# An atomic bright vector soliton as an active particle

Timo Eichmann  James R. Anglin  State Research Center OPTIMAS and Fachbereich
Physik,Technische Univerität Kaiserslautern,D-67663 Kaiserslautern, Germany

###### Abstract

Solitons in general are configurations of extended fields which move like
isolated particles. Vector bright solitons can occur in a two-component self-
attractive Bose-Einstein condensate. If the components of the condensate have
different chemical potentials, the total spin of the soliton can serve as an
internal energy depot that makes the soliton into an _active_ particle, able
to move against an external force using energy carried within the particle—if
there is a dynamical mechanism for steadily transferring energy from soliton
spin into soliton motion. Here we present such a dynamical mechanism, embed it
in an experimentally feasible way within the larger system of a spinor
condensate mean field, and show how the mechanism works to realize a solitonic
active particle. In what can be considered a toy model for the project of
going beyond toy models for active particles, we test the robustness of the
activity mechanism by exploring a range of deformations to the simplest model
for embedding the nonlinear mechanism in the condensate system.

##  I I. Introduction

One of the most elementary tasks in nature and everyday life is the transport
of objects. Nature has developed ‘motor proteins’ to achieve transport in
cells Kinesin1 ; Kinesin2 , while humans build engines for larger scale
transport. In both cases some kind of internal energy is used to perform
motional work (including work against friction). The phenomenological model of
an _active particle_ Active1 ; Active2 ; Daemon4 ; Active3 ; Active4 ; Active5
has emerged as a useful concept for modelling internally powered motion in
general. Fuel does not just automatically convert itself into work, however.
Macroscopic engines are non-trivial dynamical mechanisms, subject to laws of
thermodynamics whose emergence from microscopic mechanics is still being
studied, and the microscopic mechanisms of biological motors are not yet
understood. In this paper we contribute a concrete example of an active
particle realized within an extended environment yet described
microscopically.

A bright matter-wave soliton is a small cloud of cohesive gas Khaykovich ;
Strecker ; BStheo1 ; BStheo2 ; BStheo3  that has mechanical properties like a
particle. External force can accelerate the small cloud as a whole, and its
motion is well described with a single collective coordinate that can carry
both kinetic and potential energy. If the soliton is created in a multi-
component Bose-Einstein condensate SpinExp0 ; SpinExp1 ; SpinExp2 ; SpinExp3 ;
SpinTheo1 , it can also have internal energy. If for example the bright vector
soliton is placed in a constant magnetic field, and is composed of atoms of
one atomic species with atoms in spin states
$|-\rangle=|F^{(1)},m_{F}^{(1)}\rangle$ or
$|+\rangle=|F^{(2)},m_{F}^{(2)}\rangle$, then the soliton has a large range of
internal energies depending on the total spin of all its constituent atoms. By
the experimentally feasible technique of Rabi driving, this internal energy
can be coupled to the soliton’s motion in such a way as to realize an active
particle, within a real, extended system that can nevertheless be understood
in full dynamical detail.

###  I.1 I.1 The coupling problem

Taking data from Khaykovich , we can assume a soliton of about 6000 atoms of
7Li, having a total mass of $\sim 7\cdot 10^{-26}\text{ kg}$ and a length of
$\sim 2\text{ }\mathrm{\upmu m}$. It requires roughly $1.4\cdot 10^{-30}\text{
J}$ to lift this soliton vertically by its own length. If as in Khaykovich  we
allow the soliton to be in a magnetic field of $\sim 500\text{ G}$, a single
atom that changes its state from $|1,1\rangle$ to $|1,0\rangle$ would change
the internal energy of the soltion by $\sim 5\cdot 10^{-26}\text{ J}$, which
is enough energy to lift the soliton $\sim 3000$ times its own length against
gravity. With no less than 6000 atoms in the soliton, it is clear that the
soliton’s total spin can easily store vastly abundant energy for active
soliton motion.

It is also clearly possible to couple a vector soliton’s spin to its
collective motion. Examples using a spin-orbit coupling SolMov1  or a Rabi
coupling SolMov2  show how this can be done. The ‘active particle’ model
presents the challenge, however, of non-destructively moving a soliton over a
potentially long distance, against an external force, using an on-board energy
depot. Oscillations of the soliton as in SolMov1 , which for large Zeeman
frequency will have high frequency and small amplitude, do not meet this
challenge; neither does the limited average motion of a spreading soliton
shown in SolMov2 . Indeed the dynamical problem of coupling our vast internal
spin energy into steady motional work is not merely one of selecting an
arbitrary mechanism out of infinitely many possibilities. On the contrary, it
is not immediately clear that any such mechanisms can even exist.

The very abundance of the internal energy is the root of the problem. The
internal spin energy is large because the Zeeman frequency
$10^{-26}$J$/\hbar=10^{9}\text{ s}^{-1}$ is about a million times higher than
the kHz frequencies that are typical of motional collective modes in trapped
condensates. There is no trivial way to achieve the extreme downconversion of
splitting one quantum of energy at GHz frequency into a million kHz quanta
which can be fed efficiently into collective motion of the soliton, steadily
over a long time scale. Instead most couplings between soliton spin and
soliton motion will merely be adiabatically suppressed BOA1 ; BOA2  in the
regime of high Zeeman frequency, and do essentially nothing. The fact that
high-frequency degrees of freedom may dress and renormalize low-frequency
sectors, but not steadily pump energy into them, is after all a basic
principle throughout modern physics RG1 ; RG2 ; RG3 .

There do exist loopholes in this principle, however, and one is to trade off
length and time scales in a Chirikov resonance Chirikov ; Daemon1 . In this
paper we show how to implement such a ‘Hamiltonian daemon’ Chirikov resonance
engine using a Rabi coupling between the internal states $|+\rangle$ and
$|-\rangle$ of atoms in a vector bright soliton, by letting the Rabi coupling
depend periodically on the space coordinate along which the soliton is to be
lifted. This spatially periodic Rabi coupling will effectively act as a
ladder, up which the vector soliton can climb under its own active power.

###  I.2 I.2 Paper outline

In section II we will define this spatially periodic Rabi coupling Rabi  and
show how it provides a Chirikov resonance Chirikov  to enable energy flow from
the internal to the mechanical energy of the soliton. We will then use the
Gross-Pitaevskii equation GPE  to describe the evolution of the soliton
classically, showing numerical results for the motion of the soliton and the
time evolutions of its internal and motional energies. We use the Split-Step
Fourier method with an adaptive error control for our numerical simulations
Num1 ; Num2 .

After thus confirming that the solitonic active particle can indeed work, we
pause in section III to relate the one-dimensional Gross-Pitaevskii classical
field theory to the two-degree-of-freedom Hamiltonian daemon system of Daemon1
. This approximate mapping can be used to estimate parameter ranges for
different possible behaviors of the active soliton, and to explain some non-
trivial features of its (in general) more complex dynamics. Our results in
this section use the collective coordinate variational approach Var1 , which
is a standard technique for the analytical treatment of solitons Var2 ; Var3 ;
Var4 ; Var5 ; Var6 ; Var7 ; RabiSwitch ; ResonantControl .

Having seen how the ideal active soliton model succeeds in incorporating the
Chirikov engine in the condensate mean field, in section IV we explore
embedding the daemon mechanism in the field theory in more complex ways. In
particular we consider couplings which introduce multiple Chirikov resonances,
or broader ones. Since additional Chirikov resonances turn out to allow
soliton motions at different speeds, we also examine the possible case of
‘negative’ running speed, in which instead of being lifted against gravity the
soliton runs down a potential slope at constant speed, using its internal spin
as a brake. Finally we analyze a range of cases in which the Gross-Pitaevskii
nonlinear Schrödinger equation is no longer integrable because the scattering
lengths of the two atomic components are unequal. Even though the soliton is
then no longer a soliton in the narrowest sense, we find that the active
particle mechanism remains robust as long as a stable solitary wave still
exists.

We close in section V with a summary of our results.

