# (Invited) Two-color soliton meta-atoms and molecules

O. Melchert  S. Willms  I. Babushkin  U. Morgner  A. Demircan

###### Abstract

We present a detailed overview of the physics of two-color soliton molecules
in nonlinear waveguides, i.e. bound states of localized optical pulses which
are held together due to an incoherent interaction mechanism. The mutual
confinement, or trapping, of the subpulses, which leads to a stable
propagation of the pulse compound, is enabled by the nonlinear Kerr effect.
Special attention is paid to the description of the binding mechanism in terms
of attractive potential wells, induced by the refractive index changes of the
subpulses, exerted on one another through cross-phase modulation.
Specifically, we discuss nonlinear-photonics meta atoms, given by pulse
compounds consisting of a strong trapping pulse and a weak trapped pulse, for
which trapped states of low intensity are determined by a Schrödinger-type
eigenproblem. We discuss the rich dynamical behavior of such meta-atoms,
demonstrating that an increase of the group-velocity mismatch of both
subpulses leads to an ionization-like trapping-to-escape transition. We
further demonstrate that if both constituent pulses are of similar amplitude,
molecule-like bound-states are formed. We show that $z$-periodic amplitude
variations permit a coupling of these pulse compound to dispersive waves,
resulting in the resonant emission of Kushi-comb-like multi-frequency
radiation.

###### keywords:

Nonlinear optics , Optical solitons , Two-color soliton molecules , Resonant
radiation

††journal: Optik

[inst1]organization=Leibniz Universität Hannover, Institute of Quantum Optics,
addressline=Welfengarten 1, city=Hannover, postcode=30167, country=Germany

[inst2]organization=Leibniz Universität Hannover, Cluster of Excellence
PhoenixD, addressline=Welfengarten 1A, city=Hannover, postcode=30167,
country=Germany

###### Contents

  1. 1 Introduction
  2. 2 Model and methods
  3. 3 Nonlinear-photonics meta-atoms
    1. 3.1 Stable propagation of trapped states
    2. 3.2 Trapping-to-escape transition caused by a group-velocity mismatch
  4. 4 Two-color soliton molecules
    1. 4.1 Two-color soliton pairs
    2. 4.2 Kushi-comb-like multi-frequency radiation
  5. 5 Summary and conclusions

##  1 Introduction

The confinement of two – and possibly more – quasi co-propagating optical
pulses has been discussed in terms of various propagation settings since the
80’s of the last century, with early accounts discussing the self-confinement
multimode optical pulses in glass fibers [1], nonlinear pairing of light and
dark optical solitons [2, 3], and stability of solitons with different
polarization components in birefringent fibers [4]. A very paradigmatic
instance of self-confinement is supported by the standard nonlinear
Schrödinger equation (NSE) [5, 6]. In the integrable case, it features
localized field pulses given by solitary waves [7]. When considering two or
more quasi group-velocity matched pulses, their incoherent, cross-phase
modulation (XPM) induced mutual interaction co-determines their dynamics [1,
2, 3, 4, 8, 9, 10, 11]. For instance, in nonlinear waveguides with a single
zero-dispersion point, a soliton induces a strong refractive index barrier
that cannot be surpassed by quasi group-velocity matched waves located in a
domain of normal dispersion [12], resulting in their mutual repulsion. The
underlying interaction process is enabled by a general wave reflection
mechanism originally reported in fluid dynamics [13]. In optics this process
is referred to as push-broom effect [14], optical event horizon [15, 16], or
temporal reflection [17]. This interaction mechanism allows for a strong and
efficient control of light pulses [18, 19, 20], and has been shown to appear
naturally during the supercontinuum generation process [21, 22, 23]. When
considering waveguides that support group-velocity matched propagation of
pulses in separate domains of anomalous dispersion, their mutual interaction
is expressed in a different way: the aforementioned XPM induces attractive
potentials that hold the pulses together, enabling two-color soliton molecules
through an incoherent binding mechanism [24]; the resulting pulse compound
consists of two subpulses at vastly different center frequencies. Putting
emphasis on the frequency-domain representation of these pulse compounds lead
to observe that a soliton can in fact act as a localized trapping potential
with a discrete level spectrum [24]. Let us emphasize that in order to achieve
a strong attractive interaction between the subpulses of such pulse compounds,
group-velocity matching is crucial [25]. In terms of a modified NSE with added
fourth-order dispersion, these objects where identified as parts of a large
family of generalized dispersion Kerr solitons that can be characterized using
the concept of a meta-envelope [26]. Such pulses were recently verified
experimentally in mode-locked laser cavities [27, 28, 29]. In a complementary
approach to the multi-scales analysis presented in Ref. [26], modeling both
subpulses in terms of coupled NSEs allowed to derive a special class of two-
color soliton pairs and their meta-envelopes in closed form [30]. Let us note
that the concept of soliton molecules has meanwhile been extended to pulse
compounds with three frequency centers [31], and recently also to a number of
$J$ equally spaced frequency components [27, 32]. Further, two-color soliton
microcomb states with similar structure where also observed in the framework
of the Lugiato-Lefever equation [33, 34]. The underlying scheme is much more
general and requires quasi group-velocity mathing between different optical
pulses. This can be achieved in different settings, and can, e.g., already
been found in an early work of Hasegawa [1], where a strong incoherent XPM
interaction between different components of a multimode optical pulse has been
considered. At this point, we would also like to emphasize that these pulse
compounds are different from usual soliton molecules, which can be realized by
dispersion engineering in the framework of a standard NSE [35], characterized
by two pulses separated by a fixed temporal delay and stabilized by a phase
relation between both pulses [36].

Here, we review the rich dynamical behavior of two-color pulse compounds,
which consist of two group-velocity matched subpulses in distinct domains of
anomalous dispersion, with frequency loci separated by a vast frequency gap.
First, we will demonstrate paradigmatic propagation scenarios that demonstrate
photonic meta-atoms, arising in the limiting case where the pulse compounds
consist of an intense trapping pulse, given by a soliton, and a weak trapped
pulse. Then, we will address the case where both subpulses have similar
amplitudes, so that their mutual XPM induced confining action results in the
formation of a narrow two-color soliton molecule. Finally, we show that non-
stationary dynamics of the subpulses results in the emission of resonant
radiation, and we show how the location of the newly generated frequencies
depends on the $z$-periodic amplitude and width variations of the oscillating
soliton molecule.

The article is organized as follows. In Sec. 2 we discuss the propagation
model used for our theoretical investigations of two-color meta-atoms and
soliton molecules, and detail the numerical methods employed for their
simulation and analysis. In Sec. 3 we demonstrate the ability of solitons to
act as attractive potential wells that can host trapped states, and probe the
stability of the resulting photonic meta-atoms with respect to a group-
velocity mismatch between the trapping soliton and the trapped state. In Sec.
4 we derive a simplified model that yields simultaneous solutions for the
subpulses that make up a two-color soliton molecule and show that theses
solutions entail the two-color soliton pairs derived in Ref. [30]. We perturb
these pulse compounds by increasing their initial amplitude, which results in
periodic amplitude and width oscillations, and triggers the generation of
resonant multi-frequency radiation with a complex stucture that can be
precisely predicted theoretically. Section 5 concludes with a summary.

##  2 Model and methods

#### Propagation model

In order to study the propagation dynamics of nonlinear photonic meta-atoms
and two-color soliton molecules, we consider a modified nonlinear Schrödinger
equation (NSE) of the form

$\displaystyle
i\partial_{z}A=\left(\frac{\beta_{2}}{2}\partial_{t}^{2}-\frac{\beta_{4}}{24}\partial_{t}^{4}\right)A-\gamma|A|^{2}A,$
(1)

describing the single-mode propagation of a complex-valued field $A\equiv
A(z,t)$, on a periodic temporal domain of extent $T$ for the boundary
condition $A(z,-T/2)=A(z,T/2)$. The linear part of Eq. (1) includes higher
orders of dispersion, with $\beta_{2}>0$ (in units of $\mathrm{fs^{2}/\mu m}$)
a positive-valued group-velocity dispersion coefficient, and $\beta_{4}<0$
($\mathrm{fs^{4}/\mu m}$) a negative-valued fourth-order dispersion
coefficient. The nonlinear part of Eq. (1) includes a positive-valued scalar
nonlinear coefficient $\gamma$ ($\mathrm{W^{-1}/\mu m}$). Considering the
discrete set of angular frequency detunings
$\Omega\in\frac{2\pi}{T}\mathbb{Z}$, the transform-pair

$\displaystyle
A_{\Omega}(z)=\mathsf{F}[A(z,t)]\equiv\frac{1}{T}\int_{-T/2}^{T/2}A(z,t)\,e^{i\Omega
t}~{}{\rm{d}}t,$ (2a) $\displaystyle
A(z,t)=\mathsf{F}^{-1}[A_{\Omega}(z)]\equiv\sum_{\Omega}A_{\Omega}(z)\,e^{-i\Omega
t},$ (2b)

specifies a Fourier transform [Eq. (2a)], and the corresponding inverse [Eq.
(2b)], relating the field envelope $A(z,t)$ to the spectral envelope
$A_{\Omega}(z)$.

