††thanks: Present address: Atom Computing, Inc., Berkeley, CA,
USA.¶¶footnotetext: These authors contributed equally to this work.

# In situ observation of non-polar to strongly polar atom-ion collision
dynamics

M. Berngruber¶ 5\. Physikalisches Institut, Universität Stuttgart,
Pfaffenwaldring 57, 70569 Stuttgart, Germany  D. J. Bosworth¶
<EMAIL_ADDRESS>Zentrum für Optische Quantentechnologien,
Universität Hamburg,
Luruper Chaussee 149, 22761 Hamburg, Germany
Hamburg Centre for Ultrafast Imaging, Universität Hamburg,
Luruper Chaussee 149, 22761 Hamburg, Germany
O. A. Herrera-Sancho  5\. Physikalisches Institut, Universität Stuttgart,
Pfaffenwaldring 57, 70569 Stuttgart, Germany  Escuela de Física, Universidad
de Costa Rica, 2060 San Pedro, San José, Costa Rica  Instituto de
Investigaciones en Arte, Universidad de Costa Rica, 2060 San Pedro, San José,
Costa Rica  Centro de Investigación en Ciencias Atómicas, Nucleares y
Moleculares, Universidad de Costa Rica, 2060 San Pedro, San José, Costa Rica
V. S. V. Anasuri  5\. Physikalisches Institut, Universität Stuttgart,
Pfaffenwaldring 57, 70569 Stuttgart, Germany  N. Zuber  5\. Physikalisches
Institut, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany
F. Hummel  Max-Planck-Institute for the Physics of Complex Systems,
Nöthnitzer Straße 38, 01187 Dresden, Germany
J. Krauter  5\. Physikalisches Institut, Universität Stuttgart,
Pfaffenwaldring 57, 70569 Stuttgart, Germany  F. Meinert  5\. Physikalisches
Institut, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany
R. Löw  5\. Physikalisches Institut, Universität Stuttgart, Pfaffenwaldring
57, 70569 Stuttgart, Germany  P. Schmelcher<EMAIL_ADDRESS>Zentrum für Optische Quantentechnologien, Universität Hamburg,
Luruper Chaussee 149, 22761 Hamburg, Germany
Hamburg Centre for Ultrafast Imaging, Universität Hamburg,
Luruper Chaussee 149, 22761 Hamburg, Germany
T. Pfau<EMAIL_ADDRESS>5\. Physikalisches Institut,
Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany

###### Abstract

The onset of collision dynamics between an ion and a Rydberg atom is studied
in a regime characterized by a multitude of collision channels. These channels
arise from coupling between a non-polar Rydberg state and numerous highly
polar Stark states. The interaction potentials formed by the polar Stark
states show a substantial difference in spatial gradient compared to the non-
polar state leading to a separation of collisional timescales, which is
observed in situ. For collision energies in the range of
$k_{\textrm{B}}\cdot$$\mathrm{\SIUnitSymbolMicro}\mathrm{K}$ to
$k_{\textrm{B}}\cdot$$\mathrm{K}$, the dynamics exhibit a counter-intuitive
dependence on temperature, resulting in faster collision dynamics for cold –
initially “slow” – systems. Dipole selection rules enable us to prepare the
collision pair on the non-polar potential in a highly controlled manner, which
determines occupation of the collision channels. The experimental observations
are supported by semi-classical simulations, which model the pair state
evolution and provide evidence for tunable non-adiabatic dynamics.

Introduction — Observing, understanding and controlling individual collisions
is a prerequisite for many-body physics based on atoms or molecules.
Especially in the ultracold regime, where collisions between neutral atoms can
be engineered by Feshbach resonances, a high level of control is reached Chin
_et al._ (2010). This makes possible, for example, the study of degenerate
molecular gases De Marco  _et al._ (2019); Duda  _et al._ (2023), Feshbach
molecules Köhler  _et al._ (2006); Thalhammer  _et al._ (2006) and Efimov
physics Naidon and Endo (2017). However, when it comes to collisions between
charged and neutral particles reaching the same level of quantum control
becomes harder since the range of interactions increases, thus requiring even
lower temperatures to reach the quantum regime of scattering Weckesser  _et
al._ (2021); Hirzler  _et al._ (2022). More exotic collisions can be studied
in systems of laser-cooled Rydberg atoms, which have the advantage of showing
long-range interactions, allowing collisions to occur on larger length, slower
time and lower energy scales, making them easier to observe with spatial and
temporal resolution. Even exotic bound states between a Rydberg atom and
neutral ground state atoms forming ultralong-range molecules have been
observed Bendkowsky  _et al._ (2009); Booth  _et al._ (2015); Niederprüm  _et
al._ (2016); Fey  _et al._ (2020). Moreover, the complex Rydberg level
structure can give rise to intriguingly rich potential energy surfaces with
avoided crossings and conical intersections providing means to study effects
beyond the Born-Oppenheimer approximation Hummel  _et al._ (2023, 2021).
More recently, also systems combining Rydberg atoms and ions have become an
active field of research Engel  _et al._ (2018); Schmid  _et al._ (2018);
Gambetta  _et al._ (2020); Wang  _et al._ (2020); Deiß  _et al._ (2021);
Duspayev  _et al._ (2021); Zuber  _et al._ (2022). Here, we pursue this
direction and explore the dynamical processes that lead to a multi-channel
collision between an ion and a Rydberg atom. Instead of an ion trap we rely on
compensating electric fields to work with free floating ions in an almost net-
zero electric field environment. Our high-resolution ion microscope allows us
to study collisional dynamics with both spatial- and temporal-resolution. We
are therefore not restricted to only analysing the initial and final collision
partners, but may instead directly observe the dynamics as the collision
unfolds.

Figure 1: Ion-Rydberg collision channels. Adiabatic potential energy curves
obtained via exact diagonalisation of the electronic Hamiltonian (see
Supplementary Material). (a) Close-up around the avoided crossings near the
non-polar $\ket{129S}$ atomic Rydberg state, illustrating the presence of
initially slow (red) and fast (blue) collision channels. Counterintuitively,
occupation of the fast channels is more probable for initially slower
particles. (b) Larger-scale plot of the avoided crossings between the non-
polar $S$-state and multiple strongly polar Stark states from the neighboring
asymptotically degenerate hydrogenic manifold. The overlap of the electronic
states with the unperturbed $\ket{129S}$ atomic Rydberg state is denoted by
the colorbar. Energies are given relative to the $\ket{129S}$ atomic Rydberg
state.

Theory — The polarizability of highly excited Rydberg atoms gives rise to a
long-range charge-induced dipole interaction potential which is shown as a
function of the internulear distance $R$ for the specific case of the
$\ket{129S}$ state in Fig. 1. These potential energy curves (PECs) are
calculated by exact diagonalization of the electronic Hamiltonian
$H_{e}=H_{0}+V_{I}(R)$, where $H_{0}$ describes the unperturbed Rydberg atom
and $V_{I}$ the ion-Rydberg interaction. This interaction term can be written
in a multipole expansion, where we consider terms up to the 6th order (see
Supplementary Material). One can distinguish between two regions: the first
region is defined in the asymptotic limit for large $R$, where the potential
of the non-polar $S$-state falls off with $1/R^{4}$ due to the charge-induced
dipole interaction between the ion and the atomic Rydberg $S$-state; the
second region s found at $R\lesssim$
$33\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ for the case of
$\ket{129S}$, which is nevertheless approximately 26 times larger than the
size of the Rydberg orbit. Here, the ion-induced Stark shift becomes large
enough that strongly polar, large angular momentum states from the neighboring
$n=126$ hydrogenic manifold start to cross into the polarization potential and
form a series of avoided crossings. For Rydberg $S$-states in 87Rb, the
polarization potential strictly decreases in energy while approaching the ion
such that, the two collision partners will always be accelerated toward each
other. In the direct vicinity of these avoided crossings, the Born-Oppenheimer
(BO) approximation is no longer suitable to describe the dynamics properly.
Instead, a non-adiabatic, semi-classical model using the Landau-Zener (LZ)
formula is employed Zener and Fowler (1997). This allows to estimate the
probability for an adiabatic transition to a strongly polar state at each
crossing and thus can be used to predict the occupation of the different
collision channels. The probability $P_{ij}$ to transition non-adiabatically
from PEC $i$ to an adjacent curve $j$ is given by $P_{ij}=\exp\big{(}-2\pi
a^{2}_{ij}/(\dot{R}\,\alpha_{ij})\big{)}$ Rubbmark  _et al._ (1981), where
$a_{ij}$ is half the energy gap at the avoided crossing and $\alpha_{ij}$ is
the differential gradient between the PECs.
From this formula, it is clear that the probability of undergoing adiabatic
dynamics at a given crossing can be experimentally tuned through the relative
velocity $\dot{R}$, which is determined in an experimental setting by the atom
temperature and the additional kinetic energy acquired upon falling inward on
the polarization potential of the non-polar $S$-state. Therefore, systems with
small relative velocities have an increased probability to follow the PEC
adiabatically. In contrast, systems with high relative velocities have a
larger probability to traverse the crossing non-adiabatically and thereby
remain on the comparatively flat polarization potential. Hence, each avoided
crossing provides two collisional channels: one that is mostly populated by
systems with low kinetic energy (cold channel) and one that is mostly occupied
by high kinetic energy systems (hot channel), see Fig. 1. Interestingly, this
leads to a counter-intuitive behaviour for the overall dynamics: if a cold,
low kinetic energy system follows the PEC adiabatically, it will ultimately
reach the steep strongly polar potential and thus rapidly accelerate. This
results in a faster collision compared to a system with high kinetic energy,
which travels along the flat $S$-state potential and experiences weaker
acceleration.

Figure 2: Experimental sequence and initial state preparation.. (a)-(e)
Schematics of the experimental sequence, which consists of the following
steps: (a) two-photon ionization to create an ion, (b) two-photon Rydberg
excitation in the electric field of the ion, (c) variable interaction time,
(d) applying a small separation field which drags the ion along the optical
axis of the ion microscope, (e) ionization of the Rydberg atom and imaging of
both particles by using a large electric field. Panel (f) shows the Rydberg
atom position relative to the ion (located at the origin) for different
Rydberg laser detunings $\Delta$ from the bare $\ket{129S}$ state in absence
of an ion. (g) Histograms of the azimuthally averaged ion-Rydberg atom
distance $R$ for various detunings $\Delta$ (blue). The theoretically
calculated polarization potential is shown in white to green color code, which
indicates the overlap with the bare $\ket{129S}$ state (only overlaps
$\geqslant 0.3$ are shown).

