproducing the Dirac equation (hence the Dirac dispersion relation) has been
demonstrated in Refs. [21, 22].

The driving $f(t)$ has the general form

$f(t)=\sum_{j,q}A_{j,q}\mathrm{e}^{\mathrm{i}j\omega_{B}t}\mathrm{e}^{\mathrm{i}q\Delta
t}$ (3)

with $j\in\mathds{Z}$, $q=0,\pm 1$ and $A_{j,q}=-A_{-j-q}^{*}$ (reality
condition). For given integers $j$ the modulation resonantly couples states
centered in sites separated by $j$ lattice steps. If $q=0$, one couples states
$\varphi_{n}^{(\ell)}$ and $\varphi_{n+j}^{(\ell)}$ belonging to the same
ladder $\ell$ (_intraladder_ coupling). _Interladder_ couplings between states
$\varphi_{n}^{(g)}$ and $\varphi_{n+j}^{(e)}$ are obtained for $q=1$, or
$\varphi_{n}^{(e)}$ and $\varphi_{n+j}^{(g)}$ for $q=-1$.

With the above provisions, the general solution for the system can be
developed in the Wannier-Stark basis restricted to the subspace spanned by the
ground and first excited ladder

$\displaystyle\Psi(x,t)=$
$\displaystyle\sum_{n}\left(c_{n}(t)\mathrm{e}^{-\mathrm{i}n\omega_{B}t}\varphi_{n}^{(g)}(x)\right.$
$\displaystyle\left.+d_{n}(t)\mathrm{e}^{-\mathrm{i}\left(n\omega_{B}+\Delta\right)t}\varphi_{n}^{(e)}(x)\right)$
(4)

which holds if modulation amplitudes in $f(t)$ are low enough to avoid
projection on other eigenstates of the Hamiltonian $H_{0}$.

Plugging this form into the Schrödinger equation for $H_{0}+H_{1}(t)$ one
obtains a set of coupled differential equations for the amplitudes $c_{n}(t)$
and $d_{n}(t)$ (for details of this calculation, see the Appendix of Ref.
[21]), which reads

$\displaystyle\mathrm{i}$
$\displaystyle\frac{d}{dt}\left(\begin{array}[]{c}c_{n}(t)\\\
d_{n}(t)\end{array}\right)=\sum_{r\in\mathds{Z}}\left(\begin{array}[]{cc}T_{r}^{(gg)}&T_{r}^{(ge)}\\\
T_{r}^{(eg)}&T_{r}^{(ee)}\end{array}\right)\left(\begin{array}[]{c}c_{n+r}(t)\\\
d_{n+r}(t)\end{array}\right)$ (11)

with coupling amplitudes between sites $n$ and $n+r$ 222The translation
symmetry $\varphi_{n}^{(\ell)}(x-na)=\varphi_{0}^{(\ell)}(x)$ implies
$\left\langle\varphi_{n}^{(\ell)}\right|V\left|\varphi_{n+r}^{(\ell)}\right\rangle=\left\langle\varphi_{0}^{(\ell)}\right|V\left|\varphi_{r}^{(\ell)}\right\rangle$for
$V=\cos(2\pi x)$.

$\displaystyle T_{r}^{(gg)}$
$\displaystyle=A_{r,0}\left\langle\varphi_{0}^{(g)}\right|V\left|\varphi_{r}^{(g)}\right\rangle$
$\displaystyle T_{r}^{(ee)}$
$\displaystyle=A_{r,0}\left\langle\varphi_{0}^{(e)}\right|V\left|\varphi_{r}^{(e)}\right\rangle$
$\displaystyle T_{r}^{(ge)}$
$\displaystyle=A_{r,1}\left\langle\varphi_{0}^{(g)}\right|V\left|\varphi_{r}^{(e)}\right\rangle$
$\displaystyle T_{r}^{(eg)}$
$\displaystyle=A_{r,-1}\left\langle\varphi_{0}^{(e)}\right|V\left|\varphi_{r}^{(g)}\right\rangle.$
(12)

In the momentum representation the corresponding amplitudes
$\tilde{c}(k,t)=\sum_{n}c_{n}(t)\exp\left(-\mathrm{i}kn\right)$ (with
analogous expressions for $d_{n}$) are governed by only _two_ coupled
equations