#### Propagation constant

Using the identity $\partial_{t}^{n}\,e^{-i\Omega
t}=(-i\Omega)^{n}\,e^{-i\Omega t}$ of the spectral derivative,111 Let us note
that the “-”-sign in the bracket on the right-hand-side of the preceding
identity reflects the sign-choice of the plane-wave basis in Eqs. (2). This
has to be taken into account when using scientific computing tools such as,
e.g., Python’s scipy package [37, 38], where readily available routines for
spectral derivative exist that implement a different sign-choice for the pair
of Fourier-transforms.  the frequency-domain representation of the propagation
constant is given by the polynomial expression

$\displaystyle\beta(\Omega)=\frac{\beta_{2}}{2}\Omega^{2}+\frac{\beta_{4}}{24}\Omega^{4}.$
(3a) The frequency-dependent inverse group-velocity of a mode at detuning
$\Omega$ reads
$\displaystyle\beta_{1}(\Omega)\equiv\partial_{\Omega}\beta(\Omega)=\beta_{2}\Omega+\frac{\beta_{4}}{6}\Omega^{3},$
(3b) with group-velocity (GV) $v_{g}(\Omega)=1/\beta_{1}(\Omega)$, and the
group-velocity dispersion (GVD) is given by
$\displaystyle\beta_{2}(\Omega)\equiv\partial_{\Omega}^{2}\beta(\Omega)=\beta_{2}+\frac{\beta_{4}}{2}\Omega^{2}.$
(3c)

Subsequently, we use the parameter values $\beta_{2}=1~{}\mathrm{fs^{2}/\mu
m}$, and $\beta_{4}=-1~{}\mathrm{fs^{4}/\mu m}$, resulting in the model
dispersion characteristics shown in Fig. 1. For the nonlinear coefficient in
Eq. (1) we use $\gamma=1~{}\mathrm{W^{-1}/\mu m}$. As evident from Fig. 1(c),
the GVD profile Eq. (3c) has a concave downward shape with two zero-dispersion
points, defined by the condition
$\beta_{2}(\Omega)\stackrel{{\scriptstyle!}}{{=}}0$, located at
$\Omega_{\rm{Z1},\rm{Z2}}=\mp\sqrt{2\beta_{2}/|\beta_{4}|}=\mp\sqrt{2}\,\mathrm{rad/fs}$.
It exhibits anomalous dispersion for $\Omega<\Omega_{\rm{Z1}}$ as well as for
$\Omega>\Omega_{\rm{Z2}}$. The interjacent frequency range
$\Omega_{\rm{Z1}}<\Omega<\Omega_{\rm{Z2}}$ exhibits normal dispersion.
Inspecting the inverse group velocity shown in Fig. 1(b), it can be seen that
two frequencies are GV matched to $\Omega=0$. Due to the symmetry of the
propagation constant, these are given by the pair
$\Omega_{1}=-\Omega_{2}=-\sqrt{6\beta_{2}/|\beta_{4}|}\approx-2.828~{}\mathrm{rad/fs}$,
uniquely characterized by $\beta(\Omega_{1})=\beta(\Omega_{2})$ and indicated
by the open and filled circles in Fig. 1. In fact, for the considered
propagation constant, GV matching of three distinct modes can be realized as
long as the frequency loci in AD1 and AD2 lie within the range of frequencies
shaded in red in Fig. 1(b). Let us note that while the type of GV matching for
two optical pulses at vastly different center frequencies, supported by the
propagation constant Eq. (3a), it is methodologically different from the type
of GV matching that supports quasi co-propagation of different modes with
similar frequencies [1]. Nevertheless, both allow for quasi co-propagation of
optical pulses under different circumstances, supporting similar XPM induced
propagation effects. In our case, quasi group-velocity matched propagation of
optical pulses across a vast frequency gap is possible, enabled by a tailored
propagation constant with multiple zero-dispersion points. Further, the
considered mechanism of GV matching differs from that in Ref. [39], wherein
two pulses at the same central frequency but different polarization states
were assumed to be launched in the anomalous dispersion regime of a hollow-
core photonic crystal fiber filled with a noble gas. The mathematical
structure of Eq. (1) and the above choice of parameters yields a very basic
setting supporting the stable propagation of nonlinear photonic meta-atoms and
two-color soliton molecules. In fact, the two-parameter GVD curve shown in
Fig. 1(c) is a simplified model of the dispersion considered earlier in Ref.
[24], wherein two-color soliton molecules were first demonstrated, and is
similar to the setting considered in Ref. [26], wherein generalized dispersion
Kerr solitons were described comprehensively. However, let us note that the
phenomena reported below are not limited to the particular choice of the above
parameters and persist even in the presence of perturbations such as pulse
self-steepening [25, 30], which can be accounted for by replacing
$\gamma\rightarrow\gamma(\Omega)$ in the nonlinear part of Eq. (1), and – with
some reservation – a self-frequency shift caused by the Raman effect [31].

Figure 1: Details of the frequency-dependent propagation constant supporting
nonlinear-photonic meta-atoms and two-color soliton molecules. (a) Propagation
constant, (b) inverse group velocity, and, (c) group-velocity dispersion. In
(c), AD1 and AD2 label two distinct domains of anomalous dispersion, separated
by an extended domain of normal dispersion (labeled ND). In (a-c), the domain
of normal dispersion is shaded gray. Zero-dispersion points are labeled
$\Omega_{\rm{Z1}}$ and $\Omega_{\rm{Z2}}$. In (b), the frequency range shaded
in red allows for group-velocity matching of two modes with loci in AD1 and
AD2. Open circle (labeled $\Omega_{1}$) and filled circle (labeled
$\Omega_{2}$) indicate such a pair of group-velocity matched frequencies.

#### Propagation algorithm

For our pulse propagation simulations in terms of Eq. (1), we employ the
“Conservation quantity error” method (CQE) [40, 41]. It maintains an adaptive
$z$-propagation stepsize $h$, and uses a conservation law of the underlying
propagation equation to guide stepsize selection. Specifically, we here use
the relative error

$\displaystyle\delta_{E}(z)=\frac{|E(z+h)-E(z)|}{E(z)},$ (4)

where $E$ is the total energy, conserved by Eq. (1). Employing Parseval’s
identity for Eqs. (2) [42, 43], the total energy in the time and frequency
domains is given by

$\displaystyle
E(z)=\int_{-T/2}^{T/2}|A(z,t)|^{2}~{}{\rm{d}}t=T\sum_{\Omega}|A_{\Omega}(z)|^{2},$
(5)

with instantaneous power $|A(z,t)|^{2}$ ($\mathrm{W=J/s}$), and power spectrum
$|A_{\Omega}(z)|^{2}$ ($\mathrm{W}$). The CQE method is designed to keep the
relative error $\delta_{E}$ within the goal error range
$(0.1\,\delta_{\rm{G}},\delta_{\rm{G}})$, for a preset local goal error
$\delta_{\rm{G}}$ (throughout our numerical experiments we set
$\delta_{\rm{G}}=10^{-10}$). This is accomplished by decreasing the stepsize
$h$ when necessary while increasing $h$ when possible. To advance the field
from position $z$ to $z+h$, the CQE uses the “Fourth-order Runge-Kutta in the
interaction picture” (RK4IP) method [44]. The ability of the algorithm to
increase or decrease the stepsize is most valuable when the propagation of an
initial condition results in a rapid change of the pulse intensities over
short propagation distances. Nevertheless, if one is willing to accept an
increased running time resulting from an integration scheme with fixed
stepsize, usual split-step Fourier methods [45, 43, 46] will work similarly
well.

#### Spectrograms

To assess the time-frequency interrelations within the field $A(z,t)$ at a
selected propagation distance $z$, we use the spectrogram [47, 48, 49]

$P_{S}(t,\Omega;z)=\frac{1}{2\pi}\left|\int_{-T/2}^{T/2}A(z,t^{\prime})h(t^{\prime}-t)e^{-i\Omega
t^{\prime}}~{}{\rm d}t^{\prime}\right|^{2}.$ (6)

To localize the field in time, we use a hyperbolic-secant window function
$h(x)={\mathrm{sech}}(x/\sigma)$ with width parameter $\sigma$.