Overview of experimental sequence — The charged ion-Rydberg atom system is
realized in a laser-cooled rubidium cloud held in an optical dipole trap of a
temperature of about $20\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{K}$. In
order to minimize stray electric fields in the system, six electrodes are used
to compensate stray electric fields well below
$100\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{V}\mathrm{/}\mathrm{c}\mathrm{m}$
Veit  _et al._ (2021). In that way, the ions can be kept in position for the
time of the experiment and no ion trap is needed. An experimental block starts
with a $1\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{s}$ long ionization pulse
which incorporates a two-photon ionization process, providing just enough
energy to overcome the ionization threshold (see Fig. 2(a)). Next, a Rydberg
atom is excited in the electric field of the ion by using a
$1\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{s}$ long Rydberg excitation
pulse, again involving two laser beams (see Fig. 2(b)). The detuning $\Delta$
of the upper $480\text{\,}\mathrm{n}\mathrm{m}$ excitation laser from the bare
atomic state in zero field determines the initial radius $R_{0}$ at which the
Rydberg atom is facilitated around the ion. The corresponding effective
linewidth, $\dot{R}_{0}$ is mostly determined by the temperature of the
rubidium cloud. This blue laser illuminates the atomic cloud as a thin light
sheet in the horizontal direction, thus confining the system for highly
excited Rydberg states to a quasi-2D plane, leading to the facilitation of
Rydberg atoms located on a ring around the ion. Afterwards a variable time
$t_{\mathrm{int}}$ can be applied allowing the system to evolve freely. In
order to detect the two particles in a distinguishable way, we drag the ion
along the optical axis of the ion microscope without displacing it in the
imaging plane. To do so, two field electrodes are used to apply a weak
electric field of about
$1.1\text{\,}\mathrm{V}\mathrm{/}\mathrm{c}\mathrm{m}$, which is small enough
to not ionize the Rydberg atom (see Fig. 2(d)). In the final detection step
(Fig. 2(e)), a large electric field of
$340\text{\,}\mathrm{V}\mathrm{/}\mathrm{c}\mathrm{m}$ is applied to field-
ionize the Rydberg atom and to accelerate both particles into the ion
microscope. Due to the previous separation between the ion and the Rydberg
atom, they will arrive at different times at the detector and are therefore
easily distinguishable Zuber  _et al._ (2022); Zou  _et al._ (2023).
Results & discussion — If the interaction time $t_{\rm{int}}$ in Fig. 2(c) is
set to zero, Rydberg atoms initially excited on the flat polarization
potential can be directly detected at their original positions. Excitation to
the high angular momentum curves is ruled out due to negligible $D$-state
overlap. By scanning the detuning $\Delta$ of the Rydberg excitation we can
spectroscopically map out the resonance condition for facilitated excitation
on the interaction potential. Fig. 2(f) shows examples of averaged in situ
images of the Rydberg atom position relative to the ion, meaning that the ion
is always located at the origin. Owing to the excitation in a quasi-2D plane,
a symmetric ring can be observed with the ion microscope. The upper and lower
part are not populated due to the finite, elongated shape of the atomic cloud.
As it can be clearly seen, the distance between the ion and the Rydberg atom
decreases for larger detunings as expected from the facilitation process. Fig.
2(g) summarizes the result of such in situ images by integrating over the
azimuthal angle and showing the data as a function of the internuclear
distance $R$, which represents a direct measurement of the $\ket{129S}$ ion-
Rydberg pair state potential. The blue histogram shows the experimentally
obtained data, which is in good agreement with the calculated PEC displayed in
green.

Figure 3: Ion-Rydberg pair dynamics. Comparison of the observed relative
atomic dynamics (top row) to semi-classical LZ simulations (middle row) for
different detunings of the Rydberg laser $\Delta^{\prime}$ relative to the
outermost avoided crossing at
$R\approx$32.2\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$$. The colorbar
shows the number of counts as a fraction of the total counts at $t=0$. The
bottom row shows the population of the fast collision channels (strongly polar
states) over time for both LZ (red) and adiabatic (blue) simulations. Dashed
vertical lines (not visible for $\Delta^{\prime}=0.8$ MHz in the plotted time
interval) indicate the time $t_{e}$ at which the population has reached
$1-1/e$ (approx. 63%). $\Delta t_{e}/t_{e}$ is the relative difference in this
time between the LZ and adiabatic simulations. For details of the simulations,
see Supplementary Material.

In the next step we introduce a variable interaction time $t_{\rm{int}}>0$
between the Rydberg excitation and the detection, during which the dynamics
take place. This allows the ion-Rydberg pair to move on the interaction
potential, such that the system encounters the series of avoided crossings
shown in Fig. 1(b). The top row of Fig. 3 shows results for the observed
dynamics of the $\ket{129S}$ state at three different detunings. By
$\Delta^{\prime}$ we denote the relative detuning to the outermost avoided
crossing, such that for $\Delta^{\prime}>0$ the system is initialized outside
the fan of the hydrogenic manifold. Each panel represents an average over at
least 6500 ion-Rydberg events on the detector. The middle row shows the result
of semi-classical simulations taking into account the finite temperature,
effective laser linewidth, experimental timings as well as the geometry of the
setup. We model the observed ion-Rydberg pair dynamics by solving the pair’s
classical equation of motion along all possible collision channels. Each
resulting trajectory is assigned a weight corresponding to the probability of
following that particular channel, provided by the LZ formula (for further
details, see Supplementary Material). We observe good overall agreement
between the experimental results and the simulations. For negative
$\Delta^{\prime}$, one clearly observes faster dynamics in total due to the
transition to steep, strongly polar states.
With the help of our semi-classical model, we can better understand the
significance of non-adiabatic transitions in the dynamics by studying the
change in population of the slow and fast collision channels over time. The
bottom row of Fig. 3 shows the population of the strongly polar states over
time for our LZ simulations (red) as well as for a fully adiabatic simulation
based on individual, non-coupled Born Oppenheimer PECs (blue). For all three
relative detunings, the transition to the strongly polar curve is slower in
the LZ simulation due to non-adiabatic transitions. To better quantify this
transition we compare the time $t_{e}$ (dashed vertical lines) at which the
population of the non-polar $S$-state curve has decayed to $1/e$ in both
cases. The relative difference $\Delta t_{e}$ in $t_{e}$ between the blue and
red curves consistently decreases as the Rydberg atoms are excited further
inside the fan, which indicates that the pair’s relative motion becomes
increasingly adiabatic. This is due to the growing gap size $a_{ij}$ at the
avoided crossings at smaller $R$ (see Supplementary Material). In this way the
timescale of the dynamics can also be controlled via the laser detuning.

Figure 4: Effect of the temperature on the dynamics. Panels (a)-(c) compare
the population of the fast collision channels (strongly polar states) over
time for both LZ (red) and adiabatic (blue) simulations at different gas
temperatures $T$. Solid (dashed) lines denote an initial laser detuning of
$\Delta^{\prime}_{1}=0.06$ MHz ($\Delta^{\prime}_{2}=0.31$ MHz) relative to
the outermost avoided crossing at $R\approx$
$32.2\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ (see Fig. 1(b)). Panel
(d) shows the relative difference in $t_{e}$ between LZ and adiabatic
simulations over a range of gas temperatures. $t_{e}$ is the time at which the
polar state population reaches $1-1/e$ (approx. 63%), indicated for the case
of $T=$ $2.0\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{K}$ by the dashed
vertical lines in (b).

Non-adiabatic transitions are further influenced by the gas temperature $T$.
For high $T$, ion-Rydberg pairs have a larger probability to follow the slow
collision channel due to their initially greater relative velocity. We
illustrate this point in Fig. 4, which shows the transition to fast collision
channels over time at different temperatures and two different laser detunings
relative to the outermost avoided crossing. The relative difference in $t_{e}$
increases with $T$ for both detunings (see Fig. 4(d)), indicating that the
pairs created in hotter gases spend longer on the slow collision channel
created by the non-polar $S$-state. Pairs gain kinetic energy as they fall
inward along the polarization potential. For $\Delta^{\prime}_{1}=0.06$ MHz,
this gain in kinetic energy corresponds to approximately
$0.4\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{K}$$\cdot k_{\textrm{B}}$ of
thermal energy, whilst $\Delta^{\prime}_{2}=0.31$ MHz is equivalent to
$2.0\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{K}$$\cdot k_{\textrm{B}}$.
This additional heating accounts for the negligible change in $\Delta
t_{e}/t_{e}$ for temperatures below
$2\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{K}$ (Fig. 4(d)).
Non-adiabatic transitions will also play a more significant role in the
dynamics at higher $n$ as a result of the narrowing avoided crossings, whose
gap sizes $a_{i}$ follow a power-law decay with $n$ (see Supplementary
Material). The signal of non-adiabatic transitions can thus be enhanced by
probing dynamics at larger $n$.
Summary & outlook — In conclusion, we studied the onset of collisional
dynamics between an ion-Rydberg pair in a regime of multiple coupled channels
with different characteristic collision timescales. We were able to describe
the experimentally observed dynamics with the help of a LZ model. Further, the
simulations show that the collisional dynamics can not only be tuned by the
initial distance but should also be tunable by other parameters like the
principal quantum number.
Our work has explored the role of non-adiabatic effects in ion-Rydberg
collisions and lays the foundation for future explorations of beyond Born-
Oppenheimer physics with Rydberg atoms, such as molecular dynamics in the
presence of conical intersections. Precisely understanding non-adiabatic
couplings in complicated potential energy landscapes is also a key ingredient
to better predict the lifetime of Rydberg molecules such as macrodimers or
ion-Rydberg molecules Hummel  _et al._ (2021). In this current experimental
realization, the ion-Rydberg complex was photoassociated directly out of a
trapped gas of 87Rb. Future experiments might consider using individually
trapped atoms in a tweezer setup, which would offer more precise control over
the initial separation of the ion-Rydberg pair, thereby further improving the
starting conditions of the collision.

## Acknowledgements

This work received funding from the DFG as part of the SPP 1929 ”Giant
Interactions in Rydberg Systems (GiRyd)” [Project No. Pf 381/17-1 and No. Pf
381/17-2], and got further funding from the Cluster of Excellence “Advanced
Imaging of Matter” of the Deutsche Forschungsgemeinschaft (DFG)-EXC 2056,
Project ID No. 390715994. Furthermore we are supported by the European
Research Council (ERC) under the European Union’s Horizon 2020 research and
innovation programme (Grant Agreement No. 101019739-LongRangeFermi). F.M.
received funding from the the Federal Ministry of Education and Research
(BMBF) under the grant CiRQus. O.A.H.S. acknowledges great support from the
Alexander von Humboldt Foundation.