$\displaystyle\mathrm{i}$
$\displaystyle\frac{d}{dt}\left(\begin{array}[]{c}\tilde{c}(k,t)\\\
\tilde{d}(k,t)\end{array}\right)=\left(\begin{array}[]{cc}F_{g}(k)&F(k)\\\
F^{*}(k)&F_{e}(k)\end{array}\right)\left(\begin{array}[]{c}\tilde{c}(k,t)\\\
\tilde{d}(k,t)\end{array}\right)$ (19)

where

$F_{\ell}(k)=\sum_{r=-\infty}^{\infty}T_{r}^{(\ell\ell)}\mathrm{e}^{\mathrm{i}kr}$
(20)

with $\ell=\left\\{g,e\right\\}$, and

$F(k)=\sum_{r=-\infty}^{\infty}T_{r}^{(ge)}\mathrm{e}^{\mathrm{i}kr}.$ (21)

The functions $F_{\ell}(k)$ and $F(k)$ are Fourier series whose coefficients
are proportional to the overlap integrals $T_{r}^{\ell\ell^{\prime}}$ of the
wave functions centered at positions separated by $r$ sites but _are
controlled by the modulation coefficients_ $A_{j,q}$ [see Eqs. (12)].

Equations (19) define an effective Hamiltonian for a two-level-like system,
whose eigenvalues determine the dispersion relation $\omega(k)$:

$\left[\omega(k)-F_{g}(k)\right]\left[\omega(k)-F_{e}(k)\right]=\left|F(k)\right|^{2}.$
(22)

This dispersion relation is $2\pi$-periodic and is conventionally defined in
the interval $k\in\left[-\pi,\pi\right)$. Equation (22) is the basis of our
technique for synthesizing dispersion relations: Specifying a given dispersion
relation amounts to define the functions $F_{g}(k)$, $F_{e}(k)$ and $F(k)$
and, from Eqs. (20,21), to set conditions on the amplitudes
$T_{r}^{(\ell,\ell^{\prime})}$, which allows one, with the help of Eqs. (12),
to determine modulation parameters $A_{j,q}$ that must be used in the
modulation term $H_{1}(t)$. Examples discussed in Sec. III illustrate how a
desired dispersion relation can be obtained to a very good approximation.

##  III Applications

###  III.1 Dirac cone

A linear dispersion relation $\omega(k)=\pm c|k|$, also called Dirac cone
(corresponding to a Dirac particle of zero mass), can be obtained from Eq.
(22) with $F_{g}(k)=F_{e}(k)=0$ and $F(k)=c\left|k\right|$. This implies
$T_{r}^{(g,g)}=T_{r}^{(e,e)}=0$, that is, no intraladder couplings, and
interladder couplings given by Eq. (21). As the Fourier series of the function
$|k|$ is

$|k|=\left[\frac{\pi}{2}-\frac{4}{\pi}\sum_{r=1}^{\infty}\frac{\cos(rk)}{r^{2}}\right]$

one must have

$\sum_{r}T_{r}^{(g,e)}e^{\mathrm{i}kr}=c\left|k\right|=c\left[\frac{\pi}{2}-\frac{4}{\pi}\sum_{r}\frac{\cos(rk)}{r^{2}}\right]$

which, from Eq. (12), gives

$A_{0,1}\left\langle\varphi_{0}^{(g)}\right|V\left|\varphi_{0}^{(e)}\right\rangle=c\frac{\pi}{2}$
(23)

and

$A_{r,1}\left\langle\varphi_{0}^{(g)}\right|V\left|\varphi_{r}^{(e)}\right\rangle=\frac{c}{\pi
r^{2}}\left[\cos(r\pi)-1\right]\qquad r>0.$

The dispersion relation is thus synthesized by applying a modulation given by
Eq. (3) with coefficients:

$\displaystyle A_{r,1}=$
$\displaystyle\frac{c\pi}{2}\left\langle\varphi_{0}^{(g)}\right|V\left|\varphi_{0}^{(e)}\right\rangle^{-1}\delta_{r,0}$
$\displaystyle+\frac{c}{\pi
r^{2}}\left\\{\cos(r\pi)-1\right\\}\left\langle\varphi_{0}^{(g)}\right|V\left|\varphi_{r}^{(e)}\right\rangle^{-1}\left(1-\delta_{r0}\right).$