##  II II. Ideal Active Soliton Model

###  II.1 II.1 Setup

For our examination we assume a two-component Bose-Einstein condensate which
is confined to one spatial dimension and described by a two-component spinor
mean field $\Psi$. The dynamics of such a spinor field $\Psi$, where the atoms
have two internal states $\left|+\right\rangle$ and $\left|-\right\rangle$ and
an equal negative inter-component and intra-component scattering length, can
be calculated within the mean field approximation using the dimensionless
Gross-Pitaevskii equation (GPE) GPE ; BStheo3

$\displaystyle i\dot{\Psi}$
$\displaystyle=-\frac{1}{2}\Psi^{{}^{\prime\prime}}-\left(\Psi^{\dagger}\Psi\right)\Psi+V\Psi+M\Psi+\Gamma\Psi,$
(1)

where $\Psi=\begin{pmatrix}\Psi_{+}\\!\left(X,T\right)\\\
\Psi_{-}\\!\left(X,T\right)\end{pmatrix}$ is the dimensionless two-component
wave function, with $X$ denoting the dimensionless space coordinate, $T$ the
dimensionless time, and $V=V\\!\left(X\right)$ a space dependent external
potential. Here and in the following we will denote (where possible)
dimensionless quantities by uppercase letters, and their counterpart with
dimension by the corresponding lowercase letter. The matrices $M$ and $\Gamma$
are defined by

$\displaystyle M$ $\displaystyle=\begin{pmatrix}M_{+}&0\\\
0&M_{-}\end{pmatrix},$ $\displaystyle\Gamma=\begin{pmatrix}0&\Gamma_{+}\\\
\Gamma_{-}&0\end{pmatrix}.$ (2)

$M$ can model different chemical potentials $M_{\pm}$, which can result from
different magnetic energies or different trapping strengths of the
$\left|\pm\right\rangle$ species. $\Gamma$ models a Rabi coupling between both
components.

In adopting this simple form of the dimensionless GPE we have assumed that all
one-dimensional interaction constants are negative and equal. For some species
of condensate atoms this is naturally true to a good approximation, but if the
three-dimensional scattering lengths are unequal then adjustment can be made
by tuning the transverse confinement strengths for the different species
differently. We will use this dimensionless form of the Gross-Pitaevskii
equation (1) throughout section II, but we pause briefly here to relate it to
physical units.

####  II.1.1 (a) Physical units

For repulsively interacting condensates one typically introduces dimensionless
variables based on the so-called _healing length_ which is proportional to the
square root of the interaction constant times a typical density scale, but for
attractive interactions this is awkward; while we can easily take the absolute
value of the interaction constant, there is no ambient density scale because
an attractively interacting condensate is unstable to breaking up into bright
soliton droplets. For three-dimensional scattering length $a$, radial trapping
frequency $\omega_{\rho}$, atom mass $m$ and total atom number $N$, however,
we can always define a characteristic velocity

$\displaystyle v_{0}=\frac{2N}{\pi}\omega_{\rho}a,$ (3)

and measure time, space and the density of the wave function in units of

$\displaystyle t_{0}$ $\displaystyle=\frac{\hbar}{m}\frac{1}{v_{0}^{2}},$
$\displaystyle x_{0}$ $\displaystyle=\frac{\hbar}{m}\frac{1}{v_{0}},$
$\displaystyle\psi_{0}$ $\displaystyle=\sqrt{\frac{m}{\hbar}\frac{v_{0}}{n}},$
(4)

to achieve a description in dimensionless variables. For the experimental data
of Khaykovich , for example, we obtain $v_{0}\approx 3.6\text{ mm/s}$, i.e.
$t_{0}\approx 0.12\text{ ms}$ and $x_{0}\approx 0.43\text{ }\mathrm{\upmu m}$.

####  II.1.2 (b) Coupling terms

Without external potential and Rabi coupling, i.e. $V=\Gamma=0$, equation (1)
has solutions in form of vector bright solitons. The fact that these
hyperbolic secant solutions are exact without $V$ and $\Gamma$ is not as
important to us as the fact that these soliton forms remain excellent
approximate solutions in the case where there are non-vanishing external and
coupling potentials RabiSwitch . Without loss of generality we will choose in
the following

$\displaystyle M$ $\displaystyle=\frac{1}{2}\Omega\begin{pmatrix}1&0\\\
0&-1\end{pmatrix},$ (5)

with the real, positive parameter $\Omega$ describing the energy gap between
the two chemical potentials of the two components of the condensate. To
achieve an energy transfer from the internal to the mechanical degrees of
freedom, we choose the coupling

$\displaystyle\Gamma$
$\displaystyle=\Gamma_{0}\begin{pmatrix}0&\frac{1}{2}\exp\\!\left(iKX\right)\\\
\frac{1}{2}\exp\\!\left(-iKX\right)&0\end{pmatrix},$ (6)

with the real, positive parameter $K$ describing the spatial period of the
Rabi coupling with strength $\Gamma_{0}$ and the external potential

$\displaystyle V\\!\left(X\right)=GX,$ (7)

with the real, positive parameter $G$ describing the strength of the linear
potential against which the soliton is to be lifted.

###  II.2 II.2 Numerical results

We solve (1) numerically, starting with a bright soliton which has an almost
vanishing second component and an initial speed $v=1.2v_{0}$:

$\displaystyle\Psi_{+}\\!\left(X,T=0\right)$
$\displaystyle=\frac{1}{2}\sqrt{0.998}\,\mathrm{sech}\\!\left(\frac{1}{2}X\right)\exp\\!\left(1.2iX\right)$
$\displaystyle\Psi_{-}\\!\left(X,T=0\right)$
$\displaystyle=\frac{1}{2}\sqrt{0.002}\,\mathrm{sech}\\!\left(\frac{1}{2}X\right)\exp\\!\left(1.2iX\right).$
(8)

It is important that this initial speed is greater than the critical speed
$v_{\text{c}}=\frac{\Omega}{K}v_{0}$ (see section III.2), but any other
initial speeds above $v_{\text{c}}$ would produce qualitatively similar
evolution. We take $K=\Omega=1$,
$\Gamma_{0}=\pi/\left(800\,\mathrm{csch}\\!\left(\pi\right)\right)$ and
$G=1/1600$ as system parameters; these precise values have no qualitative
significance for the evolution, but their orders of magnitude do (again see
section III.2).

Figure 1:  The expectation value $x/x_{0}$ of the soliton plotted over time
for the parameters given in the text (solid), and for the uncoupled case
(dashed). While the uncoupled case shows a ballistic curve the velocity stays
about constant if the coupling is apparent. The inset shows the density of the
solitonic field around $t/t_{0}=1700$.

The behavior of the soliton can be concluded from Fig. 1, where the
expectation value of the space coordinate

$\displaystyle\langle
X\rangle=\int\limits_{-\infty}^{\infty}\\!\mathrm{d}X\,\Psi^{\dagger}X\Psi$
(9)

is plotted over time $T$ for the former described parameters and initial
fields as solid, black line. The uncouple case with the same parameters,
except $\Gamma_{0}=0$, is shown as dashed line. While in the uncoupled case
the soliton shows a ballistic curve, the position of the soliton increases
linearly in time with coupling. Keeping a constant speed greater zero means
also that the soliton keeps its kinetic energy, while increasing its potential
energy linearly. Since the energy of the total system is constant, this energy
can only originate from the internal Zeeman energy. The inset in Fig. 1 shows
the density $\Psi^{\dagger}\Psi$ at $T\approx 1700$ for about a range of $40$
units of $X$ and $30$ units of $T$. There are no changes in the density
notable.

To demonstrate that the soliton indeed shows the dramatic behavior of
transferring energy from the internal degrees of freedom into motional work,
we directly examine the time evolution of the different terms in the system’s
energy. The Gross-Pitaevskii equation (1) corresponds to the Hamiltonian
density

$\displaystyle\mathcal{H}=\frac{1}{2}\left(\Psi^{\dagger}\right)^{\prime}\Psi^{\prime}-\frac{1}{2}\left|\Psi^{\dagger}\Psi\right|^{2}+\Psi^{\dagger}V\Psi+\Psi^{\dagger}\left(M+\Gamma\right)\Psi.$
(10)

Within this total energy we can identify the various kinetic, potential,
internal (‘fuel’), nonlinear and Rabi coupling terms:

$\displaystyle H_{\text{kin}}$
$\displaystyle=\frac{1}{2}\int\limits_{-\infty}^{\infty}\\!\mathrm{d}X\,\left(\Psi^{\dagger}\right)^{\prime}\Psi^{\prime},$
(11) $\displaystyle H_{\text{pot}}$
$\displaystyle=\int\limits_{-\infty}^{\infty}\\!\mathrm{d}X\,\Psi^{\dagger}V\Psi,$
(12) $\displaystyle H_{\text{fuel}}$
$\displaystyle=\int\limits_{-\infty}^{\infty}\\!\mathrm{d}X\,\Psi^{\dagger}M\Psi,$
(13) $\displaystyle H_{\text{nl}}$
$\displaystyle=-\frac{1}{2}\int\limits_{-\infty}^{\infty}\\!\mathrm{d}X\,\left|\Psi^{\dagger}\Psi\right|^{2},$
(14) $\displaystyle H_{\Gamma}$
$\displaystyle=\int\limits_{-\infty}^{\infty}\\!\mathrm{d}X\,\Psi^{\dagger}\Gamma\Psi,$
(15)

and denote the corresponding physical energies as $E_{x}=\hbar\omega H_{x}$
for $\omega=\Omega/t_{0}$. We define further the total mechanical energy of
the soliton’s motional degree of freedom as