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##  Appendix A Supplementary Material

###  A.1 A — Experimental sequence

The experimental sequence starts with the preparation of a cold 87Rb cloud.
The atoms originate from an effusive oven source and are subsequentially
cooled by employing a Zeeman slower and trapped in a magneto-optical trap
(MOT) in a separate MOT-chamber. After a $15\text{\,}\mathrm{m}\mathrm{s}$
long compressed MOT phase followed by a $25\text{\,}\mathrm{m}\mathrm{s}$
optical molasses phase, the atoms are transferred to a movable dipole trap
which transports the atoms into the science chamber below the ion microscope.
At this point the atoms have a temperature of about
$20\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{K}$. In such a sample several
thousand experiments can be performed before a new atomic sample has to be
loaded.
A single experimental block starts with the creation of an ion by using a two-
photon ionization process. The lower transition is realized by a
$420\text{\,}\mathrm{n}\mathrm{m}$ laser beam which is detuned by
$80\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ from the intermediate
$\ket{6P_{3/2},F=3}$ state. The beam is shone into the atomic cloud in the
horizontal direction and has a 1/$e^{2}$ waist of
$w_{420}=$7\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$$. The upper laser
is operated at a wavelength of $1010\text{\,}\mathrm{n}\mathrm{m}$, providing
just enough energy to overcome the ionization threshold. By shining this laser
vertically in the experimental chamber with a small 1/$e^{2}$ waist of
$w_{1010}=$3.2\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$$ one can create
a reasonably good ion spot. Both lasers are simultaneously on for
$1\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{s}$. In the subsequent step the
Rydberg atom is excited in the electric field of the ion, which also takes
place within $1\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{s}$. Here, a two-
photon process is employed once-again, involving a
$780\text{\,}\mathrm{n}\mathrm{m}$ laser,
$250\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ detuned from the intermediate
$\ket{5P_{3/2},F=3}$ state. The second laser is operated at around
$480\text{\,}\mathrm{n}\mathrm{m}$, depending on the desired target Rydberg
state. While the $780\text{\,}\mathrm{n}\mathrm{m}$ laser is much larger than
the atomic cloud and shone in from below, the
$480\text{\,}\mathrm{n}\mathrm{m}$ beam forms a thin light sheet and is
introduced horizontally into the chamber. Thus, the system can be approximated
by quasi-2D plane for large Rydberg states.
In the following step a variable interaction time $t_{\mathrm{int}}$ can be
introduced before the two particles are detected. To distinguish them, a small
electric field is applied along the $z$-axis, which drags the ion along the
optical axis of the ion microscope but does not yet ionize the Rydberg atom.
When the Rydberg atom is field-ionized in the next step by applying the large
extraction field of $340\text{\,}\mathrm{V}\mathrm{/}\mathrm{c}\mathrm{m}$,
the ion and the Rydberg atom have different starting positions and therefore
will arrive at different times on the detector.

Figure 5: Scaling of Landau-Zener transition parameters. (a) Transition
probability as a function of the relative internuclear speed $\dot{R}$ for a
selection of avoided crossings in the ion-129$S$ Rydberg system, where the
crossings are indexed such that 1 refers to the outermost avoided crossing.
(b),(c) Scaling of the gap size $a_{i}$ and differential gradient $\alpha_{i}$
for each crossing in the ion-129$S$ Rydberg system. Solid lines represent fits
to exponentially decaying functions of the form $y=A\exp(-bx)$. We find fit
values of $b=0.258\pm 0.002$ and $b=0.031\pm 0.001$ for $a_{i}$ and
$\alpha_{i}$, respectively. (d),(e) Scaling of the gap size $a_{1}$ and
differential gradient $\alpha_{1}$ of the outermost avoided crossing with
varying principal quantum number $n$. Solid lines represent fits to power-law-
decaying functions of the form $y=Ax^{-b}$. We find fit values of $b=3.422\pm
0.075$ and $b=5.641\pm 0.0.009$ for $a_{1}$ and $\alpha_{1}$, respectively. In
(b)-(e), the fits have a root-mean-square error of less than 10%.

###  A.2 B — Details on the Landau-Zener model for ion-Rydberg pair dynamics

We employ a semi-classical model for the ion-Rydberg pair dynamics, in which
the pair’s classical motion is determined along the $N$ relevant open
collision channels. In the case of $\ket{129S}$, $N=34$. The open channels are
obtained from the adiabatic potential energy curves (PECs) of the system’s
electronic Hamiltonian $H_{e}$, which reads:

$~{}H_{e}=H_{0}+V_{I}(R).$ (1)

Here, $H_{0}$ denotes the unperturbed Rydberg atom and $V_{I}$ describes the
interaction between the ion and the Rydberg atom. We are interested in
ultralong-range internuclear separations $R$ at which the overlap between the
species’ charge distributions vanishes, enabling $V_{I}$ to be approximated
using a multipole expansion Duspayev  _et al._ (2021); Deiß  _et al._ (2021):

$V_{I}(R)=-\sum_{\lambda=1}^{\infty}\sqrt{\frac{4\pi}{2\lambda+1}}\,\frac{r^{\lambda}}{R^{\lambda+1}}Y_{\lambda}^{0}(\theta,\phi),$
(2)

where $r$ is the position of the Rydberg electron with respect to the core,
$Y_{\lambda}^{0}(\theta,\phi)$ are spherical harmonics which are functions of
the Rydberg electron’s angular position $(\theta,\phi)$ and $\lambda$ denotes
the order of the multipole expansion. The above expansion in (2) is valid in
the regime $R\gg r$. The adiabatic PEC $\\{\varepsilon_{i}(R)\\}$ are
determined by evaluating eigenvalues of the electronic Hamiltonian in Eq. (1)
over a range of internuclear separations $R$ using the pairinteraction program
Weber  _et al._ (2017), which truncates the series in Eq. (2) at $\lambda=6$.
We are particularly interested in adiabatic PEC which correspond
asymptotically to weakly-polar atomic Rydberg states. At large internuclear
separations, the non-polar $\ket{nS}$ state acquires an induced dipole moment
and the leading-order correction to its energy is given by
$\varepsilon_{S}\propto-n^{7}/R^{4}$. In the Rydberg series of 87Rb, the
quantum-defect-split $\ket{nS}$ state lies below the $n-3$ degenerate
hydrogenic manifold. The states in this manifold have an angular momentum of
$l>3$ and acquire a permanent dipole moment at large $R$ due to $l$-mixing,
which makes them strongly polar and gives them a leading-order energy
correction of $\varepsilon_{l}\propto\pm n^{2}/R^{2}$.
The competing $R$-scaling of $\varepsilon_{S}$ and $\varepsilon_{L}$ means
that the attractive branches of the Stark-split high-$l$ states eventually
become near-degenerate with the $\ket{nS}$ state at sufficiently low $R$,
giving rise to a series of avoided crossings (see Fig. 1(b) in the main text).
We refer to this region of internuclear separations $R_{c}$ as the Inglis-
Teller limit, which was defined originally in studies on plasmas as the point
at which Stark broadening is sufficiently strong to mix states of different
$n$ (Gallagher, 1994, p. 75). Based on the asymptotic behaviour of the non-
polar and polar PEC in our system, the onset of this regime should scale
roughly as $R_{c}\propto n^{5/2}$, which is equivalent to a critical electric
field strength of $E\propto n^{-5}$ and agrees with earlier results Inglis and
Teller (1939).
In the vicinity of these avoided crossings, the adiabatic Born-Oppenheimer
approximation is no longer strictly valid and in general it becomes necessary
to employ a coupled-channel formalism for modelling the vibrational dynamics
Köppel  _et al._ (1984), often at greater computational cost. In contrast, the
Landau-Zener (LZ) formula Zener and Fowler (1997) provides a straightforward
semi-classical approach for modelling the dynamics of vibrational degrees of
freedom in the vicinity of avoided crossings and has been previously applied
within the context of ultralong-range Rydberg molecules for predicting
electronic transitions due to non-adiabatic couplings Schlagmüller  _et al._
(2016). The LZ formula states that for a wavepacket moving with speed
$\dot{R}$ toward an avoided crossing between two coupled PECs
$\varepsilon_{i}(R)$ and $\varepsilon_{j}(R)$, the system may undergo a non-
adiabatic transition between the channels with probability

$P_{ij}=\exp\bigg{(}-2\pi\frac{a^{2}_{ij}}{\dot{R}\,\alpha_{ij}}\bigg{)}.$ (3)

Here $a_{ij}$ is half the energy gap at the avoided crossing and $\alpha_{ij}$
is the differential gradient between the curves. For brevity, we re-write Eq.
(3) in terms of a single index $i$ for each avoided crossing
$P_{ij}\rightarrow P_{i}$, where $i=1$ corresponds to the outermost (i.e.
large $R$) avoided crossing in the ion-Rydberg PEC.
As shown in Fig. 5(a) for Rydberg atoms excited to the $\ket{129S}$ state, the
ion-Rydberg pair has a finite probability to transition between PECs at the
avoided crossings, which opens up a multitude of potential collision channels.
The transition probability rises with increasing relative velocity $\dot{R}$
and decreases drastically for avoided crossings at smaller internuclear
separations. This is primarily due to the increasing gap size $a_{i}$ at small
$R$, which can be seen in Fig. 5(b). The quantities featured in Eq. (3) show
significant variation with principal quantum number $n$. The variation in the
gap size and gradient at the outermost crossing ($i=1$) with $n$ is shown in
Fig. 5(d) and 5(e). We expect that the quadratic depedence of $a_{i}$ in Eq.
(3) will ensure that its variation dominates the change in transition
probabilities compared to variations in the differential gradient
$\alpha_{i}$. Thus, the probability for non-adiabatic transitions will
increase with $n$.
We model the dynamics of the ion-Rydberg pair starting on the $\ket{129S}$ PEC
and account for non-adiabatic transitions between collision channels via the
LZ formula. The initial conditions $R(0)=R_{0}$ and $\dot{R}(0)=\dot{R}_{0}$
of the pair’s equation of motion are determined from the Rydberg laser
detuning $\Delta$ and the gas temperature $T$, respectively. To ensure a fair
comparison with the experimental measurements, it is necessary to account for
the finite effective linewidth of the Rydberg excitation laser $\delta$, which
broadens the initial separation distribution of the ion-Rydberg pairs
$\\{R_{0}\\}$. In addition, various time delays are present during the
preparation and extraction steps of the experimental sequence due to the
finite operation time of the excitation lasers and the time of flight of the
ions to the multi-channel plate detector, which have also been accounted for
in our simulations.