Experimentally, this kind of modulation can be easily created by an arbitrary-
wave generator. The overlap integrals
$\left\langle\varphi_{0}^{(g)}\right|V\left|\varphi_{r}^{(e)}\right\rangle$
depend on the lattice parameters $V_{0}$ and $F$ and go to zero rather fast
with the site distance $r$, implying that the modulation amplitude at higher
frequencies must increase rapidly, eventually breaking the slow modulation
condition implied by Eq. (4). Fortunately, for typical lattice parameters,
keeping only $r\leq 3$ terms yet gives a rather good approximation of a Dirac
cone, as shown Fig. 1 (_top_), except at the tip of the cone at $k=0$ which
needs higher harmonics to be well reproduced.

A way to overcome this limitation is to include in Eq. (23) a frequency offset
parameter $b$ such that $\omega(k)=\pm c\left(|k|+b\right)$, which is readily
done by setting

$A_{0,1}=c\left(b+\frac{\pi}{2}\right)\left\langle\varphi_{0}^{(g)}\right|V\left|\varphi_{0}^{(g)}\right\rangle^{-1}.$

The parameter $b$ controls the gap between the two half-cones: If $b>0$ a gap
is opened, and if $b<0$ one obtains two intersecting Dirac cones. Figure 1
(_bottom_) illustrates this case for $b=-1$ and shows perfect linear behavior
in the vicinity of the Dirac points whose position $k=\pm b$ is controlled by
the applied modulation. We have thus obtained flexible way to generate Dirac-
cone-type dispersion relations, allowing one to open a gap or to merge the
cones.

Figure 1: Linear dispersion relation. _Top_ : Dirac cone ($c=1$); _Bottom_ :
intersecting Dirac cones ($c=1,b=-1$). Only terms $0\leq r\leq 3$ were kept in
the Fourier series (red circles). The analytical expressions are shown for
comparison (black solid lines).

###  III.2 The superfluid dispersion relation, the roton and the maxon

Superfluidity is a purely quantum behavior appearing notably in liquid Helium
and quantum atomic gases. Its simplest mathematical treatment relies on the
Bogoliubov correction to the mean-field Gross-Pitaevskii equation [39, 28, 3],
which leads to the well-known Bogoliubov dispersion relation

$\omega(k)=\sqrt{\frac{k^{2}}{2M}\left(\frac{\hbar^{2}k^{2}}{2M}+2gn\right)}$
(24)

where $M$ is the atom mass, $g$ a parameter characterizing the binary atomic
interaction in the Gross-Pitaevskii equation and $n$ the atomic density (the
approximation is valid in the limit of low density and temperature). This
dispersion is characterized by a “phononic”, i.e. linear $\omega(k)\propto k$
dependence at small $k$ $(k\ll\sqrt{gnM}/\hbar$) and a “free particle” part
$\omega(k)\propto k^{2}$ at large $k$.

Following the same ideas as in the preceding section, Eq. (24) can be obtained
from the general form of the dispersion relation with $F_{g}(k)=F_{e}(k)=0$
and

$F(k)=c\left(\left|k\right|+\alpha k^{2}\right)$ (25)

where $\alpha$ is an imaginary number ($\alpha=\mathrm{i}|\alpha|$), leading,
through Eq. (22), to the desired dispersion

$\omega(k)=c\sqrt{k^{2}\left(1+\left|\alpha\right|^{2}k^{2}\right)}$ (26)

Beyond the Bogoliubov approach, phenomenological arguments by Landau [1]
concerning the superfluidity of the strongly-interacting liquid 4He supported
the existence in the dispersion relation of a minimum $\omega(k_{r})$ at some
$k_{r}$, called the _roton_ 333The roton minimum determines Landau’s critical
velocity given by slope of the dispersion relation, thus setting a gap
responsible for the superfluidity. which, by continuity, implies the existence
of a maximum $\omega(k_{m})$ with $k_{m}<k_{r}$, called the _maxon_. Roton and
maxon features have recently been observed experimentally using a Bose-
Einstein condensate in a modulated flat lattice [41], in Erbium condensates
with interactions controlled by Feshbach resonances [42, 43] and in acoustic
metamaterials [44].