$\displaystyle E_{\text{mech}}=E_{\text{kin}}+E_{\text{pot}}.$ (16)

Figure 2:  Evolution of the different energy terms of the two-component mean
field system (blue) and the collective coordinate model (red, see section
III), and the velocity $v/v_{0}$. (a) The energy in the fuel term decreases
linearly during the running phase while the mechanical energy increases
linearly, such that the sum of both is about constant. The change of the
energy in the coupling term (b) and in the non-linear term (c) are two orders
of magnitude below the former energy changes—negligible in comparison. The
velocity $v/v_{0}$ (d) shows oscillations around a constant value during the
running phase. Outside the running phase it decreases linearly.

Energies corresponding to (11)-(15) are plotted in Fig. 2 over time (blue
curves). It can be seen in (a) that the total change in the fuel energy
$E_{\text{fuel}}$ and in the mechanical energy $E_{\text{mech}}$ is of order
$\hbar\omega$, and that the sum of both is approximately constant. (We have
shifted the origin of $E_{\text{mech}}$ for convenience.) The maximal changes
in the coupling energy $E_{\Gamma}$ (b) and the energy of the non-linear term
$E_{\text{nl}}$ (c) are in contrast only of order $0.01\hbar\omega$,
insignificant compared to the energy transfer between $E_{\text{pot}}$ and
$E_{\text{fuel}}$. What the couplings are therefore doing is sustaining this
steady energy transfer from $E_{\text{fuel}}$ to $E_{\text{pot}}$, rather than
directly contributing their own energy to the motion. The velocity of the
soliton can be calculated as

$\displaystyle
v/v_{0}=\int\limits_{-\infty}^{\infty}\\!\mathrm{d}X\,\Psi^{\dagger}\Psi^{\prime},$
(17)

and is shown in Fig. 2(d). During the time at which the energy transfer
happens the velocity stays about constant. Outside this time the velocity
decreases linearly, as one expects from a particle in a linear potential.

###  II.3 II.3 Persistence of solitary wave form

The one-dimensional nonlinear Schrödinger equation with $V=\Gamma=0$ is
integrable and the sech-form of our initial state (8) is a _bright soliton_
solution which maintains its spatial profile exactly as it moves at arbitrary
speed. With the addition of our Rabi coupling term $H_{\Gamma}$ and the
potential gradient, the system is (presumably) no longer integrable. Our
initial state nonetheless evolves as a stable solitary wave, very close to its
initial form. We will therefore continue to refer to this moving structure as
‘the soliton’, for the sake of brevity.

Figure 3:  The density for the first (second) component plotted in blue (red),
for the three different times $T=0$ (a), $T=700$ (b) and $T=1600$ (c). It can
be seen that the amplitude of the first component decreases while the
amplitude of the second increases which leads to a lower fuel energy. The
shape stays the same during the evolution.

Figure 3 shows snapshots of the density
$\rho_{\pm}=\left|\Psi_{\pm}\right|^{2}$ of the first (second) component of
the soliton in blue (red) for the times $T=0$ (a), $T=700$ (b) and $T=1600$
(c). As mentioned above, the initial state has very small density $\rho_{-}$
in the second component. It can be seen that during the process the density
$\rho_{+}$ decreases while $\rho_{-}$ increases, which leads to the decrease
of the fuel energy $E_{\text{fuel}}$; this is the energy which lifts the
soliton against the external force. It can also be seen from Fig. 3 that both
density profiles remain qualitatively quite similar (though changing in
relative size) over the whole evolution, during which the soliton moves many
times its own width. How exact can this statement be made?

The general form of a bright soliton for our GPE with $V,\Gamma\to 0$ is

$\displaystyle\Psi_{\pm}^{\text{bs}}\\!\left(X,T\right)=\frac{1}{2}\sqrt{N_{\pm}}\,\mathrm{sech}\\!\left(\alpha_{\pm}\left(X-Q\right)\right)e^{i\Phi_{\pm}},$
(18)

where $Q$ depends linearly on time, $\Phi_{\pm}$ depend on time and space,
$N_{\pm}\in\left[0,1\right]$ (such that $N_{+}+N_{-}=1$) are constant
parameters, and $\alpha_{\pm}=\frac{1}{2}$. From the evolution that we have
just seen in Fig. 3, it is clear that $N_{\pm}$ do not remain constant once
$\Gamma,V\not=0$. The small but finite change in the non-linear energy
$E_{\text{nl}}$ seen in Fig. 2(c) suggests that $\alpha_{\pm}$ may not be
exactly constant, either. The question is how far the time-dependent
$\Psi_{\pm}$ can still be closely approximated by _some_
$\Psi_{\pm}^{\text{bs}}\\!\left(X,T\right)$, for some time-dependent set of
parameters. We will find that a close approximation is indeed possible within
this bright soliton family, allowing us to pursue a variational model in
section III.

To show this we fit the densities $f_{\pm}=|\Psi_{\pm}^{\text{bs}}|^{2}$ to
the numerically obtained densities, using $N_{\pm}$, $Q$, and $\alpha_{\pm}$
as tunable fit parameters. As a measure for the deviation of the exactly
evolving solution from the soliton form, we define the error
$\delta=\delta\\!\left(T\right)$ between the density
$\rho_{\pm}=\left|\Psi_{\pm}\\!\left(X,T\right)\right|^{2}$ and the fitted
$f_{\pm}=\left|\Psi_{\pm}^{\text{bs}}\\!\left(X,T\right)\right|^{2}$ as

$\displaystyle\delta\\!\left(T\right)=\sqrt{\int\limits_{-\infty}^{\infty}\\!\mathrm{d}X\,\left(\left|\rho_{+}-f_{+}\right|^{2}+\left|\rho_{-}-f_{-}\right|^{2}\right)}.$
(19)

We determine our time-dependent fit parameters as those which minimize this
error $\delta\\!\left(T\right)$ at each instant $T$.

Figure 4:  Deviation over time between the numerically exact densities and the
soliton ansatz densities for (18) with instantaneous best-fit parameters.
Panel (a) shows the error when $\alpha_{\pm}$ are not fitted but held at
$\alpha_{\pm}=1/2$; panel (b) shows the error when $\alpha_{\pm}$ are fitted
as well.

Figure 4(b) shows $\delta$ plotted over time. The deviation between the exact
and fitted-soliton densities remains small, confirming that the densities keep
close to the form (18). Figure 4(a) shows that the error is not worsened much
by leaving out $\alpha_{\pm}$ as fit parameters and simply keeping
$\alpha_{\pm}=\frac{1}{2}$. As shown in Fig. 5, the properly fitted
$\alpha_{\pm}(T)$ do show some understandable trends over longer times, as
well as fluctuations on shorter time scales. Substantial departures of
$\alpha_{\pm}$ from $1/2$ only occur at the earliest and latest times,
however, when the corresponding $\rho_{\pm}$ is small.

The error involved in replacing $\alpha_{\pm}$ with $1/2$ does not seem,
therefore, to represent a significant distortion of the overall $\Psi_{\pm}$.
In section III we will accordingly pursue a time-dependent variational
description of our soliton with $\alpha_{\pm}=1/2$; the expected gain in
accuracy from including $\alpha_{\pm}$ as a variational parameter does not
appear to be worth the increased complexity.

Figure 5:  Width parameters $\alpha_{+}$ ($\alpha_{-}$) plotted in blue (red)
over time. The total evolution is shown in (a), while (b) and (c) are
enlargements at early and at intermediate times. In general the $+$ component
in the soliton slowly becomes narrower during the evolution, and the $-$
component broadens after a quick early narrowing; this occurs while the total
atomic population also migrates from the $+$ spin component to the $-$, but it
is not what we would expect for a vector bright soliton without $\Gamma$ or
$G$, since with equal scattering lengths the width of a bright soliton depends
only on the total number $N$, not the distribution in spin. When the density
of a component is low, large variations in the corresponding width are visible
(b). Otherwise the widths show rapid oscillations with a small amplitude (c).

