# Features of the motion of ultracold atoms in quasiperiodic potentials

Ivan Dynnikov1 and Andrei Maltsev2 1 V.A. Steklov Mathematical Institute of
Russian Academy of Sciences
119991 Moscow, Gubkina str. 8
2 L.D. Landau Institute for Theoretical Physics of Russian Academy of Sciences
142432 Chernogolovka, pr. Ak. Semenova 1A

###### Abstract

We consider here quasiperiodic potentials on the plane, which can serve as a
‘‘transitional link’’ between ordered (periodic) and chaotic (random)
potentials. As can be shown, in almost any family of quasiperiodic potentials
depending on a certain set of parameters, it is possible to distinguish a set
(in the parameter space) where, according to a certain criterion, potentials
with features of ordered potentials arise, and a set where we have potentials
with features of random potentials. These sets complement each other in the
complete parameter space, and each of them has its own specific structure. The
difference between ‘‘ordered’’ and ‘‘chaotic’’ potentials will manifest
itself, in particular, in the transport properties at different energies,
which we consider here in relation to systems of ultracold atoms. It should be
noted here that the transport properties of particles in the considered
potentials can be accompanied by the phenomena of ‘‘partial integrability’’
inherent in two-dimensional Hamiltonian systems.

##  I Introduction

In this paper, we consider applications of relatively recent results in the
theory of quasiperiodic functions on the plane to the dynamics of ultracold
atoms in quasiperiodic potentials. The general theory of quasiperiodic
functions, the origins of which go back to the studies of H. Bohr and A.S.
Besicovich (see, for example Bohr ; Besicovitch ), is currently a classical
area of mathematics and mathematical physics. In different papers, actually,
one can find slightly different definitions of a quasiperiodic function. Here
we will call a quasiperiodic function on the plane any function $\,f(x,y)\,$,
obtained by restricting a smooth $d$-periodic function
$\,F(z^{1},\dots,z^{d})\,$ to a generic affine embedding
$\,\mathbb{R}^{2}\,\subset\,\mathbb{R}^{d}\,$. In this case, the dimension
$\,d\,$ will be called the number of quasiperiods of the corresponding
quasiperiodic function on the plane.

The questions we consider here are mainly related to the geometry of the
regions defined by the relation

$f(x,y)\,\,\,\leqslant\,\,\,\epsilon_{0}$ (I.1)

for different energy values $\,\epsilon_{0}\,$. The function $\,f(x,y)\,$ will
play the role of a potential in which the two-dimensional dynamics of a
particle is considered. It is easy to see that if the dynamics is purely
classical, then such regions play the role of ‘‘accessibility regions’’ in
which a particle with energy $\,\epsilon_{0}\,$ may appear. As we will see
below, the complexity of the motion of a particle, inherent in the general
case of any Hamiltonian dynamics, can add up to the complexity of the geometry
of the corresponding ‘‘regions of accessibility’’, which may have certain
‘‘chaotic properties’’.

In addition to the geometry of the ‘‘accessibility regions’’ for a particle
with a given energy, we will also be interested in describing the changes in
this geometry when changing the parameters that define the function
$\,f(x,y)\,$ in real systems. In particular, it may be of interest to control
the dynamics of a gas of particles in quasiperiodic potentials by changing
such parameters.

It is easy to see that the geometry of the domains (I.1) is in fact closely
related to the geometry of the level lines

$f(x,y)\,\,\,=\,\,\,\epsilon_{0}$ (I.2)

of the function $\,f(x,y)\,$, bounding these domains for the corresponding
values of $\,\epsilon_{0}\,$. We can also say that each of the regions (I.1)
is the union of the level lines (I.2) for all
$\,\epsilon_{0}^{\prime}\,\leqslant\,\epsilon_{0}\,$.

The general problem of describing the geometry of the level lines of
quasiperiodic functions on the plane was set by S.P. Novikov in the early
1980s (see MultValAnMorseTheory ) and then was actively studied in his
topological school (zorich1 ; dynn1992 ; Tsarev ; dynn1 ; zorich2 ; DynnBuDA ;
dynn2 ; dynn3 ). In its original setting, it was focused on the study of the
geometry of the level lines of functions with three quasiperiods, which is
actually equivalent to describing the geometry of the intersection lines of an
arbitrary 3-periodic two-dimensional surface in $\,\mathbb{R}^{3}\,$ by planes
of a given direction. In this setting, Novikov’s problem is most directly
related to the problem of describing the dynamics of electrons in metals with
complex Fermi surfaces, which, in turn, plays an important role in describing
galvanomagnetic phenomena in metals (see for example lifkag1 ; lifkag2 ;
lifkag3 ; Kittel ; etm ; Abrikosov ; KaganovPeschansky ). We note here that
the results obtained in the study of Novikov’s problem in this case turn out
to be very important for the description of galvanomagnetic phenomena in
metals in the most general situation. In particular, it is possible to define
nontrivial topological numbers observable in the conductivity of normal metals
with complex Fermi surfaces in strong enough magnetic fields (PismaZhETF ). In
addition, a complete classification of possible types of dynamics
(trajectories) of electrons on the Fermi surface allowed to give also a
complete description of various asymptotic behaviors of conductivity in metals
with arbitrary Fermi surfaces in the limit of strong magnetic fields (see, for
example UFN ; BullBrazMathSoc ; JournStatPhys ).

In general, Novikov’s problem for an arbitrary number of quasiperiods is
closely related to the theory of foliations and the theory of dynamical
systems on manifolds. In particular, this connection plays an important role
in the study of Novikov’s problem in the case of four quasi-periods (see
NovKvazFunc ; DynNov ), which allows to obtain a number of fundamental results
for this case.

In this paper, we try to use, in the most complete way, the results obtained
so far in the study of Novikov’s problem to describe the main features of the
particle dynamics in the potentials associated with this problem. In addition,
the analysis of the results obtained for Novikov’s problem allows, in fact, to
propose a criterion dividing all quasiperiodic potentials (with three or more
quasiperiods) into two types, namely, potentials with topologically regular
level lines and potentials with chaotic level lines. If we consider
quasiperiodic potentials as a model of the transition between periodic and
random potentials, then the potentials of the first type can be classified as
potentials in which the dynamics retains some topological integrability, and
potentials of the second type as potentials in which the dynamics is close to
the dynamics in random potentials. In this sense, only potentials of the
second type can represent models of random potentials.

In fact, quasiperiodic potentials arise quite often as members of a family
(and this will be so in this work), i.e. we usually have a whole family of
functions $\,V(x,y,{\bf U})\,$, depending smoothly on a number of additional
parameters $\,{\bf U}=(U^{1},\dots,U^{N})\,$. In this situation, it is
important that the subsets of the parameter space corresponding to potentials
of different types have, as a rule, very specific structure. Namely,
potentials of the first type, being stable with respect to small variations of
parameters, form an open subset in the complete space of parameters. Moreover,
this subset is a union of a (finite or countable) number of pairwise disjoint
regions (stability zones) each of which is determined by its individual value
of some topological invariant (a set of integers). On the contrary, potentials
of the second type form a subset of fractal type complementary to the set
described above.

It can be seen, therefore, that in order to create quasiperiodic potentials
that are most alike to truly random potentials, it is necessary to find a
rather complicated set in the parameter space defining the family of
quasiperiodic potentials under consideration. It can also be noted here that
the larger the number of quasiperiods the richer becomes the structure of this
set.

We believe that the most convenient systems for the experimental study of
dynamics in quasiperiodic potentials are systems of ultracold atoms in optical
traps, where such potentials can be easily constructed by superposition of
several standing waves. It must be said that, despite the special technique
for creating such potentials, they, in fact, have all the features of
Novikov’s problem in the general setting, so such systems allow us to study
all essential aspects of the problem under consideration.

As we have already said, here we study the dynamics of particles of different
energies in the potentials described above. Thus, the presented results will
be directly related to the description of transport phenomena in the limit of
almost noninteracting atoms in traps with quasiperiodic potentials. This
limit, in fact, arises quite often in the case of low concentration of
(neutral) atoms trapped in a trap, as well as small radius of their
interaction. In our case, we should naturally require that the mean free path
of the atoms is greater than the typical length at which the global geometric
features of their trajectories manifest themselves. What is also essential, as
is known, two-dimensional Hamiltonian systems have, as a rule, very special
dynamics (see Ott ; LikhtLib ), being integrable at sufficiently low energy
levels and passing to chaotic regimes with increasing energy. In our case, we
will be able to observe how this circumstance agrees with the geometry of the
level lines of our potentials.

##  II General analytical results. Regular and random potentials.

As is well known, the most common method for creating external potentials for
atoms in optical traps is a superposition of standing waves from additional
laser sources (see, for example Letokhov1968 ; LetokhovBook ; Bloch ;
BlochDalibardZwerger ; GreinerFolling ). In the leading approximation, such
potentials usually have a finite number of harmonics, i.e., they can be
represented as the sum of a finite number of sinusoidal potentials. Despite
this circumstance, as we will see below, in the situations we are considering,
such potentials already have sufficient complexity, and to describe the atomic
dynamics in them, it is necessary to describe the complete picture that arises
in the study of the general Novikov’s problem.

It is easy to see that the problem under consideration is rapidly becoming
more complicated with an increase in the number of quasiperiods. As we have
already said, rigorous analytical results exist at the moment only for the
case of three and four quasiperiods. For comparison, it is convenient here to
briefly consider also the situations of ‘‘one’’ and ‘‘two’’ quasiperiods
corresponding to periodic potentials depending on a single coordinate, and
doubly periodic potentials on the plane, respectively.

In our situation, the case of ‘‘one’’ quasiperiod will correspond in reality
to the presence of a very simple potential, which can often be written
approximately in the form

$V(x,y)\,\,\,=\,\,\,V_{0}\,\,\sin kx\,\,,$ (II.1)

where $\,T\,=\,2\pi/k\,$ is the ordinary period of the potential along the $x$
axis. The creation of potentials of this type is the simplest one from the
technical point of view, and, certainly, such potentials are widely used in
systems of cold atoms.