#### Incoherently coupled pulse pairs

To facilitate a simplified description of two-color pulse compounds in the
form

$\displaystyle
A(z,t)=A_{1}(z,t)\,e^{-i\Omega_{1}t}+A_{2}(z,t)\,e^{-i\Omega_{2}t},$ (7)

in which two quasi group-velocity-matched subpulses $A_{1}\equiv A_{1}(z,t)$
and $A_{2}\equiv A_{2}(z,t)$ exist at the frequency gap
$\Omega_{\rm{gap}}=|\Omega_{2}-\Omega_{1}|$, it is convenient to consider the
two coupled nonlinear Schrödinger equations (CNSEs) [6, 10, 11]

$\displaystyle
i\partial_{z}\,A_{1}+\beta_{0}^{\prime}\,A_{1}-i\beta_{1}^{\prime}\partial_{t}\,A_{1}-\frac{\beta_{2}^{\prime}}{2}\partial_{t}^{2}\,A_{1}+\gamma^{\prime}\left(|A_{1}|^{2}+2|A_{2}|^{2}\right)A_{1}=0,$
(8a) $\displaystyle
i\partial_{z}\,A_{2}+\beta_{0}^{\prime\prime}\,A_{2}-i\beta_{1}^{\prime\prime}\partial_{t}\,A_{2}-\frac{\beta_{2}^{\prime\prime}}{2}\partial_{t}^{2}\,A_{2}+\gamma^{\prime\prime}\left(|A_{2}|^{2}+2|A_{1}|^{2}\right)A_{2}=0.$
(8b)

The parameters in Eqs. (8) are related to Eqs. (3) through
$\beta_{0}^{\prime}=\beta(\Omega_{1})$,
$\beta_{0}^{\prime\prime}=\beta(\Omega_{2})$,
$\beta_{1}^{\prime}=\beta_{1}(\Omega_{1})$,
$\beta_{1}^{\prime\prime}=\beta_{1}(\Omega_{2})$,
$\beta_{2}^{\prime}=\beta_{2}(\Omega_{1})$,
$\beta_{2}^{\prime\prime}=\beta_{2}(\Omega_{2})$, and,
$\gamma^{\prime}=\gamma^{\prime\prime}=\gamma$. The mismatch of inverse GV for
both subpulses is given by
$\Delta\beta_{1}\equiv|\beta_{1}^{\prime\prime}-\beta_{1}^{\prime}|$. For
specific choices of the detunings $\Omega_{1}$ and $\Omega_{2}$, exact GV
matching, signaled by $\Delta\beta_{1}=0$, can be achieved. In contrast to Eq.
(1), the incoherently coupled Eqs. (8) neglect higher-orders of dispersion
within their linear parts, as well as rapidly varying four-wave-mixing terms
within their nonlinear parts. The mutual interaction of both subpulses is
taken into account via XPM. As evident from Eq. (8a), pulse $A_{1}$ can be
viewed as being exposed to a total potential field of the form
$V_{1}\equiv\gamma^{\prime}(|A_{1}|^{2}+2|A_{2}|^{2})$, entailing the effects
of SPM and XPM. Likewise, $A_{2}$ is exposed to the potential field
$V_{2}\equiv\gamma^{\prime\prime}(|A_{2}|^{2}+2|A_{1}|^{2})$. As we will show
in Sects. (3), (4), the potential fields $V_{1}$ and $V_{2}$ yield attractive
potentials that enable the mutual trapping of both subpulses. Subsequently we
take $\Omega_{1}$ and $\Omega_{2}$ as indicated in Fig. 1, so that the above
parameters are given by
$\beta_{0}^{\prime}=\beta_{0}^{\prime\prime}=1.33~{}\mathrm{\mu m^{-1}}$,
$\beta_{1}^{\prime}=\beta_{1}^{\prime\prime}=0$,
$\beta_{2}^{\prime}=\beta_{2}^{\prime\prime}=-2~{}\mathrm{fs^{2}/\mu m}$, and,
$\gamma^{\prime}=\gamma^{\prime\prime}=1~{}\mathrm{W^{-1}/\mu m}$. For a more
general description of simultaneous solutions in the form of Eq. (7), we will
continue to refer to the nonlinear coefficients in Eqs. (8) as
$\gamma^{\prime}$ [Eq. (8a)] and $\gamma^{\prime\prime}$ [Eq. (8b)]. In
addition, the scalar factors
$\beta_{0}^{\prime}=\beta_{0}^{\prime\prime}\equiv\beta_{0}$ can be removed by
a common linear transformation $A_{1,2}\rightarrow A_{1,2}e^{i\beta_{0}z}$,
which does not affect the $z$-propagation dynamics of the interacting pulses.

Let us note that, in general, higher-orders of dispersion within a modified
NSE can cause a solitary wave to shed resonant radiation [50], and can result
in a modification of its group-velocity [50, 51]. These types of perturbations
are neglected by Eqs. (8), which can be justified in the limit where the
subpulse separation $\Omega_{\rm{gap}}$ is large and their spectra are
sufficiently narrow. Moreover, in case of a frequency dependent coefficient
function $\gamma(\Omega)$, $\gamma^{\prime}=\gamma(\Omega_{1})$ and
$\gamma^{\prime\prime}=\gamma(\Omega_{2})$ in Eqs. (8). Let us point out that,
in the presence of a linear variation of $\gamma$, a solitary wave exhibits a
further modification of its group-velocity [52], an effect neglected by Eqs.
(8). It is important to bear these perturbation effects in mind when comparing
results based on Eqs. (8) to numerical simulations in terms of the full model
Eq. (1).

We can relate the above trapping mechanism for two-color pulse compounds to
the mechanism enabling the self-confinement of a multimode optical pulses in a
multimode fiber, discussed by Hasegawa as early as 1980 [1]. Therein, Hasegawa
considered a propagation equation of the nonlinear Schrödinger type for a
multimodal pulse, where the nonlinear change of the refractive index, felt by
an individual mode, depends on the total intensity of the multimodal pulse.
This results in coupled equations for the different modes, wherein an
individual mode perceives the intensity of the total pulse as a potential
field. If the considered mode is subject to anomalous dispersion, the
potential is attractive. Based on the expectation that if the velocity
mismatch between a given mode and the potential is smaller than the escape
velocity, the potential has the ability to trap the mode, he derived a
condition for self-confinement of the multimode pulse. While the results in
Ref. [1] are valid for multimodal optical pulses composed of possibly many
modes, the simplified modeling approach given by Eqs. (8) considers only two
subpulses. Meanwhile, an extension of the above approach to pulse compounds
with three and more subpulses has been accomplished [31, 53].

Given the ansatz for two-color pulse compounds in the form of Eq. (7), initial
conditions $A_{0}(t)\equiv A(z=0,t)$ that specify nonlinear photonic meta-
atoms and two-color soliton molecules in terms of the subpulses $A_{1}$ and
$A_{2}$ are different in some respects and are discussed separately in Sect.
3, and Sect. 4. Subsequently, we demonstrate the self-consistent
$z$-propagation dynamics of these pulse compound, originally reported in Refs.
[24, 26, 30, 54, 55], as well as their breakup in response to sufficiently
large GV mismatches between both subpulses, originally reported in Ref. [25],
in terms of numerical simulations governed by the full model Eq. (1). These
numerical results demonstrate several theoretical findings reported by
Hasegawa [1], applied to the concept of two-color pulse compounds.

In passing, let us stress that coupled equations of the form of Eqs. (8)
comprise a much-used theoretical instrument for studying mutually bound
solitons [56, 57, 58, 59, 3, 60, 61, 62].

##  3 Nonlinear-photonics meta-atoms

#### Description of stationary trapped states

Subsequently we look for stationary solutions in the form of Eq. (7) under the
additional constraint ${\rm{max}}(|A_{2}|)\ll{\rm{max}}(|A_{1}|)$. This allow
to decouple Eqs. (8) and enables direct optical analogues of quantum
mechanical bound-states [63, 24, 54]. Therefore, we assume the resulting two-
color pulse compounds to consist of a strong trapping pulse, given by a
solitary wave (S) at detuning $\Omega_{\rm{S}}\equiv\Omega_{1}$, and a weak
trapped pulse (TR) at detuning $\Omega_{\rm{TR}}\equiv\Omega_{2}$. For the
solitary wave part of the total pulse we neglect the XPM contribution in the
nonlinear part of Eq. (8a) and assume

$\displaystyle
A_{1}(z,t)=U_{\rm{S}}(t)\,e^{i\kappa^{\prime}z},\quad\text{with}\quad
U_{\rm{S}}(t)=\sqrt{P_{0}}\,{\rm{sech}}\left(\frac{t}{t_{0}}\right),$ (9)

wherein $P_{0}=|\beta_{2}^{\prime}|/(\gamma^{\prime}t_{0}^{2})$, and
$\kappa^{\prime}=\beta_{0}+\gamma^{\prime}P_{0}/2$. Neglecting the SPM
contribution in the nonlinear part of Eq. (8b) and making the ansatz

$\displaystyle A_{2}(z,t)=\phi(t)\,e^{i\kappa^{\prime\prime}z},$ (10)

the envelope $\phi(t)$ of a weak stationary trapped state is determined by the
Schrödinger type eigenvalue problem

$\displaystyle\left(-\frac{|\beta_{2}^{\prime\prime}|}{2}\frac{{\rm{d}}^{2}}{{\rm{d}}t^{2}}+V_{S}(t)\right)\,\phi_{n}(t)=\kappa_{n}\,\phi_{n}(t).$
(11)