##  III III. Variational approximation for the active bright soliton

###  III.1 III.1 Evolution of the parameters

Our active solitons remain bright vector solitons to a good approximation, and
it is well known that bright solitons behave very much like particles BStheo1
; BStheo2 . This makes it an obvious approach to try to describe the motion of
our active soliton more simply in terms of a few collective coordinates, by
means of a time-dependent variational approximation. This variational approach
is a standard technique for the treatment of solitons RabiSwitch ;
ResonantControl ; Var1 ; Var2 ; Var3 ; Var4 ; Var5 ; Var6 ; Var7 .

We begin with the variational Ansatz discussed in section II above:

$\displaystyle\Psi^{\text{bs}}_{\pm}\\!\left(X,T\right)=\frac{1}{2}\sqrt{N_{\pm}}\,\mathrm{sech}\\!\left(\frac{1}{2}\left(X-Q\right)\right)$
(20) $\displaystyle\cdot e^{-iP^{2}T}e^{iP\left(X-Q\right)}e^{i\Phi_{\pm}}\;,$

where now we allow the Ansatz parameters
$Q\\!\left(T\right),P\\!\left(T\right),N_{\pm}\\!\left(T\right)$ and
$\Phi_{\pm}\\!\left(T\right)$ to be time-dependent. We then insert this Ansatz
into the Lagrangian for the Gross-Pitaevskii equation, to produce a Lagrangian
that depends only on our Ansatz parameters (and their time derivatives):

$\displaystyle L$ $\displaystyle\\!\left(Q,P,N_{\pm},\Phi_{\pm}\right)$ (21)
$\displaystyle=\int\limits_{-\infty}^{\infty}\\!\mathrm{d}X\,\left\\{\mathrm{Im}\left(\dot{\Psi}_{\text{bs}}^{\dagger}\Psi_{\text{bs}}\right)-\mathcal{H}\\!\left(\Psi_{\text{bs}},\Psi^{\prime}_{\text{bs}}\right)\right\\},$

with $\Psi_{\text{bs}}=\begin{pmatrix}\Psi_{+}^{\text{bs}}\\\
\Psi_{-}^{\text{bs}}\end{pmatrix}$ from (20).

Noting that $N_{+}+N_{-}=1$ is exactly conserved under the GPE, and that the
canonically conjugate common global phase of $\Psi_{\pm}$ is a trivial cyclic
variable, we can reduce the number of our parameters by defining

$\displaystyle\Delta$ $\displaystyle=\Phi_{+}-\Phi_{-},$ $\displaystyle I$
$\displaystyle=\frac{1}{2}\left(N_{+}-N_{-}\right).$ (22)

This leaves us with two mechanical coordinates $\left(Q,P\right)$ to be the
position and momentum of the motional degree of freedom of the whole soliton,
and $\left(\Delta,I\right)$ as internal coordinates describing its total spin
degree of freedom as an active particle’s internal energy depot. In particular
$I$ is directly proportional to the fuel energy $E_{\text{fuel}}=\hbar\omega
I$; $I=1/2$ represents a fully stocked energy depot while at $I=-1/2$ the
depot is empty.

With these definitions the Euler-Lagrange equations of motion for our
variational Lagrangian read

$\displaystyle\dot{Q}$ $\displaystyle=P,$ (23) $\displaystyle\dot{P}$
$\displaystyle=-G+K\Gamma_{K}\sqrt{\frac{1}{4}-I^{2}}\sin\\!\left(KQ-\Delta\right),$
(24) $\displaystyle\dot{\Delta}$
$\displaystyle=\Omega-\Gamma_{K}\frac{I}{\sqrt{\frac{1}{4}-I^{2}}}\cos\\!\left(KQ-\Delta\right),$
(25) $\displaystyle\dot{I}$
$\displaystyle=-\Gamma_{K}\sqrt{\frac{1}{4}-I^{2}}\sin\\!\left(KQ-\Delta\right),$
(26)

with the one additional definition, from the Fourier transform

$\int_{-\infty}^{\infty}\\!dz\,\frac{e^{ikz}}{\cosh^{2}(z/2)}=\frac{4\pi
k}{\sinh\pi k}\;,$ (27)

of

$\displaystyle\Gamma_{K}$ $\displaystyle=\Gamma_{0}\pi
K\,\mathrm{csch}\\!\left(\pi K\right),$ (28)

whereby $\Gamma_{K}\leq\Gamma_{0}$ for all $K$. These equations of motion can
also be identified as the canonical equations of motion for the Hamiltonian

$\displaystyle H=\frac{1}{2}P^{2}+GQ+\Omega
I+\Gamma_{K}\sqrt{\frac{1}{4}-I^{2}}\cos\\!\left(KQ-\Delta\right),$ (29)

if we assume $\left(Q,P\right)$ and $\left(\Delta,I\right)$ to be pairs of
conjugate coordinates. The several energy terms in (29) are therefore the
energies that were plotted in red in Fig. 2 for comparison with the Gross-
Pitaevskii energies (11)-(13), (15) that were plotted in blue. As we saw in
Fig. 2(a), the global evolution of the fuel energy $E_{\text{fuel}}$ and
mechanical energy $E_{\text{mech}}$ is well reproduced by the reduced
variational model. Also the variational parameter $P\\!\left(t\right)$, shown
as red line in Fig. 2(d), approximates the velocity of the soliton. We can
therefore seek to understand the behavior of the active bright soliton in
terms of this much simpler effective Hamiltonian (29), which has only two
degrees of freedom instead of the continuum of the Gross-Pitaevskii order
parameter fields.

The inital state (8) for the GPE evolution that we showed in section II above
corresponds to the initial parameters

$\displaystyle Q(0)$ $\displaystyle=0,$ $\displaystyle P(0)$
$\displaystyle=1.2,$ $\displaystyle\Delta(0)$ $\displaystyle=0,$
$\displaystyle I(0)$ $\displaystyle=0.498$ (30)

in this reduced representation.

###  III.2 III.2 The ‘Hamiltonian daemon’

The Hamiltonian (29) has in fact been studied at some length already Daemon1 ;
Daemon2 ; Daemon4 ; it has been labelled a ‘Hamiltonian daemon’ in allusion to
Maxwell’s Demon and to small autonomous Unix processes (daemons). Indeed the
present paper was motivated largely as a deliberate attempt to embed this
Hamiltonian daemon into a Gross-Pitaevskii system. We refer readers to Daemon1
; Daemon2 ; Daemon4  for a fuller discussion of the Hamiltonian itself, but
here we briefly review some of its properties that are important for the
behavior of the active soliton.

####  III.2.1 (a) The Chirikov resonance

First of all we can understand the Chirikov resonance that allows secular
energy transfer from high frequency into steady motion. The reason why we need
a Chirikov resonance is that generically $\Delta$ has a rapid linear
dependence on the dimensionless time $T$, $\Delta\sim\Omega T$ with $\Omega$
large because it is the Zeeman frequency in dimensionless form. This
generically makes the $\cos(KQ-\Delta)$ term in the Hamiltonian (29) oscillate
rapidly, and thus time-average to zero so that the fuel and motional degrees
of freedom adiabatically decouple.

The Chirikov resonance occurs, however, if the soliton moves at nearly the
(dimensionless) speed $P_{c}=\Omega/K$. At this special speed we have
$Q\sim\Omega T/K$, so that $\cos(KQ-\Delta)$ no longer oscillates rapidly
around zero but becomes nearly constant instead. The Rabi coupling term, which
is represented in the variational $H$ as the
$\Gamma_{K}\sqrt{\frac{1}{4}-I^{2}}\cos\\!\left(KQ-\Delta\right)$ term, can
therefore have a secular effect.

What is not generic for Chirikov resonances, but occurs in this particular
‘daemon’ case, is that the secular effect of this interaction term which
appears at the Chirikov resonance $P=P_{c}$ is to _keep_ $P$ close to $P_{c}$,
so that the resonance actually sustains itself, allowing secular energy
transfer to continue for a long time. If the (dimensionless) wave number $K$
of the Rabi coupling’s spatial periodicity is large enough, $K$ can
effectively ‘gear down’ a high frequency $\Omega$ to an arbitrarily slow
steady speed $P_{c}=\Omega/K$.