The level lines of potential (II.1), obviously, represent vertical straight
lines, and the motion of atoms in such a potential occurs in straight vertical
stripes at $\,E<V_{0}\,$. In fact, it is easy to see in this situation, that
the motion of an atom is confined to a vertical strip provided

$E\,-\,p_{y}^{2}/2m\,\,\,<\,\,\,V_{0},$

and in this case, the atom performs periodic oscillations along the $x$ axis
and moves uniformly along the $y$ axis. If the above condition is violated,
the atom obviously moves along a periodic trajectory having nonzero mean
inclination with respect to the $y$ axis. It is easy to see that similar
conditions can also be written for a uniformly moving coordinate system, which
also allows to give a similar description of the motion of atoms in potentials
of the form

$V(x,y,t)=V_{0}\,\sin\bigl{(}k(x-ut)\bigr{)}.$

It can be seen here that at sufficiently low atomic energies and the value of
the velocity $u$, the moving potential carries out a complete
‘‘transportation’’ of the atomic gas along the $x$ axis. In the general case,
with a significant spread of energies, the moving potential allows only a
partial ‘‘transportation’’ of the atomic gas.

The geometry of the level lines of a doubly periodic potential on the plane
(with two independent periods $\,{\bf e}_{1}\,$ and $\,{\bf e}_{2}$) also has
a relatively simple description. As in the case of ‘‘one’’ quasiperiod, the
values of the potential $\,V(x,y)\,$ lie here in some closed segment
$\,[V_{\min},V_{\max}]\,$. It is easy to see that, for generic potentials, the
level lines of the potential are closed for values of $\,E\,$ sufficiently
close to $\,V_{\min}\,$ or $\,V_{\max}\,$. In the first case, however, closed
level lines bound areas of lower potential values, while in the second case
they bound areas of larger values. In the generic case, we have here two
different values $\,V_{1},V_{2}\,$:

$V_{\min}\,\,<\,\,V_{1}\,\,<\,\,V_{2}\,\,<\,\,V_{\max},$

such that for all fixed values of $\,V(x,y)$ lying in the interval
$\,(V_{1},V_{2})\,$ the corresponding levels contain open (non-closed)
components. All open level components (lines) of the potential $\,V(x,y)\,$
are, in this case, periodic curves having the same mean direction in the plane
(for all levels in the interval $\,(V_{1},V_{2})$). The average direction of
the open level lines can be any integer direction, i.e. a direction given by a
vector of the form

${\bf l}\,\,\,=\,\,\,n_{1}\,{\bf e}_{1}\,\,+\,\,n_{2}\,{\bf e}_{2}$

with some integers $\,n_{1},n_{2}\,$. Note that it is natural to demand the
numbers $\,n_{1},n_{2}\,$ to be relatively prime, given up to a common sign.

All open level lines for a given value of the potential can be divided into a
finite number of families, such that all lines in one family pass into each
other when shifted by some period of $\,V(x,y)\,$. The number of such
different families is always even (although it can vary within the interval
$\,(V_{1},V_{2})$ for sufficiently complex potentials).

Thus, it can be seen that the motion of atoms in generic periodic potentials
occurs in bounded regions at $\,V_{\min}\,<\,\epsilon_{0}\,<\,V_{1}\,$, in
periodic stripes (as well as, possibly, isolated bounded regions) at
$\,V_{1}\,<\,\epsilon_{0}\,<\,V_{2}\,$, in the plane with excluded bounded
domains (and possibly isolated bounded areas) at
$\,V_{2}\,<\,\epsilon_{0}\,<\,V_{\max}\,$ and in the whole plane at
$\,\epsilon_{0}\,>\,V_{\max}\,$. In the case of adiabatic shifts of the
potential in the plane, the atoms in the bounded regions shift with them,
while the atoms in the periodic stripes move along with the stripes only when
the shift is in the direction perpendicular to the stripes. Here, in contrast
to the case of ‘‘one’’ quasiperiod, however, the shift of the potential along
the direction of the stripes also affects the motion of atoms, since the
stripes now have a nontrivial shape.

For non-generic periodic potentials (for example, having elements of
rotational symmetry), the values $\,V_{1}\,$ and $\,V_{2}\,$ may coincide.
Such potentials do not have open level lines, but at the level
$\,V_{1}=V_{2}\,$ they have a singular net (Fig. 1), separating the areas of
smaller values of $\,V(x,y)\,$ from the areas of its larger values. Atoms move
in bounded regions at $\,V_{\min}\,<\,\epsilon_{0}\,<\,V_{1}=V_{2}\,$ and in
the entire plane with excluded bounded regions (and, possibly, isolated
bounded domains) at $\,V_{1}=V_{2}\,<\,\epsilon_{0}\,<\,V_{\max}\,$.

Figure 1: Periodic net of singular level lines separating areas of the lower
values from areas of the larger values of a non-generic periodic potential.

Methods for creating periodic potentials in a plane naturally imply the
presence of a certain finite number of parameters describing such potentials.
For example, when making a potential by using a superposition of two
sinusoidal standing waves (with or without generation of higher harmonics),
such parameters can be the orientations of both sinusoids, their amplitudes,
periods, positions of the maxima, possibly the angle between their
polarizations and the relative phase shift. It is easy to see that in this
case the shift of the maxima of any of the sinusoids actually leads to a shift
of the resultant potential as a whole and does not qualitatively change the
picture we are considering.

As for more general (continuous) variations of the parameters described above,
one important common feature can be noted here. Namely, generic potentials
form an open set, i.e. the relation $\,V_{2}>V_{1}\,$ is stable with respect
to an arbitrary small change in parameters. On the contrary, the condition
$\,V_{1}=V_{2}\,$ is unstable and can break down under an arbitrarily small
variation of the parameters of the general form. In addition, the numbers
$\,(n_{1},n_{2})\,$, relating the average directions of the open level lines
to the periods of the potential, are also locally stable (although $\,{\bf
e}_{1}\,$ and $\,{\bf e}_{2}\,$ can change with a change in parameters), and
can change only when the potential passes through a non-generic situation
($V_{1}=V_{2}$). Thus, it can be seen that the complete space of parameters
can generally be divided into regions corresponding to different integer pairs
$\,(n_{1},n_{2})\,$ and separated by boundaries on which the relation
$\,V_{1}=V_{2}\,$ holds.

Here we would also like to note that despite the relatively simple description
of the ‘‘accessibility regions’’ in periodic potentials at any energy values,
the conservative dynamics of cold atoms in such potentials can have very
nontrivial properties (see, for example HemmSchroppHansch ; HennequinVerkerk1
; HennequinVerkerk2 ).

We now turn to considering potentials with larger numbers of quasiperiods,
which are the main subject of our work.

The creation of quasiperiodic potentials in systems of ultracold atoms using
superposition of standing waves has also attracted interest from both the
theoretical and the experimental points of view. In particular, such
potentials were considered both in the case of three-dimensional (see, for
example, Guidoni1 ) and in the case of two-dimensional (see Guidoni2 ;
SanchezPalenciaSantos ; ViebSbrosCartYuSchneid ; GautierYaoSanchezPalencia )
optical lattices for atoms trapped in magneto-optical traps (see also a review
on the creation, confinement and monitoring of the behavior of gases of
ultracold atoms, including that in potentials of various shapes, in the work
Guidoni3 ). It can also be noted that quasiperiodic (quasicrystalline)
structures in two-dimensional systems of interacting ultracold atoms can arise
even in the absence of special external modulation (see, for example,
GopMartinDemler ).

As we have said above, here we are interested in quasiperiodic potentials for
two-dimensional systems of ultracold atoms. As we have also already noted, we
will especially focus here on the cases of 3 and 4 quasi-periods, for which
profound analytical results are known to date. In this section, we will simply
formulate the rigorous analytical results obtained so far for such potentials.
In the next section, we will consider in more detail the features of the
geometry arising here, with specific examples.

As we have already said, Novikov’s problem for the case of three quasiperiods,
which we consider here in the most detail, is currently most thoroughly
studied. In the considered method of creating potentials in the plane, this
situation corresponds to the potentials obtained by superimposing three
sinusoidal standing waves oriented at different angles (with or without the
presence of higher harmonics) (Fig. 2).

Figure 2: Superposition of three standing waves in a plane with the formation
of a potential with three quasi-periods (schematically). (The vectors
$\,\bm{\eta}_{i}\,$ indicate the directions of the wave fronts, and the
vectors $\,{\bf a}_{i}\,$ are the shifts between the maxima of their
amplitudes).

It should be noted here, of course, that a superposition of three (or more)
standing waves in lattices of cold atoms can be used to create not only
quasiperiodic, but also interesting periodic potentials in two-dimensional
systems (in particular, such a scheme was proposed in the work DuanDemlerLukin
to create hexagonal (honeycomb) lattices and in KagomeLattice  to create tri-
hexagonal (Kagome) lattices). In this case, the wave numbers of the
corresponding waves must, generally speaking, satisfy a number of special
additional conditions. In our situation, we will assume that the wave numbers
are independent parameters in the space of the considered potentials. In the
general case, the potentials we are considering will not have exact periods in
the plane, which can arise only for special values of the parameters.

As we have already said, the results presented below will be based only on
quasiperiodic properties of the potentials arising in our case, therefore,
many additional details of their origin will not, in fact, play any essential
role.

It is easy to see that a potential generated by three sinusoidal waves (and
higher harmonics)

$V({\bf r})\,\,\,=\,\,\,\sum_{i=1}^{3}\,V_{i}\cos({\bf k}_{(i)}{\bf
r}+\delta_{i})\,\,\,+\,\,\,\dots\,\,,$

represents a restriction of the periodic in $\,{\mathbb{R}^{3}}\,$ function

$V(X^{1},X^{2},X^{3})\,\,\,=\,\,\,\sum_{i=1}^{3}\,V_{i}\cos
X^{i}\,\,\,+\,\,\,\dots\,\,$

to an affine embedding $\,\mathbb{R}^{2}\rightarrow\mathbb{R}^{3}$, given by
the formulas

$\left(\begin{array}[]{c}x\cr
y\end{array}\right)\quad\rightarrow\quad\left(\begin{array}[]{c}k_{(1)}^{1}x+k_{(1)}^{2}y+\delta_{1}\cr
k_{(2)}^{1}x+k_{(2)}^{2}y+\delta_{2}\cr
k_{(3)}^{1}x+k_{(3)}^{2}y+\delta_{3}\end{array}\right)\,=\,\left(\begin{array}[]{c}{\bf
k}_{(1)}{\bf r}+\delta_{1}\cr{\bf k}_{(2)}{\bf r}+\delta_{2}\cr{\bf
k}_{(3)}{\bf r}+\delta_{3}\end{array}\right)$ (II.2)

It can be noted here that the potentials differing only in shifts of the
maxima of the standing waves ($\delta_{i}$) correspond to the same function
$\,V(X^{1},X^{2},X^{3})\,$ in $\,{\mathbb{R}^{3}}$, and their difference is
due only to a change in the affine embedding
$\,\mathbb{R}^{2}\rightarrow\mathbb{R}^{3}$, under which the plane
$\,\mathbb{R}^{2}\,$ is shifted in $\,{\mathbb{R}^{3}}\,$ keeping its
direction fixed.

