# Loading and spatially-resolved characterization of a cold atomic ensemble
inside a hollow-core fiber

Thorsten Peters  Institute of Applied Physics, Technical University of
Darmstadt, Hochschulstrasse 6, 64289 Darmstadt, Germany  Leonid P. Yatsenko
Institute of Physics, National Academy of Science of Ukraine, Nauky Ave. 46,
Kyiv 03028, Ukraine  Thomas Halfmann  Institute of Applied Physics, Technical
University of Darmstadt, Hochschulstrasse 6, 64289 Darmstadt, Germany

###### Abstract

We present a thorough experimental investigation of the loading process of
laser-cooled atoms from a magneto-optical trap into an optical dipole trap
located inside a hollow-core photonic bandgap fiber, followed by propagation
of the atoms therein. This, e.g., serves to identify limits to the loading
efficiency and thus optical depth which is a key parameter for applications in
quantum information technology. Although only limited access in 1D is
available to probe atoms inside such a fiber, we demonstrate that a detailed
spatially-resolved characterization of the loading and trapping process along
the fiber axis is possible by appropriate modification of probing techniques
combined with theoretical analysis. Specifically, we demonstrate the loading
of up to $2.1\times 10^{5}$ atoms with a transfer efficiency of $2.1~{}\%$
during the course of $50$ ms and a peak loading rate of $4.7\times 10^{3}$
atoms ms-1 resulting in a peak atomic number density on the order of
$10^{12}~{}$cm-3. Furthermore, we determine the evolution of the spatial
density (profile) and ensemble temperature as it approaches its steady-state
value of $T=1400~{}\upmu$K, as well as loss rates, axial velocity and
acceleration. The spatial resolution along the fiber axis reaches a few
millimeters, which is much smaller than the typical fiber length in
experiments. We compare our results to other fiber-based as well as free-space
optical dipole traps and discuss the potential for further improvements.

††preprint: APS/123-QED

##  I Introduction

Since its proposal by Ol’shanii et al. Ol’Shanii  _et al._ (1993) laser
guiding of atoms into or through hollow-core optical fibers Debord  _et al._
(2019) has received increasing experimental interest over the past years Renn
_et al._ (1995, 1996); Ito  _et al._ (1996); Renn  _et al._ (1997); Dall  _et
al._ (1999); Müller  _et al._ (2000); Christensen  _et al._ (2008); Bajcsy
_et al._ (2009, 2011); Takekoshi and Knize (2007); Vorrath  _et al._ (2010);
Peyronel  _et al._ (2012); Blatt  _et al._ (2014); Langbecker  _et al._
(2017); Okaba  _et al._ (2014); Langbecker  _et al._ (2018); Hilton  _et al._
(2019, 2018); Xin  _et al._ (2018); Noaman  _et al._ (2018); Xin  _et al._
(2019); Yoon and Bajcsy (2019); Yoon  _et al._ (2019); Okaba  _et al._ (2019);
Hilton  _et al._ (2020); Wang  _et al._ (2020); Leong  _et al._ (2020a); Li
_et al._ (2020); Leong  _et al._ (2020b) This is due to their potential for
controlled guiding of atoms over macroscopic distances for atom interferometry
Vorrath  _et al._ (2010); Xin  _et al._ (2018) and quantum optics Bajcsy  _et
al._ (2009); Noh and Angelakis (2017). The latter application requires long
coherence time Xin  _et al._ (2019) and strong light-matter coupling, i.e.,
transverse tightly confined atomic ensembles of large optical depth
$d_{opt}=\sigma_{0}nL$ Blatt  _et al._ (2014), where $\sigma_{0}$ is the
atomic absorption cross-section, $n$ is the atomic number density, and $L$ is
the length of the ensemble. Therefore, efforts have been devoted in recent
years to load laser-cooled atoms from a magneto-optical trap (MOT) located at
some distance from the fiber into their hollow core Christensen  _et al._
(2008); Bajcsy  _et al._ (2009, 2011); Takekoshi and Knize (2007); Vorrath
_et al._ (2010); Peyronel  _et al._ (2012); Blatt  _et al._ (2014); Okaba  _et
al._ (2014); Langbecker  _et al._ (2017, 2018); Hilton  _et al._ (2018, 2019);
Noaman  _et al._ (2018); Xin  _et al._ (2018, 2019); Yoon and Bajcsy (2019);
Yoon  _et al._ (2019); Okaba  _et al._ (2019); Hilton  _et al._ (2020); Wang
_et al._ (2020); Leong  _et al._ (2020a); Li  _et al._ (2020); Leong  _et al._
(2020b) by use of a standing- or traveling-wave optical dipole trap Grimm  _et
al._ (2000).