####  III.2.2 (b) Parameter regimes

As well as understanding how the active soliton can basically work, we can
identify the parameter regimes in which the soliton’s nontrivially driven
motion represents a reasonable form of active particle. First of all we can
see that the soliton can only be active if

$G<K\Gamma_{K}/2\;,$ (31)

since otherwise $\dot{P}$ will always be negative according to (24), and
therefore cannot remain near the Chirikov resonance at $P=P_{c}$ for any long
time. In effect $K\Gamma_{K}/2$ represents the largest force which the active
particle’s ‘motor’ can exert; an external downward force stronger than this
limit will prevent upward motion, in the same way that a car with limited
engine torque cannot climb too steep a hill.

A second condition for an active particle is what we were able to achieve in
section II: steady motion over a long time because a mechanism is slowly
transferring a large amount of energy from fuel into work, while the energy of
the mechanism itself remains comparatively small at all times. This requires
the regime

$\Gamma_{K}\ll\Omega\;.$ (32)

Finally, for small $\Gamma_{K}$ the Chirikov resonance is narrow and can only
sustain itself once $P$ is quite close to $P_{c}$. The active soliton must
therefore somehow be given the initial kinetic energy $P_{c}^{2}/2$ before the
depot energy $\Omega I$ can be exploited for motion. To make this initial
energy investment worthwhile, an active soliton should be able to hold at
least that much energy in its depot, requiring

$P_{c}^{2}<2\Omega$ (33)

and hence

$\Omega<2K^{2}\;.$ (34)

Since any one of our four dimensionless parameters $G,\Gamma_{K},\Omega$ and
$K$ can effectively be set to one by rescaling the dimensionless time $T$, the
inequality hierarchy

$\frac{2G}{K}<\Gamma_{K}\ll\Omega<2K^{2}$ (35)

fully defines the regime in which this daemon-soliton can reasonably
considered as an active particle. As noted in Daemon1 ; Daemon2 ; Daemon4 ,
the strong inequality which is the middle condition in (35) implies a time
scale hierarchy in the evolution under (29) which allows adiabatic methods to
be applied, clarifying some otherwise rather complicated nonlinear dynamics.
For reference, parameters used in section II satisfy the hierarchy (35) with

$\frac{1}{800}<\frac{\pi^{2}}{800}\ll 1<2\;.$ (36)

####  III.2.3 (c) Performance limits

The total time $T_{\text{c}}$ during which the soliton can remain active
before all of its fuel has been expended ($I$ falls from $+1/2$ to $-1/2$) can
be estimated as $\Omega$ (maximum total fuel energy) divided by power needed
to sustain speed $P_{\text{c}}=\Omega/K$ against external force $G$. This
yields

$\displaystyle T_{\text{c}}$ $\displaystyle=\frac{K}{G},$ $\displaystyle
X_{\text{c}}$ $\displaystyle=P_{\text{c}}T_{\text{c}}=\frac{\Omega}{G}\;.$
(37)

If we use the data from Khaykovich  to translate our parameters from section
II into physical units, we find a critical velocity $P_{\text{c}}v_{0}\approx
3.6\text{ mm}/\text{s}$, active motion duration $T_{\text{c}}t_{0}\approx
0.2\text{ s}$, and total height raised $X_{\text{c}}x_{0}\approx 0.7\text{
mm}$. These appear not unreasonable as experimental dimensions; they fall well
short of the potential maximum height to which Zeeman energy might carry a
bright vector soliton, but with soliton widths in microns, a million times the
soliton width might require an extravagantly large vacuum chamber.

####  III.2.4 (d) Daemon ‘ignition’

Because the evolution of the Hamiltonian daemon is indeed quite complicated,
the accuracy of the collective coordinate model in representing the active
soliton can be limited in some subtle ways. For example in Fig. 6 we show two
cases with $K=\Omega=1$ and $G=\pi^{2}/1600$ as in section II, but with either
$\Gamma_{K}=\pi^{2}/800$ as in section II or $\Gamma_{K}=\pi^{2}/3200$, four
times smaller than in section II. Figure 6(a) shows the relative change of the
fuel energy $\delta E_{\text{fuel}}/\left(\hbar\omega\right)$ between the
start of the process, and after an evolution of time $\delta T=2300$ for
different initial angle $KQ_{0}-\Delta_{0}$. Analogously Fig. 6(b) show the
relative change in the mechanical energy. The small-$\Gamma_{K}$ cases are
shown as solid lines, blue for the field system and red for the variational
system. The big-$\Gamma_{K}$ cases are shown as dashed black lines. The field
and variational calculation yield only not notable deviations from each other
in this case. While the dashed big-$\Gamma_{K}$ solutions show the same
behavior as could be seen before (the energy is transferred from
$H_{\text{fuel}}$ to $H_{\text{mech}}$), the solutions for the smaller
$\Gamma_{K}$ do not show this behavior on the whole range of initial values.

With smaller $\Gamma_{K}$ the Chirikov resonance is narrower, and the system
does not always get captured into the resonant region of phase space Liouville
. Whether or not active motion occurs depends in this small-$\Gamma_{K}$ case
on the initial value of the angle $\Delta_{0}$. This phenomenon of state-
dependent ‘ignition’ of the active phase occurs qualitatively in both the
reduced variational model and the full Gross-Pitaevskii description, but as
Fig. 6 shows the non-ignition effect is different, when it occurs, in the full
theory and in the variational approximation.

When not all of the spin energy is successfully consumed for motional work,
the variational approximation predicts that very little of it will be used in
any way—the ‘motor’ simply fails to start. In these cases where the
variational approximation predicts little change in spin energy, however, the
full GPE evolution shows some significant loss of $E_{\text{fuel}}$. Closer
examination shows that what is occurring in these cases is that correspondence
between the field theory and the variational model breaks down, because
passing through the Chirikov resonance makes the soliton partly break up; spin
energy is used to unbind part of the soliton, rather than to keep the whole
soliton moving against the external force. It is perhaps counter-intuitive
that this happens for _weaker_ $\Gamma_{K}$ coupling, while the stronger
coupling keeps the soliton together reliably, as well as reliably using its
spin energy as an active particle. The reason for this is that the bigger
coupling forces the system to get caught in the Chirikov resonance. If the
system is not caught in the resonance, which can only happen for the
small-$\Gamma_{K}$ case, the effect of the Rabi coupling is rather that it
transfers a part of the soliton from the + component to the - component of the
condensate, while increasing the momentum only of the transferred part, which
will hence leave the soliton.

Figure 6:  (a) The relative change in the fuel energy and (b) in the
mechanical energy during the process plotted over the initial angle
$KQ_{0}-\Delta_{0}$. For a weak coupling $\Gamma_{K}$ two regions can be
identified for the particle system (red) and the field system (blue). In the
one region almost all energy of the fuel gets transferred to the mechanical
degrees of freedom of the soliton, in the other region much less energy gets
transferred. This effect does not occur for a stronger coupling (dashed black
line, both cases), for which complete energy transfer occurs for all initial
angles.

This example shows that although our simple picture of the vector bright
soliton as an active particle certainly can apply well, with the simple daemon
model as an accurate variational approximation, yet there can also be cases
where the soliton fails to remain robust under the spatially periodic Rabi
coupling and the external force. Interesting as it may be to pursue a detailed
investigation of exactly what happens in the particular case of Fig. 6, we
will leave this for future work; we conclude the present paper by looking more
broadly at how our active particle model may break down, or survive, when the
condensate system is generalized in various ways.

##  IV IV. More general active soliton models

###  IV.1 IV.1 Other Rabi couplings

In (6) we introduced a Rabi coupling that was tailored specifically to realize
the Chirikov resonance of the Hamiltonian daemon for an atomic soliton. To
consider engineered Hamiltonians that do not involve Chirikov resonances at
all would give this paper an infinite scope, but we can consider a wider range
of qualitatively similar Chirikov resonances. In particular in this subsection
we modify our assumption that the Rabi coupling is perfectly periodic in space
with a single positive wave number $K$. Instead we will examine a case with
multiple simultaneous $K\to K_{n}$; a case with negative $K$, in which the
active soliton harvests internal energy from the external force instead of
working against it; and a case where the periodicity of the Rabi coupling is
somewhat disordered, so that $K$ has a finite statistical width.