Therein, the solitary wave enters as a stationary attractive potential well
$V_{\rm{S}}(t)=-2\gamma^{\prime\prime}P_{0}\,{\rm{sech}}^{2}(t/t_{0})$. Hence,
as pointed out above and discussed in the context of multimode optical pulses
in glass fibers in Ref. [1], a weak pulse can be attracted by the intensity of
the entire pulse if it exists in a domain of anomalous dispersion. Due to
$\beta_{2}^{\prime\prime}<0$, this condition is met in the considered case. In
analogy to the ${\mathrm{sech}}^{2}$-potential in one-dimensional quantum
scattering theory we may equivalently write the solitary-wave induced
potential as [63]

$\displaystyle
V_{\rm{S}}(t)=-\nu\,(\nu+1)\,\frac{|\beta_{2}^{\prime\prime}|}{2t_{0}^{2}}\,{\mathrm{sech}}^{2}\left(\frac{t}{t_{0}}\right),\quad\text{with}\quad\nu=-\frac{1}{2}+\left(\frac{1}{4}+4\left|\frac{\gamma^{\prime\prime}}{\gamma^{\prime}}\frac{\beta_{2}^{\prime}}{\beta_{2}^{\prime\prime}}\right|\right)^{1/2}.$
(12)

Moreover, due to the particular shape of the trapping potential, the
eigenvalue problem Eq. (11) can even be solved exactly [63, 64]. The number of
trapped states of the potential in Eq. (12) is given by
$N_{\rm{TR}}=\lfloor\nu\rfloor+1$, where $\lfloor\nu\rfloor$ is the integer
part of the strength-parameter $\nu$. From the analogy to the quantum
mechanical scattering problem [64], the real-valued wavenumber eigenvalues can
directly be stated as

$\displaystyle\kappa_{n}=-\frac{|\beta_{2}^{\prime\prime}|}{2t_{0}^{2}}\,(\nu-n)^{2},\quad\text{for}\quad
n=0,\ldots,\lfloor\nu\rfloor.$ (13)

For a given value of $n$, they are related to Eq. (10) through
$\kappa^{\prime\prime}=\beta_{0}-\kappa_{n}$. To each eigenvalue corresponds
an eigenfunction $\phi_{n}$ with $n$ zeros, specifying the $(n+1)$-th
fundamental solution of the eigenvalue problem Eq. (11). These solutions
constitute the weak trapped states of the potential $V_{\rm{S}}$. Referring to
the Gaussian hypergeometric function as ${{}_{2}F_{1}}$ [65], and abbreviating
$a_{n}=\tfrac{1}{2}(1+n)$ and $b_{n}=\tfrac{1}{2}(2\nu+1-n)$, they can be
stated in closed form as [64]

$\displaystyle\phi_{n}(t)$
$\displaystyle=\begin{cases}\cosh^{\nu+1}\left(\frac{t}{t_{0}}\right)~{}{{}_{2}F_{1}}\left[a_{n},b_{n};\frac{1}{2};-\sinh^{2}\left(\frac{t}{t_{0}}\right)\right],&\text{for
even $n$},\\\
\cosh^{\nu+1}\left(\frac{t}{t_{0}}\right)~{}\sinh\left(\frac{t}{t_{0}}\right)~{}{{}_{2}F_{1}}\left[a_{n}+\frac{1}{2},b_{n}+\frac{1}{2};\frac{3}{2};-\sinh^{2}\left(\frac{t}{t_{0}}\right)\right],&\text{for
odd $n$}.\\\ \end{cases}$ (14)

Let us note that, as evident from the potential strength parameter $\nu$ in
Eq. (12), the number $N_{\rm{TR}}$ of trapped states is uniquely defined by
the four parameters $\beta_{2}^{\prime}$, $\beta_{2}^{\prime\prime}$,
$\gamma^{\prime}$, and $\gamma^{\prime\prime}$. It is not affected by the
duration $t_{0}$ of the trapping potential, which, according to Eq. (13),
codetermines the value of the wavenumber eigenvalue of a fundamental solution.

#### Analogy to quantum mechanics

The eigenvalue problem Eq. (11) suggests an analogy to quantum mechanics,
wherein a fundamental solution $\phi_{n}$ represents the wavefunction of a
fictitious particle of mass $m=|\beta_{2}^{\prime\prime}|^{-1}$, confined to a
localized, ${\rm{sech}}^{2}$-shaped trapping potential $V_{\rm{S}}$. The
discrete variable $n=0,\ldots,\lfloor\nu\rfloor$ resembles a principal quantum
number that labels solutions with distinct wavenumbers, and the number of
trapped state $N_{\rm{TR}}$ is similar to an atomic number. Consequently, a
bare soliton, with none of its trapped states occupied, resembles the nucleus
of an one-dimensional atom. By this analogy, a soliton along with its trapped
states represents a nonlinear-photonics meta-atom.

Figure 2: Solitary-wave induced potential well exhibiting two trapped states.
(a) Trapping potential $V_{\rm{S}}$, wavenumber eigenvalues $\kappa_{n}$, and
squared magnitude $|\phi_{n}|^{2}$ of trapped state eigenfunctions for
$n=0,1$. (b) Dispersion profile $D_{\rm{TS}}(\Omega)$ in the vicinity of the
trapped state center frequency $\Omega_{\rm{TS}}$, (c) Time-domain propagation
dynamics of the soliton and its lowest lying trapped state for $n=0$. The
propagation distance is scaled by the soliton period
$z_{0}=(\pi/2)(t_{0}^{2}/{\beta_{2}^{\prime}})\approx 50~{}\mathrm{\mu m}$.
(d) Corresponding spectrum. The inverse Fourier transform of the part of the
spectrum enclosed by the box (labeled A) in (d) is shown in the box (labeled
A) in (c), providing a filtered view of the trapped state while leaving out
the soliton part of the total pulse. (e) Spectrogram of the total pulse at
$z/z_{0}=45$ for $\sigma=8~{}\mathrm{fs}$. (f,g,h) Same as (c,d,e) for the
trapped state with $n=1$. (i,j,k) Same as (c,d,e) for a superposition of both
trapped states. Movies of the propagation dynamics are provided as
supplementary material under Ref. [66].

###  3.1 Stable propagation of trapped states

Subsequently, we discuss the propagation dynamics of a nonlinear-photonics
meta-atom with the ability to host two trapped states. More precisely, we
consider an example for $\Omega_{\rm{S}}=-2.828\,\mathrm{rad/fs}$ and
$t_{0}=8\,\mathrm{fs}$, with $\Omega_{\rm{TR}}=2.828\,\mathrm{rad/fs}$ and
$\nu\approx 1.566$. The resulting trapping potential and both its trapped
states are shown in Fig. 2(a). In this case, the wavenumber eigenvalues are
$(\kappa_{0},\,\kappa_{1})=(-0.0382,-0.0050)\,\mathrm{\mu m^{-1}}$, and the
corresponding fundamental solutions take the simple form

$\displaystyle\phi_{0}(t)$
$\displaystyle={\mathrm{sech}}^{\nu}\left(\frac{t}{t_{0}}\right),\quad\text{and},$
(15a) $\displaystyle\phi_{1}(t)$
$\displaystyle={\mathrm{sech}}^{\nu-1}\left(\frac{t}{t_{0}}\right)\,{\mathrm{tanh}}\left(\frac{t}{t_{0}}\right).$
(15b)

As evident in Fig. 2(b), in the vicinity of $\Omega_{\mathrm{TR}}$ and due to
$\kappa^{\prime\prime}>0$ [Eq. (10)], a finite wavenumber-gap separates each
trapped state from linear waves bound to the dispersion curve
$D_{\rm{TR}}(\Omega)\equiv\beta(\Omega)-\beta(\Omega_{\rm{TR}})-\beta_{1}(\Omega_{\rm{TR}})(\Omega-\Omega_{\rm{TR}})<0$.
Therefore, we expect that trapped states composed by Eqs. (15) propagate in a
stable manner. For the lowest lying trapped state, having order $n=0$, this is
demonstrated in Figs. 2(c,d). These figures summarize pulse propagation
simulations in terms of the modified NSE (1), using an initial condition of
the form of Eq. (7) with $A_{1}$ as in Eq. (9), and $A_{2}$ as in Eq. (10)
with $\phi(t)=\sqrt{10^{-7}P_{0}}\,\phi_{0}(t)$. In the time-domain
propagation dynamics, shown in Fig. 2(c), a small drift of the soliton, caused
by higher orders of dispersion at $\Omega_{\rm{S}}$ [see Fig. 1], is accounted
for by shifting to a moving frame of reference with time coordinate
$\tau=t-\tilde{\beta}_{1}z$ and $\tilde{\beta}_{1}=0.00637~{}\mathrm{fs/\mu
m}$. In Fig. 2(d), the vast frequency gap between the soliton and the trapped
state is clearly visible. By means of an inverse Fourier transform of the
frequency components belonging to the trapped state [box labeled A in Fig.
2(d)], an unhindered “filtered view” of the time-domain propagation dynamics
of the trapped state is possible [box labeled A in Fig. 2(c)]. A spectrogram,
providing a time-frequency view of the field at $z/z_{0}=45$, is shown in Fig.
2(d). The stable propagation of a trapped state with $n=1$ for
$\phi(t)=\sqrt{10^{-7}P_{0}}\,\phi_{1}(t)$, is detailed in Figs. 2(f-h).
Finally, the simultaneous propagation of a superposition of both trapped
states in the form $\phi(t)=\sqrt{10^{-7}P_{0}}\,[\phi_{0}(t)+5\phi_{1}(t)]$
is shown in Figs. 2(i-k). The $z$-periodicity of the beating pattern visible
in the time-domain propagation dynamics in Fig. 2(i), is a result of the
different wavenumber eigenvalues of the trapped states, and is determined by
$z_{\rm{p}}=2\pi/|\kappa_{1}-\kappa_{0}|\approx 189~{}\mathrm{\mu m}$
($z_{\rm{p}}/z_{0}\approx 3.8$). Thus, the coherent superposition of trapped
states exhibits Rabi-type oscillations, similar to bound state dependent
revival times in the quantum recurrence of wave packets [67, 68].