As we have already said, even quasiperiodic potentials having a small number
of harmonics are already sufficiently complex to observe all aspects of the
general Novikov’s problem for three quasiperiods; for this reason, we will
consider the problem described above on the grounds of the general results for
functions with three quasiperiods. Below we will formulate, in the most
convenient form for us, a number of fundamental results for the general
Novikov’s problem following from the works zorich1 ; dynn1992 ; Tsarev ; dynn1
; zorich2 ; DynnBuDA ; dynn2 ; dynn3 . Similar statements for artificially
created potentials in the plane, in fact, were given in the work JMathPhys ,
where electron transport phenomena in such potentials in the presence of a
strong magnetic field were studied. Note, however, that in the work JMathPhys
, the main role was played by the geometry of the level lines of quasiperiodic
potentials, rather than by that of the regions of lower values, which we
consider here.

It is convenient for us to start with the remark that, although this does not
happen in the general case, the potentials obtained by the method considered
here can also be periodic. This situation arises whenever the corresponding
plane $\,\mathbb{R}^{2}\subset\mathbb{R}^{3}\,$ is integral (rational), that
is, contains two independent integer vectors in $\,{\mathbb{R}^{3}}\,$. It is
easy to see that the corresponding potentials are everywhere dense among all
potentials of our interest, while their periodic properties are determined by
the values of the parameters $\,({\bf k}_{1},{\bf k}_{2},{\bf k}_{3})\,$. To
describe the level lines of such potentials, all previously made statements
about periodic potentials can be used. However, it should be noted here right
away that the overwhelming majority of these potentials will have very large
(in absolute value) periods $\,{\bf e}_{1}\,$ and $\,{\bf e}_{2}\,$. As a
consequence, the above-described properties of the level lines of such
potentials will be observed only on very large scales, while on smaller scales
their level lines may have completely different nontrivial properties that are
more important for describing the observed experimental data. As a result, for
potentials with large periods, as a rule, more informative will be more
general statements about the level lines of quasiperiodic potentials, which we
cite below.

At the same time, it turns out that the role of the everywhere dense set of
periodic potentials in the parameter space is, in fact, extremely important in
the study of Novikov’s problem for the quasiperiodic potentials we are
considering. Below we will formulate an extremely important statement
concerning small deformations of periodic potentials and following from the
results of works zorich1 ; dynn1992 . Note here that in works zorich1 ;
dynn1992  it is actually assumed that certain regularity conditions (of
general position) are fulfilled, which we do not formulate here in detail,
assuming that they are always satisfied for arising in reality physical
potentials (Conditions ). In this case, in the simplest form, the corollaries
we need from zorich1 ; dynn1992  can be formulated as follows:

Let some complete set of parameters

${\bf U}_{0}\,=\,({\bf k}_{(1)}^{0},{\bf k}_{(2)}^{0},{\bf
k}_{(3)}^{0},V_{1}^{0},V_{2}^{0},V_{3}^{0},\dots)$

correspond to some periodic potential in the plane. Then there is an open
neighborhood $\,\Omega\,$ of the point $\,{\bf U}_{0}\,$ (‘‘stability zone’’)
in the parameter space, such that for any $\,{\bf U}\in\,\Omega\,$ the
following holds:

A1. All open level lines of the corresponding two-dimensional potential
$\,V(x,y,{\bf U})\,$ lie in straight strips of finite width, passing through
them (Fig. 3);

A2. The mean direction $\,{\bf l}({\bf U})\,$ of strips, containing open level
lines of the potential $\,V(x,y,{\bf U})\,$, is defined in the entire region
$\,\Omega\,$, for some (irreducible) integer triple $\,(m^{1},m^{2},m^{3})\,$,
by the relation

$\left(m^{1}{\bf k}_{(1)}\,+\,m^{2}{\bf k}_{(2)}\,+\,m^{3}{\bf
k}_{(3)}\,,\,{\bf l}({\bf U})\right)\,\,\,=\,\,\,0$ (II.3)

Note here that, for embeddings (II.2) of maximal irrationality degree,
relation (II.3) uniquely determines (up to the sign) the triple
$\,(m^{1},m^{2},m^{3})\,$.

Figure 3: An open level line of a quasiperiodic potential lying in a straight
strip of finite width and passing through it (schematically).

The triples $\,(m^{1},m^{2},m^{3})\,$ have, actually, a topological origin and
can also be introduced in another way. Namely, we recall that among the
parameters of our potentials, there are the positions of the maxima of the
standing waves used to create the potential. In the picture we are now
considering, the shift of an individual standing wave is no longer equivalent
to a simple shift of the resulting potential and represents a somewhat more
complex transformation. It is also easy to see that the shift of the front of
a standing wave (perpendicular to itself) by the period of this wave is
equivalent to the identical transformation. In general, the full set of all
such transformations forms a three-parameter group ($\mathbb{T}^{3}$)
containing simple shifts as an algebraic subgroup. At the same time, simple
shifts form an everywhere dense set in the considered group of transformations
for generic potentials; therefore, all such potentials associated with the
described transformations are, in a sense, mutually related. In particular,
such potentials have similar level lines for any energy value
$\,\epsilon_{0}\,$.

If we now consider a generic potential $\,V(x,y,{\bf U})\,$ corresponding to
some values $\,{\bf U}\in\Omega\,$, and fix a level $\,\epsilon_{0}\,$,
corresponding to the appearance of open level lines, then we can follow the
change of any of these lines for each of the three continuous shifts of the
standing wave fronts (in the direction of the phase growth) for the
corresponding periods. In this case, each of the continuous shifts will
correspond to the movement of an open level line in the plane (possibly with
merging with it and separation from it of closed level lines in the process of
movement), as a result of which it will finally transform into some other
level line of the same potential (Fig. 4). It can be shown in this case that
the number of positions $\,N\,$, by which the line is shifted, is the same for
all open lines on this level, and is determined only by the selected
transformation.

Figure 4: The shift of regular open level lines of a quasiperiodic potential
under a shift of the maxima of one of the standing waves by a full period in
the direction of the phase growth.

The triplet of numbers $\,(N^{1},N^{2},N^{3})\,$, determined at three
successive shifts of the first, second and third waves, respectively, can be
represented as

$(N^{1},N^{2},N^{3})\,\,\,=\,\,\,M\,(m^{1},m^{2},m^{3})\,\,\,,$

where $\,M\in\mathbb{Z}\,$ and $\,(m^{1},m^{2},m^{3})\,$ is an irreducible
integer triple. The number $\,M\,$ is always even and also has a topological
origin (the number of ‘‘equivalence classes’’ of open level lines in the
plane), and the triple $\,(m^{1},m^{2},m^{3})\,$ coincides with the one
introduced earlier up to a sign.

Thus, we can see that in the space of our parameters we can distinguish a set
of ‘‘stability zones’’ $\,\Omega_{{\bf m},M}$, parameterized by integer
triples $\,{\bf m}\,=\,(m^{1},m^{2},m^{3})\,$ (and also by numbers $\,M$). The
numbers $\,(m^{1},m^{2},m^{3})\,$, generally speaking, do not run through the
entire set of integer (relatively prime) triples, but their number is
generally infinite, and they can take arbitrarily large values. Each
‘‘stability zone’’ represents some open region in the parameter space with a
piecewise smooth boundary (the boundaries of different zones can be adjacent
to each other). The set of ‘‘stability zones’’ is a rather rich structure in
the parameter space, in particular, the ‘‘stability zones’’ contain all the
values of parameters corresponding to the emergence of periodic potentials
$\,V(x,y,{\bf U})\,$ (see Conditions ).

It can be seen that the above description of open level lines of potentials
arising in the zones $\,\Omega_{{\bf m},M}$ is rather simple and very
informative (especially in the case of small values of
$\,(m^{1},m^{2},m^{3})$). In particular, it gives much more information about
the behavior of level lines of arising in $\,\Omega_{{\bf m},M}\,$ periodic
potentials with large periods than can be obtained from the fact of their
periodicity. As the values of $\,(m^{1},m^{2},m^{3})\,$ grow, the sizes of the
stability zones $\,\Omega_{{\bf m},M}\,$ decrease, and this also makes the
strips containing open level lines (and the corresponding areas of lower
values) wider. Along with this, the above description also refers to larger
and larger scales in the plane, giving less detailed information about the
geometry of the level lines at smaller scales (Fig. 5). As we will see below,
in this situation, the behavior of the level lines (on small scales) can no
longer be described in such a simple way and has much more complex (chaotic)
properties. As we will also see, in limiting cases such behavior can lead to
completely chaotic behavior of the level lines of quasiperiodic potentials,
which has complex chaotic properties at all scales.

Figure 5: Complication of the geometry of ‘‘topologically regular’’ open level
lines of a quasiperiodic potential with increasing values of
$\,(m^{1},m^{2},m^{3})\,$ (schematically).

Based on the properties of the level lines of quasiperiodic potentials in
stability zones, it can be seen here that the corresponding regions
$\,V(x,y,{\bf U})\,<\,\epsilon_{0}\,$ should also possess the same properties
if the level lines $\,V(x,y,{\bf U})\,=\,\epsilon_{0}\,$ are open (Fig. 6).

Figure 6: The region of smaller values of a quasiperiodic potential, lying in
a straight strip of finite width and passing through it (schematically).

Namely, if the level $\,\epsilon_{0}\,$ contains open level lines, then any
open connected domain $\,V(x,y,{\bf U})\,<\,\epsilon_{0}\,$ also lies in a
straight strip of finite width and passes through it. The mean direction of
such a strip, obviously, coincides with the mean direction of the open level
lines and is given by equation (II.3). We note here that now both the level
lines and the regions of lower values of the potential are no longer periodic
for the general values of the parameters. As can be seen, however, properties
(A1) - (A2) give here a certain analogy with the case of periodic potentials.

It is easy to see that transport phenomena in a potential with parameters
lying in one of the stability zones can have an explicit anisotropy. This
property should, as a rule, be observed if the ensemble of particles placed in
such a potential contains particles with energies corresponding to the
emergence of open level lines of the potential. It can be seen, in fact, that
such anisotropy can also be observed under more general assumptions, in
particular, in systems of strongly interacting particles or in the
hydrodynamic approximation.