After pioneering work on laser-guiding of atoms through hollow fibers which
had to rely on lossy glass capillaries of large core diameters Renn  _et al._
(1995, 1996); Ito  _et al._ (1996); Renn  _et al._ (1997); Dall  _et al._
(1999); Müller  _et al._ (2000), later experiments employed single-mode, low-
loss photonic cyrstal fibers Russell (2003) due to their more advantageous
properties. Nowadays, basically two types of fibers are applied. Kagome-
structured fibers Couny  _et al._ (2006) and hollow-core photonic bandgap
fibers (HCPBGFs) Cregan  _et al._ (1999). The first type of fiber provides
single-mode guidance of light over a large wavelength range, low
birefringence, and low numerical aperture for core diameters of several ten
micrometers. The large bandwidth and low birefringence make Kagome fibers
ideally suited for far-off-resonance guiding with on-resonance probing and
controlled manipulation of different atomic species. Their low numerical
aperture, and thus slowly diverging laser beams, allow for direct transport
from the distant MOT into the fiber core Okaba  _et al._ (2014); Hilton  _et
al._ (2018); Langbecker  _et al._ (2018); Xin  _et al._ (2018); Hilton  _et
al._ (2019, 2020); Wang  _et al._ (2020); Leong  _et al._ (2020a); Li  _et
al._ (2020). Experimental and theoretical loading studies using Kagome fibers
were reported recently Langbecker  _et al._ (2018); Hilton  _et al._ (2018);
Wang  _et al._ (2020). A drawback of these type of fibers is their relatively
large core diameter, resulting in weaker light-matter coupling strengths,
compared to HCPBGFs. HCPBGFs, on the other hand, are available with core
diameters below $10~{}\upmu$m to enable stronger light-matter coupling and
suffer less from micro-lensing Noaman  _et al._ (2018); Sulzbach  _et al._
(2019), but provide smaller guiding bandgaps and larger, uncontrolled
birefringence Wegmuller  _et al._ (2005). Nonetheless, efficient loading of
atoms into such fibers Christensen  _et al._ (2008); Bajcsy  _et al._ (2009);
Blatt  _et al._ (2014) and quantum optics experiments Bajcsy  _et al._ (2009);
Blatt  _et al._ (2016), even down to the quantum level Peters  _et al._
(2020), were demonstrated recently. The larger numerical aperture of HCPBGFs
usually prohibits a direct transfer of atoms from the MOT into the fiber, as
the trapping potential (proportional to the laser intensity) quickly decreases
outside of the fiber. Thus, a different approach to loading has to be chosen.

So far, only loading via free-fall, magnetic guiding, and hollow-beam guiding
was investigated experimentally by Bajcsy et al. Bajcsy  _et al._ (2011) for a
HCPBGF. The more efficient loading technique, using a dark funnel to guide the
atoms into the fiber Blatt  _et al._ (2014, 2016); Peters  _et al._ (2020),
has however not yet been investigated thoroughly. Furthermore, spatially-
resolved detection within such fiber has not been demonstrated so far. Such
information is however important for optimizing the loading efficiency and
minimizing losses within the fiber. We report in the following on a detailed
experimental study of loading atoms from a MOT into a far-off-resonance trap
(FORT) Chu  _et al._ (1986); Miller  _et al._ (1993) located inside a HCPBGF
by using a dark funnel guide. Although FORTs in free space have been studied
for years (see, e.g., Miller  _et al._ (1993); Kuppens  _et al._ (2000); Grimm
_et al._ (2000)) and a variety of probing techniques are available, there are
some important differences compared to a FORT located inside a hollow-core
fiber. Loading and probing is only possible along the direction of the fiber
axis and the axial extension of the FORT can be four orders of magnitude
larger than the radial extension. The atoms are therefore basically untrapped
along the fiber axis and propagation of the ensemble becomes relevant. We will
present experimental data along with comparison to theoretical models to
characterize the system–even spatially-resolved along the fiber axis.
Specifically, we determine the loading efficiency into the fiber, and loss
rates, temperature, velocity as well as the spatial density distribution
inside the fiber. This allows us to identify limits to the loading efficiency
and possible improvements as well as to compare our loading scheme to others.