Figure 6: Dynamics for different principal quantum numbers $\bm{n}$. (a) The
population of the strongly polar high-$l$ states is plotted on a rescaled time
axis, where $\beta(n)$ is chosen in such a way that the dynamics on the bare
$S$-state potential collapse on the $n=70$ curve for all $n$. (b) For better
comparison the arrival time of the first crossing is set to zero. (c)
Transition probability as a function of $n$. Colors represent different
avoided crossings - ranging from the outermost crossing (blue) to the
innermost crossing (orange). Probabilities corresponding to the same crossing
are connected by lines to guide the eye.

###  A.3 C — n-scaling of non-adiabatic transition probabilities

We now investigate the tunability of the LZ transition probabilities with
principal quantum number $n$. For the initial separation we choose a fixed
distance of $33\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$, such that for
all $n$ the Rydberg atom is excited onto the $\ket{nS}$ PEC before it crosses
into the fan of Stark-split strongly polar states. For simplicity, we set the
initial velocity and the effective laser linewidth to zero and consider a one-
dimensional case.
The duration of the dynamics varies considerably with $n$ due to the changing
length scale and gradient of the PEC (see Fig. 5(d) and 5(e)). Consequently,
to be able to easily compare the relative adiabaticity of the different
dynamics, we rescale the time-axis by a factor $e^{\beta}$, where $\beta$ is
chosen for each $n$ in such a way that the dynamics on the bare $S$-state
potential collapse onto the same trajectory. Here, we choose $n=70$ as the
reference trajectory. Therefore, $\beta(n{=}70)=0$ and we find that $\beta$
strictly increases for larger $n$, reaching $\beta(n{=}130)\approx 2.2$ for
the largest considered principal quantum number.
Fig. 6(a) shows the fraction of systems that populate the strongly polar
high-$l$ states as a function of the rescaled time, where each point
corresponds to the time at which a crossing is reached. For higher $n$, the
avoided crossings are reached earlier and as a result the population starts to
increase at earlier times. In Fig. 6(b), the arrival time at the first
crossing is set to zero for all curves, revealing the difference of the
adiabaticity inside the fan of the hydrogenic manifold for different $n$. This
is mainly dominated by the density of the crossings since, the LZ probability
at each individual crossing does not change drastically with $n$ for the first
10 crossings, as can be seen in Fig.6(c)). In this figure,, we plot the LZ
probabilities for all individual crossings, starting with the outermost
crossing shown in blue to the deepest considered crossing shown in orange. The
deviation from the overall trend at $n=130$ is due to the fact that the atoms
are initialized very close to the first crossing for this state.

Figure 7: Ion-Rydberg pair dynamics. Comparison of the observed (projected)
relative atomic dynamics $R^{\prime}(t)$ (top row) to semi-classical
simulations (middle row) for different detunings of the Rydberg laser
$\Delta^{\prime}$ relative to the $\ket{70S}$ atomic Rydberg state. The bottom
row shows the population of the fast collision channels over time for both LZ
(red) and adiabatic (blue) simulations. Dashed vertical lines indicate the
time $t_{e}$ at which the population has reached $1-1/e$ (approx. 63%).
$\Delta t_{e}/t_{e}$ is the relative difference in this time between the LZ
and adiabatic simulations.

###  A.4 D — Ion-Rydberg pair dynamics for the $\ket{70S}$ state

We also observed ion-Rydberg dynamics for the $\ket{70S}$ atomic Rydberg state
and the results are shown in Fig. 7. For $\ket{70S}$, the Inglis-Teller regime
manifests at smaller internuclear separations than $\ket{129S}$ (cf.
$R_{c}\approx$ $6.6\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ to
$R_{c}\approx$ $32.2\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$) due to
the lower principal quantum number of the Rydberg excitation. Moreover, ion-
Rydberg collision pairs at $\ket{70S}$ have fewer open channels than at
$\ket{129S}$ due to the smaller number of degenerate high-$l$ states in the
neighboring $n=67$ manifold.
Additionally, in contrast to the measurements taken for $\ket{129S}$, no 2D
confinement was employed during the experimental sequence and consequently the
dynamics unfold in 3D. As a result, the measurements show the projection of
the internuclear separation $R$ onto the micro-channel plate, denoted by
$R^{\prime}$, instead of the true value of $R$. This has the consequence that
the ion-Rydberg separation distribution is extremely broad – even for time
$t=0$ – as can be seen in the top and middle rows of Fig. 7.
As the ion-Rydberg pairs travel inward to small internuclear separations, they
will eventually reach an extreme regime in which the Stark-splitting of
neighboring and next-neighboring hydrogenic manifolds becomes far greater than
their field-free energetic separation. Thus, the electronic states of the
Rydberg atom become highly mixed and an accurate description of the physics
becomes increasingly difficult due to the sheer multitude of open channels. We
do not aim at a description of the dynamics at such small internuclear
separations and instead focus on capturing the physics at intermediate
separations at the onset of the Inglis-Teller regime. Therefore, we introduced
a short-distance cut-off to the simulations which removes particles once they
have crossed a certain threshold separation, equal to
$0.1\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ for the case of
$\ket{70S}$.
When comparing the LZ and adiabatic simulations in the bottom row of Fig. 7,
we see that the absolute difference in the transition times $\Delta t_{e}$ to
the strongly polar curves decreases with decreasing detuning $\Delta^{\prime}$
– indicating, as expected, the increased frequency of non-adiabatic
transitions at higher detuning. However, unlike the simulations carried out
for $\ket{129S}$, there is no clear trend in the relative difference of these
transition times. We attribute this partially to the effective linewidth of
the excitation laser $\delta=$ $3.4\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$,
which leads to broadening of the initial separation distribution $\\{R_{0}\\}$
of the collision pairs. Whilst this broadening is also present at
$\ket{129S}$, the effect is amplified in the case of $\ket{70S}$ due to the
smaller length scales at which the dynamics occur.


# Semiflexible polymer enclosed in a 3D compact domain

Pavel Castro-Villarreal 1,∗, and J. E. Ramírez 2,∗

###### Abstract

## 1

The conformational states of a semiflexible polymer enclosed in a volume
$V:=\ell^{3}$ are studied as stochastic realizations of paths using the
stochastic curvature approach developed in [Rev. E 100, 012503 (2019)], in the
regime whenever $3\ell/\ell_{p}>1$, where $\ell_{p}$ is the persistence
length. The cases of a semiflexible polymer enclosed in a cube and sphere are
considered. In these cases, we explore the Spakowitz-Wang type polymer shape
transition, where the critical persistence length distinguishes between an
oscillating and a monotonic phase at the level of the mean-square end-to-end
distance. This shape transition provides evidence of a universal signature of
the behavior of a semiflexible polymer confined in a compact domain.

##  2 Keywords:

semiflexible polymer, stochastic curvature, shape transition, critical
persistence length, mean square end-to-end distance.

##  3 Introduction

Semiflexible polymers is a term coined to understand a variety of physical
systems that involve linear molecules. The most popular polymers are
industrial plastics, like polyethylene or polystyrene, with various
applications in daily life [1, 2]. Another prominent example is the DNA
compacted in the nucleus of cells or viral DNA/RNA packed in capsids [3, 4].
These last examples are of particular interest since they are confined
semiflexible polymers. Indeed, biopolymers’ functionality is ruled by their
conformation, which in turn is considerably modified in the geometrically
confined or crowded environment inside the cell [5, 6, 7].

A common well-known theoretical framework used to describe the fundamental
properties of a semiflexible polymer is the well-known worm-like chain model
(WLC), which pictures a polymer as a thin wire with a flexibility given by its
bending rigidity constant $\alpha$ [8]. The central quantity in this model is
the persistence length defined by $2\alpha/(k_{B}T(d-1))$ [9, 10], being $d$
the space dimension, however, here we simply use
$\ell_{p}:=\alpha/(k_{B}T)$111For the sake of notation, it is hidden the
dimension of the space in the persistence length definition. In those cases
where an explicit dependence on the dimension is needed, it should be
adequately scaled by the factor $2/(d-1)$. , which is the characteristic
length along the chain over which the directional correlation between segments
disappears. $k_{B}T$ is the thermal energy, with $k_{B}$ and $T$ the Boltzmann
constant and $T$ the bath temperature, respectively. [11].

In the absence of thermal fluctuations, when $\alpha\gg k_{B}T$, the
conformations of the polymer are well understood through different curve
configurations determined by variational principles [12, 13]. For the WLC
model, the bending energy functional is given by

$H[\mathbf{R}]=\frac{\alpha}{2}\int ds\kappa^{2}(s),$ (1)

where ${\bf R}(s)$ is a polymer configuration and $\kappa(s)$ is the curvature
of the chain, with $s$ the arc-length parameter. Additional terms can be added
to the Hamiltonian to account for other effects including multibody
interactions, external fields, and constraints on the chain dimensions [14,
15]. When the thermal fluctuations are relevant $\alpha\simeq k_{B}T$, it is
usual to introduce a statistical mechanics description. Since $H[\mathbf{R}]$
represents the bending energy for a curve configuration ${\bf R}$, the most
natural approach is to defined the canonical probability density

$\displaystyle\mathcal{P}\left(\bm{R}\right)\mathcal{D}\bm{R}:=\frac{1}{\mathcal{Z}_{\rm
c}}\exp\left(-\frac{\ell_{p}}{2}\int
ds\kappa^{2}\left(s\right)\right)\mathcal{D}\bm{R},$ (2)

where $\mathcal{Z}_{\rm c}$ is the canonical partition function, and
$\mathcal{D}{\bf R}$ is an appropriate functional measure. In this
description, the theory turns out to be a one-dimensional statistical field
theory. Nonetheless, the theory is not easy to tackle since
$\kappa\left(s\right)$ acquires non-linear terms in ${\bf R}$. To avoid this
difficulty, a different perspective was introduced by Saito’s et al. [8],
where it was studied the following probability density function

$\displaystyle\mathcal{P}\left(\bm{T}\right)\mathcal{D}\bm{T}:=\frac{1}{\mathcal{Z}_{\rm
s}}\exp\left(-\frac{\ell_{p}}{2}\int
ds\kappa^{2}\left(s\right)\right)\mathcal{D}\bm{T},$ (3)

instead of Eq. (2). Here $\mathcal{Z}_{\rm s}$ is the Saito’s partition
function and $\mathcal{D}{\bf T}$ is an appropriate functional measure for the
tangent direction of a given polymer configuration ${\bf R}$. The Saito’s
partition function can be solved since one has $\kappa^{2}(s)=\left({d{\bf
T}(s)/ds}\right)^{2}$, thus one can relate $\mathcal{Z}_{\rm s}$ with the
Feynman’s partition function for a quantum particle in the spherical surface
described by ${\bf T}^{2}=1$. For the cases when the semiflexible polymer is
in an open Euclidean space, the Saito’s approach works very well. For
instance, it reproduces the standard results of Kratky-Porod [16], among other
results [8]. However, for the cases when the semiflexible polymer is confined
to a bounded region of the space the Saito’s approach is difficult to use,
with some exceptional cases like the situation for semiflexible polymers
confined to a spherical shell [14].