Let us note that, bearing in mind that the number of bound states
$N_{\rm{TR}}$ is determined by the potential strength parameter $\nu$ in Eq.
(12), a setup with a different number of bound states can be obtained as well.
This is possible by fixing $\Omega_{\rm{S}}$ at some other feasible value,
resulting in a different group-velocity matched detuning $\Omega_{\rm{TR}}$,
implying different values of the parameters $\beta_{2}^{\prime}$,
$\beta_{2}^{\prime\prime}$, $\gamma^{\prime}$, and $\gamma^{\prime\prime}$.
For example, keeping $t_{0}=8\,\mathrm{fs}$ but choosing
$\Omega_{\rm{S}}=-2.75~{}\mathrm{rad/fs}$ yields $\nu\approx 3.1$, resulting
in a potential well with the ability to host $N_{\rm{TR}}=4$ trapped states.
In such a case, however, phase-matched transfer of energy from the trapped
states to dispersive waves within the domain of normal dispersion can be
efficient [69].

###  3.2 Trapping-to-escape transition caused by a group-velocity mismatch

In the context of multimodal pulses in glass fibers in Ref. [1], the
attraction of a wave packet by a potential well, created by the total pulse,
was illustrated in terms of the kinetic equations of a fictitious particle
associated with the wave packet. From a classical mechanics point of view, in
order to ensure trapping of the wave packet by the total pulse, the velocity
mismatch between the particle and the potential needs to be smaller than the
escape velocity of the potential. Based on this view, and for a given velocity
mismatch, the critical value of the total pulse intensity, required to achieve
self-confinement, was determined [1]. In the presented work, pulse propagation
simulations, such as those reported in Fig. 2, comprise a complementary
approach to study the considered XPM induced attraction effect. Specifically,
by keeping the detuning of the soliton fixed at $\Omega_{\rm{S}}=\Omega_{1}$,
but shifting the detuning of the trapped pulse to
$\Omega_{\rm{TR}}=\Omega_{2}+\Delta\Omega$, we can enforce a group-velocity
mismatch between both pulses and probe the stability of the meta-atom. For
$\Delta\Omega>0$ it is
$\beta_{1}(\Omega_{\rm{S}})>\beta_{1}(\Omega_{\rm{TR}})$, see Fig. 1(b). Thus,
in a reference frame in which the soliton is stationary, the trapped state
will initially have the propensity to move towards smaller times. This is
demonstrated in Figs. 3(a,b) for the center frequency shift
$\Delta\Omega=0.05~{}\mathrm{rad/fs}$. To assess the fraction of energy of the
trapped state that is retained within the soliton induced potential well, we
consider the quantity

$\displaystyle
e_{\rm{TR}}(z)\equiv\frac{E_{\rm{TR}}(z)}{E_{\rm{TR}}(0)},\quad\text{with}\quad
E_{\rm{TR}}(z)=\int_{-10\,t_{0}}^{10\,t_{0}}|\phi(z,\tau)|^{2}~{}{\rm{d}}\tau.$
(16)

As evident from Fig. 3(e), at $\Delta\Omega=0.05~{}\mathrm{rad/fs}$, the
trapped state is kept almost entirely within the well, i.e.
$e_{\rm{TR}}\approx 1$. In contrast, at $\Delta\Omega=0.25~{}\mathrm{rad/fs}$,
a major share of the trapped pulse escapes the well during the initial
propagation stage [Figs. 1(c,d)], indicated by the small value
$e_{\rm{TR}}\approx 0.3$ [Fig. 3(e)]. Let us note that, when viewing the
considered pulse compounds as meta-atoms, the quantity $1-e_{\rm{TR}}(z)$
specifies the fraction of trapped energy that is radiated away, resembling an
ionization probability for quantum mechanical atoms. A parameter study,
detailing the dependence of $e_{\rm{TR}}$ as function of the center frequency
shift $\Delta\Omega$ is summarized in Fig. 3(f). The transition from trapping
to escape can be supplemented by an entirely classical picture similar as in
Ref. [1]: from a classical point of view we might expect that a particle,
initially located at the center of the well, remains confined to the well if
its “classical” kinetic energy
$T_{\rm{kin}}=\tfrac{1}{2}m\Delta\beta_{1}^{2}=\tfrac{1}{2}|\beta_{2}^{\prime\prime}|^{-1}\left[\beta_{1}(\Omega_{\rm{S}})-\beta_{1}(\Omega_{\rm{TR}})\right]^{2}$
does not exceed the well depth $V_{0}=2\gamma^{\prime\prime}P_{0}$. As evident
from Fig. 3, the findings based on this classical picture complement the
results obtained in terms of direct simulations of the modified NSE (1) very
well. The above results clearly demonstrate the limits of stability of
nonlinear photonics meta-atoms with respect to a group-velocity mismatch
between the trapping soliton and the trapped state. These findings are
consistent with our previous results on the break-up dynamics of two-color
pulse compounds [25].

Figure 3: Characterization of the transition from trapping to escape. (a)
Time-domain propagation dynamics of the soliton and its lowest lying trapped
state ($n=0$) shifted from $\Omega_{\rm{TS}}$ to
$\Omega_{\rm{TS}}+\Delta\Omega$ for $\Delta\Omega=0.05~{}\mathrm{(rad/fs)}$.
(b) Corresponding spectrum. The inverse Fourier transform of the part of the
spectrum enclosed by the box (labeled A) in (b) is shown in the box (labeled
A) in (a). This provides a filtered view of the trapped state with the benefit
of leaving out the soliton part of the total pulse. (c,d) Same as (a,b) for
$\Delta\Omega=0.25~{}\mathrm{(rad/fs)}$. (e) Fraction of trapped energy as
function of the propagation distance. (f) Fraction of trapped energy as
function of the trapped state center frequency shift. Secondary ordinate shows
the potential depth ($V_{0}$) as well as the kinetic energy $T_{\rm{kin}}$ of
the fictitious classical particle. Parameter range in which the particle
cannot escape the well is shaded gray. Movies of the propagation dynamics
shown in (a-d) are provided as supplementary material under Ref. [66].  Figure
4: Transition from trapping to tightly bound, molecule-like two-color pulse
compounds. (a-c) Time-domain propagation dynamics arising from an initial
condition of the form $\phi(t)=r\sqrt{P_{0}}\,\phi_{0}(t)$ (see text). (a)
Trapped state for amplitude parameter $r=0.3$, (b) $r=0.7$, and, (c) $r=1$.
(d-f) Solutions of the coupled ODEs (18), fitted to functions of the form
$U_{m}=U_{0,m}{\rm{sech}}^{\nu_{m}}(t/t_{m})$, for $m=1,2$. (d) Scaled pulse
amplitudes $u_{n}=U_{0,m}/\sqrt{P_{0}}$, (e) pulse durations $t_{n}$, and, (f)
pulse shape exponents $\nu_{m}$, $m=1,2$. In (d), $\tilde{u}_{2}$ indicates
the peak amplitude of a fundamental nonlinear Schrödinger soliton with
wavenumber $\kappa_{2}$.