All of these cases still have

$\displaystyle\Gamma$ $\displaystyle=\Gamma_{0}\begin{pmatrix}0&F(X)\\\
F^{*}(X)&0\end{pmatrix},$ (38)

for some spatially varying function $F=F\\!\left(X\right)$. Letting
$\tilde{F}$ denote the Fourier transform of $F$, we can express our spatially
dependent Rabi coupling as

$\displaystyle
H_{\Gamma}\equiv\int\limits_{-\infty}^{\infty}\\!\mathrm{d}X\,\Psi_{\text{bs}}^{\dagger}\Gamma\Psi_{\text{bs}}$
$\displaystyle=\frac{\Gamma_{0}}{2}\sqrt{\left(\frac{1}{2}\right)^{2}-I^{2}}\text{Re}\\!\left(e^{i\Delta}\int\limits_{-\infty}^{\infty}\\!\mathrm{d}X\,{F}\\!\left(X\right)\mathrm{sech}^{2}\\!\left(\frac{X-Q}{2}\right)\right)$
$\displaystyle=\Gamma_{0}\sqrt{\left(\frac{1}{2}\right)^{2}-I^{2}}\;\text{Re}\\!\left(e^{i\Delta}\int\limits_{-\infty}^{\infty}\\!\mathrm{d}Y\,\frac{Y\tilde{F}\\!\left(Y\right)}{\sinh\\!\left(\pi
Y\right)}e^{-iQY}\right),$ (39)

again using the Fourier transform of $\mathrm{sech}^{2}$. Where our original
ansatz (6) made $\tilde{F}$ a delta function at $Y=K$, we now explore some
more general cases. We find that the mechanism of energy transfer through the
self-sustaining Chirikov resonance of the Hamiltonian daemon does not only
occur in our original special case, but persists much more generally.
Embedding the Hamiltonian daemon in the spinor condensate mean field does
introduce generalisations and corrections to the most basic daemon
Hamiltonian, however.

####  IV.1.1 (a) A coupling with 3 isolated resonances

Since the complex exponential $F=e^{iKX}$ in our original model explicitly
broke time-reversal invariance, one might well worry that the active soliton
can only work when this important discrete symmetry is broken by hand. If
instead we take $F\\!\left(X\right)=2\cos^{2}\\!\left(KX/2\right)$, however,
we maintain time-reversal invariance with the real Hamiltonian term

$\displaystyle
H_{\Gamma}=\sqrt{\frac{1}{4}-I^{2}}(2\Gamma_{0}\cos\\!\left(\Delta\right)+\Gamma_{K}\cos\\!\left(KQ-\Delta\right)$
$\displaystyle+\Gamma_{K}\cos\\!\left(KQ+\Delta\right))\;.$ (40)

Our previous ‘$K$’ Chirikov resonance is now joined by similar ‘$-K$’ and
‘$K=0$’ resonances: a term in $H_{\Gamma}$ is now time-independent when
$\dot{Q}=P$ is close to $-P_{c}=-\Omega/K$ or close to zero. The presence of
additional well separated resonances turns out to have little effect, however,
because with $\Gamma_{0}\ll\Omega$ each term in $H_{\Gamma}$ can only have
significant effect when $P$ is close to its critical speed. The soliton cannot
be close to more than one critical speed simultaneously, because the three are
separated by $\Omega/K$, and so effectively this time-reversal-invariant model
is nearly equivalent to our original model, as long as the soliton speed is
not close to either zero or $-P_{c}$.

Figure 7 shows numerical evolutions with this coupling (IV.1.1), but otherwise
with all parameters and initial conditions the same as in section II. In the
Figure both the full GPE field theory (blue) and the corresponding variational
approximation (red) are shown. The behaviors of the fuel energy
$E_{\text{fuel}}$, the mechanical energy $E_{\text{mech}}$, and the velocity
$v$ are qualitatively very much the same as for our original single ’$K$’
resonance: energy is slowly drained from the fuel term and converted into
mechanical energy.

The new terms in the Hamiltonian introduce additional Chirikov resonances;
that is, these new terms would be resonant perturbations of the system at
speeds zero or $-P_{c}$. As long as the active soliton has $P$ near $P_{c}$,
the additional terms are far from being resonant. Their effects on the motion
of the soliton near $P_{c}$ are therefore not the dramatic qualitative change
of a resonant perturbation, but only ‘dressing’: the non-resonant terms can be
adiabatically eliminated and replaced, in an adiabatic effective Hamiltonian
with renormalized terms. Such an effect can be seen in Fig. 7, for example.
When the exact soliton velocity $v/v_{0}$ from (17) is compared with the
variational parameter $P(T)$, we see that there is a small linear downward
trend in the exact velocity which is not captured by the variational model.

Figure 7:  Evolution of the fuel and mechanical energy ((a) and (b)) and the
velocity $v/v_{0}$(c), for the case of multiple resonances. The blue (red)
lines correspond to the field (particle) system. While there are rapid
oscillations on top of the global evolution, the global behavior is almost the
same as for a single resonance.

Such a downward trend in the active particle speed, as its fuel $I$ is
depleted, can be incorporated into the Hamiltonian daemon by adding a
quadratic term $bI^{2}$ for some $b>0$ to the daemon Hamiltonian (29). Doing
that effectively makes $\Omega\to\Omega+2bI$ and thus $P_{c}->P_{c}+(2b/K)I$,
decreasing as $I$ decreases. Within the full mean field theory this effect can
easily arise, because the vector bright soliton distorts somewhat in form
during the active phase, as we have already seen in the non-trivial behavior
of the $\alpha_{\pm}$ variational parameters shown in Fig. 5. What evidently
occurs with the additional couplings of (IV.1.1) is that the soliton slightly
distorts in an $I$-dependent way, such that the nonlinear and kinetic terms in
the energy contribute an effective $bI^{2}$ energy term.

####  IV.1.2 (b) Working as a brake

We have just seen that for a more general $H_{\Gamma}$ the active soliton’s
speed may be generalized away from the single sharp $P_{c}$ of the simplest
model. We can now study a more drastic speed alternative for the active
soliton while still using the same three-resonance $H_{\Gamma}$ (IV.1.1). With
time-reversal symmetry now restored, we can find a case in which the ’$-K$’
resonance generates the major effect of holding the soliton’s speed nearly
steady as it moves with the external force instead of against it. In other
words the Chirikov mechanism which previously worked as a motor now works as a
brake. Since the internal energy $I$ is now raised instead of being depleted,
we could also consider this evolution as a microscopically described case of
energy harvesting by an active particle.

Initial conditions which lead to this motion are

$\displaystyle\Psi_{+}\\!\left(X,T=0\right)$
$\displaystyle=\frac{1}{2}\sqrt{0.002}\,\mathrm{sech}\\!\left(\frac{1}{2}X\right)\exp\\!\left(-0.8iX\right)$
$\displaystyle\Psi_{-}\\!\left(X,T=0\right)$
$\displaystyle=\frac{1}{2}\sqrt{0.998}\,\mathrm{sech}\\!\left(\frac{1}{2}X\right)\exp\\!\left(-0.8iX\right).$
(41)

These initial fields mean that $I\\!\left(T=0\right)=-0.498$, _i.e._ there is
initially almost no energy in the fuel term, and the soliton starts with the
negative velocity $-0.8v_{0}$, falling down the linear potential. Due to the
potential gradient the soliton will linearly increase its momentum, until it
reaches $P\approx-1$. It then becomes trapped into the ’$-K$’ resonance,
locking its momentum close to $-P_{c}=\frac{\Omega}{-K}=-1$. The mechanical
energy which is lost as the soliton moves down the potential gradient without
correspondingly accelerating is transferred into the internal depot of spin
energy. This behavior can be seen in Fig. 8, which can be directly compared to
Fig. 7.

Figure 8:  Evolution of the fuel and mechanical energy ((a) and (b)) and the
velocity (c) for the case, where the velocity locks to a negative value. The
field (particle) system is plotted in blue (red). Instead of getting
accelerated the soliton keeps its negative velocity, while the energy of the
fuel term is increased.

Much as in Fig. 7, a small but systematic discrepancy between the variational
approximation and the full field theory can be seen in Fig. 8. The active
soliton not only avoids accelerating despite the external force, but actually
decelerates slightly (in the negative direction, _i.e._ becomes slightly less
negative). The same form of $bI^{2}$ modification to (29) can account for this
residual deceleration, but now we will need $b<0$, so that
$P_{c}=(\Omega+2bI)/(-K)$ decreases in absolute value as $I$ rises. Evidently
the way in which soliton deformation dresses the effective daemon Hamiltonian
depends on which Chirikov coupling is actually resonant.