This paper is organized as follows: In Sec. II we give an overview of the
experimental system and describe the probing techniques applied in our work.
In Sec. III.1 we study the atomic ensemble inside the fiber with a focus on
spatially-resolved measurements in both radial and axial dimensions. Next, we
study the number and propagation of atoms loaded from the MOT into the FORT
for various parameters with a focus on efficient loading in Sec. III.2.
Finally, we present a summary and outlook in Sec. IV.

##  II Experimental details

Details on the experimental setup and sequence can be found in our previous
work using a HCPBGF Blatt  _et al._ (2014, 2016); Peters  _et al._ (2020).
Thus, we here only briefly summarize the setup.

###  II.1 Experimental setup and sequence

Figure 1:  (a) Schematic experimental setup. DM: dichroic mirror; aHWP:
achromatic half wave plate; PBS: polarizing beam splitter: BS: beam splitter;
APD: avalanche photodiode; SPCM: single-photon counting module. (b) Coupling
scheme. (c) Scanning electron micrograph of the HCPBGF and false color
intensity profile of the FORT.

We first load laser-cooled 87Rb atoms into a cigar-shaped vapor cell MOT
located about $5.5$ mm above the tip of the vertically mounted HCPBGF (NKT
Photonics HC800-02) (see Fig. 1). The HCPBGF has a length of $14$ cm and a
core diameter of $d_{c}\sim 7~{}\upmu$m. The $1/e$ intensity mode field
diameter inside the fiber is around $5.5~{}\upmu$m. After the MOT is loaded
for about 1 s yielding typically around $10^{7}$ atoms, we shift the cloud
down towards the HCPBGF tip during the next $40$ ms by applying a magnetic
offset field in the vertical direction. Simultaneously, we transfer the atoms
from the bright MOT into a dark funnel guide Blatt  _et al._ (2014) by
switching off the MOT repumper and ramping up the quadrupole gradient of the
MOT coils which compresses the atom cloud towards the HCPBGF. Also, the
trapping detuning is increased to $-30~{}$MHz for optimized cooling during the
transfer. The dark funnel guide consists of two nearly orthogonally aligned
repumper beams with a funnel-shaped shadow in their center, termed dark funnel
repumper (DFR) in the following. This dark funnel is aligned from the side
onto the HCPBGF. In order to improve the density above the HCPBGF and confine
the atoms in $F=1$ we use a depumper beam traveling upwards through the HCPBGF
Anderson  _et al._ (1994). The depumper is locked to the cross-over transition
$5^{2}S_{1/2}\,F=2\rightarrow 5^{2}P_{3/2}\,F^{\prime}=2,3$. As the effective
transition light-shift by the FORT is estimated to be near $+130~{}$MHz for
our typical FORT depth, the depumper is almost resonant within and slightly
above the HCPBGF, where loading and guiding is relevant. All trapping, MOT
repumper, and DFR beams have diameters of nearly 20 mm thus covering the whole
region of MOT and HCPBGF tip where trapping and guiding is relevant. After the
cloud has reached the fiber tip we hold it for $10~{}$ms in place. Here, the
atoms are transferred into a FORT of depth $T_{F}\lesssim 4.3~{}$mK. The FORT
system consists of two independent unstabilized laser diodes operating near
820 nm and of orthogonal linear polarizations, combined at a polarizing beam
splitter. Using a FORT system operating at multiple frequencies was shown to
improve the loading efficiency into the HCPBGF as compared to single-mode
operation Blatt  _et al._ (2014). As the FORT beam quickly diverges above the
HCPBGF, its trapping potential decreases quickly with the square of the
distance from the fiber tip. A simulation of the spatial trapping potential
shows that the atoms have to be brought within a distance of around
$200~{}\upmu$m of the fiber tip for the FORT being strong enough to guide the
atoms into the core. Thus, cooling and compression of the atom cloud above the
HCPBGF is crucial for obtaining a good loading efficiency. Once the atoms are
inside the fiber they can be probed as described in the next section. In order
to prevent detection of atoms that might still be left above the HCPBGF during
probing, we use a strong repumper beam above the fiber to purge the volume
through which the probe beam propagates. This confines the population in $F=2$
above the HCPBGF, whereas we always probe the atoms in $F=1$ inside the fiber.
Except for the repumper, all laser beams are temporally modulated by acousto-
optic modulators (AOMs).