##  4 Two-color soliton molecules

#### Seeding of tightly bound two-color pulse compounds

When considering initial conditions of the form of Eq. (7), with $A_{1}$ a
fundamental nonlinear Schrödinger soliton as in Eq. (9), and $A_{2}$ a trapped
state as in Eq. (10) with $\phi(t)=r\,\sqrt{P_{0}}\,\phi_{0}(t)$, the XPM
contribution of the weak trapped pulse onto the trapping soliton can be
heightened by increasing the parameter $r$. This is demonstrated in Figs.
4(a-c), where pulse propagation simulations in terms of the modified NSE (1)
are shown for different values of $r$, significantly larger than those
considered in the preceding section. Especially for larger values of $r$
[Figs. 4(b,c)], the intensity exhibits the following dynamics: the mutual
confining action of XPM results in a contraction of both subpulses, prompting
the formation of a narrow localized pulse compound. A similar effect has
previously been suggested by Hasegawa for multimode optical pulses in glass
fibers in Ref. [1], where he writes “[…] as many modes are trapped, the peak
intensity of the packet increases quite analogously to a gravitational
instability, resulting in a further contraction of the packet.” (Ref. [1], p.
417). The results shown in Figs. 4(a-c) demonstrate this effect in the context
of two-color pulse compounds in nonlinear fibers or waveguides with two zero-
dispersion points. Let us note that, for $r\approx 1$, initial conditions as
pointed out above directly generate tightly bound, mutually confined two-color
pulse compounds. They are accompanied by radiation, emanating from the
localized state upon propagation, and can exhibit internal dynamics
reminiscent molecular vibrations [24, 25, 70, 31]. However, such a seeding
procedure generates two-color pulse compounds in a largely uncontrolled
manner. For completeness, we have observed the formation of similar localized
pulse compounds when taking trapped state initial conditions of the form
$\phi(t)=r\,\sqrt{P_{0}}\,\phi_{1}(t)$ for large enough $r$.

#### Simultaneous solutions of the coupled equations

We can surpass the above seeding approach by directly searching for
simultaneous solitary-wave solutions of the coupled nonlinear Eqs. (8) beyond
the linear limit discussed in Sect. 3. Substituting an ansatz for two
subpulses, labeled $m=1,2$, in the form of

$\displaystyle
A_{m}(z,t)=U_{m}(t)\,e^{i(\beta_{0}+\kappa_{m})z},\quad\text{with}\quad m=1,2$
(17)

into Eqs. (8), yields two coupled ordinary differential equations (ODEs) of
second order

$\displaystyle\ddot{U}_{1}-\frac{2}{\beta_{2}^{\prime}}\left[\gamma^{\prime}\left(|U_{1}|^{2}+2|U_{2}|^{2}\right)-\kappa_{1}\right]U_{1}=0,$
(18a)
$\displaystyle\ddot{U}_{2}-\frac{2}{\beta_{2}^{\prime\prime}}\left[\gamma^{\prime\prime}\left(|U_{2}|^{2}+2|U_{1}|^{2}\right)-\kappa_{2}\right]U_{2}=0,$
(18b)

for two real-valued envelopes $U_{m}\equiv U_{m}(t)$, $m=1,2$, with dots
denoting derivatives with respect to time. Under suitable conditions,
solitary-wave solutions for the coupled nonlinear Eqs. (18) can be specified
analytically [71, 72, 60, 73, 30]. Approximate solutions based on
parameterized trial functions can be found, e.g., in terms of a variational
approach [74]. In order to obtain simultaneous solutions $U_{1}(t)$, and
$U_{2}(t)$ under more general conditions, Eqs. (18) need to be solved
numerically. This can be achieved, e.g., by spectral renormalization methods
[75, 76, 77, 78], shooting methods [8, 9], squared operator methods [79],
conjugate gradient methods [80, 81], $z$-propagation adapted imaginary-time
evolution methods [82, 83], or Newton-type methods [84]. Here, in order to
solve for simultaneous solutions of the ODEs (18), we employ a Newton method
that is based on a boundary value Runge-Kutta algorithm [85]. So as to
systematically obtain solutions $U_{1}(t)$ and $U_{2}(t)$, we keep five of the
six parameters that enter Eqs. (18) fixed. Therefore we set
$\beta_{2}^{\prime}$, $\beta_{2}^{\prime\prime}$, $\gamma^{\prime}$, and
$\gamma^{\prime\prime}$ to the values considered througout the preceding
section, and preset the wavenumber
$\kappa_{1}=|\beta_{2}^{\prime}|(2t_{0}^{2})^{-1}\approx 0.0156~{}\mathrm{\mu
m^{-1}}$ of a fundamental nonlinear Schrödinger soliton with
$t_{0}=8\,\mathrm{fs}$ in Eq. (18a). We then sweep the remaining paramter
$\kappa_{2}$ over the wavenumber range $(0.002,0.05)~{}\mathrm{\mu m^{-1}}$,
enclosing the value of $\kappa_{1}$. We start the parameter sweep at
$\kappa_{2}=0.05~{}\mathrm{\mu m^{-1}}$, which vastly exceeds the wavenumber
eigenvalue of the lowest lying trapped state solution at $0.0382~{}\mathrm{\mu
m^{-1}}$. Above this value, we expect $U_{2}$ to vanish, and $U_{1}$ to yield
a fundamental soliton $U_{1}(t)=\sqrt{P_{0}}\,{\mathrm{sech}}(t/t_{0})$ with
$P_{0}=|\beta_{2}^{\prime}|(\gamma^{\prime}\,t_{0}^{2})^{-1}$. We set initial
trial functions for $U_{1}$ and $U_{2}$ with parity similar to the soliton and
the lowest lying trapped state, and continue the obtained solutions to smaller
values of $\kappa_{2}$. The results of this paramter sweep are summarized in
Figs. 4(d-f). We find that all solutions can be parameterized in the form
$U_{m}(t)=U_{0,m}\,{\mathrm{sech}}^{\nu_{m}}(t/t_{m})$, with pulse peak
amplitudes $U_{0,m}$ [Fig. 4(d)], pulse durations $t_{m}$ [Fig. 4(e)], and
pulse shape exponents $\nu_{m}$ [Fig. 4(f)], for $m=1,2$. In agreement with
the results reported in sect. 3.1, we find that a weak nonzero solution
$U_{2}$ with $t_{2}=8~{}\mathrm{fs}$ and $\nu_{2}\approx 1.55$ originates at
$\kappa_{2}\approx 0.038~{}\mathrm{\mu m^{-1}}$. For
$\kappa_{2}<0.038~{}\mathrm{\mu m^{-1}}$, the peak amplitude of the subpulse
$m=1$ continuously decreases while that for $m=2$ increases. Below
$\kappa_{2}\approx 0.007\mathrm{\mu m^{-1}}$, subpulse $U_{1}$ vanishes and
$U_{2}$ describes a fundamental soliton with pulse shape paramter $\nu_{2}=1$
and wavenumber $\kappa_{2}$. To facilitate intuition, we included the
amplitude of a free soliton with wavenumber $\kappa_{2}$, i.e. peak amplitude
$\tilde{U}_{0,2}=\sqrt{2\kappa_{2}/\gamma^{\prime\prime}}$, in Fig. 4(d). Let
us note that the intermediate parameter range $0.007~{}\mathrm{\mu
m^{-1}}<\kappa_{2}<0.038~{}\mathrm{\mu m^{-1}}$ bears tightly coupled pulse
compounds, characterized by subulse amplitudes with similar peak heights, see
Fig. 4(d).

Figure 5:  Resonant radiation of two-color soliton molecules. (a,b) Stationary
propagation of a soliton molecule with subpulse loci at
$\Omega_{1}=-\Omega_{2}=2.828~{}\mathrm{rad/fs}$. (a) Time-domain propagation
dynamics. The inset shows a close-up view of
$|A(z,\tau)|^{2}/{\rm{max}}(|A(0,\tau)|^{2})$ in the range $\tau=-20\ldots
20~{}\mathrm{fs}$ and $z/z_{0}=2\ldots 10$. (b) Corresponding spectrum. (c,d)
Same as (a,b) for soliton molecule order $N=1.8$. Horizontal dashed line in
(d) indicates $z/z_{0}=29.2$. (e) Dispersion profile and grapical solution of
the resonance conditions Eqs. (22) for $z$-oscillation periods of order
$m=-10\ldots 1$. (f) Spectrum at $z/z_{0}=29.2$ (g-l) Same as (a-f) for a
soliton molecule with subpulse loci at $\Omega_{1}=-2.674~{}\mathrm{rad/fs}$
and $\Omega_{2}=2.134~{}\mathrm{rad/fs}$. In (i,j) the soliton molecule order
is $N=1.6$. Horizontal dashed line in (j) indicates $z/z_{0}=28.6$. The time-
domain propagation dynamics is shown in a moving frame of reference where
$\tau=t-\tilde{\beta}_{1}z$. In (a,c) $\tilde{\beta}_{1}=0~{}\mathrm{fs/\mu
m}$. In (g,i) $\tilde{\beta}_{1}=0.509~{}\mathrm{rad/fs}$. Propagation
distance is scaled by $z_{0}=32~{}\mathrm{\mu m}$. Movies of the propagation
dynamics shown in (a-d) and (g-j) are provided as supplementary material under
Ref. [66].