For semiflexible polymers in plane space, an alternative theoretical approach
to the above formalisms was introduced in [17]. This consists of postulating
that each conformational realization of any polymer in the plane is described
by a stochastic path satisfying the stochastic Frenet equations, defined by
$\frac{d}{ds}\mathbf{R}(s)=\mathbf{T}(s)$, and
$\frac{d}{ds}\mathbf{T}(s)=\kappa(s){\bf N}\left(s\right)$, where
$\mathbf{R}(s)$ is the configuration of the polymer, $\mathbf{T}(s)$ is the
tangent vector to the curve describing the chain at $s$, ${\bf
N}(s):=\mathbf{\epsilon}\mathbf{T}(s)$ is the normal stochastic unit vector,
with $\mathbf{\epsilon}$ a rotation by an angle of $\pi/2$, and $\kappa(s)$ is
the stochastic curvature that satisfies the following probability density
function

$\displaystyle\mathcal{P}\left(\kappa\right)\mathcal{D}\kappa:=\frac{1}{\mathcal{Z}_{\rm
s-c}}\exp\left(-\frac{\ell_{p}}{2}\int
ds\kappa^{2}\left(s\right)\right)\mathcal{D}\kappa,$ (4)

where $\mathcal{Z}_{\rm s-c}$ is the partition function in the stochastic
curvature formalism, and $\mathcal{D}\kappa$ is an appropriate measure for the
curvature. This in particular, implies a white noise-like structure, i.e.,
$\langle\kappa(s)\rangle=0$ and
$\langle\kappa(s)\kappa(s^{\prime})\rangle=\delta(s-s^{\prime})/\ell_{p}$
[17]. This theoretical framework successfully explains, by first principles,
the Kratky-Porod results for free chains confined into an open 2D-plane.
Moreover, it correctly describes the mean square end-to-end distance for
semiflexible polymers confined to a square box, a key descriptor of the
statistical behavior of a polymer chain.

In the present work we carry out an extension of the stochastic curvature
approach for semiflexible polymers in the three-dimensional space
$\mathbb{R}^{3}$. In particular, we analyze the conformational states of a
semiflexible polymer enclosed in a bounded region in three-dimensional space.
This polymer is in a thermal bath with a uniform temperature. The shapes
adopted by the polymer are studied through the mean-square end-to-end distance
as a function of the polymer total length as well as its persistence length.
In particular, we analyze the cases of a polymer confined to a cube of side
$a$ and a sphere of radius $R$.

The plan of this paper is as follows. In Sect. 4, we introduce the stochastic
Frenet equations for the semiflexible polymers in three-dimensional spaces,
and by using standard procedure we derive a corresponding Fokker-Planck
equation. In particular, the Kratky-Porod result for polymers in a 3d open
space is obtained. Sect. 5 contains the derivation of the mean square end-to-
end distance for semiflexible polymers confined to a compact domain. In Sect.
6, we present the analysis of the mean square end-to-end distance for the
cases when the compact domain correspond with a cube of side $a$, and a sphere
of radius $R$. Finally, Sect. 7 contains our concluding remarks.

##  4 Preliminary notation and semiflexible polymers in 3D

Let us consider a polymer in a three-dimensional Euclidean space
$\mathbb{R}^{3}$ as a space curve $\gamma$, ${\bf
R}:I\subset\mathbb{R}\to\mathbb{R}^{3}$, parametrized by an arc-length, $s$.
For each point $s\in I$, a Frenet-Serret trihedron can be defined in terms of
the vector basis $\\{{\bf T}(s),{\bf N}(s),{\bf B}(s)\\}$, where ${\bf
T}(s)=d{\bf R}/ds$ is the tangent vector, whereas ${\bf N}(s)$ and ${\bf
B}(s)$ are the normal and bi-normal vector, respectively. It is well known
that each regular curve $\gamma$ satisfies the Frenet-Serret structure
equations, namely, $d{\bf T}/ds=\kappa(s){\bf N}$, $d{\bf N}/ds=-\kappa(s){\bf
T}-\tau(s){\bf B}$ and $d{\bf B}/ds=\tau(s){\bf N}$, where $\kappa(s)$ and
$\tau(s)$ are the curvature and the torsion of the space curve. In addition,
the fundamental theorem of space curves estates that given continuous
functions $\kappa(s)$ and $\tau(s)$ one can determine the shape curve
uniquely, up to a Euclidean rigid motion [18].

###  4.1 Stochastic curvature approach in 3D

In order to study the conformational states of a semiflexible polymer, we
adapt the stochastic curvature approach introduced in [17] to the case of
semiflexible polymers in 3D Euclidean space. For the 2D Euclidean space, the
formalism starts by postulating that each conformational realization of any
polymer is described by a stochastic path satisfying the stochastic Frenet
equations. In the 3D case, it is enough to consider the following stochastic
equations

$\displaystyle\frac{d}{ds}{\bf R}(s)$ $\displaystyle=$ $\displaystyle{\bf
T}(s),$ (5a) $\displaystyle\frac{d}{ds}{\bf T}(s)$ $\displaystyle=$
$\displaystyle\mathbb{P}_{\bf T}\bm{\kappa}(s),$ (5b)

where ${\bf R}(s)$, ${\bf T}(s)$ and $\bm{\kappa}(s)$ are now random
variables. $\bm{\kappa}(s)$ is named here stochastic vectorial curvature.
Also, it has been introduced a normal projection operator $\mathbb{P}_{\bf
T}=\bm{1}-{\bf T}\otimes{\bf T}$ such that ${\bf T}(s)\cdot\frac{d}{ds}{\bf
T}(s)=0$. According to these equations, it can be shown that $\left|{\bf
T}(s)\right|$ is a constant that can be fixed to unit, where
$\left|~{}~{}\right|$ is the standard 3D Euclidean norm. The remaining
geometrical notions also turn into random variables as follows. The stochastic
curvature is defined by $\kappa(s):=\left|\bm{\kappa}(s)\right|$. The
stochastic normal and bi-normal vectors are defined by ${\bf
N}(s):=\bm{\kappa}(s)/\kappa(s)$ and ${\bf B}(s):={\bf
T}(s)\times\bm{\kappa}(s)/\kappa(s)$, respectively, where $\kappa(s)$ is the
stochastic curvature. In addition, the stochastic torsion is defined with the
equation $\tau(s):={\bf N}(s)\cdot\frac{d}{ds}{\bf B}(s)$.

In addition to the stochastic equations (5), the random variable
$\bm{\kappa}(s)$ is distributed according to the probability density function

$\displaystyle\mathcal{P}\left(\bm{\kappa}\right)\mathcal{D}\bm{\kappa}:=\frac{1}{\mathcal{Z}_{\rm
s-c}}\exp\left(-\beta
H\left[{\bm{\kappa}}\right]\right)\mathcal{D}\bm{\kappa},$ (6)

where $H\left[\bm{\kappa}\right]=\frac{\alpha}{2}\int\bm{\kappa}^{2}ds$ is the
bending energy, and $\alpha$ is the bending rigidity modulus. This energy
functional corresponds to the continuous form of the WLC model [8]. Also, in
Eq. (6) $\mathcal{Z}_{\rm s-c}$ is an appropriate normalization constant,
$\mathcal{D}{\bm{\kappa}}:=\prod_{i=1}^{3}\mathcal{D}\kappa_{i}$ is a
functional measure, and $\beta=1/k_{B}T$ is the inverse of the thermal energy.
The Gaussian structure of the probability density implies the zero mean
$\left<\kappa_{i}(s)\right>=0$, and following $3D$ fluctuation theorem

$\displaystyle\left<\kappa_{i}(s)\kappa_{j}(s^{\prime})\right>=\frac{1}{\ell_{p}}\delta_{ij}\delta(s-s^{\prime}),$
(7)

where $\kappa_{i}(s)$ is the $i-$th component of the stochastic vectorial
curvature $\bm{\kappa}(s)$.

###  4.2 From Frenet-Serret stochastic equations to Hermans-Ullman equation in
3D

In this section, we present the Fokker-Planck formalism corresponding to the
stochastic equations (5). This description allows us to determine an equation
for the probability density function associated to the position and direction
of the endings of the polymer $P\left({\bf R},{\bf T}\left.\right|{\bf
R}^{\prime},{\bf T}^{\prime};s\right)=\left<\delta\left({\bf R}-{\bf
R}(s)\right)\delta\left({\bf T}-{\bf T}(s)\right)\right>$, where ${\bf R}$ and
${\bf R}^{\prime}$ are the ending positions of the polymer, and ${\bf T}$ and
${\bf T}^{\prime}$ are the corresponding directions. The parameter $s$ is the
polymer length.

Now, the stochastic Frenet-Serret equations (5) can be identified with a
multi-dimensional stochastic differential equation in the Stratonovich
perspective, thus applying the standard procedure [19], we find the following
Fokker-Planck type equation

$\displaystyle\frac{\partial P}{\partial s}+\nabla\cdot\left({\bf
T}~{}P\right)=\frac{1}{2\ell_{p}}\Delta_{g}P,$ (8)

where ${\bf T}$ is identified with the unit normal vector on $S^{2}$, thus
satisfies the condition ${\bf T}^{2}=1$. The operator $\Delta_{g}$ is the
Laplace-Beltrami of the sphere $S^{2}$. Similarly, as the situation for
semiflexible polymers confined to a plane space [17], this equation is exactly
the same as the one obtained by Hermans and Ullman in 1952 [20], where the
heuristic parameter they included can now be identify exactly with
$1/(2\ell_{p})$. In addition, we can make a contact with the Saito’s approach
[8] by considering the marginal probability density function

$\displaystyle\mathcal{Z}_{\rm s}\left({\bf T},{\bf
T}^{\prime},s\right)\propto\int d^{3}{\bf R}d^{3}{\bf R}^{\prime}P\left({\bf
R},{\bf T}\left.\right|{\bf R}^{\prime},{\bf T}^{\prime},s\right).$ (9)

Using the Hermans-Ullman equation, we can show that $\mathcal{Z}_{\rm c}$
satisfies a diffusion equation on a spherical surface with diffusion
coefficient equal to $1/(2\ell_{p})$ [8], that is,

$\displaystyle\frac{\partial\mathcal{Z}_{\rm c}}{\partial
s}=\frac{1}{2\ell_{p}}\Delta_{S^{2}}\mathcal{Z}_{\rm s}.$ (10)

An immediate consequence of the above equation is the exponential decay of the
correlation function between the two ending directions $C(L):=\left<{\bf
T}(L)\cdot{\bf T}(0)\right>=\exp\left(-L/\ell_{p}\right)$, where $L$ is the
polymer length. Indeed, this expectation value satisfies the following
equation
$\frac{d}{ds}C(s)=\frac{1}{2\ell_{p}}\frac{1}{4\pi}\int_{S^{2}}d\Omega\left({\bf
T}(s)\cdot{\bf T}(s^{\prime})\right)\Delta_{S^{2}}\mathbb{Z}$, where $d\Omega$
is the solid angle and $4\pi$ is a normalization constant. Now, we can
integrate twice by parts the r.h.s. of last equation and since $S^{2}$ is a
compact manifold the boundary terms vanish. Also, using $\Delta_{S^{2}}{\bf
T}=-\frac{2}{R^{2}}{\bf T}$, it is found that the correlation function
satisfies the ordinary differential equation
$\frac{d}{ds}C(s)=-\frac{2}{R^{2}}C(s)$. Now, we solve this equation using the
initial condition $C(s^{\prime}=0)=1$, and the length of the polymer set up by
$s=L$.