A dispersion relation presenting a roton and maxon can be synthesized by
choosing $\alpha$ as a _complex_ number, thus

$\omega(k)=c\sqrt{k^{2}\left(1+\left(\alpha+\alpha^{*}\right)|k|+\left|\alpha\right|^{2}k^{2}\right)}.$
(27)

Equation (21) then leads to

$T_{r}^{ge}=A_{r,1}\left\langle\varphi_{0}^{g}\right|V\left|\varphi_{r}^{e}\right\rangle=\frac{c}{\pi}\intop_{0}^{\pi}\cos\left(rk\right)\left(\left|k\right|+\alpha
k^{2}\right),$

from which we obtain the modulation amplitudes

$A_{r,1}=c\left(\frac{\pi}{2}+\alpha\frac{\pi^{2}}{3}\right)\left\langle\varphi_{0}^{(g)}\right|V\left|\varphi_{0}^{(e)}\right\rangle^{-1}\delta_{r0}+\frac{c}{\pi
r^{2}}\left[(2\pi\alpha+1)\cos(r\pi)-1\right]\left\langle\varphi_{0}^{(g)}\right|V\left|\varphi_{r}^{(e)}\right\rangle^{-1}\left(1-\delta_{r0}\right).$
(28)

The fact that $\left|A_{r,1}\right|\propto r^{-2}$ (for $r\neq 0$) implies
that the series has a rather fast convergence, and thus only a few terms
suffice to give a good approximation of the desired dispersion relation.
Figure (2) shows the resulting Bogoliubov dispersion relation with the roton-
maxon features obtained for $\alpha=-0.5+0.1\mathrm{i}$, compared to Eq. (27).

Figure 2: Synthetic dispersion relation displaying a maxon and a roton,
generated by the Fourier series Eq. (28) (blue circles), restricted to
$|r|\leq 3$, compared to Eq. (27) with parameters $c=1$ and
$\alpha=-0.5+0.1\mathrm{i}$ (black solid line).

###  III.3 Scanning the dispersion relation

A natural question is, how to obtain experimental information on the shape of
the dispersion relation? We show below that this can be done by preparing an
initial wave packet and monitoring its average position $\left\langle
X\right\rangle(t)$, in the modulated lattice while applying a slow phase chirp
to the modulation Eq. (3); we thus use a generalized modulation form
$\sum_{j}A_{j,q}\exp\left(\mathrm{i}\left[j\left(\omega_{B}t+\phi(t)\right)+q\Delta
t\right]\right)$, where $\phi(t)$ is an arbitrary phase modulation. All the
developments of Sec. II remain valid except that, for a slow variation of
$\phi(t)$, the amplitudes become $A_{j,0}\exp\left(\mathrm{i}j\phi(t)\right).$
Equations (12) show that phase modulation $\phi(t)$ is then imprinted in the
coupling coefficients $T_{r}$, which finally results in shifting $k\rightarrow
k(t)=k+\phi(t)$ in Eqs. (20,21). Changing the phase $\phi$ turns out to be
equivalent to change $k$ 444That is, $\phi$ acts as a (time-dependent)
quasimomentum., resulting in slowly varying functions $F(k(t))$ and
$F_{\ell}\left(k(t)\right)$ and thus $\omega\left(k(t)\right)$.

We illustrate this idea in the simple case of an intraladder model with only
the ground ladder $\ell=0$ [i.e. $q=0$ in Eq. (2)]. In this case we have
$\omega(k)=F_{g}\left[k+\phi(t)\right]$, and a slow linear chirp $\phi=\gamma
t$ ($\gamma\ll\omega_{B}$) results in $\left\langle
X\right\rangle(t)=\left.\omega(k)\right|_{k=k(t)}/\gamma$ (see appendix A). We
choose here the example of an analytical dispersion relation

$\omega(k)=a\left[1-\cos(2k)\right]+b\sin(2k)+c\sin k$ (29)

which exhibits a roton-like behavior as shown in Fig. 3 (full black line). We
performed a numerical simulation of the full Schrödinger equation
corresponding to the Hamiltonian $H=H_{0}+H_{1}(t)$ (with a suitable choice of
modulation parameters) with an initial Gaussian wave packet whose width in
$k-$space is small compared to the extent of the dispersion relation (or more
precisely to its typical length scale). The time evolution of the wave packet
is then registered while the phase is linearly swept $\phi(t)=\gamma t$. The
evolution of the mean position $\left\langle X\right\rangle(t)$ is shown in
Fig. 3 (blue circles) and shows a good agreement with the analytical
expression Eq. (29), validating our proposal.