Here, however, we must immediately make an important remark about transport
phenomena in the case we are considering. Namely, in stability zones with
large values of the numbers $\,(m^{1},m^{2},m^{3})\,$ the width of the strips
containing the domains $\,V(x,y,{\bf U})\,<\,\epsilon_{0}\,$ (in the presence
of open level lines at $\,V(x,y,{\bf U})\,=\,\epsilon_{0}$) becomes rather
large, and the shape of such regions becomes more and more complex, which is
(very schematically) shown at Fig. 6. As a consequence of this, the motion of
particles in such regions becomes more and more complicated, gradually
acquiring the features of wandering in a random potential.

We also note here that the above-described movement of the open level lines
(and the ‘‘regions of accessibility’’ restricted by them) under the shifts of
the maxima of each of the standing waves depends in the most significant way
on the values $\,(m^{1},m^{2},m^{3})\,$. In particular, the speed of movement
of the ‘‘accessibility regions’’ at large values of $\,(m^{1},m^{2},m^{3})\,$
can significantly exceed the speed of the maxima of standing waves. The latter
circumstance can, in fact, play an essential role for many questions of the
transport of atoms in optical lattices (see, for example, LPGARMNMCA  and the
references therein). Moreover, such a movement, generally speaking, produces
an incomplete transportation of atoms in the corresponding direction due to
separation of some closed areas from the ‘‘accessibility areas’’ and joining
of others to them in the process of their movement.

We also point out here one more important circumstance. Namely, as we have
already said, the presence of a stability zone in the space of parameters
means, naturally, the preservation of the picture determined by it under
sufficiently small variations of the parameters. In fact, as follows from the
topological considerations, this picture is also stable under much more
general variations of the potential, in particular, those arising on finite
scales provided that they are small enough. Thus, it can be seen that the
above description of the geometry of the regions $\,V(x,y,{\bf
U})\,<\,\epsilon_{0}\,$, as well as the features of transport phenomena, is
stable under disturbances or ‘‘defects’’ of a sufficiently small magnitude.
This circumstance is also important in the presence of additional (non-
quasiperiodic) slowly varying potentials, which are also often present in
experimental techniques. The above description (with the same numbers
$\,(m^{1},m^{2},m^{3})$) is preserved in general in regions where the maximum
change of such potentials does not exceed the energy interval of the existence
of open level lines of the potential $\,V(x,y,{\bf U})\,$. It should be said
here that the estimate of the admissible variations of the potential
$\,V(x,y,{\bf U})\,$ also rapidly decreases with increase in numbers
$\,(m^{1},m^{2},m^{3})\,$.

Despite the fact that the set of all stability zones $\,\Omega_{{\bf m},M}\,$
forms an open cover of an everywhere dense set in the parameter space, it,
generally speaking, does not cover it entirely, and quasiperiodic potentials
with three quasi-periods can have, as we said, level lines that are more
complex than those described above (Tsarev ; dynn2 ). It can be said,
nevertheless, that the ‘‘regular’’ situation described above is, in a sense,
basic for the case of three quasi-periods, while the more complex behavior of
the level lines requires a special construction of the corresponding
potential. For a more complete description of the picture arising in the
general case, we present here, based on the results of works dynn1 ; dynn3 , a
number of important statements about the structure of the level lines of
potentials with three quasiperiods of the most general form. In fact, together
with the above statements, the results presented below represent, in a certain
sense, a complete theory of the level lines of potentials with three
quasiperiods in the plane.

We note here at once that all the potentials obtained from 3-periodic
functions by means of an embedding $\,\mathbb{R}^{2}\subset\mathbb{R}^{3}\,$
can in fact be divided into 3 types. Namely, first of all, as we have already
seen, for certain values of $\,({\bf k}_{(1)},{\bf k}_{(2)},{\bf k}_{(3)})\,$
the corresponding potential $\,V(x,y,{\bf U})\,$ may actually turn out to be
doubly periodic. Here we call such potentials type I potentials. The second
possibility is that the potential $\,V(x,y,{\bf U})\,$, while not being doubly
periodic, still has one (up to a factor) period in the plane
$\,\mathbb{R}^{2}\,$. Such potentials will be called type II potentials. Like
potentials of type I, they arise on an everywhere dense set of zero measure in
the space of parameters $\,({\bf k}_{(1)},{\bf k}_{(2)},{\bf k}_{(3)})\,$.
Finally, into the family of potentials of type III we include potentials that
do not have exact periods in the plane $\,\mathbb{R}^{2}\,$. In our case, only
potentials of type III are in general position and correspond to a set of the
full measure in the parameter space. In the statements about the level lines
of quasiperiodic potentials formulated below, we will assume that the
corresponding potentials are of type II or III, since potentials of type I
have already been actually considered above. Under the assumptions made, the
following statements can be formulated.

Let $\,V(x,y,{\bf U})\,$ be a potential with three quasi-periods taking values
in the interval $\,[V_{\min}({\bf U}),\,V_{\max}({\bf U})]\,$. Then:

B1. Open level lines of $\,V(x,y,{\bf U})\,$ exist in a connected interval
$\,[V_{1}({\bf U}),\,V_{2}({\bf U})]\,$,

$V_{\min}({\bf U})\,\,\,<\,\,\,V_{1}({\bf U})\,\,\,\leqslant\,\,\,V_{2}({\bf
U})\,\,\,<\,\,\,V_{\max}({\bf U})\,\,\,,$

which can degenerate to a single point $\,V_{0}({\bf U})=V_{1}({\bf
U})=V_{2}({\bf U})\,$.

B2. Whenever open level lines appear in a finite interval $\,[V_{1}({\bf
U}),\,V_{2}({\bf U})]\,$, they have the properties (A1)-(A2), presented above.

B3. In the case when the interval $\,[V_{1}({\bf U}),\,V_{2}({\bf U})]\,$
shrinks to a single point $\,V_{0}({\bf U})=V_{1}({\bf U})=V_{2}({\bf U})\,$,
the open level lines arising at the corresponding level can either satisfy
conditions (A1)-(A2) (this occurs at the boundaries of the stability zones
$\,\Omega_{{\bf m},M}$), or have more complex chaotic behavior (this occurs at
the accumulation points of an infinite number of zones $\,\Omega_{{\bf
m},M}\,$ with unboundedly increasing values of $\,(m^{1},m^{2},m^{3})$).

It can be seen, therefore, that the appearance of open level lines in
‘‘stability zones’’ does not at all resemble the similar phenomenon for truly
random potentials, where, as a rule, open level lines arise at a single energy
level (if we consider random potentials from the point of view of the
percolation theory, see e.g. Stauffer ; Essam ). Also, the potentials arising
at the boundaries of the ‘‘stability zones’’, although they have open level
lines only for one value of $\,\epsilon_{0}\,$, are not very suitable for the
role of random potentials due to the ‘‘too regular behavior’’ of their open
level lines. Therefore, it can be seen that it is natural to consider only
potentials with chaotic level lines as models of a random potential. As we
said above, in the case of three quasi-periods, such potentials always arise
at accumulation points of ‘‘stability zones’’ with an increasingly complex
geometry of open level lines, so that there is always a passage from the
‘‘regular’’ behaviour to the ‘‘chaotic’’ one in this case.

It should also be noted here that chaotic level lines arising in the case of
type II potentials are very different from those of type III potentials.
Namely, chaotic level lines of potentials of type II always have the form of
curves with an asymptotic direction in the plane $\,\mathbb{R}^{2}\,$ (see
dynn2 ). Thus, such level lines resemble to some extent ‘‘regular’’ level
lines described above, and, in general, pass through the plane along some
fixed direction. The difference between the two cases lies in the fact that
the deviations of a chaotic level line of a type II potential in the direction
perpendicular to the asymptotic one are not necessarily bounded, so the line
may not be enclosed in any straight strip of finite width. In this case, the
same can be said also about the corresponding regions of lower values of the
corresponding potential

$V(x,y,{\bf U})\,\,\leqslant\,\,V_{0}\,\,\,,$

namely, they have the form of ‘‘strips’’ passing through plane in some fixed
direction. The width of these strips, however, can vary indefinitely in their
different parts, and they cannot be enclosed in straight strips of a fixed
width. When the boundary value $\,\epsilon_{0}\,$ is shifted downward by an
arbitrarily small amount, the region of lower values of the corresponding
potential consists of strongly elongated bounded regions, and when
$\,\epsilon_{0}\,$ is shifted upward, this region becomes the entire plane
with strongly elongated bounded regions removed (and, possibly, with bounded
regions not associated with the main component and lying inside the excluded
regions added).

One may wonder, therefore, to what extent type II potentials with chaotic
level lines can be considered as a model of a random potential. In a sense,
they can be considered as an intermediate case between ‘‘regular’’ and
‘‘chaotic’’ potentials.

Chaotic level lines arising in the case of type III potentials are much more
complex and ‘‘sweep’’ the entire plane $\,\mathbb{R}^{2}\,$ in a chaotic
manner (Fig. 7). A similar behavior is exhibited in this case also by the
regions of the smaller values $\,V(x,y,{\bf U})\,\,\leqslant\,\,V_{0}\,$. When
the boundary value $\,\epsilon_{0}\,$ is shifted downward by an arbitrarily
small amount, the region of lower values of the corresponding potential splits
into rather complicated bounded regions, and when $\,\epsilon_{0}\,$ is
shifted upward, this region becomes the entire plane from which bounded areas
of complex shape are excluded (and, possibly, additional bounded areas that
are not associated with the main component and lying inside the excluded areas
added). An important circumstance here is that the linear dimensions of such
regions grow according to a power law when approaching the value of
$\,\epsilon_{0}\,$ with a fractional exponent
($\sim|\epsilon-\epsilon_{0}|^{-\alpha}$), which makes such potentials alike
random potentials (see, for example, Riedel ; Trugman ). It should be noted,
however, that here, in the general case, a certain anisotropy can be observed,
namely, the presence of two different growth degrees $\,\alpha\,$ and
$\,\beta\,$, in a certain direction in the plane and in the one perpendicular
to it ($0<\alpha,\beta<1$). In general, the stochastic properties of such
level lines are quite complex and are currently the subject of intensive
research (see, for example dynn2 ; dynn3 ; ZhETF2 ; zorich3 ; DeLeo1 ; DeLeo2
; DeLeoPhysLettA ; DeLeoPhysB ; DeLeo3 ; DeLeoDynnikov1 ; DeLeoDynnikov2 ;
Skripchenko1 ; Skripchenko2 ; DynnSkrip1 ; DynnSkrip2 ; AvilaHubSkrip1 ;
AvilaHubSkrip2 ; TrMIAN ).