###  II.2 Probing techniques

####  II.2.1 Time-resolved optical pumping

The number of atoms inside the HCPBGF, $N_{a}$, can be inferred, e.g., by
nonlinear saturation measurements Bajcsy  _et al._ (2011) or spectroscopic
studies followed by comparison to a simulation Hilton  _et al._ (2018). A
simpler method infers $N_{a}$ from time-resolved optical pumping (TROP) Blatt
_et al._ (2014).

Here, population is first prepared in the lower ground state $F=1$. Then, the
optical trap is quickly switched off and a strong pump laser, tuned, e.g., to
the transition $F=1\rightarrow F^{\prime}=2$ [see Fig. 2(a)],

Figure 2:  (a) Level scheme for TROP on the D1 line. (b) Transmission for TROP
with $N_{a}=2.1(2)\times 10^{5}$ atoms ($\square$) and without atoms ($\circ$)
inside the HCPBGF. The vertical axis is calibrated to show the actual power
inside the HCPBGF.

is sent through the fiber and detected by an avalanche photodiode (APD). This
beam optically pumps the population into $F=2$ where it does not interact with
the pump anymore. The medium thus becomes transparent. If the APD is
calibrated and the transmission from the fiber to the APD is known, the
difference of the temporally-resolved transmissions without and with atoms is
thus a measure for the energy absorbed by the atoms until they are in the
uncoupled level $F=2$. With the branching ratio from $F^{\prime}=2$ into
$F=1,2$, here 0.5, the number of photons an atom absorbs on the average before
it ends up in $F=2$ is 2. From the known energy of a photon, $hc/\lambda$, we
therefore get $N_{a}$. The result of such measurement is shown in Fig. 2(b).
We use this technique to study the loading efficiency for various parameters
as we will discuss in the following sections.

####  II.2.2 Electromagnetically induced transparency

The other technique suitable to extract information from within the HCPBGF is
electromagnetically induced transparency (EIT) Harris (1997); Fleischhauer
_et al._ (2005). Here, a weak probe and a strong control field couple two
long-lived states $|1,2\rangle$ via a decaying intermediate state $|3\rangle$,
e.g., in a $\Lambda-$scheme. The strong control field renders the previously
opaque medium transparent for the weak probe within a frequency window around
two-photon resonance between the two long-lived states. The width of the EIT
window $\Delta\omega_{EIT}=\Omega_{c}^{2}/(\Gamma\sqrt{d_{opt}})$ depends on
the control Rabi frequency $\Omega_{c}$, the intermediate state decay rate
$\Gamma$ and the optical depth $d_{opt}$ of the atomic ensemble. The depth of
the EIT window depends on the ground state dephasing rate $\gamma_{21}$
between states $|1,2\rangle$ relative to $\Delta\omega_{EIT}$. This enables
determination of $\gamma_{21}$ from EIT spectra Perrella  _et al._ (2012). For
$\Lambda-$type coupling of nearly energetically degenerate ground states and
collinear probe and control beams with wavevectors $\bm{k}_{p,c}$ the two-
photon resonance is not very susceptible to atomic motion, as the
corresponding Doppler shift
$\left(\bm{k}_{p}-\bm{k}_{c}\right)\cdot\bm{v}_{a}$ is very small, where
$\bm{v}_{a}$ is the atomic velocity along the laser beam direction. This
allows for determining the effective dephasing rate $\gamma_{21}$ comprised of
transit-time broadening, two-photon linewidth, and inhomogeneous transition
shifts due to the radially varying control field intensity Blatt  _et al._
(2016). For simplicity we do not discriminate here between reversible
dephasing processes, e.g., due to a spatial magnetic field gradient and
irreversible decoherence, e.g., due to transit-time broadening. For anti-
collinear probe and control beams, however, the two-photon resonance is very
susceptible to atomic motion as the two-photon Doppler shifts is now
$\left(\bm{k}_{p}-\bm{k}_{c}\right)\cdot\bm{v}_{a}\approx
2\cdot\bm{k}_{p,c}\cdot\bm{v}_{a}$. This can be used in ultracold media to
obtain information on the atomic velocity (distribution) along the laser beam
direction Peters  _et al._ (2012). Thus, EIT can be used to obtain information
on the magnetic field (gradient) and atomic velocities inside the HCPBGF,
depending on the configuration used.