Figure 3: Detecting the dispersion relation. The solid black line dispersion
relation is obtained from $\omega(k)=F_{g}\left[k+\phi(t)\right]$ with
$F_{g}(k)=-0.81\left[1-\cos(2k)\right]-0.5\sin(2k)+2\sin k$ and $F_{e}=F=0$.
The blue circles show the average wave packet position $\left\langle
X\right\rangle(t)$ (with an adjusted vertical scale factor) and the chirp
ratio $\gamma=\pi\times 10^{-4}\omega_{B}$.

##  IV Dispersion relations in two dimensions

The strategy developed in Sec. II can be generalized to higher dimensions. We
give here two examples: the creation of 2D Dirac points and of a Lieb lattice
dispersion relation with a flat band.

###  IV.1 Creating, moving and the merging of Dirac cones

Manipulation of Dirac points both in optical lattices [20] and crystals [46]
is an active research subject. The creation of a 2D optical lattice simulating
Dirac physics was discussed in Ref. [22]. In brief, the corresponding
dispersion relations can be synthesized in a 2D modulated tilted lattice.
Hamiltonian (2) is easily translated into 2D (the definition of the
dimensionless units is analogous to the 1D case):

$H_{0_{2D}}=\frac{p_{x}^{2}+p_{y}^{2}}{2m^{*}}-V_{0}\left(\cos(2\pi
x)+\cos(2\pi y)\right)+\omega_{B}^{(x)}x+\omega_{B}^{(y)}y.$ (30)

Since the above Hamiltonian is separable, all Wannier-Stark properties
described in Sec. II trivially generalize to the present case. We consider
only ground ladder eigenstates formed by the tensorial product
$\varphi_{n}^{(x)}(x)\varphi_{m}^{(y)}(y)$ with
$\varphi_{s}^{(i)}(x_{i})=\varphi_{0}^{(i)}(x_{i}-sx_{0_{i}})$,
$i=\left\\{x,y\right\\},s=\left\\{n,m\right\\}$ with eigenenergies
$E_{0}+n\omega_{B}^{(x)}+m\omega_{B}^{(y)}$ .

The controlled dynamics for Dirac system we are interested in is generated by
adding a perturbation

$\displaystyle H_{1_{2D}}(x,y,t)=$ $\displaystyle\cos\left(\pi
x\right)\left[V_{x}\mathrm{e}^{\mathrm{i}\left(2\omega_{B}^{(x)}t+\phi_{x}\right)}+\mathrm{c.c.}+V_{0}\right]$
$\displaystyle+$ $\displaystyle\left[V_{y}\cos\left(2\pi
y\right)\mathrm{e}^{\mathrm{i}\left(\omega_{B}^{(y)}t+\phi_{y}\right)}+\mathrm{c.c.}\right]$
(31)

depending on the arbitrary phases $\phi_{x,y}$. The general solution
(restricted to the ground ladder) can be written, in analogy with Eq. (4),

$\Psi(x,y,t)=\sum_{n,m}c_{n,m}(t)\varphi_{n}^{(x)}(x)\varphi_{m}^{(y)}(y).$
(32)

Substituting the above solution in Schrödinger’s equation for the total
Hamiltonian $H_{0_{2D}}+H_{1_{2D}}$, one obtains the following set of coupled
equation (in the resonant approximation)