###  4.3 Modified telegrapher equation

As in the situation of the two-dimensional case [17], we carry out a
multipolar decomposition for HU equation in 3D. This consists of expanding the
probability density function $P\left({\bf R},{\bf T}\left.\right|{\bf
R}^{\prime},{\bf T}^{\prime};s\right)$ in a linear combination of the
cartesian tensor basis elements $1$, $T_{i}$,
$T_{i}T_{j}-\frac{1}{3}\delta_{ij}$,
$T_{i}T_{j}T_{k}-\frac{1}{5}\delta_{\left(ij\right.}T_{\left.k\right)}$,
$\cdots$, where the symbols $(ijk)$ means symetrization of the indices
$i,j,k$, that is,
$\delta_{\left(ij\right.}T_{\left.k\right)}=\delta_{ij}T_{k}+\delta_{jk}T_{i}+\delta_{ki}T_{j}$
whose expansion coefficients are hydrodynamic-like tensor fields. These
tensors are $\rho({\bf R},s)$, meaning by the manner how the ending positions
are distributed in the space; $\mathbb{P}\left({\bf R},s\right)$, meaning as
the local average of the polymer direction; $\mathbb{Q}_{ij}({\bf R},s)$,
pointing the way how the directions are correlated along the points of the
space, etc. These tensors are the moments associated to the cartesian tensor
basis, e.g. $\mathbb{P}_{i}=\int\frac{d\Omega}{4\pi}T_{i}P\left({\bf R},{\bf
T},s\right)$. These fields satisfy the following hierarchy equations

$\displaystyle\frac{\partial\rho({\bf R},s)}{\partial s}$ $\displaystyle=$
$\displaystyle-\partial_{i}\mathbb{P}^{i}\left({\bf R},s\right),$ (11)
$\displaystyle\frac{\partial\mathbb{P}_{i}({\bf R},s)}{\partial s}$
$\displaystyle=$ $\displaystyle-\frac{1}{\ell_{p}}\,\mathbb{P}_{i}({\bf
R},s)-\frac{1}{3}\partial_{i}\rho({\bf
R},s)-\partial^{j}\mathbb{Q}_{ij}\left({\bf R},s\right),$ (12)
$\displaystyle\frac{\partial\mathbb{Q}_{ij}\left({\bf R},s\right)}{\partial
s}$ $\displaystyle=$
$\displaystyle-\frac{3}{\ell_{p}}\mathbb{Q}_{ij}\left({\bf
R},s\right)-\frac{1}{5}\mathbb{T}_{ij}\left({\bf
R},s\right)-\partial^{k}\mathbb{R}_{ijk}\left({\bf R},s\right),$ (13)

where
$\mathbb{T}^{ij}=\partial^{i}\mathbb{P}^{j}+\partial^{j}\mathbb{P}^{i}-\frac{2\delta^{ij}}{3}\partial_{k}\mathbb{P}^{k}$.

Now, by combining Eqs. (11) and (12) we can obtain a modified telegrapher
equation

$\displaystyle\frac{\partial^{2}\rho\left({\bf R},s\right)}{\partial
s^{2}}+\frac{1}{\ell_{p}}\frac{\partial\rho\left({\bf R},s\right)}{\partial
s}=\frac{1}{3}\nabla^{2}\rho\left({\bf
R},s\right)+\partial_{i}\partial_{j}\mathbb{Q}^{ij}\left({\bf R},s\right),$
(14)

where $\nabla^{2}$ is the 3D Laplacian. In a mean-field point of view one can
consider the preceding equation as an equation for the probability density
function $\rho\left({\bf R},s\right)$ under the presence of a mean-field
$\mathbb{Q}_{ij}\left({\bf R},s\right)$. In particular,
$\mathbb{Q}_{ij}\left({\bf R},s\right)$ does not play any role for the mean-
square end-to-end distance for a semiflexible polymer in the open Euclidean 3D
space. Indeed, let us defined the end-to-end distance as $\delta{\bf R}:={\bf
R}-{\bf R}^{\prime}$, thus the mean-square end-to-end distance is given by

$\displaystyle\left<\delta{\bf
R}^{2}\right>_{\mathcal{D}}\equiv\int_{\mathcal{D}\times\mathcal{D}}\rho\left(\left.{\bf
R}\right|{\bf R}^{\prime},s\right)\delta{\bf R}^{2}d^{3}{\bf R}d^{3}{\bf
R}^{\prime}.$ (15)

Now, we implement the same procedure used in [17] to calculate the mean-square
end-to-end distance in the open three-dimensional space
$\mathcal{D}=\mathbb{R}^{3}$, where it is use the modify telegrapher equation
(14) and the traceless property of $\mathbb{Q}_{ij}\left({\bf R},s\right)$. We
can reproduce the standard Kratky-Porod [16] result for a semiflexible polymer
in the three-dimensional space [20, 16]

$\displaystyle\left<\delta{\bf
R}^{2}\right>_{\mathbb{R}^{3}}=2\ell_{p}L-2\ell_{p}^{2}\left(1-\exp\left(-\frac{L}{\ell_{p}}\right)\right),$
(16)

with the typical well-known asymptotics limits: diffusive regime
$\left<\delta{\bf R}^{2}\right>\simeq 2\ell_{p}L$ for $L\gg\ell_{p}$, and
ballistic regime $\left<\delta{\bf R}^{2}\right>\simeq L^{2}$ for
$L\ll\ell_{p}$.

##  5 Semiflexible polymer in a compact domain

In this section, we apply the hierarchy equations developed in the previous
section in order to the determine the conformational states of a semiflexible
polymer confined to a compact volume domain of size $V$. From the hierarchy
Eqs. (12) and (13), the tensors $\mathbb{P}_{i}({\bf R},s)$ and
$\mathbb{Q}_{ij}({\bf R},s)$ damp out as $e^{-L/\ell_{p}}$ and
$e^{-3L/\ell_{p}}$, respectively. Furthermore, if we consider that the
semiflexible polymer is enclosed in a compact volume $V:=\ell^{3}$, with a
typical length $\ell$, thus as long as we consider cases when $3\ell/\ell_{p}$
is far from 1, we may assume that $\mathbb{Q}_{ij}({\bf R},s)$ is uniformly
distributed. This condition corresponds to truncate the hierarchy equations at
the second level, that is, the only equations that survive in this
approximation are Eqs. (11) and (12).

In the latter situation, the distribution, $\rho({\bf R},s)$ of the endings of
the semiflexible polymer is describes through the following telegrapher’s
equation

$\displaystyle\frac{\partial^{2}\rho\left({\bf R},s\right)}{\partial
s^{2}}+\frac{1}{\ell_{p}}\frac{\partial\rho\left({\bf R},s\right)}{\partial
s}=\frac{1}{3}\nabla^{2}\rho\left({\bf R},s\right),$ (17)

that satisfies the initial conditions

$\displaystyle\lim_{s\to 0}\rho\left({\bf R}\right.\left|{\bf
R}^{\prime},s\right)$ $\displaystyle=$ $\displaystyle\delta^{(3)}\left({\bf
R}-{\bf R}^{\prime}\right),$ (18) $\displaystyle\lim_{s\to
0}\frac{\partial\rho\left({\bf R}\right.\left|{\bf
R}^{\prime},s\right)}{\partial s}$ $\displaystyle=$ $\displaystyle 0.$ (19)

The condition (18) means that the polymers’ ends coincide when the polymer
length is zero, whereas (19) means that the polymer length does not change
spontaneously. In addition, since the polymer is enclosed in the compact
domain $\mathcal{D}$ of volume $V(\mathcal{D})$, we also impose a Neumann
boundary condition

$\displaystyle\left.\nabla\rho\left({\bf R}\right.\left|{\bf
R}^{\prime},s\right)\right|_{{\bf R},{\bf R}^{\prime}\in\partial
D}=0,~{}~{}~{}\forall s,$ (20)

where $\partial D$ is a surface bounding the domain $\mathcal{D}$. This
boundary condition means that the polymer does not cross the boundary neither
wrap the domain. The procedure to obtain a solution of the above telegrapher’s
equation (17) is identically to the one developed in [17]. We just have to
take into account the right factors and the dimensionality considerations. In
this sense, the probability density function is given by

$\displaystyle\rho\left(\left.{\bf R}\right|{\bf
R}^{\prime};s\right)=\frac{1}{V({\mathcal{D}})}\sum_{{\bf k}\in
I}G\left(\frac{s}{2\ell_{p}},\frac{4\ell^{2}_{p}}{3}\lambda_{{\bf
k}}\right)\psi^{\dagger}_{{\bf k}}\left({\bf R}\right)\psi_{{\bf k}}\left({\bf
R}^{\prime}\right)$ (21)

where we recall from [17]

$\displaystyle
G(v,w)=e^{-v}\left[\cosh\left(v\sqrt{1-w}\right)+\frac{\sinh\left(v\sqrt{1-w}\right)}{\sqrt{1-w}}\right],$
(22)

and $\\{\psi_{\bf k}\\}$ and $\\{\lambda_{{\bf k}}\\}$ are a complete set of
orthonormal eigenfunctions and a set of corresponding eigenvalues of the
Laplace operator $-\nabla^{2}$ in $\mathbb{R}^{3}$. Notice that each
$\psi_{\bf k}({\bf R})$ must satisfy the Neumann boundary equation
$\left.\nabla\psi_{{\bf k}}\right|_{{\bf R}\in\partial\mathcal{D}}=0$. In
addition, it is known [21, 22] that for Neumann boundary Laplacian eigenvalue
problem there is a zero eigenvalue $\lambda_{0}=0$ corresponding to a positive
eigenfunction given by $\psi_{0}={1/\sqrt{V}}$.