###  4.1 Two-color soliton pairs

Upon closely assessing the results shown in Figs. 4(d-f), we find that at
$\kappa_{2}=0.0156~{}\mathrm{\mu m^{-1}}$, a pair of matching solutions with
plain hyperbolic-secant shape $U_{m}(t)=U_{0,m}\,{\mathrm{sech}}(t/t_{0})$,
$m=1,2$, is attained. This can be traced back to the uniformity of Eqs. (18a)
and (18b) for the considered set of parameters. Formally, by assuming
$\kappa\equiv\kappa_{1}=\kappa_{2}$ and $U\equiv U_{1}=U_{2}$, both equations
take the form of a standard NSE with modified paramters

$\displaystyle-\frac{\beta_{2}^{\prime}}{2}\frac{{\rm{d}}^{2}}{{\rm{d}}t^{2}}U(t)+3\gamma^{\prime}|U(t)|^{2}U(t)=\kappa
U(t),$ (19)

where, for convenience only, we used the parameters of Eq. (18a). The real-
valued pulse envelope $U$ should therefore be identified by the peak intensity
$\tilde{P}_{0}=|\beta_{2}^{\prime}|(3\gamma^{\prime}t_{0}^{2})^{-1}$, and thus
$u_{1}=u_{2}=\sqrt{1/3}\approx 0.57$ in Fig. 4(d). Hence, at
$\kappa_{2}=0.0156~{}\mathrm{\mu m^{-1}}$, both subpulses resemble true two-
color _soliton_ pairs: the pulse envelopes $U_{1}$ and $U_{2}$ both specify a
fundamental NSE soliton; for each pulse, its binding partner modifies the
nonlinear coefficient of the underlying NSE through XPM, helping the pulse
sustain its shape. Consequently, both pulses can only persist conjointly as a
bonding unit. This special case is consistent with a description of two-color
pulse compounds in terms of incoherently coupled pulses [30]. By considering
the ansatz Eq. (7), we can plug in the obtained pulse envelopes for $U_{1}$
and $U_{2}$ and resubstitute the parameters that define the propagation
constant in sect. 2 to obtain

$\displaystyle
A(z,t)=F(z,t)\,\cos\left(\sqrt{\frac{6\beta_{2}}{|\beta_{4}|}}t\right)\,e^{-i\beta_{0}z},\quad\text{with}\quad
F(z,t)=\sqrt{\frac{8\beta_{2}}{3\gamma
t_{0}^{2}}}\,{\rm{sech}}\left(\frac{t}{t_{0}}\right)\,e^{i\kappa
z},\quad\text{and}\quad\kappa=\frac{\beta_{2}}{t_{0}^{2}}.$ (20)

Let us note that $F$ is equivalent to the fundamental meta-soliton obtained in
Ref. [26], which becomes evident when substituting
$\epsilon=t_{0}^{-1}[3\beta_{2}/(2|\beta_{4}|)]^{-1/2}$ and
$\mu_{0}\epsilon^{2}=\beta_{2}/t_{0}^{2}$. This fundamental meta-soliton was
first formulated by Tam _et al._ , when studying stationary solutions for the
modified NSE (1) by putting emphasis on the time-domain representation of the
field in terms of a multi-scales analysis [26]. This unveiled a large
superfamily of solitons, now referred to as generalized dispersion Kerr
solitons. We would like to point out that within the presented approach, i.e.
by putting emphasis on the frequency-domain representation of two-color pulse
compounds, the fundamental meta-soliton is derived with great ease.
Furthermore, both approaches complement each other very well. We should note
that the above two-color soliton pairs resemble vector solitons studied in the
context of birefringent optical fibers [86, 87, 88, 61, 89, 90]. The
stationary propagation of the two-color soliton pair defined by Eq. (20) in
terms of the modified NSE (1) is demonstrated in Figs. 5(a,b). The inset in
Fig. 5(a) provides a close-up view onto the localized pulse, indicating
interference fringes with period $\Delta
t\approx\sqrt{|\beta_{4}|/(6\beta_{2}\pi^{2})}\approx 1.3~{}\mathrm{fs}$ that
are due to the cosine in Eq. (20). These interference fringes appear
stationary since the propagation scenarion exhibits the symmetry
$\beta(\Omega_{1})=\beta(\Omega_{2})$ and $\kappa_{1}=\kappa_{2}$. A
spectrogram of the propagation scenario at $z/z_{0}=29.17$ is shown in Fig.
6(a). A small amount of residual radiation can be seen to lie right on the
curve $\beta_{1}(\Omega)z$, given by the short-dashed line in Fig. 6(a). It
was emitted by the pulse compound during the initial propagation stage and is
caused by the presence of higher orders of dispersion at the individual
subpulse loci, which were neglected in the simplified description leading to
Eqs. (20).

###  4.2 Kushi-comb-like multi-frequency radiation

Previously, it was shown that $z$-periodic amplitude and width oscillations of
two-color soliton molecules can be excited in a systematic manner by
increasing their initial peak amplitude by some factor $N$ according to
$F(z,t)\leftarrow NF(z,t)$ [26, 55]. In analogy to usual nonlinear Schrödinger
solitons, values $N>1$ define higher order metasolitons. Recently, we have
performed a comprehensive analysis of the amplitude oscillations of such
higher order metasolitons, indicating that with increasing $N$, the number of
spatial Fourier-modes needed to characterize their periodic peak-intensity
variation, increases [55]. In other words, with increasing strength of
perturbation of a soliton molecule, its dynamics changes from harmonic to
nonlinear oscillations.

#### Degenerate multi-frequency radiation

To demonstrate amplitude and width oscillations, we show the propagation
dynamics of a symmetric soliton molecule of order $N=1.8$, based on the two-
color soliton pair (20), in Figs. 5(c,d). As can be seen from the time-domain
dynamics in Fig. 5(c), the localized pulse exhibits periodic amplitude and
width variations [close-up view in Fig. 5(c)], and emits radiation along
either direction along the coordinate $t$ in a symmetric fashion. Quite
similar dynamics where obtained using the seeding approach in Figs. 4(b,c).
The oscillation of the soliton molecule is also clearly visible in the
spectrum shown in Fig. 5(d). As evident from Fig. 5(f), at $z/z_{0}\approx
29.17$ it exhibits comb-like bands of frequencies in the vicinity of the
subpulse loci $\Omega_{1}$ and $\Omega_{2}$. The location of these newly
generated frequencies can be understood by extending existing approaches for
the derivation of resonance conditions [91, 92, 93, 94, 95] to two-color pulse
compounds [70, 55]. Below, we summarize these resonance conditions, which
where obtained by assuming a dynamically evolving pulse compound of the form
[70]

$U_{m}(z,t)=\sum_{\ell}C_{m\ell}(t)\,\exp\left[i\left(\kappa_{m}+K_{\ell}\right)z\right],\quad\text{with}\quad
m\in(1,2),~{}\ell\in\mathbb{Z}.$ (21)

In Eq. (21), $C_{m\ell}$ are expansions coefficients, and $\kappa_{m}$
indicate wavenumbers that govern the $z$-propagation of each subpulse. The
wavenumbers of the higher harmonics of the $z$-oscillation period are
$K_{\ell}=2\pi\ell/\Lambda$, with $\Lambda$ refering to the $z$-oscillation
wavelength of the pulse compound and $\ell$ labeling the corresponding order.
Based on this ansatz, the resonance conditions

$\displaystyle D_{m}(\Omega_{RR})-\kappa_{m}=K_{\ell},\quad\text{with}\quad
m\in(1,2),~{}\ell\in\mathbb{Z},\quad\text{and}$ (22a) $\displaystyle
D_{m}(\Omega_{RR})-2\kappa_{m}+\kappa_{m^{\prime}}=K_{\ell},\quad\text{with}\quad
m,m^{\prime}\in(1,2),~{}m\neq m^{\prime},~{}\ell\in\mathbb{Z},$ (22b)

with dispersion profiles
$D_{m}(\Omega)\equiv\beta(\Omega)-\beta(\Omega_{m})-\beta_{1}(\Omega)(\Omega-\Omega_{m})$
for $m=1,2$, can be derived [70]. In Eqs. (22), $\Omega_{RR}$ specifies those
frequencies at which resonant radiation (RR) is excited. While Eq. (22a)
defines resonance conditions for the generation of Cherenkov radiation by each
subpulse, Eq. (22b) defines additional resonance conditions indicative of
four-wave mixing (FWM) processes involving both subpulses.

For the considered soliton molecule of order $N=1.8$, we find $\Lambda\approx
63~{}\mathrm{\mu m}\approx 2z_{0}$ [with $z_{0}=32~{}\mathrm{\mu m}$, see Fig.
5(d)]. In this case, the aforementioned symmetry $\kappa_{1}=\kappa_{2}$
renders Eqs. (22a) and (22b) degenerate. As evident from the graphical
solution of Eqs. (22) in Fig. 5(e), the resonance conditions predict the newly
generated frequencies in Fig. 5(f) very well. A spectrogram of the propagation
scenario at $z/z_{0}=29.17$ is shown in Fig. 6(b). Therein, the multi-peaked
spectral bands, at which the oscillating soliton molecule sheds radiation, are
reminiscent of the shape of traditional Japanese Kushi combs.