$\displaystyle\mathrm{i}\frac{dc_{n,m}}{dt}=$
$\displaystyle(-1)^{n}\left[T_{x}\mathrm{e}^{\mathrm{i}\phi_{x}}c_{n+2,m}+T_{x}\mathrm{e}^{-\mathrm{i}\phi_{x}}c_{n-2,m}+T_{0}c_{n,m}\right]$
$\displaystyle+T_{y}\mathrm{e}^{\mathrm{i}\phi_{y}}c_{n,m+1}+T_{y}\mathrm{e}^{-\mathrm{i}\phi_{y}}c_{n,m-1}$

with couplings

$\displaystyle T_{x}$
$\displaystyle=V_{x}\left\langle\varphi_{0}^{(x)}\right|\cos\left(\pi
x\right)\left|\varphi_{2}^{(x)}\right\rangle$ $\displaystyle T_{y}$
$\displaystyle=V_{y}\left\langle\varphi_{0}^{(y)}\right|\cos\left(2\pi
y\right)\left|\varphi_{1}^{(y)}\right\rangle$ $\displaystyle T_{0}$
$\displaystyle=V_{0}\left\langle\varphi_{0}^{(x)}\right|\cos\left(\pi
x\right)\left|\varphi_{0}^{(x)}\right\rangle$

where the factor $\cos(\pi x)$ in Eq. (31) introduces a parity-dependent
factor $(-1)^{n}$ which results in a system of two coupled sublattices
corresponding to sites where $n$ is odd or even. In the reciprocal space
$(k_{x},k_{y})$, we define two-dimensional Fourier amplitudes,

$\tilde{c}(k_{x},k_{y},t)=\sum_{n\textrm{
even}}\sum_{m}c_{nm}(t)\mathrm{e}^{-\mathrm{i}nk_{x}}\mathrm{e}^{-\mathrm{i}mk_{y}}$

for $n$ even, and, equivalently, for $n$ odd, $\tilde{d}(k_{x},k_{y},t)$.
These amplitudes can be written as a two-component spinor
$\left[\psi\right]=\left(\tilde{c}(k_{x},k_{y},t),\tilde{d}(k_{x},k_{y},t)\right){}^{\mathsf{T}}$.
The Hamiltonian projected in the $k$-space turns out to be

$\left[\begin{array}[]{cc}T_{0}+T_{x}\cos(2k_{x}+\phi_{x})&T_{y}\cos(k_{y}+\phi_{y})\\\
T_{y}\cos(k_{y}+\phi_{y})&-T_{0}-T_{x}\cos(2k_{x}+\phi_{x})\end{array}\right]$
(33)

for which the dispersion relation is

$\displaystyle\omega(k_{x},k_{y})=$
$\displaystyle\pm\left\\{\left[T_{0}+T_{x}\cos(2k_{x}+\phi_{x})\right]^{2}\right.$
$\displaystyle\left.+\left[T_{y}\cos(k_{y}+\phi_{y})\right]^{2}\right\\}^{1/2}.$
(34)

In the $k_{x,y}\rightarrow 0$ limit, choosing $\phi_{x}=\phi_{y}=\pi/2$ and
$T_{0}=0$, a Dirac equation (for a free particle) is obtained with the
dispersion relation

$\omega(k_{x},k_{y})=\pm\sqrt{4T_{x}^{2}k_{x}^{2}+T_{y}^{2}k_{y}^{2}}$

which is a Dirac cone (anisotropic if $2T_{x}\neq T_{y}$).

Another interesting situation is obtained from Eq. (34) with $\phi_{x}=0$,
$\phi_{y}=\pi/2$ and $T_{0}\neq 0$ . Then, the dispersion relation is, for
small $k_{y}$

$\omega(k_{x},k_{y})\approx\pm\sqrt{\left[T_{0}+T_{x}\cos(2k_{x})\right]^{2}+T_{y}^{2}k_{y}^{2}}$
(35)

Depending on parameters $T_{0}$ and $T_{x}$, different structures are obtained
(assuming here $T_{0}$ and $T_{x}$ of opposite sign without loss of
generality). As shown in Fig. 4, when $\left|T_{0}\right|<\left|T_{x}\right|$,
there are two Dirac cones centered at the positions $k_{y}=0$ and
$k_{x}=\pm(1/2)\cos^{-1}\left(\left|T_{0}/T_{x}\right|\right)$. These cones
moves towards each other as the value of $\left|T_{0}/T_{x}\right|$ increases,
and finally coalesce when $T_{0}=-T_{x}$ giving a dispersion relation
$\omega(k_{x},k_{y})\approx\pm\sqrt{4T_{x}^{2}k_{x}^{4}+T_{y}^{2}k_{y}^{2}}$
in the vicinity of the point $k_{x},k_{y}=0$, which is thus a hybrid point
with a linear dependence in the $y$ direction (corresponding to a free Dirac
particle, or phonon) and quadratic dependence in $x$ (corresponding to a free
non-relativistic particle). A gap opens for
$\left|T_{0}\right|>\left|T_{x}\right|$, potentially leading to topological
effects.