The randomness of the level lines of a potential formed by three standing
waves is invariant under shifts of the phases $\,\delta_{i}\,$ (i.e., the
positions of the standing wave maxima) with the remaining parameters fixed. In
this case, the value of $\,V_{0}\,$, as well as the geometric features of the
chaotic level lines (in particular, the degrees $\,\alpha\,$ and $\,\beta$),
are preserved in a natural way under such shifts. The change in the
‘‘accessibility areas’’ $\,V(x,y)\leqslant V_{0}\,$ during the shifts of the
maxima of the standing waves is accompanied here by their rather complex
movement, as well as numerous rearrangements at their boundaries. Generally
speaking, the transportation of atomic gas with an adiabatic change in the
positions of the maxima of standing waves in this situation must be computed
separately for each such potential.

In this work, we will try to represent the described level lines and the
corresponding areas of lower potential values in the most visual way. In
addition, as we said above, we will be interested in the dynamics of ultracold
atoms in the regions we have described. Especially interesting in this case,
in our opinion, is the superposition of the properties of chaotic dynamics
itself and the chaotic properties of the ‘‘accessibility regions’’ for this
dynamics.

Figure 7: A ‘‘chaotic’’ open level line of a potential with three quasi-
periods (schematically).

Here we would like to note one more important property of the ‘‘chaotic’’
level lines of potentials $\,V(x,y,{\bf U})\,$ with three quasi-periods,
namely, the presence of their segments where the level line (as well as the
corresponding region of smaller values $\,V(x,y,{\bf
U})\,\,\leqslant\,\,V_{0}$) passes ‘‘very close’’ to itself (Fig. 8). More
precisely, by considering larger and larger areas in the plane, we can find
segments of such a level line that are arbitrarily close to each other. As a
consequence, when considering the semiclassical dynamics of atoms with
energies close to the corresponding level $\,V_{0}\,$, it is always necessary
to consider also the effects of tunneling from one part of the ‘‘accessibility
region’’ to another near such places.

Figure 8: Tunneling effects between different parts of the classical
‘‘accessibility regions’’ as they approach each other, for potentials with
three quasi-periods.

Another consequence of the above circumstance is that, in contrast to the
situation in stability zones, here the global geometry of the domains
$\,V(x,y,{\bf U})\,\,\leqslant\,\,V_{0}\,$ is unstable with respect to
arbitrarily small local variations of the potential $\,V(x,y)\,$ and can vary
greatly (on large scales) in the presence of arbitrarily small perturbations
or defects. As a consequence, the transport properties of particles in such
potentials can also strongly depend on the presence of such defects in the
plane of the potential. In the presence of additional slowly varying
potentials, for example, of the type

$V(x,y)\,\,\,=\,\,\,a\,x^{2}\,\,\,,\quad a\,\rightarrow\,0\,\,\,,$

the global geometry of open level lines and the corresponding regions of lower
values of the resulting potential will most often resemble the corresponding
geometry for a smooth potential on large scales and behave in a typical
‘‘chaotic’’ manner on small scales.

According to the general conjecture of S.P. Novikov, in the case of three
quasi-periods, the emergence of potentials with chaotic level lines can occur
only at a set of measure zero and, moreover, of the fractal codimension
strictly greater than one, in the full space of parameters. It can be seen,
therefore, that experimental construction of a potential with three quasi-
periods which is close in properties to a random one in the sense described
above requires a very special choice of parameters specifying such a
potential. Fig. 9 gives an example of the location of stability zones in the
space of essential parameters under the restriction of the potential

$\cos\,X^{1}\,\,+\,\,\cos\,X^{2}\,\,+\,\,\cos\,X^{3}$

to two-dimensional planes for all possible affine embeddings
$\,\mathbb{R}^{2}\rightarrow\mathbb{R}^{3}\,$. In this case, only the
direction of the embedding is essential, which can be specified by a unit
vector in $\,\mathbb{R}^{3}\,$ orthogonal to the corresponding plane
$\,\mathbb{R}^{2}\,$. The endpoints of such vectors lie on the unit sphere
$\,\mathbb{S}^{2}\,$, and thus all stability zones can be viewed as regions on
the unit sphere. It can be seen that the union of (an infinite number of) such
regions defines a rather complex set on the sphere, and the complement to it
has the properties of a fractal. We also note here that Novikov’s conjecture
has not yet been rigorously proven, although it has been confirmed in a number
of serious numerical experiments.

Figure 9: Stability zones for quasiperiodic potentials in the plane obtained
by restricting the potential $\,\cos X^{1}\,+\,\cos X^{2}\,+\,\cos X^{3}\,$ to
all possible affine embeddings $\,\mathbb{R}^{2}\rightarrow\mathbb{R}^{3}\,$
(DeLeoPhysLettA ; DeLeoPhysB ). The zones are domains on the unit sphere
formed by the endpoints of the unit vectors orthogonal to the direction of the
embedding.

Let us now formulate the analytical results known to date for potentials with
four quasiperiods. It must be said at once that Novikov’s problem for the case
of four quasi-periods is more complicated in comparison with the case of three
quasi-periods. At the same time, the case of four quasi-periods may turn out
to be very important in the formulation we are considering in connection with
the problem of modulation of two-dimensional quasicrystals in systems of cold
atoms.

In the setting we are considering, potentials with four quasi-periods are
obtained as a result of a superposition of four independent sinusoidal
standing waves (possibly with the generation of higher harmonics, Fig. 10) and
can be written in the following general form

$V({\bf r})\,\,\,=\,\,\,\sum_{i=1}^{4}\,V_{i}\cos({\bf k}_{(i)}{\bf
r}+\delta_{i})\,\,\,+\,\,\,\dots$

Figure 10: Superposition of four standing waves in the plane with the
formation of a potential with four quasi-periods (schematically). (The vectors
$\,\bm{\eta}_{i}\,$ indicate the directions of the wave fronts, and the
vectors $\,{\bf a}_{i}\,$ are the shifts between the maxima of their
amplitudes).

The quasiperiodic properties of the potential are determined by the parameters
$\,({\bf k}_{(1)},{\bf k}_{(2)},{\bf k}_{(3)},{\bf k}_{(4)})\,$ specifying the
affine embedding $\,\mathbb{R}^{2}\rightarrow\mathbb{R}^{4}\,$, according to
the formulas

$\left(\begin{array}[]{c}x\cr
y\end{array}\right)\quad\rightarrow\quad\left(\begin{array}[]{c}k_{(1)}^{1}x+k_{(1)}^{2}y+\delta_{1}\cr
k_{(2)}^{1}x+k_{(2)}^{2}y+\delta_{2}\cr
k_{(3)}^{1}x+k_{(3)}^{2}y+\delta_{3}\cr
k_{(4)}^{1}x+k_{(4)}^{2}y+\delta_{4}\end{array}\right)\,=\,\left(\begin{array}[]{c}{\bf
k}_{(1)}{\bf r}+\delta_{1}\cr{\bf k}_{(2)}{\bf r}+\delta_{2}\cr{\bf
k}_{(3)}{\bf r}+\delta_{3}\cr{\bf k}_{(4)}{\bf
r}+\delta_{4}\end{array}\right).$

We present here some simplified consequences from the results of NovKvazFunc ;
DynNov , which are analogous to the first of the statements formulated above
for the case of three quasi-periods. As before, we will not consider here in
detail all the regularity conditions (of general position) imposed in the
works NovKvazFunc ; DynNov , and we will assume that they are always satisfied
for real potentials. Then, from the results of NovKvazFunc ; DynNov , the
following statement follows:

The complete space of parameters

${\bf U}_{0}\,=\,({\bf k}_{(1)}^{0},{\bf k}_{(2)}^{0},{\bf k}_{(3)}^{0},{\bf
k}_{(4)}^{0},V_{1}^{0},V_{2}^{0},V_{3}^{0},V_{4}^{0},\dots)\,\,\,,$

defining potentials with four quasi-periods contains an everywhere dense set
of open domains (‘‘stability zones’’ $\,\Omega$) such that in each of the
domains $\,\Omega\,$ the following holds:

C1. open level lines of $\,V(x,y,{\bf U})\,$ exist in a finite connected
energy interval $\,[V_{1}({\bf U}),V_{2}({\bf U})]$:

$V_{\min}({\bf U})\,\,\,<\,\,\,V_{1}({\bf U})\,\,\,\leqslant\,\,\,V_{2}({\bf
U})\,\,\,<\,\,\,V_{\max}({\bf U})\,\,\,;$

C2. all open level lines of $\,V(x,y,{\bf U})\,$ lie in straight strips of
finite width and pass through them (Fig. 3);

C3. the mean direction $\,{\bf l}({\bf U})\,$ of strips containing open level
lines of the potential $\,V(x,y,{\bf U})\,$ is defined in the entire domain
$\,\Omega\,$ by some (irreducible) integer quadruple
$\,(m^{1},m^{2},m^{3},m^{4})\,$ by the relation

$\left(m^{1}{\bf k}_{(1)}\,+\,m^{2}{\bf k}_{(2)}\,+\,m^{3}{\bf
k}_{(3)}\,+\,m^{4}{\bf k}_{(4)}\,,\,{\bf l}({\bf U})\right)\,\,\,=\,\,\,0.$

As in the case of three quasiperiods, the quadruples
$\,(m^{1},m^{2},m^{3},m^{4})\,$ are actually of topological origin and can be
defined similarly to the earlier definition of the triples
$(m^{1},m^{2},m^{3})$ given in terms of the shift transformations of the
parameters $\,\delta_{i}\,$.

Also, as in the case of three quasi-periods, the union of the stability zones
(in the general case) does not cover here the entire parameter space. The
complement to this union forms a complex set parametrizing potentials with
chaotic level lines. It must be said that both the features of the chaotic
behavior of the level lines and the structure of the corresponding set in the
space of parameters almost have not been studied to date. In particular, it
can be expected that the set of parameters corresponding to potentials with
chaotic level lines has a nonzero measure here. Thus, in families of
potentials with four quasi-periods, the construction of potentials with the
properties of truly random potentials might be simpler from the experimental
point of view. Note that, as in the case of three quasi-periods, each
‘‘chaotic’’ potential here is the limit of getting more and more complicated
‘‘regular’’ potentials due to the accumulation of an infinite number of
‘‘stability zones’’ near the point $\,{\bf U}_{0}\,$ defining this potential.