#### Non-degenerate multi-frequency radiation

Let us note that, due to the wide variety of two-color pulse compounds with
different substructure, their emission spectra manifest in various forms. For
example, considering a pair of group-velocity matched detunings different from
the one considered above, the degeneracy among Eqs. (22) can be lifted.
Subsequently we take $\Omega_{1}=-2.674~{}\mathrm{rad/fs}$ and
$\Omega_{2}=2.134~{}\mathrm{rad/fs}$, for which
$\beta_{1}^{\prime}=0.514~{}\mathrm{fs/\mu m}$,
$\beta_{1}^{\prime\prime}=0.514~{}\mathrm{fs/\mu m}$,
$\beta_{2}^{\prime}=-2.576~{}\mathrm{fs^{2}/\mu m}$, and,
$\beta_{2}^{\prime\prime}=-1.278~{}\mathrm{fs^{2}/\mu m}$. In terms of the
coupled ODEs (18) we then determine a pair of simultaneous solutions which
specify the initial condition

$\displaystyle
A_{0}(t)=U_{0,1}\,{\rm{sech}}^{\nu_{1}}\left(\frac{t}{t_{1}}\right)\,e^{-i\Omega_{1}t}+U_{0,2}\,{\rm{sech}}^{\nu_{2}}\left(\frac{t}{t_{2}}\right)\,e^{-i\Omega_{2}t},$
(23)

with parameters $U_{0,1}=0.050~{}\mathrm{\sqrt{W}}$,
$U_{0,2}=0.141~{}\mathrm{\sqrt{W}}$, $t_{1}=7.207~{}\mathrm{fs}$,
$t_{2}=7.271~{}\mathrm{fs}$, $\nu_{1}=0.901$, and $\nu_{2}=1.022$. The
stationary propagation of this soliton molecule with non-identical subpulses
is shown in Fig. 4(g,h). As a consequence of the broken subpulse-symmetry, the
interference fringes that characterize the pulse compound are not stationary
any more [close-up view in Fig. 5(g)]. The fact that the pulse compound
remains localized, despite its envelope exhibiting a non-stationary profile,
might be the reason why no such objects could be found using a time-domain
based Newton conjugate-gradient method [26]. Next, we increase the order of
this soliton molecule to $N=1.6$, resulting in the propagation dynamics with
$z$-oscillation period $\Lambda\approx 106~{}\mathrm{\mu m}\approx 3.3z_{0}$
shown in Figs. 5(i,j). In this case, a pronounced mulit-peaked spectral band
of frequencies within the domain of normal dispersion is excited [see Figs.
5(j,l)]. These newly generated frequencies can be linked to multi-frequency
Cherenkov radiation emitted by the subpulse at $\Omega_{2}$, as can be seen
from the graphical solution of the resonance conditions (22a), shown in Fig.
5(k). Let us note that similar coupling phenomena of localized states to the
continuum have have earlier been observed for solitons in periodic dispersion
profiles [93], oscillating bound solitons in twin-core fibers [94], and
dissipative solitons in nonlinear microring resonators [95]. A further band of
frequencies, excited in the vicinity of $\Omega\approx 3.5~{}\mathrm{rad/fs}$
can be attributed to FWM-resonances described by Eq. (22b). A spectrogram of
the propagation scenario at $z/z_{0}=28.6$ is shown in Fig. 6(c), unveiling
that the resonant radiation emanates from the oscillating soliton molecule in
a pulse-wise fashion.

Figure 6:  Spectrograms of selected soliton molecules. (a) Two-color soliton
pair of Figs. 5(a,b) at $z/z_{0}=29.17$, computed using
$\sigma=30~{}\mathrm{fs}$ in Eq. (6). (b) Oscillating, symmetric soliton
molecule of Figs. 5(c,d) at $z/z_{0}=29.17$ for $\sigma=30~{}\mathrm{fs}$,
showing many narrowly spaced resonances reminiscent of the shape of
traditional Japanese Kushi combs. (c) Oscillating, non-symmetric soliton
molecule of Figs. 5(i,j) at $z/z_{0}=28.6$ for $\sigma=20~{}\mathrm{fs}$. The
pulse-wise emission of radiation, synchronized with the periodic amplitude and
width variations of the pulse compound, is clearly visible. In (a-c), the
short-dashed line shows $\beta_{1}(\Omega)\,z$, indicating the delimiting
temporal position of a mode at detuning $\Omega$, emitted at $z=0$. Movies of
the propagation dynamics are provided as supplementary material under Ref.
[66].

##  5 Summary and conclusions

In summary, we have discussed several aspects of the $z$-propagation of two-
color pulse compounds in a modified NSE with positive group-velocity
dispersion coefficient and negative fourth-order dispersion coefficient.
Therefore, we considered the interaction dynamics of two pulses in distinct
domains of anomalous dispersion, group-velocity matched despite a large
frequency gap.

We have demonstrated that their mutual confining action can manifest itself in
different forms, depending on the relative strength of SPM and XPM felt by
each pulse. In the limiting case where the resulting bound states consist of a
strong trapping pulse, given by a soliton, and a weak trapped pulse, we have
shown that optical analogues of quantum mechanical bound states can be
realized that are determined by a Schrödinger-type eigenvalue problem [24].
The resulting photonic meta-atoms even support Rabi-type oscillations of its
trapped states, similar to the recurrence dynamics of wave packets in quantum
wells [67]. We further probed the limits of stability of these meta-atoms by
imposing a group-velocity mismatch between the trapping soliton and the
trapped pulse. With increasing strength of perturbation, parts of the trapped
state escapes the soliton, similar in effect to the ionization of quantum
mechanical atoms. These findings complement our earlier results on the break-
up dynamics of two-color pulse compounds [95].

For the more general case where the mutual confining action between the pulses
is dominated by XPM, we have discussed a simplified modeling approach,
allowing to determine simultaneous solutions for the bound pair of pulses. The
resulting solutions feature the above meta-atoms as limiting cases when the
disparity of the subpulse amplitudes is large. Further, by exploiting
symmetries of the underlying propagation model, a special class of solutions,
forming true two-color soliton pairs [30], was characterized in closed form.
This special class of solutions, referred to as generalized dispersion Kerr
solitons, has also been derived in Ref. [26]. We have presented numerical
results demonstrating the complex propagation dynamics of such pulse
compounds, which we here referred to as two-color soliton molecules.
Specifically, we have shown that soliton molecules exhibit highly robust
vibrational characteristics, a behavior that is difficult to achieve in a
conservative NSE system. These non-stationary, $z$-periodic dynamics of the
subpulses triggers the emission of resonant radiation. The location of the
resulting multi-peaked spectral bands can be precisely predicted by means of
phase-matching conditions [70, 55]. Due to the manifold of soliton molecules
with different substructure, their emission spectra manifest in various
complex forms. Most notably, if the oscillating soliton molecule consists of a
pair of identical subpulses, inherent symmetries lead to degeneracies in the
resonance spectrum, causing their spectrogram trace to resemble the shape of
Japanese Kushi combs. Additional perturbations lift existing degeneracies and
result in more complex emission spectra which are characterized by distinct
spectral bands that can be separately linked to resonant Cherenkov radiation
and additional four-wave mixing processes. The occurrence of such multi-
frequency radiation, especially in the degenerate form, comprises a
fundamental phenomenon in nonlinear waveguides with multiple zero-dispersion
points and sheds light onto the puzzling propagation dynamics of two-frequency
pulse compounds, resembling the generation of radiation by vibrating
molecules.

Finally, let us note that we recently extended the range of systems in which
such two-color pulse compounds are expected to exist. Therefore, we considered
waveguides with a single zero-dispersion point and frequency dependent
nolinearity with a zero-nonlinearity point [96, 97]. In such waveguides,
soliton dynamics in a domain of normal dispersion can be achieved by a
negative nonlinearity [98, 99]. In the corresponding description of pulse
compounds in terms of the simplified model (8), having $\beta_{2}^{\prime}<0$
and $\beta_{2}^{\prime\prime}>0$ then requires $\gamma^{\prime}>0$ and
$\gamma^{\prime\prime}<0$, and the potential well in the eigenproblem
corresponding to Eq. (11) is ensured by $\gamma^{\prime\prime}<0$ [54]. We
studied the above binding mechanism for incoherently coupled two-color pulse
compounds in such waveguides, demonstrating meta-atoms and molecule-like bound
states of pulses that persist in the presence of the Raman effect [31, 54],
allowing to understand the complex propagation dynamics observed in a recent
study on higher-order soliton evolution in a photonic crystal fiber with one
zero-dispersion point and frequency dependent nonlinearity [100].

## Acknowledgements

We acknowledge support from the Deutsche Forschungsgemeinschaft (DFG) under
Germany’s Excellence Strategy within the Cluster of Excellence PhoenixD
(Photonics, Optics, and Engineering – Innovation Across Disciplines) (EXC
2122, projectID 390833453).

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