Figure 4: Dirac cones. We plot the positive root of Eq. (35) for parameters:
$T_{x}=T_{y}=3$ and $T_{0}=-2.5$ with two Dirac points.

###  IV.2 The Lieb lattice

In this section, we consider the synthesis of a Lieb lattice dispersion
relation [5, 6, 7], using a form of Eq. (31) which results in an effective
spin $S=1$ with a flat band. The idea is sketched in Fig. 5: it essentially
consists in creating different types of sites: $A,$ $B$, coupled with coupling
$T_{x}$, $B$ and $C$ coupled with $T_{y}$, while sites $D$ are completely
uncoupled and thus dynamically irrelevant.

Figure 5: Synthesis of a Lieb lattice. The unit cell is indicated by the
dashed-dotted square. Sites $D$ ($n$ and $m$ odd) are uncoupled to their
neighbors. The other sites $A,$$B$ and $C$ are resonantly coupled to their
neighbors: $B$ to $C$ with amplitude $T_{y}$ along the lines of $m$ even; $A$
to $B$ with amplitude $T_{x}$ along the lines of $n$ even.

This selective coupling can be obtained with the following perturbation

$\displaystyle H_{1_{2D}}(x,y,t)=\cos\left(2\pi x\right)$
$\displaystyle\left[V_{x}^{(1)}+V_{x}^{(2)}\cos(\pi
y)\right]\cos\left(\omega_{B}^{(x)}t\right)+\cos\left(2\pi
y\right)\left[V_{y}^{(1)}+V_{y}^{(2)}\cos(\pi
x)\right]\cos\left(\omega_{B}^{(y)}t\right).$

Substituting the general solution of Eq. (32) in Schrödinger’s equation for
the Hamiltonian $H_{0_{2D}}+H_{1_{2D}}$ one obtains the following set of
coupled equation (in the resonant approximation)

$\displaystyle\mathrm{i}\frac{dc_{n,m}}{dt}=$ $\displaystyle
T_{x}\left[1+(-1)^{m}\right]\left(c_{n+1,m}+c_{n-1,m}\right)$
$\displaystyle+T_{y}\left[1+(-1)^{n}\right]\left(c_{n,m+1}+c_{n,m-1}\right)$
(36)

with couplings

$\displaystyle T_{x}$
$\displaystyle=\frac{1}{2}V_{x}^{(1)}\left\langle\varphi_{0}^{(x)}\right|\cos\left(2\pi
x\right)\left|\varphi_{1}^{(x)}\right\rangle$ $\displaystyle T_{y}$
$\displaystyle=\frac{1}{2}V_{y}^{(1)}\left\langle\varphi_{0}^{(y)}\right|\cos\left(2\pi
y\right)\left|\varphi_{1}^{(y)}\right\rangle$

provided that the modulation amplitudes obey the following relations

$\displaystyle V_{x}^{(1)}$ $\displaystyle=$ $\displaystyle
V_{x}^{(2)}\left\langle\varphi_{0}^{(y)}\right|\cos\left(\pi
y\right)\left|\varphi^{(y)}\right\rangle$ $\displaystyle V_{y}^{(1)}$
$\displaystyle=$ $\displaystyle
V_{y}^{(2)}\left\langle\varphi_{0}^{(x)}\right|\cos\left(\pi
x\right)\left|\varphi^{(x)}\right\rangle.$

As expected, Eqs. (36) show that the complex amplitudes $c_{n,m}$ with $n$ and
$m$ odd (corresponding to $D$ sites) are dynamically inert. These amplitudes
are therefore irrelevant and, as shown in Fig. 5, the effective unit cell
contains only three relevant sites, which is the characteristic of Lieb
lattices.