Now, using (21), the mean-square end-to-end distance $\left<\delta{\bf
R}^{2}\right>_{\mathcal{D}}$ can be computed in the standard fashion by

$\displaystyle\left<\left(\delta{\bf
R}\right)^{2}\right>_{\mathcal{D}}=\sum_{{\bf k}\in
I}a_{k}G\left(\frac{s}{2\ell_{p}},\frac{4\ell^{2}_{p}}{3}\lambda_{{\bf
k}}\right),$ (23)

where the coefficients $a_{k}$ are obtained from

$\displaystyle
a_{k}=\frac{1}{V(\mathcal{D})}\int_{\mathcal{D}\times\mathcal{D}}\left({\bf
R}-{\bf R}^{\prime}\right)^{2}\psi^{\dagger}_{{\bf k}}({\bf R})\psi_{{\bf
k}}({\bf R}^{\prime})d^{3}{\bf R}d^{3}{\bf R}^{\prime}.$ (24)

We can have a further simplification since after squaring the end-to-end
distance inside the last integral. It is not difficult to see that the square
terms ${\bf R}^{2}$ and ${\bf R}^{\prime 2}$ in $({\bf R}-{\bf
R}^{\prime})^{2}$ only the zero mode contribute, thus we have

$\displaystyle\left<\left(\delta{\bf
R}\right)^{2}\right>_{\mathcal{D}}=2\sigma^{2}\left({\bf
R}\right)-\frac{2}{V\left(\mathcal{D}\right)}\sum_{{\bf k}\neq 0}{\bf
r}^{*}_{{\bf k}}\cdot{\bf r}_{{\bf
k}}~{}G\left(\frac{s}{2\ell_{p}},\frac{4\ell^{2}_{p}}{3}\lambda_{{\bf
k}}\right),$ (25)

where $\sigma^{2}\left({\bf R}\right):=\left<{\bf R}^{2}\right>_{g}-\left<{\bf
R}\right>^{2}_{g}$ is called mean-square end position,
$\left<\cdots\right>_{g}:=\frac{1}{V\left(\mathcal{D}\right)}\int_{\mathcal{D}}d^{3}{\bf
R}\cdots$ is termed geometric average, and the factor ${\bf r}_{{\bf
k}}:=\int_{\mathcal{D}}{\bf R}\psi_{{\bf k}}\left({\bf R}\right)d^{3}{\bf R}$
for ${\bf k}\neq 0$. The factor ${\bf r}_{{\bf k}}$ can be written in a
simpler form for Neumann boundary conditions, since $\psi_{{\bf
k}}=-\frac{1}{\lambda_{\bf k}}\nabla^{2}\psi_{{\bf k}}$ and by integrating out
by parts, this factor is expressed in terms of a boundary integral

$\displaystyle{\bf r}_{{\bf k}}=\frac{1}{\lambda_{\bf
k}}\oint_{\partial\mathcal{D}}dS~{}{\bf n}\psi_{\bf k}\left({\bf
R}_{S}\right),$ (26)

where ${\bf R}_{S}\in\partial\mathcal{D}$ and $dS$ is the area element of
$\partial\mathcal{D}$. Since the function $G\left(v,w\right)$ decays
exponentially as the polymer length gets larger values, we can convince
ourselves that twice the mean-square end position corresponds to a saturation
value for the mean-square end-to-end distance. An additional property of ${\bf
r}_{\bf k}$ is the identity

$\displaystyle\frac{1}{V\left(\mathcal{D}\right)}\sum_{{\bf k}\neq 0}{\bf
r}_{k}^{*}{\bf r}_{\bf k}=\sigma^{2}\left({\bf R}\right).$ (27)

This identity can be proved using the completeness relation of the
eigenfunctions, that is, $\sum_{\bf k}\psi^{*}_{\bf k}\left({\bf
R}\right)\psi_{\bf k}\left({\bf R}^{\prime}\right)=\delta^{(3)}\left({\bf
R}^{\prime}-{\bf R}\right)$. This identity allows us to proved that in general
$\left<\left(\delta{\bf R}\right)^{2}\right>_{\mathcal{D}}$ starts at zero.

##  6 Results

###  6.1 Semiflexible polymer enclosed by a cube surface

In this section we provide results for the mean-square end-to-end distance for
a semiflexible polymer enclosed inside of a cube domain. All the problem is
reduced to solve the Neumann eigenvalue problem $-\nabla^{2}\psi=\lambda\psi$
with Neumann boundary condition when the compact domain is
$\mathcal{C}:=\left\\{(x,y,z)\in\mathbb{R}^{3}:0\leq x\leq a,0\leq y\leq
a,0\leq z\leq a\right\\}$ is a cube of side $a$ in the positive octant. This
problem is widely studied in different mathematical physics problems [21, 23].
The eigenfunctions in this case can be given by

$\displaystyle\psi_{\bf k}\left({\bf
R}\right)=\frac{N_{nmp}}{a^{3/2}}\cos\left(\frac{\pi
n}{a}x\right)\cos\left(\frac{\pi m}{a}y\right)\cos\left(\frac{\pi
p}{a}z\right),$ (28)

where $x,y$ and $z$ are the standard cartesian coordinates, and ${\bf
R}=(x,y,z)$ is the usual vector position. The eigenfunctions are enumerated by
the collective index $nmp$, with $n,m,p=0,1,2,\cdots$. $N_{nmp}$ is a
normalization constant with respect to the volume of the cube
$V(\mathcal{D})=a^{3}$, whose values are given by $N_{000}=1$;
$N_{n00}=N_{0n0}=N_{00n}=\sqrt{2}$, for $n\neq 0$;
$N_{np0}=N_{n0p}=N_{0np}=2$, for $n,p\neq 0$; and $N_{npm}=2\sqrt{2}$, for
$n,m,p\neq 0$. The eigenvalues of the Laplacian are given by $\lambda_{{\bf
k}}={\bf k}^{2}$, where ${\bf k}=\left(\frac{\pi n}{a},\frac{\pi
m}{a},\frac{\pi p}{a}\right)$. Now, we proceed to calculate ${\bf r}_{{\bf
k}}$ using its definition, that is, ${\bf r}_{{\bf k}}=\int_{\mathcal{C}}{\bf
R}\psi_{\bf k}\left({\bf R}\right)d^{3}{\bf R}$. The three components are
given by

$\displaystyle\left({\bf r}_{\bf k}\right)_{x}$ $\displaystyle=$
$\displaystyle-\sqrt{2}\frac{a^{5/2}}{n^{2}\pi^{2}}\left(1-(-1)^{n}\right)\delta_{m0}\delta_{p0}$
$\displaystyle\left({\bf r}_{\bf k}\right)_{y}$ $\displaystyle=$
$\displaystyle-\sqrt{2}\frac{a^{5/2}}{m^{2}\pi^{2}}\left(1-(-1)^{m}\right)\delta_{n0}\delta_{p0}$
$\displaystyle\left({\bf r}_{\bf k}\right)_{z}$ $\displaystyle=$
$\displaystyle-\sqrt{2}\frac{a^{5/2}}{p^{2}\pi^{2}}\left(1-(-1)^{p}\right)\delta_{n0}\delta_{m0}$
(29)

In the following, we use the general expression (25) for the mean-square end-
to-end distance. The mean-square end position can be easily calculated
$\sigma^{2}\left({\bf R}\right)=\frac{a^{2}}{4}$. Since the Kronecker deltas
in ${\bf r}_{\bf k}$ each contribution of $\left({\bf r}_{\bf k}\right)_{i}$
is the same, thus taking into account the correct counting factor the mean
square end-to-end distance is

$\displaystyle\left<\delta{\bf
R}^{2}\right>_{\mathcal{C}}=\frac{a^{2}}{2}-24a^{2}\sum_{k=1}^{\infty}\frac{\left(1-(-1)^{k}\right)}{k^{4}\pi^{4}}G\left(\frac{s}{2\ell_{p}},\frac{4}{3}\left(\frac{\ell_{p}}{a}\right)^{2}\pi^{2}k^{2}\right).$
(30)

Following, the same line of argument performed in [17], it is observed that
$24\sum_{k=1}^{\infty}\frac{\left(1-(-1)^{k}\right)}{k^{4}\pi^{4}}=\frac{1}{2}$
consistently with (27), thus up to a numerical error of $10^{-2}$, we claim
that

$\displaystyle\frac{\left<\delta{\bf R}^{2}\right>_{\mathcal{C}}}{a^{2}}$
$\displaystyle\simeq$
$\displaystyle\frac{1}{2}-\frac{1}{2}\exp\left(-\frac{L}{2\ell_{p}}\right)\left\\{\cosh\left[\frac{L}{2\ell_{p}}\left(1-\frac{4\pi^{2}}{3}\frac{\ell^{2}_{p}}{a^{2}}\right)^{\frac{1}{2}}\right]\right.$
(31) $\displaystyle+$
$\displaystyle\left.\left(1-\frac{4\pi^{2}}{3}\frac{\ell^{2}_{p}}{a^{2}}\right)^{-\frac{1}{2}}\sinh\left[\frac{L}{2\ell_{p}}\left(1-\frac{4\pi^{2}}{3}\frac{\ell^{2}_{p}}{a^{2}}\right)^{\frac{1}{2}}\right]\right\\}.$

Let us remark that for any fixed value of $a$, the r.h.s of (31), as a
function of $L$, shows the existence of a critical persistence length,
$\ell_{p}^{*}=\sqrt{3}a/(2\pi)$ such that for all values of
$\ell_{p}>\ell_{p}^{*}$ it exhibits an oscillating behavior, whereas for
$\ell_{p}<\ell_{p}^{*}$ it is monotonically increasing. In Fig. (1), we show
the behavior of the mean-square end-to-end distance versus the length of the
polymer for several values of the persistence length below and above
$\ell_{p}^{*}$. Moreover, we also show sketches of conformational states
corresponding to the monotonous and oscillating behaviors of the mean square
end-to-end distance. In addition, it is noticeably that the same mathematical
structure as the mean-square end-to-end distance found by Spakowitz and Wang
[14] for semiflexible polymers wrapping a spherical shell, and recently for
semiflexible polymers confined to a square box [17].

Figure 1: Monotonous and oscillating behaviors of the mean square end-to-end
distance (Eq. (30)) of polymers with $\ell_{p}$ below [a)] and above [b)] the
critical persistence length $\ell^{*}_{p}=\sqrt{3}/(2\pi)a$ in cubic
confinement. Inside the plotting area we sketch the conformational states of
each class of polymers.