####  IV.1.3 (c) Finite-width resonance

As a last example in our exploration of more general $F(X)$ coupling profiles,
we return to the original case with only one Chirikov resonance. We now
suppose, however, that the Fourier transform of $F$ is not a sharp delta
function at $K$, but has a small finite width $\sigma$, because the Rabi drive
does not extend over all space, but instead has a Gaussian envelope with a
large width $1/\sigma$:

$\displaystyle
F\\!\left(x\right)=\frac{1}{\sqrt{2\pi}}\exp\\!\left(-iKX\right)\exp\\!\left(-\frac{1}{2}\sigma^{2}X^{2}\right)\;.$
(42)

Since this means that the active motion of the soliton can only occur within
the Gaussian envelope, we are interested in $\sigma$ small enough that
$1/\sigma$ still allows room for a reasonable amount of active motion. Below
we will compare two cases: $\sigma=0.001$ and $\sigma=0.002$, for both of
which $1/\sigma$ is on the order of the total travel distances that we saw in
section II.

For such small $\sigma$ the spatial modulation of $F(X)$ is slow on the $1/K$
scale over which the Chirikov resonance motor effect operates, and so we can
anticipate behavior much like what we have previously seen for the single-
resonance model, except with a $\Gamma_{K}$ that slowly changes as the soliton
moves. Quantitatively, we expect that the active motion can only persist as
long as the instantaneous coupling strength $\Gamma_{K}$ is above the $2G/K$
threshold from (35), so that the maximum possible force from the Rabi coupling
can overcome the external force $G$.

To test these predictions and also continue our comparisons between GPE
evolution and the variational approximation with two degrees of freedom, we
should ideally define $H_{\Gamma}$ using the integral in (IV.1). Unfortunately
this integral has no convenient analytical form. We therefore instead take a
more transparent $H_{\Gamma}$ with the same Gaussian form as $F(X)$ itself,
and allow for modification due to averaging over the soliton width by using
fitting parameters $\alpha$ and $\beta$:

$\displaystyle
H_{\Gamma}=\Gamma_{K}(Q)\sqrt{\frac{1}{4}-I^{2}}\cos\\!\left(K_{0}Q-\Delta\right),$
(43)

where we define

$\displaystyle\Gamma_{K}(Q)$
$\displaystyle=\Gamma_{0}\frac{\alpha}{\sqrt{1+2\beta\sigma^{2}}}\exp\\!\left(-\frac{\beta
K^{2}}{1+2\beta\sigma^{2}}\right)\cdot$
$\displaystyle\qquad\qquad\cdot\exp\\!\left(-\frac{\sigma^{2}Q^{2}}{2\left(1+2\beta\sigma^{2}\right)}\right),$
(44) $\displaystyle K_{0}$ $\displaystyle=\frac{K}{1+2\beta\sigma^{2}}\;.$
(45)

Best fits to the ideal $H_{\Gamma}$ from the (IV.1) integral are for
$\alpha\approx 0.969$ and $\beta\approx 1.269$.

Figure 9 shows in (a) and (b) the evolution of the fuel energy and velocity of
the soliton according to both the GPE and the variational approximation, for
the two different resonance widths $\sigma=0.002$ (dashed) and $\sigma=0.001$
(solid). All system parameters and initial values are otherwise the same as
for the evolutions in section II. The soliton thus begins at $X=0$, in the
middle of the Rabi coupling’s Gaussian envelope, at a speed slightly above the
critical speed for active motion $P_{c}$. Decelerating under the external
force, its speed soon falls to $P_{c}$ and active motion begins. With these
two particular resonance widths the now $Q$-dependent coupling $\Gamma_{K}(Q)$
drops below the critical value $2G/K_{0}$ before the active motion would
otherwise stop for lack of internal energy.

Figure 9:  Evolution of the fuel energy
$E_{\text{fuel}}/\left(\hbar\omega\right)$ (a) and the velocity of the soliton
(b) for the case of a finite resonance. The blue (red) curves correspond to
the field (particle) system, the solid (dashed) curves to the narrow (wide)
resonance width. The begin of the evolution is the same as for a sharp
resonance. The process stops when the effective coupling strength
$\Gamma_{\text{eff}}$ drops to the critical value $\Gamma_{\text{crit}}$ (c)
even though the fuel has not yet been drained completely.

For $\sigma=0.002$ this happens at about $T\approx 720$, and the active phase
of motion indeed ceases then, with the fuel energy holding constant after this
point, and velocity decreasing linearly under the external force. For the
narrower resonance width $\sigma=0.001$, the local coupling constant
$\Gamma_{K}(Q)$ reaches the critical value at $T\approx 1600$, but the active
phase ends noticeably earlier, at $T\approx 1400$, when $\Gamma_{K}(Q)\doteq
2.8G/K_{0}$. It must be remembered, however, that the condition
$\Gamma_{K}>2G/K$ is the requirement for active motion _at any possible_ $I$.
The force exerted by the Rabi coupling is also proportional to
$\sqrt{(1/4)-I^{2}}$, so active motion for $|I|>0$ in general requires higher
$\Gamma_{K}$. For the spatially narrower Rabi drive envelope with
$\sigma=0.002$, the too-small-$\Gamma_{K}$ threshold is crossed sooner, when
$I$ is close to zero, while for the spatially wider case $\sigma=0.001$ the
active motion continues until $I$ is near $-0.3$. At that point the
$\sqrt{(1/4)-I^{2}}$ factor and the smaller $\Gamma_{K}(Q)$ together leave the
Rabi drive too weak to overcome the external force, and active motion stops.

Overall our investigations of different forms of spatially dependent Rabi
coupling support the view that although the Chirikov resonance mechanism of
the active vector bright soliton is somewhat complicated, it is not especially
fragile. The many possible perturbations and parameters of the larger field
theory model can impose many deformations and renormalizations of the simple
Hamiltonian daemon model, which can effect its behavior significantly. They do
not destroy its operation entirely, however; on the contrary, the basic active
particle behavior can robustly persist.

###  IV.2 IV.2 Different scattering lengths

The form (1) of the one-dimensional Gross-Pitaevskii equation does not hold if
the three-dimensional inter-species scattering lengths $a_{\pm\pm^{\prime}}$
and radial trapping frequencies $\omega_{\rho\pm}$, which together determine
the effective one-dimensional nonlinear coupling matrix, are not all equal for
the two species. While one coupling constant can be set equal to one by
scaling, in general the other three are independent and our GPE (1) must be
generalized by replacing

$\displaystyle\left(\Psi^{\dagger}\Psi\right)\Psi\to\begin{pmatrix}\left(\left|\Psi_{+}\right|^{2}+A^{\prime}\left|\Psi_{-}\right|^{2}\right)\Psi_{+}\\\
\left(A^{\prime}\left|\Psi_{+}\right|^{2}+A\left|\Psi_{-}\right|^{2}\right)\Psi_{-}\end{pmatrix}$
(46)

for some real constants $A$ and $A^{\prime}$. By modifying the internal states
of the atoms with external fields, the three-dimensional scattering lengths
can be controlled experimentally to a great extent; by further making the
potentials which confine the gas to one dimension spin-dependent, the mean-
field coupling constants in one dimension can further be modified. It may not
be experimentally easy to achieve $A=A^{\prime}=1$, however, and so we ask
what effects general $A$ and $A^{\prime}$ may have on the active bright vector
soliton.

####  IV.2.1 (a) Variational Hamiltonian

The main effect of general $A$ and $A^{\prime}$ is that the soliton is no
longer a soliton in the strict sense, since the general two-component
nonlinear Schrödinger equation is no longer integrable. Similar solitary wave
solutions still exist for a range of $A$ and $A^{\prime}$, however, and we
will continue to refer to these as ‘the soliton’ for brevity. Their wave
function profile is no longer exactly a hyperbolic cosecant, and in general
the widths of the solitary wave are not the same for both components
$\Psi_{\pm}$. For simplicity, however, we will continue to compare numerical
GPE evolution with a variational approximation based on the same ansatz (20)
that we used before, with hyperbolic cosecants of the same fixed width.