When passing to potentials with larger numbers of quasi-periods given by the
embeddings

$\left(\begin{array}[]{c}x\cr
y\end{array}\right)\,\,\,\rightarrow\,\,\,\left(\begin{array}[]{c}k_{(1)}^{1}x+k_{(1)}^{2}y+\delta_{1}\cr\vdots\cr
k_{(d)}^{1}x+k_{(d)}^{2}y+\delta_{d}\end{array}\right)\,=\,\left(\begin{array}[]{c}{\bf
k}_{(1)}{\bf r}+\delta_{1}\cr\vdots\cr{\bf k}_{(d)}{\bf
r}+\delta_{d}\end{array}\right)\,\,\,,$

it should be noted right away that there are currently no rigorous analytic
results for the case $\,d>4\,$. It can be seen, however, that in this case
too, quasiperiodic potentials can have a ‘‘regular’’ behavior of open level
lines. In general, we will say that a potential with $\,d\,$ quasiperiods has
regular open level lines if the following conditions are satisfied:

D1. Open level lines of $\,V(x,y)\,$ exist in a finite connected energy
interval

$V_{\min}\,\,\,<\,\,\,V_{1}\,\,\,\leqslant\,\,\,V(x,y)\,\,\,\leqslant\,\,\,V_{2}\,\,\,<\,\,\,V_{\max}\,\,\,;$

D2. All open level lines of $\,V(x,y,{\bf U})\,$ lie in straight strips of
finite width and pass through them (Fig. 3);

D3. The mean direction $\,{\bf l}\,$ of strips containing open level lines of
the potential $\,V(x,y)\,$ is determined, for some (irreducible) integer
vector $\,(m^{1},\dots,m^{d})\,$, by the relation

$\left(m^{1}{\bf k}_{(1)}\,+\dots\,+\,m^{d}{\bf k}_{(d)}\,,\,{\bf
l}\right)\,\,\,=\,\,\,0.$

Potentials with $\,d\,$ quasi-periods with regular behavior of open level
lines arise, in particular, whenever they are formed by means of sufficiently
small (quasiperiodic) additions to potentials with a smaller number of quasi-
periods (and regular behavior of open level lines). But in reality, such
situations are not limited to this, and it is possible to construct a huge
number of examples of very complex potentials with a large number of quasi-
periods and regular behavior of open level lines in the sense described above.

It can be shown that in the most general case there is a huge set of
situations when for a generic potential from some family of quasiperiodic
potentials $\,V(x,y,{\bf U})\,$ properties (D1)-(D3) take place and, moreover,
such a situation is locally stable. In this case we have a stability zone
$\,\Omega_{(m^{1},\dots,m^{d})}\,$ in the space of parameters, such that
conditions (D1)-(D3) are satisfied at all its points for some values
$\,V_{1}({\bf U})\,$ and $\,V_{2}({\bf U})\,$ and the same
$\,(m^{1},\dots,m^{d})\,$. The boundaries of the zone
$\,\Omega_{(m^{1},\dots,m^{d})}\,$ are actually determined then by the
condition $\,V_{1}({\bf U})\,=\,V_{2}({\bf U})\,$.

Returning to the considered method of creating quasiperiodic potentials in
systems of cold atoms (a superposition of standing waves), we can show, as
before, that the numbers $\,(m^{1},\dots,m^{d})\,$ can be determined in a
purely topological way (by successive shifts of the maxima of the standing
waves by a period and observation of the corresponding shifts of open level
lines).

In the general case, it can be stated that when creating quasiperiodic
potentials it is natural to divide them, according to the behavior of their
level lines, into potentials that retain certain properties of ordered
potentials, and potentials approaching random potentials. The potentials of
the first type are stable and arise in some open regions in the space of their
parameters, while each of these regions is determined by the respective value
of the topological invariant—an integer vector $\,(m^{1},\dots,m^{d})\,$.
Potentials of the second type are unstable and arise in rather complex sets of
fractal type (the complements to the union of domains
$\,\Omega_{(m^{1},\dots,m^{d})}$). To construct potentials with the properties
of random potentials, it is therefore necessary to fix a set of domains
$\,\Omega_{(m^{1},\dots,m^{d})}\,$ in the parameter space. Considering the
potentials in the complement, it can be expected that the corresponding
potentials with a large number of quasi-periods can, in fact, serve as one of
the models of random potentials, since the complexity of the behavior of their
open level lines increases very rapidly with an increase in the number of
quasi-periods. In conclusion of this section, we note that the emergence of
random potentials has also been considered in systems of optical lattices for
ultracold atoms (see, for example HorCourtGryn ; Boiron ).

##  III Particle dynamics in potentials of different types

The main subject of this section is the dynamics of ultracold atoms in a two-
dimensional plane in the presence of a quasiperiodic potential $\,V(x,y)\,$.
In the leading approximation, such dynamics can be considered in a classical
way provided it is assumed that the atoms have fairly well-defined
trajectories in $\,\mathbb{R}^{2}\,$. An exception in this case may be the
motion of atoms at special segments of trajectories, where quantum tunneling
from one segment of the trajectory to another can play an important role. As
we have already said, we will be primarily interested in the dynamics of atoms
with energies corresponding to the presence of open level lines of the
considered potentials. To apply the semiclassical description, we must
therefore assume the relation $\,h/\sqrt{2MV}\ll a\,$, where $\,a\,$ is is the
typical value of the periods of standing waves used in experiment. In our
situation we assume also that a sufficiently large number of atoms in the
ensemble have energies corresponding to the presence of open level lines,
which also implies the relation $\,T\simeq V\,$. Thus, in our case, we must
also put $\,h/\sqrt{2MT}\ll a\,$. In the general case, the forces acting on
atoms in optical traps can contain both conservative and dissipative parts
(see, for example GordonAshkin ; DalibardCohen-Tannoudji ; MinoginLetokhov ;
MechanicalAction ). We will consider here the classical dynamics of
noninteracting atoms in the nondissipative limit, which gives a good
approximation to the real dynamics of heavy atoms in many important
situations.

As can be seen in this case, the features of the geometry of quasiperiodic
potentials should manifest themselves most of all in the dynamics of particles
at energies lying in the intervals of the presence of open level lines of the
potential. Here we will focus specifically on the study of such dynamics. It
is easy to see that the features of this dynamics should naturally manifest
themselves in the transport properties of an ultracold gas in the presence of
particles of the corresponding energies in the ensemble.

As we saw in the previous section, quasiperiodic potentials can in fact be
divided into two types according to the behavior of their open level lines. At
the same time, this division is most directly related to the dynamics of
particles in such potentials, since it determines the geometry of the
‘‘regions of accessibility’’ for particles with certain energies. As a
consequence of this, one can expect a difference in the dynamics of particles
in the potentials of these two types, which is observable in the study of
systems of ultracold atoms.

As we mentioned above, dynamics in two-dimensional Hamiltonian systems has one
more feature, namely, it is integrable at low energy levels and becomes
chaotic at high energies (see Ott ; LikhtLib ). At intermediate energy levels,
the phase space of a system is divided into regions where the integrable case
takes place, and domains where chaotic dynamics occurs. In particular, one can
observe the effects of gradual chaotization of dynamics, when a particle can
‘‘stick’’ to invariant two-dimensional tori for a long time, performing rare
‘‘jumps’’ (Levy flights) between different tori, as well as other similar
effects. This feature of the dynamics has been studied in systems of cold
atoms both for free atoms moving in periodic potentials (see HennequinVerkerk1
; HennequinVerkerk2 ), and in the presence of additional interaction of the
atom motion with internal degrees of freedom (see ArgonovPrants ; Prants2020
).

In this work, we do not consider the internal degrees of freedom of atoms and
focus only on the motion of the atom as a whole. As we have already said, we
will be interested here in the dynamics of atoms with energies corresponding
to the appearance of open level lines at the corresponding potentials. This is
the area in which it is especially interesting for us to observe the
combination of different modes (integrability, its complication, and
transition to chaotic dynamics) in different parts of the phase space. It can
be immediately noted that, since open level lines of a quasiperiodic potential
also appear at some ‘‘intermediate’’ (between the minimum and maximum) values,
the ‘‘gradual chaotization’’ of dynamics described above will often arise
precisely near such level lines. It can also be said that such chaotization
must, of course, correlate with the nontrivial geometry of the ‘‘accessible
regions’’ of the motion of atoms in the potentials we are considering. In
particular, for developed chaotization, the motion of atoms should be close to
diffusion, so that the transport properties of an atomic gas are determined by
the diffusion of atoms in regions of a given geometry. Under conditions of
‘‘intermediate’’ chaotization, one can expect that the geometry of the
accessibility regions significantly affects the Levy flights between two-
dimensional tori, which, of course, is also determinant when considering the
transport properties of an atomic gas. In addition, in the situation under
consideration, two-dimensional tori corresponding to integrable dynamics can
cut (three-dimensional) manifolds of constant energy in a nontrivial way, so
that we can also observe non-integrable dynamics localized in different parts
of the ‘‘accessibility regions’’. The geometry of such areas and their
location is, of course, also related to the general geometry of the ‘‘regions
of accessibility’’ for a given particle energy. On the whole, as it is easy to
see, in the model we are considering, it is the diffusion or Levy flight
regimes that determine the transport properties of an atomic gas that can make
it possible to observe the differences between the ‘‘regular’’ and ‘‘chaotic’’
quasiperiodic potentials introduced above.

Below we give the results of a numerical study of the classical dynamics of
particles in the potentials consedered here. As a model, we consider here
potentials with three quasiperiods obtained by the restrictions of the
potential

$V\left(\cos X^{1}\,+\,\cos X^{2}\,+\,\cos X^{3}\right)$

to the planes defined by different embeddings (II.2). It must be said that,
from the point of view of Novikov’s problem, this family of potentials
contains absolutely all situations that can occur for potentials with three
quasi-periods; namely, we will find here both stable ‘‘regular’’ potentials
with very different features of the geometry of open level lines, and a rich
set of ‘‘chaotic’’ potentials. It is worth noting just one feature of the
potentials from this set, which is as follows. For any ‘‘regular’’ potential
arising in the described family, the interval of existence of open level lines
is symmetric with respect to zero, i.e. we always have the relation
$\,V_{1}({\bf U})\,=\,-\,V_{2}({\bf U})\,$ for the values $\,V_{1}({\bf U})\,$
and $\,V_{2}({\bf U})\,$ introduced above. Similarly, for all ‘‘chaotic’’
potentials of this family, open level lines arise exactly at the zero energy
level ($V_{0}=0$). This feature is specific for this family of potentials and,
generally speaking, does not take place in the most general case. In all other
respects, the geometric properties of the level lines of potentials from the
above family reflect the most general situation. Since, as we have already
said, we are interested in the dynamics of particles in the areas of the
appearance of open potential level lines, we will often study such a dynamics
here at the value $\,\epsilon=0\,$.