###  6.2 Semiflexible polymer enclosed by a spherical surface

In this section we provide results for the mean-square end-to-end distance for
a semiflexible polymer enclosed inside of a spherical domain. All the problem
is reduced to solve the Neumann eigenvalue problem
$-\nabla^{2}\psi=\lambda\psi$ with Neumann boundary condition when the compact
domain is $\mathcal{B}:=\left\\{{\bf r}\in\mathbb{R}^{3}:{\bf r}^{2}\leq
R^{2}\right\\}$ is a center ball of radius $R$. This problem is widely studied
in different mathematical physics problems [21, 23]. The eigenfunctions in
this case can be given in terms of spherical Bessel functions
$j_{\ell}\left(x\right)$ and spherical harmonic functions $Y_{\ell
m}\left(\theta,\varphi\right)$,

$\displaystyle\psi_{\ell mk}\left(r,\theta,\varphi\right)=N_{\ell
k}~{}j_{\ell}\left(\alpha_{\ell k}\frac{r}{R}\right)Y_{\ell
m}\left(\theta,\varphi\right)$ (32)

where $r,\theta$ and $\varphi$ are the standard spherical coordinates. The
factor $N_{\ell k}$ is a normalization constant with respect to the volume of
the ball $\mathcal{B}$, given by

$\displaystyle N_{\ell k}=\frac{\sqrt{2}}{R^{3/2}}\frac{\alpha_{\ell
k}}{j_{\ell}\left(\alpha_{\ell k}\right)\left(\alpha_{\ell
k}^{2}-\ell\left(\ell+1\right)\right)^{1/2}}.$ (33)

The coefficients $\alpha_{\ell k}$ are the roots of $\partial
j_{\ell}\left(x\right)/\partial x$, which by using the identity $\ell
j_{\ell-1}\left(x\right)-\left(\ell+1\right)j_{\ell+1}=\left(2\ell+1\right)\partial
j_{\ell}\left(x\right)/\partial x$, that satisfies the equation $\ell
j_{\ell-1}\left(\alpha_{\ell
k}\right)=\left(\ell+1\right)j_{\ell+1}\left(\alpha_{\ell k}\right)$. The
eigenfunctions are enumerated by the collective index $\ell mk$, with
$\ell=0,1,2,\cdots$ countings the order of spherical Bessel functions,
$m=-\ell,-\ell+1,\cdots,\ell$, and $k=1,2,3,\cdots$ counting zeros. The
eigenvalues of the Laplacian are given by $\lambda_{\ell mk}={\alpha^{2}_{\ell
k}/R^{2}}$, which are independent of the numbers $m$. Now, we proceed to
calculate ${\bf r}_{\ell mk}$ using (26). It is enough to calculate
$\oint_{S^{2}}dS~{}{\bf n}~{}Y_{\ell m}\left(\theta,\varphi\right)$, since
${\bf n}\propto Y_{1m}$, thus $\oint_{S^{2}}dS~{}{\bf n}~{}Y_{1,\pm
1}\left(\theta,\varphi\right)=-\sqrt{\frac{2\pi}{3}}R^{2}\left(\pm
1,i,0\right)$ and $\oint_{S^{2}}dS~{}{\bf
n}~{}Y_{1,0}\left(\theta,\varphi\right)=2\sqrt{\frac{\pi}{3}}R^{2}\left(0,0,1\right)$.
Now, we call $\alpha_{1k}:=\alpha_{k}$, then using (33) one has

$\displaystyle{\bf r}_{1,\pm 1,k}$ $\displaystyle=$
$\displaystyle-2\sqrt{\frac{\pi}{3}}\frac{R^{5/2}}{\alpha_{k}\left(\alpha^{2}_{k}-2\right)^{1/2}}\left(\pm
1,i,0\right),$ (34) $\displaystyle{\bf r}_{1,0,k}$ $\displaystyle=$
$\displaystyle
2\sqrt{\frac{2\pi}{3}}\frac{R^{5/2}}{\alpha_{k}\left(\alpha^{2}_{k}-2\right)^{1/2}}\left(0,0,1\right),$
(35)

where roots $\\{\alpha_{k}\\}$ satisfy the equation
$j_{0}\left(\alpha_{k}\right)=2j_{2}\left(\alpha_{k}\right)$. Using explicit
functions of the spherical Bessel functions the root condition is
$F(\alpha_{k})=0$, where

$\displaystyle F\left(x\right)=\left(\frac{x^{2}}{2}-1\right)\sin x+x\cos x.$
(36)

In the following, we use the general expression (25) for the mean-square end-
to-end distance. We calculate the mean-square end position,
$\sigma^{2}\left({\bf R}\right)=\frac{3}{5}R^{2}$, and use the factors ${\bf
r}_{\ell mk}$, thus the mean square end-to-end distance is

$\displaystyle\left<\delta{\bf
R}^{2}\right>_{\mathcal{B}}=\frac{6}{5}R^{2}-12R^{2}\sum_{k=1}^{\infty}\frac{1}{\alpha^{2}_{k}(\alpha_{k}^{2}-2)}G\left(\frac{s}{2\ell_{p}},\frac{4}{3}\left(\frac{\ell_{p}}{R}\right)^{2}\alpha^{2}_{k}\right).$
(37)

Following, the same line of argument performed in [17], we observe numerically
that $12\sum_{k=1}^{N}\frac{1}{\alpha^{2}_{k}\left(\alpha_{k}^{2}-2\right)}\to
6/5$ as $N$ increases, this is consistent with Eq. (27). Thus up to a
numerical error $10^{-2}$, we claim that

$\displaystyle\frac{\left<\delta{\bf R}^{2}\right>_{\mathcal{B}}}{R^{2}}$
$\displaystyle\simeq$
$\displaystyle\frac{6}{5}-\frac{6}{5}\exp\left(-\frac{L}{2\ell_{p}}\right)\left\\{\cosh\left[\frac{L}{2\ell_{p}}\left(1-\frac{4\alpha_{1}^{2}}{3}\frac{\ell^{2}_{p}}{R^{2}}\right)^{\frac{1}{2}}\right]\right.$
(38) $\displaystyle+$
$\displaystyle\left.\left(1-\frac{4\alpha_{1}^{2}}{3}\frac{\ell^{2}_{p}}{R^{2}}\right)^{-\frac{1}{2}}\sinh\left[\frac{L}{2\ell_{p}}\left(1-\frac{4\alpha_{1}^{2}}{3}\frac{\ell^{2}_{p}}{R^{2}}\right)^{\frac{1}{2}}\right]\right\\}.$

Let us remark that for any fixed value of $R$, the r.h.s of (38), as a
function of $L$, shows the existence of a critical persistence length,
$\ell_{p}^{*}=\sqrt{3}R/(2\alpha_{1})$, with $\alpha_{1}\simeq 2.08158$
according to (36), such that for all values of $\ell_{p}>\ell_{p}^{*}$ it
exhibits an oscillating behavior, whereas for $\ell_{p}<\ell_{p}^{*}$ it is
monotonically increasing. In Fig. (2), we show the behavior of the mean-square
end-to-end distance versus the length of the polymer for several values of the
persistence length below and above $\ell_{p}^{*}$. Moreover, we also show
sketches of conformational states corresponding to the monotonous and
oscillating behaviors of the mean square end-to-end distance. In addition, it
is noticeably that the same mathematical structure as the mean-square end-to-
end distance found by Spakowitz and Wang [14] for semiflexible polymers
wrapping a spherical shell, and recently for semiflexible polymers confined to
a square box [17].

Figure 2: Monotonous and oscillating behaviors of the mean square end-to-end
distance (Eq. (37)) of polymers with $\ell_{p}$ below [a)] and above [b)] the
critical persistence length $\ell^{*}_{p}=\sqrt{3}R/(2\alpha_{1})$ in
spherical confinement. Inside the plotting area we sketch the conformational
states of each class of polymers.

##  7 Concluding remarks

In this work we carry out an extension of the stochastic curvature formalism
introduced in [17] to analyze the conformational states of a semiflexible
polymer in a thermal bath for the cases when the polymer is in the open space
$\mathbb{R}^{3}$, and when is in a bounded domain
$\mathcal{D}\subset\mathbb{R}^{3}$. The basic idea of formalism in the 3D case
is followed by two postulates, that is, that each conformational state
corresponds to the realization of a path described by the stochastic Frenet-
Serret equations (5), to introduce an stochastic curvature vector
$\bm{k}\left(s\right)$, and a second postulate that gives the manner how
$\bm{\kappa}(s)$ is distributed according to the thermal fluctuations.

In the case of a polymer in an open space $\mathbb{R}^{3}$, the standard
Kratky-Porod formula for polymers is reproduced in three dimensions [16],
while when the polymer is confined to a space bounded region
$\mathcal{D}\subset\mathbb{R}^{3}$ the conformational states shows the
existence of a critical persistence length $\ell_{p}^{*}$ such that for all
values of $\ell_{p}>\ell_{p}^{*}$ the mean square distance from end-to-end
exhibits an oscillating behavior, while for $\ell_{p}<\ell_{p}^{*}$ it
exhibits a monotonic behavior in both cases of a cubic region and a spherical
region . Furthermore, for each value of $\ell_{p}$, the function converges to
the twice of the mean-square end position $\sigma^{2}\left({\bf R}\right)$,
that is, twice the variance of ${\bf R}^{2}$ with respect to the volume of the
domain. The critical persistence length, therefore, distinguishes two
conformational behaviors of the semiflexible polymer in the bound domain. On
one hand, polymers with persistence length below the critical value have a
conformation similar to a Brownian random path. On the other hand, polymers
with persistence length above the critical value adopt smooth conformations.
In addition, it is highlighted that the mean square end-to-end distance
exhibits the same mathematical form for the discussed cases along with the
manuscript (see Eq.(31) and Eq. (38)), and with the results reported for a
polymer enclosed to a square box and rolling up a spherical surface [17, 24].
Nevertheless, the values-difference of saturation and the critical persistence
length reflects the particular geometric nature of the compact domain,
including the dimensionality of the space. Note the particular mathematical
expression in our work is due to the probability density function of the
polymer’s ends which is governed by a modified telegrapher equation. As a
consequence of this resemblance, it can be concluded that the shape transition
from oscillating to monotonous conformational states provides furthermore
evidence of a universal signature for a semiflexible polymer enclosed in
compact space.

## Conflict of Interest Statement

The authors declare that the research was conducted in the absence of any
commercial or financial relationships that could be construed as a potential
conflict of interest.

## Author Contributions

Both authors contributed to the formulation of the method, and the writing of
the manuscript. JER contributed in the numerical analysis that provide the
figures, while PCV in the mathematical calculations.

## Funding

## Acknowledgments

P.C.V. acknowledges financial support CONACyT. .

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