Figure 1: Schematic of the system. (a) Three interacting atoms
in a one-dimensional harmonic trap. (b) Representation of the
refractive index in the x − y plane. (c) Schematic of the fiber.
fiber is described by a wave propagation equation which
is Schr¨odinger-like and often called the Fock-Leontovich
equation [7, 8]. The longitudinal dimension along the
fiber plays the role of time and the inhomogeneous re-
fractive GRIN index profile of the fiber plays the role of
an external potential. We will show below that the thin
metallic slabs can play the role of the contact interac-
tions between the atoms and that by properly designing
the spatial profile of the incident laser beam it is possi-
ble to select the statistics of the atoms emulated, that
is, if they resemble bosons, fermions, or mixtures.
We emphasize here that the characterization of the
modes guided by the GRIN fiber with three thin metal-
lic slabs is of interest in itself for the optics community,
independently of the analogy with the quantum sys-
tem of three atoms. Graded-index fibers are multimode
fibers, that is, they can propagate several modes [9–
11]. There is a recent revival in the interest in these
kinds of fibers, as they have been identified as very
versatile systems to study spatio-temporal non-linear
effects [12, 13]. A non-comprehensive list of recent
works include the observation of optical solitons and
soliton self-frequency lifting [14], the generation of ul-
trashort pulses and even supercontinuum [15], or beam
self-cleaning [16, 17]. However, the description of pulse
propagation in these fibers is rather difficult, as it must
include both the three spatial dimensions and time to
correctly capture the non-linear dynamics of multiple
co-propagating modes (for a simplified model see [18]).
Yet, GRIN fibers represent an ideal set-up for a va-
riety of phenomena in complexity science, due to the
collective dynamics associated with the interplay be-
tween disorder, dissipation, and non-linearity [16]. Here
we do not consider spatio-temporal dynamics or non-
linearities, as we detail later. However, multimode
GRIN fibers with thin metallic slabs allow for both to
be included in future work.
As our model is an example of an analogy between
a classical and a quantum system, an inferred property
for the target optical system from the source quantum
system is the existence of classical entanglement [19–
22]. Classical entanglement occurs in a wide variety
in optical systems, is not restricted to those described
by the Fock-Leontovich equation, and often includes
polarization degrees of freedom [23, 24]. It has been
proposed that a better name for this property is clas-
sical non-separability [25], because the classical target
system lacks the potential non-locality of quantum sys-
tems with entanglement [24]. It is also worth stressing
that classical entanglement cannot be used as a resource
for applications in quantum information theory. In our
system non-locality is associated with a measurement
of an entangled system, which when taken in one re-
gion of space dictates the outcome in another region.
In this sense one can distiniguish two types of entan-
glement [19, 20, 23, 26]: (i) intersystem entanglement
(or true-multiparticle entanglement) and (ii) intrasys-
tem entanglement (between different degrees of freedom
of a single particle). It is commonly accepted that in-
tersystem entanglement can only occur in quantum sys-
tems as it can lead to non-locality. The examples of
classical non-separability found in literature are mostly
associated with two different degrees of freedom of the
same particle, and a remarkable realization classically
non-separable states with three degrees of freedom were
done using path, polarization and transverse modes [27].
Below we show how in the system we introduce here
classical non-separability between different particles oc-
curs in a scalar system. In this sense it is an analo-
gous to type (i) entanglement (intersystem), but as the
measurement problem remains, it does not lead to non-
locality. We note that there is a set of works where the
goal is to use classical fields to reproduce non-classical
correlations between different measurements [28–31], in-
cluding simulations of Bell-like inequalities [32, 33].
Our manuscript is structured as followed. In Section 2
we detail the characteristics of the fiber under study.
We perform a full modal analysis of it and classify the
modes according to the rotational discrete symmetry of
the system. The analogy with the atomic system is con-
structed in Section 3 and we show how the wave function
can be interpreted as giving information of the order-
ing of the particles. In Section 4 we discuss the non-
separability of the classical states and in Section 5 we
conclude by laying out possible applications and further
developments of this system. Two appendixes provide
supplementary details about the symmetry methods we
employ and about the Bose-Fermi mapping.
2 Optical system: GRIN fiber with three
thin metallic slabs
The paraxial propagation of a monochromatic optical
beam of constant polarization along a fiber with an in-
Accepted in Quantum 2019-11-15, click title to verify. Published under CC-BY 4.0. 2