In the same spirit as in Sec. IV.1 we define three amplitudes in $k$-space

$\tilde{c}(k_{x},k_{y},t)=\sum_{n\textrm{ even }}\sum_{m\textrm{ even
}}c_{nm}(t)\mathrm{e}^{-\mathrm{i}nk_{x}}\mathrm{e}^{-\mathrm{i}mk_{y}},$

and $\tilde{d}(k_{x},k_{y},t)$ for $m$ even, $n$ odd, and
$\tilde{f}(k_{x},k_{y},t)$ for $m$ odd, $n$ even. The time-evolution in
$k$-space for the three-component spinor
$\left[\psi\right]=\left(\tilde{c}(k_{x},k_{y},t),\tilde{d}(k_{x},k_{y},t),\tilde{f}(k_{x},k_{y},t)\right){}^{\mathsf{T}}$
obeys

$\mathrm{i}\frac{d\left[\psi\right]}{dt}=\left[\begin{array}[]{ccc}0&4T_{y}\cos
k_{y}&4T_{x}\cos k_{x}\\\ 4T_{y}\cos k_{y}&0&0\\\ 4T_{x}\cos
k_{x}&0&0\end{array}\right]\left[\psi\right]$

leading to the well-known three-band dispersion relations of the Lieb lattice:

$\displaystyle\omega(k_{x},k_{y})$ $\displaystyle=$ $\displaystyle 0$
$\displaystyle\omega(k_{x},k_{y})$ $\displaystyle=$
$\displaystyle\pm\sqrt{\left(4T_{x}\cos k_{x}\right)^{2}+\left(4T_{y}\cos
k_{y}\right)^{2}}$ (37)

where the first relation corresponds to the flat band and the second one to
two symmetric bands, and is anisotropic if $T_{x}\neq T_{y}$, as shown in Fig.
6.

Figure 6: Lieb lattice dispersion relation, Eq. (37). Parameters are
$T_{x}=1.5$, $T_{y}=1$.

##  V Conclusion

We introduced in this work a technique allowing the generation of arbitrary
dispersion relations in a tilted modulated lattice, which we illustrated
through several important examples: the 1D Dirac phononic dispersion relation,
the Bogoliubov dispersion relation, the Landau superfluid relation, with the
maxon and the roton features. We proposed a simple way to experimentally
detect the dispersion relation by adding a slow chirp to the modulation and
measuring an easily accessible quantity, namely the average position of the
wave packet. Finally, we illustrated a generation of Dirac points in 2
dimensions and the generation of a flat band in a Lieb lattice. The technique
introduced in the present work thus appears as an efficient, state-of-the-art
experimentally feasible, way to synthesize lattice systems with arbitrary
dispersion relations, thus mimic a large variety of important condensed matter
systems.

###### Acknowledgements.

This work was supported by Agence Nationale de la Recherche through MANYLOK
project (Grant No. ANR-18-CE30-0017), the Labex CEMPI (Grant No.
ANR-11-LABX-0007-01), the Ministry of Higher Education and Research, Hauts-de-
France Council and European Regional Development Fund (ERDF) through the
Contrat de Projets État-Région (CPER Photonics for Society, P4S).

##  Appendix A Dynamics in a chirped lattice

In this appendix we show how the introduction of a chirp allows to extract the
shape of the dispersion relation from the measurement of the average wave
packet position evolution.

Consider a smooth wave packet in a system obeying a dispersion relation
$\omega(k,t)$. Its average position $\left\langle X\right\rangle(t)$ can be
written as

$\left\langle X\right\rangle(t)=\left\langle
X\right\rangle(t=0)+\int_{0}^{t}v_{G}(t^{\prime})dt^{\prime}$

where $v_{G}(t)=d\omega/dk$ is the group velocity.

An adiabatic chirp $\varphi(t)=\gamma t$ corresponds to $k(t)=k_{0}+\gamma t$,
and thus, by integration

$\left\langle X\right\rangle(t)=\left\langle
X\right\rangle(t=0)+\left[\omega\left(k(t)\right)-\omega\left(k_{0}\right)\right]/\gamma$

with $\left\langle X\right\rangle(t=0)=0$. Choosing $k_{0}$ such that
$\omega(k_{0})=0$ this results in the simple expression:

$\left\langle X\right\rangle(t)=\omega\left(k(t)\right)/\gamma.$

Hence, measuring of the temporal evolution of the average position of the wave
packet directly gives the shape of dispersion relation.

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