Thus, all potentials will have the form

$V(x,y)\,\,\,=\,\,\,\cos\left(a_{1}x+b_{1}y+c_{1}\right)\,\,+\\\
+\,\,\cos\left(a_{2}x+b_{2}y+c_{2}\right)\,\,+\,\,\cos\left(a_{3}x+b_{3}y+c_{3}\right)$
(III.1)

with some coefficients $\,a_{i}$, $\,b_{i}$ and $\,c_{i}$. In fact, since the
choice of the direction of the coordinate axes in the plane
$\,\mathbb{R}^{2}\,$ will not matter much to us, we will assume that the axis
$\,x\,$ coincides with the intersection line of $\,\mathbb{R}^{2}\,$ with the
plane $\,(X^{1},X^{2})\,$ in $\,\mathbb{R}^{3}\,$. So we can always put here
$\,a_{3}=0\,$.

As we have already mentioned above, the type of the potential is determined in
our case only by the direction of the embedding
$\,\mathbb{R}^{2}\rightarrow\mathbb{R}^{3}\,$ and can be obtained from the
diagram shown in Fig. 9. It can be seen that by selecting the embedding
parameters, we can easily implement any of the situations interesting for us.

To compare the dynamics in potentials of different types, we present here the
results for three potentials, the first two of which have ‘‘regular’’ open
level lines and belong to rather large stability zones in Fig. 9, and the
third has ‘‘chaotic’’ open level lines, which appear only at the zero energy
value.

Our first series of computations refers to the potential with coefficients

$a_{1}\,=\,-0.12251993420338196,\\\ b_{1}\,=\,-0.2250221718850486,\\\
c_{1}\,=\,1.5505542426422338,\\\ a_{2}\,=\,0.9924660526802913,\\\
b_{2}\,=\,-0.02777898711920925,\\\ c_{3}\,=\,0.12374024573075965,\\\
a_{3}\,=\,0,\\\ b_{3}\,=\,0.9739575709622912,\\\ c_{3}\,=\,3.1548761694687415$
(III.2)

(Fig. 11), which lies inside the largest zone in Fig. 9 with topological
numbers $\,(m^{1},m^{2},m^{3})\,=\,(1,0,0)\,$. It must be said that we are
intentionally considering separately a potential from this zone here, since
the latter actually differs somewhat from the other zones. Namely, in addition
to the above-mentioned ‘‘partial’’ integrability inherent in two-dimensional
Hamiltonian systems, here there is an additional closeness to the integrable
situation, due to the fact that the center of this zone corresponds to a
potential integrable at all energies, which is

$V(x,y)\,\,\,=\,\,\,\cos x\,\,+\,\,\cos y$

Figure 11: The potential (III.2) with ‘‘regular’’ open level lines from the
largest stability zone in Fig. 9. (The filled areas correspond to the values
$\,V(x,y)\leqslant 0$).

As a consequence of this, integrable dynamics can arise here for richer sets
of initial conditions than it can for potentials from other zones. As we will
actually see, this assumption is confirmed; in particular, we failed to find
here a well-pronounced diffusion dynamics at the level $\,\epsilon=0\,$. This
dynamics arises only with a noticeable increase in the particle energy. At the
same time, the emergence of potentials from this particular zone is the most
expected in the experiment in comparison with others due to the significant
size of this zone.

As we have already said, we restrict ourselves here to considering three-
dimensional manifolds in the phase space by fixing the total energy of a
particle. The potential we are considering has open level lines in rather wide
energy interval

$-0.7493\,\,\,\leqslant\,\,\,V\,\,\,\leqslant\,\,\,0.7493$

(approximately). We first consider the dynamics of particles with energy
$\,\epsilon=0\,$. As we said above, we can expect here the appearance of large
regions where the dynamics is in fact integrable and occurs on two-dimensional
tori embedded in the phase space. This is exactly what happens at the level
$\,\epsilon=0\,$, moreover, by carefully choosing the initial data, one can
find the regions of their values, where the dynamics on the tori corresponds
to a rather simple motion in the coordinate space (Fig. 12).

Figure 12: Examples of invariant tori corresponding to a relatively simple
dynamics of atoms in the ‘‘regular’’ potential (III.2) at zero total energy.

By varying the initial data at the level $\,\epsilon=0\,$ (for the same
potential), one can, however, discover (smaller) regions in which the geometry
of invariant tori becomes more and more complicated, which also leads to
complication of the motion of particles in the coordinate space (Fig. 13).

Figure 13: Examples of invariant tori defining more complex dynamics of atoms
in the ‘‘regular’’ potential (III.2) at zero total energy.

Choosing the initial data in an even more special way (at the level
$\,\epsilon=0$), one can also see even more complicated surgery of invariant
tori, where particles ‘‘stick’’ for a rather long time to simpler invariant
tori and ‘‘jump’’ from one of these tori to another (Levy flights) at certain
times (Fig. 14).

Figure 14: Levy flights between ‘‘close’’ and ‘‘distant’’ tori and the
complication of invariant tori when the initial conditions change in the case
of the ‘‘regular’’ potential (III.2) at zero total particle energy.

The regimes shown in Fig. 12–14, correspond to the dynamics of particles with
zero total energy. As we have already said, for this potential we failed to
find diffusion regimes at $\,\epsilon=0\,$; however, they appear with
increasing particle energy (Fig. 15). In fact, a clearly pronounced diffusion
behavior arises here at energies for which the open level lines of the
potential have already disappeared, and the region of accessibility extends
infinitely in two dimensions. It is interesting that the diffusion dynamics
here retains, nevertheless, a pronounced anisotropy, preserving the memory of
the mean direction of the open level lines of the potential. It can also be
noted here that even at these levels there remain quite a lot of invariant
tori, and the diffusion dynamics has, at the same time, the form of Levy
flights with ‘‘sticking’’ to invariant tori. As we have already said, this
behavior is apparently characteristic only of potentials from the zone with
$\,(m^{1},m^{2},m^{3})\,=\,(1,0,0)\,$ (and the zones identical with it) due to
the circumstances mentioned above. In particular, we will present below a
description of the dynamics in a potential from another large stability zone,
which, apparently, is characteristic for most potentials with ‘‘regular’’
level lines.

Figure 15: Accessibility regions and transition to the diffusion regime in the
case of the ‘‘regular’’ potential (III.2) with increasing particle energy
($\epsilon=0.5$ and $\epsilon=0.98$).

With a further increase in energy, the motion of particles in the potential
gradually changes from diffusion motion to ballistic motion. We should note
here, however, that ballistic motion in quasiperiodic potentials also has,
apparently, noticeable features. In particular, for potentials from the family
under consideration, even at rather high energies, a fairly large part of the
phase volume is occupied by ballistic trajectories of certain directions. For
the potential we are considering, three main directions can be distinguished
at once, namely the directions of the level lines of the three cosines in the
formula (III.1), along which ballistic motion occurs already at sufficiently
low energies (Fig. 16). With increasing energy, the number of such directions
increases and ballistic trajectories with directions other than the three
indicated ones can also be observed.

In addition to the ‘‘purely’’ ballistic trajectories described above, one can
also observe trajectories consisting of long ballistic sections of the
indicated directions, connected by short sections ‘‘switching’’ between two
directions (Fig. 17). Like the ‘‘purely’’ ballistic trajectories, such
trajectories should also introduce specific features into transport phenomena
at the corresponding particle energies. It can be seen, however, that with an
increase in the number of the corresponding ballistic directions, as well as
the complication of the geometry of the ‘‘quasi-ballistic’’ trajectories, it
will be more and more difficult to detect such properties.

Figure 16: The appearance of ballistic trajectories in the ‘‘regular’’
potential (III.2) with increasing particle energy ($\epsilon=4$).

Figure 17: ‘‘Almost’’ ballistic trajectories in the case of a ‘‘regular’’
potential (III.2) ($\epsilon=4$).

Thus, it can be seen that transport phenomena caused by the ballistic motion
of atoms also reveal the geometric structure of quasiperiodic potentials. In
comparison with the geometry of open level lines, however, this structure is
simpler and is directly related to the harmonics generating the potential. As
we will see below, this property of ballistic motion manifests itself in fact
for potentials of all types, and, in this sense, the corresponding transport
phenomena almost do not distinguish between ‘‘regular’’ quasiperiodic
potentials from ‘‘chaotic’’ ones. Also, in contrast to the case of diffusion
motion, which gives a well-observed contribution to the transport processes
associated with the geometry of open level lines, the experimental observation
of transport contributions from ballistic trajectories of stable directions
can be more complicated due to the addition of a large number of such
contributions at high energies. On the other hand, ballistic motion in
quasiperiodic potentials is also, apparently, a fundamental property of such
potentials, in particular, ballistic directions also play an important role in
quantum dynamics in potentials of this type (see ViebSbrosCartYuSchneid ).

As we have already noted above, diffusion dynamics, as well as dynamics
including distant jumps between different types of localized dynamics, which
give us the most information about the type and topological parameters of a
potential, in this example arise mainly at energies lying above the interval
of existence of open level lines. In this case, however, such dynamics keeps
the ‘‘memory’’ of the geometry of open level lines of the potential and gives
a strongly anisotropic contribution of the corresponding direction to
transport phenomena at these energies. This feature, as we have already said
above, is apparently associated with ‘‘additional reasons’’ for the appearance
of integrable dynamics in this stability zone, leading to an increase in the
phase volume filled with such dynamics.

The second series of our computations refers to the potential with
coefficients

$a_{1}\,=\,-0.6194151736623348,\\\ b_{1}\,=\,-0.44502823229775823,\\\
c_{1}\,=\,1.4421279589366298,\\\ a_{2}\,=\,0.7850635914605004,\\\
b_{2}\,=\,-0.3511272752829312,\\\ c_{3}\,=\,0.8986352554278761,\\\
a_{3}\,=\,0,\\\ b_{3}\,=\,0.8238079321117983,\\\
c_{3}\,=\,2.3379002628621635,$ (III.3)

lying inside the zone in Fig. 9 with topological numbers
$\,(m^{1},m^{2},m^{3})\,=\,(1,1,1)\,$ (Fig. 18). Like the previous one,
potential (III.3) has a fairly large energy interval

$-0.7548\,\,\,\leqslant\,\,\,V\,\,\,\leqslant\,\,\,0.7548\,,$

containing open level lines of the potential.

Figure 18: Potential (III.3) with ‘‘regular’’ open level lines from the
stability zone in Fig. 9 corresponding to the topological numbers
$\,(m^{1},m^{2},m^{3})\,=\,(1,1,1)\,$. (The filled areas correspond to the
values $\,V(x,y)\leqslant 0$).