This simpler variational ansatz leaves most of our effective variational
Hamiltonian (29) unaffected by the new set of scattering lengths, even though
a more accurate ansatz would include indirect effects due to the modification
of the solitary wave shapes. Even without taking those changes into account,
however, additional $I$ dependences appear in the variational $H$ through the
nonlinear term in the Gross-Pitaevskii Hamiltonian density, in which we must
replace $\left|\Psi^{\dagger}\Psi\right|^{2}$ in (10) with

$\displaystyle\frac{1}{2}\left|\Psi^{\dagger}\Psi\right|^{2}$ (47)
$\displaystyle\to\frac{1}{2}\left(\left|\Psi_{+}^{*}\Psi_{+}\right|^{2}+A\left|\Psi_{-}^{*}\Psi_{-}\right|^{2}+2A^{\prime}\left|\Psi_{+}\right|^{2}\left|\Psi_{-}\right|^{2}\right)\;.$

Using the integral

$\int_{-\infty}^{\infty}\\!dz\,\mathrm{sech}^{4}(z)\equiv\int_{-\infty}^{\infty}\\!d(\tanh
z)\,(1-\tanh^{2}z)=\frac{4}{3}$ (48)

we see that we now have

$\displaystyle H=$ $\displaystyle\frac{1}{2}P^{2}+GQ+\left(\tilde{\Omega}+\nu
I\right)I-$ (49)
$\displaystyle-\Gamma_{K}\sqrt{\frac{1}{4}-I^{2}}\cos\\!\left(KQ-\Delta\right),$

with

$\displaystyle\tilde{\Omega}$ $\displaystyle=\Omega+\frac{A-1}{12},$
$\displaystyle\nu$ $\displaystyle=\frac{2A^{\prime}-A-1}{12}\;.$ (50)

Our understanding of Hamiltonian (29) remains essentially intact, therefore,
if we introduce a new, $I$-dependent, critical velocity

$\displaystyle P_{\text{c}}\\!\left(I\right)=\frac{\tilde{\Omega}+\nu I}{K},$
(51)

which thus changes during the evolution.

The question is whether this straightforward generalization of our Hamiltonian
daemon model, or even a more sophisticated generalization that takes into
account the deformations of the soliton, can actually approximate the GPE
evolution well, when we have general inter-component interactions that break
the integrability of the nonlinear Schrödinger equation. As an illustration of
what can happen in a moderate departure from the simplest case, we again
repeat our evolution from the initial state of section II, and with all the
same parameters, except that now we use $A=2$ and $A^{\prime}=0.5$ instead of
$A=A^{\prime}=1$.

Figure 10:  Evolution of the fuel energy
$E_{\text{fuel}}/\left(\hbar\omega\right)$ (a) and the velocity of the soliton
(b), for the case where the critical velocity $P_{c}(I)$ changes during the
evolution. The numerical GPE evolution is plotted in blue, the variational
approximation in red. The GPE evolution can be understood quite well from the
variational approximation; its error is initially negligible and although it
later becomes significant it does not become drastic.

In Fig. 10 we see for this case that the active phase of energy transfer
indeed begins when the soliton has decelerated from its high initial velocity
to the critical velocity $P_{c}(I)$ predicted by the variational approximation
for the initial fuel level ($I$ close to +1/2). The gradual increase of the
active soliton’s velocity is also accurately given by the variational
approximation, up until about half-way through the plotted evolution. From
this point on the variational approximation becomes less accurate as the
exactly evolving soliton accelerates more rapidly than the variational model
predicts. The variational approximation thus ultimately overestimates the
total duration of the active phase, because it underestimates the active
soliton’s power consumption at its higher final speeds. The variational
approximation does correctly predict the final fuel level.

As Fig. 11 shows, the onset of discrepancy between the variational model and
the GPE evolution is clearly due to increasing distortion of the soliton
profile away from the fixed equal widths that the simple variational model
assumes. Figure 11(c) indicates that a more general hyperbolic secant form
with independently variable $\alpha_{\pm}$ width parameters should still
describe the active soliton well, but the simple equal-width ansatz breaks
down.

Figure 11:  (a) The fit parameters $\alpha_{+}$ ($\alpha_{-}$) in blue (red),
computed as in Fig. 5 for the case with unequal interaction constants $A=2$
and $A^{\prime}=0.5$ that was shown in Fig. 10. (b) The deviation between the
density of the exactly evolving fields, and the soliton form for fixed
$\alpha_{\pm}$, is significantly larger for $t/t_{0}>1000$ than the error for
optimally fitted $\alpha_{\pm}$, shown in (c).

For nonintegrable cases like this particular case of $A$ and $A^{\prime}$ it
might therefore be worth pursuing a more general variational approximation
with $\alpha_{\pm}$ as additional variational parameters; we leave this
exercise for future work, simply noting that the active bright vector soliton
indeed remains qualitatively robust against even significant changes to its
nonlinear interaction strengths. Just how far this robustness extends will be
our final investigation in this paper.

####  IV.2.2 (b) Stability range of the active soliton

For more extreme deviations from $A=A^{\prime}=1$, solitary wave solutions to
the vector nonlinear Schrödinger equation may not only deform but even break
up, as atoms escape from the solitary wave. Insofar as a solitary wave does
still persist, even partially, it is then a related but independent question,
whether the solitary wave can move as an active particle by consuming the
Zeeman energy of its internal spin depot. Solitary waves in different regimes
have long been studied, so our goal in this final portion of our paper is to
relate the second question to the first.

To do this we numerically evolved our same initial soliton configuration,
until the time $T=5000$ after which any active motion should be complete,
under a range of different $A$ and $A^{\prime}$ cases. In each case we
computed how much energy was transferred from spin into motional energy, as
well as what fraction of the initial atoms remained in the soliton. We assumed
that some kind of active motion must be occurring if any significant fraction
of the spin energy is drained; to assess how many atoms remain in the soliton,
even if the soliton is distorted, we fit the final mean-field density profile
to a hyperbolic secant with independent amplitudes, widths, and centers for
the two components. We then defined the fraction of atoms remaining in the
bright soliton at $T=5000$—the _retention fraction_ —as the Gross-Pitaevskii
norm of this fitted hyperbolic secant profile, divided by the exact initial
norm. This procedure offers a reasonable measure of how well the ‘soliton’
holds together, as long as the norm which is lost from the soliton tends to
spread out in low-density noise that does not significantly affect the best
fit $\mathrm{sech}^{2}$ profile; it would not be accurate if the soliton held
together but deformed in shape radically away from $\mathrm{sech}^{2}$, so
that the $\mathrm{sech}^{2}$ fit represented the surviving solitary wave
poorly. Checking particular cases seems to show, however, that this does not
occur, and that the $\mathrm{sech}^{2}$ fitting procedure does provide a good
measure of soliton survival.

In Fig. 12 we show the results in the plane of positive $A$ and $A^{\prime}$,
logarithmically. We indicate soliton survival with black contours of atom
retention fraction, and fuel energy consumption with color. The message of the
Figure in general is clear: the soliton can essentially remain active as long
as it can remain stable at all.

Figure 12:  Stability of the soliton and amount of energy transfer for
different scattering parameter $A$ and $A^{\prime}$. The black lines are
contours (from outer to inner) of $70\%$, $80\%$, $90\%$, and $95\%$ atom
retention by the soliton, as defined in the text. The ‘soliton’ is a stable
solitary wave for a wide range of $A$ and $A^{\prime}$. The color scale
linearly indicates the amount of fuel energy that was transferred into
mechanical energy over the course of the evolution to $T=5000$. The energy
transfer occurs over the whole range in which the soliton is stable.

##  V V. Conclusion

In this paper we have used numerical evolution in Gross-Pitaveskii mean field
theory to show that the internal energy of a vector bright soliton in a two-
component self-attractive one-dimensional Bose-Einstein condensate can be used
to lift the soliton against an external force. The internal energy of the
soliton is realized by different chemical potentials for the two components of
the Bose-Einstein condensate. The energy transfer from the internal energy to
the potential energy is induced by an off-resonant Rabi coupling whose
strength varies periodically in space.

This one-dimensional field theory can be approximated at least reasonably well
with a variational ansatz of two degrees of freedom. The active-particle
behavior of the soliton can be understood from the much simpler Hamiltonian of
the variational approximation, which reveals that the spatially periodic Rabi
coupling effectively implements the ‘Hamiltonian daemon’ realization of a
combustion engine analog, based on a Chirikov resonance. This basic mechanism
has been shown to be affected non-trivially by the complications that are
possible in the realistic physical representation of the active particle as an
atomic bright vector soliton.

The mechanism has nonetheless been shown to be qualitatively robust in a wide
range of conditions. Atomic bright solitons are certainly a highly artificial
dynamical system, but they are more complicated than minimal Hamiltonian
models: they are complicated enough to be experimentally realizable. Our study
can thus be considered as a toy model for the project of extending microscopic
understanding of active particles beyond toy models and into real systems.

The authors acknowledge support from State Research Center OPTIMAS and the
Deutsche Forschungsgemein- schaft (DFG) through SFB/TR185 (OSCAR), Project No.
277625399.

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