Here, in fact, to observe most of the described regimes, it is sufficient to
study the dynamics of particles at the energy $\,\epsilon=0\,$. In particular,
we can also observe here the presence of invariant tori of varying complexity
(Fig. 19), as well as regions in the phase space, separated by such tori (Fig.
20).

Figure 19: Examples of invariant tori corresponding to the dynamics of atoms
in the case of the ‘‘regular’’ potential (III.3) at zero total energy. Figure
20: Non-integrable dynamics in a region separated from the rest of the phase
space in the case of the ‘‘regular’’ potential (III.3) at zero total energy.

Here one can also observe such a phenomenon as incomplete separation of the
energy level by invariant tori, when there are ‘‘gaps’’ between the tori,
allowing the trajectory to leave an ‘‘almost isolated’’ region at certain
times. This situation is presented in the coordinate space by long wandering
paths of particles in certain areas with very rare transitions (Levy flights)
between them (Fig. 21).

Figure 21: Rare Levy flights in the case of the ‘‘regular’’ potential (III.3)
at zero total particle energy.

Finally, in certain areas of the initial data at zero energy level, one can
also observe here much more complex Levy flights (Fig. 22), turning into
diffusion modes (Fig. 23). As we have already said, we expect that the
presence of clearly pronounced diffusion dynamics among other regimes in the
interval of the existence of open level lines is in fact a general phenomenon
for ‘‘regular’’ quasiperiodic potentials, if there are no special reasons
suppressing such dynamics (as in the previous case). As it is easy to see, the
geometry of the accessible regions for particles of fixed energy has here the
most direct influence on the geometry of Levy flights and diffusion dynamics.

Figure 22: Frequent Levy flights in the case of the ‘‘regular’’ potential
(III.3) at zero total particle energy. Figure 23: Diffusion dynamics in the
case of the ‘‘regular’’ potential (III.3) at zero total particle energy.

As in the previous case, a further increase in energy leads to the appearance
of ballistic trajectories in the considered potential. The main stable
directions of such trajectories (Fig. 24) are also determined here simply by
the directions of the level lines of cosines present in (III.1), and are not
related, in fact, with the type of arising potential. Ballistic trajectories
of stable directions, as we have already noted above, occupy a finite phase
volume at a fixed energy level.

With a further increase in the particle energy, the number of stable
directions of ballistic trajectories increases. In addition, as in the
previous case, a lot of ‘‘quasi-ballistic’’ trajectories also arise here, with
rather long ballistic segments joined by short transitions between them (Fig.
25). As in the previous case, it can be noted that the geometric features of
the contribution of ballistic trajectories to transport phenomena become more
and more ‘‘blurred’’ with an increase in the number of stable directions of
such trajectories, as well as with a complication of the geometry of ‘‘quasi-
ballistic’’ trajectories. As for determining the type of potential, as well as
its stable topological parameters (numbers $\,(m^{1},m^{2},m^{3})$), they, as
in the previous case, are best determined by the contribution of diffusion
trajectories, as well as trajectories containing long ‘‘hops’’ (Levy flights)
between segments of almost integrable or localized dynamics. It can be noted
that in this example (in contrast to the previous one), the corresponding
dynamics arises mostly in the interval of the existence of open level lines of
the potential. We expect that this property should actually manifest itself
for most types of ‘‘regular’’ potentials (in the absence of additional reasons
for increasing the phase volume occupied by the integrable dynamics) created
by the method under consideration.

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Figure 24: Ballistic trajectories of the ‘‘main’’ directions in the case of
the ‘‘regular’’ potential (III.3) ($\epsilon=4$).

Figure 25: ‘‘Almost’’ ballistic trajectories in the case of the ‘‘regular’’
potential (III.3) ($\epsilon=4$).

Our last series of computations refers to the potential with coefficients

$a_{1}\,=\,-0.6190763027420052,\\\ b_{1}\,=\,-0.2572674789786692,\\\
c_{1}\,=\,1.311209111211166,\\\ a_{2}\,=\,0.7853308419916342,\\\
b_{2}\,=\,-0.20280395367888493,\\\ c_{3}\,=\,0.8662242771884692,\\\
a_{3}\,=\,0,\\\ b_{3}\,=\,0.9448195598272575,\\\ c_{3}\,=\,2.950743051151684$
(III.4)

having ‘‘chaotic’’ level lines (Fig. 26). In this case, open level lines, as
we have already said, exist only at the value $\,V_{0}=0\,$.

Figure 26: Potential (III.4) having ‘‘chaotic’’ open level lines. (The filled
areas correspond to the values $\,V(x,y)\leqslant 0$).

As in the previous two cases, in this case, at the energy level
$\,\epsilon=0\,$ we can also see regions corresponding to the motion along
tori of relatively simple (Fig. 27) as well as more complex geometry (Fig.
28).

Figure 27: Examples of invariant tori corresponding to a relatively simple
dynamics of atoms in the case of the ‘‘chaotic’’ potential (III.4) at zero
total energy.

Figure 28: Examples of invariant tori defining more complex dynamics of atoms
in the case of the ‘‘chaotic’’ potential (III.4) at zero total energy.

In addition, in certain regions of initial data, one can observe non-
integrable dynamics, which is bounded, however, by some invariant tori in the
manifold $\,\epsilon=0\,$. Such dynamics is also easily distinguishable from
other types when it is projected onto the coordinate space (Fig. 29).

Figure 29: Non-integrable dynamics bounded by invariant tori in the phase
space in the case of the ‘‘chaotic’’ potential (III.4) at zero total energy.

As in the previous cases, for certain initial conditions at the zero energy
level, one can also observe the ‘‘sticking’’ of the particle trajectory to
invariant tori for a rather long time, interspersed with Levy flights at
certain moments (Fig. 30).

Figure 30: Levy flights in the case of the ‘‘chaotic’’ potential (III.4) at
zero total energy.

Also, as in the previous case, by changing the initial data, we can achieve
the complication of the geometry of the tori and the transition to the
diffusion dynamics of particles (Fig. 31). Diffusion dynamics here is also
limited to the ‘‘region of accessibility’’, which now has a completely
different geometry and itself has, in a sense, ‘‘diffusion properties’’. It
must be said that, apparently, in the classical limit, the transport
properties of particles at the zero energy level here are close to the
transport properties of localized (although in large regions) particles, since
the probability of distant diffusion in the region under consideration is very
small. It can be noted, however, that in the case of three quasi-periods, such
regions always contain segments of the boundary that are very close to each
other, where quantum tunneling should be possible. In this case, it is quantum
tunneling that, apparently, should play an important role for transport
phenomena at $\,\epsilon=0$.

Figure 31: Transition to diffusion dynamics under a change of the initial data
in the case of the ‘‘chaotic’’ potential (III.4) at zero total particle
energy.

In general, with a gradual increase in the energy of particles in the
ensemble, the transport properties of an atomic gas in the described potential
should (in the classical limit) change significantly when particles with
positive energies appear in the ensemble. Indeed, with increasing energy, the
accessibility regions expand and become non-simply connected, in contrast to
the case $\,\epsilon=0\,$. We can say that, in a certain sense, such regions
have the property of ‘‘percolation’’. At the same time, they retain for some
time also certain ‘‘diffusion’’ form, which should manifest itself in the
transport properties of an atomic gas. As we can see in Fig. 32, the diffusion
properties of the particle dynamics in such potentials rapidly increase with
an increase in the value of $\,\epsilon$. As the energy of the particles
decreases, the ‘‘accessible regions’’ become bounded regions in the plane.
Thus, one can see here a noticeable difference from the potentials with
‘‘regular’’ level lines, where the picture does not change significantly when
the particle energy is varied near zero.

Figure 32: Accessibility regions and diffusion dynamics in the case of the
‘‘chaotic’’ potential (III.4) at positive particle energies ($\epsilon=0.64$
and $\epsilon=0.72$).

With a significant increase in energy, ballistic trajectories with stable
directions appear for the potential (III.4) as well (Fig. 33). As in the
previous two cases, the main stable directions are the directions of the level
lines of the cosines constituting the potential (III.4) according to formula
(III.1). The corresponding trajectories appear at the lowest energy levels;
with a further increase in energy, the number of such directions increases. As
in the previous two cases, the final phase volume is also occupied by ‘‘quasi-
ballistic’’ trajectories, consisting of long segments of ballistic
trajectories joined by short intermediate segments (Fig. 34). As we have
already said above, to determine the geometric features of a ‘‘chaotic’’
potential, as in the ‘‘regular’’ case, it seems most appropriate to study the
contribution of the ‘‘diffusion’’ and ‘‘jumping’’ trajectories appearing at
‘‘intermediate’’ energy values.

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Figure 33: The appearance of ballistic trajectories in the case of the
‘‘chaotic’’ potential (III.4) with increasing particle energy ($\epsilon=5$).

Figure 34: ‘‘Almost’’ ballistic trajectories in case of the chaotic potential
(III.4) ($\epsilon=4$).

##  IV Conclusion

The paper considers questions related to the geometry of quasiperiodic
potentials on a plane, the methods of their creation, their dependence on
control parameters, and the dynamics of semiclassical particles in such
potentials. The main considerations are related to the situation of the
emergence of such potentials in systems of ultracold atoms in magneto-optical
traps, although our results are actually valid for the most general types of
quasiperiodic potentials. It is shown that, in the general case, quasiperiodic
potentials on the plane can be naturally divided into two main classes,
namely, potentials with a ‘‘regular’’ behavior of open level lines and
potentials with a ‘‘chaotic’’ behavior of open level lines. In each family of
quasiperiodic potentials, depending smoothly on some set of parameters,
potentials from these classes are parametrized by sets having different
structures, complementing each other in the full space of parameters. Namely,
the first set has the form of a union of (countably many) regions with
piecewise smooth boundaries, while the second has fractal properties.
According to the behavior of open level lines, the former potentials can be
attributed the ‘‘regular’’ type (approaching periodic potentials), while the
latter can be considered as a model of random potentials. The non-dissipative
dynamics of ultracold atoms in the considered potentials is integrable at
lower energy levels, gradually becoming chaotic with increasing the energy of
atoms. As a rule, in the interval of the existence of open level lines of the
potential, both types (integrable and chaotic) of dynamics are present, and
the properties of the chaotic dynamics substantially depend on the geometry of
the potential level lines. The study of the transport properties of an atomic
gas in quasiperiodic potentials in the presence of particles with the
corresponding energies in the ensemble can thus allow observing the
differences between potentials of both types and also provide more detailed
information on the geometry of their open level lines.

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