##  III Experimental studies

In the following experimental studies we demonstrate how to gain information
on the atoms as they propagate through the HCPBGF. This allows us to identify
limits to the loading efficiency, loss rates, and coherence time, which are
important parameters for applications. Unlike in free-space setups, optical
access from the side is not possible for a hollow-core fiber. Therefore,
information on the number, position and velocity of the atoms, temperature,
radial distribution, and magnetic field (gradient) have to be obtained from
measurements performed along the fiber axis only. A detailed characterization
of the FORT potential can be found in App. A, as it is required for a
quantitative analysis of the following results.

###  III.1  Spatially-resolved characterization of the atomic ensemble

Typically, HCPBGFs are loaded with the aim to maximize the number of atoms
inside the fiber as to create the highest possible $d_{opt}$ or to guide atoms
with losses from the FORT as small as possible. This is achieved when the
fiber can be filled over a length as large as possible. However, experimental
studies of the confined atomic ensemble thereafter only yield an average over
the ensemble length of several centimeters (up to the length of the HCPBGF; 14
cm in our case). This impedes gaining information on, e.g., possible magnetic
field gradients along the fiber, the axial atomic density profile etc. Also
the radial distribution of atoms is of interest. It relates to the temperature
of the atoms inside the FORT and affects inhomogeneous broadening as the
atomic density profile width is typically comparable to the mode field
diameter of the laser beams. We thus describe in the following methods to
extract such information. To this end, we load the HCPBGF only with a short
pulse of atoms and interrogate them as a function of time $t_{if}$ they have
propagated inside the fiber. We do this by switching on the FORT for only a
short period of 1-3 ms for loading. This creates an atomic pulse of 1-3 mm
length, as can be estimated from the most probable initial velocity $v_{0}\sim
1$ m/s due to the FORT depth $T_{F}$ (see Sec. III.1.2). This is significantly
shorter than typical fiber lengths of several centimeters used in experiments.
Probing the atoms as a function of time therefore yields information at
different positions inside the HCPBGF.

####  III.1.1 Determination of the temperature inside the FORT

We start by determining the temperature of the atomic ensemble inside the
HCPBGF which affects the radial density profile according to Eq. (7) and thus
inhomogeneous broadening. This can be done by employing a time-of-flight (TOF)
measurement, e.g., via a release-and-recapture technique as shown by Bajcsy et
al. Bajcsy  _et al._ (2011). Here, we simply measure the atom number by TROP
after the FORT has been switched off. If the pumping time is much shorter than
the time the atoms reach the fiber wall, we verified that this simple method
yields the same temperature as the release-and-recapture technique. We assume
that the atoms are not detectable once they reached the fiber wall Renn  _et
al._ (1997); Ghosh  _et al._ (2006). The result of such measurement, taken at
$t=1050~{}$ms, i.e., 5 ms after loading has ended, is shown in Fig. 3(a).

Figure 3:  (a,c,d) Time-of-flight measurements to determine the ensemble
temperature inside the HCPBGF. The normalized atom number ($\circ$) is plotted
vs. the time after the FORT has been switched off. The red solid lines show
the results of a calculation based on a Gaussian atomic density distribution
[see Eq. (1)]. For maximum loading of the fiber (a) we determine the
temperature as $T=1.0(2)~{}$mK and a radial ensemble $1/e$ half width of
$\sigma_{a}=1.1~{}\upmu$m for $T_{F}=3.8~{}$mK. Due to the finite optical
pumping time we included a moving average over a $\sim 1~{}\upmu$s period in
the calculation. For a pulsed loading of $3$ ms only, the results are shown
for times $t_{if}=1.5(15)~{}$ms (c) and $t_{if}=21.5(15)~{}$ms (d),
respectively. The temperatures obtained are $T=2.7(2)~{}$mK (c) and
$T=1.0(4)~{}$mK (d). In (b) we plotted the obtained temperatures ($\circ$) vs.
the time $t_{if}$. The temperature from (a) is shown for reference ($\square$
and gray-shaded area).

Here, the fiber was loaded to its maximum by keeping the FORT on during the
transfer process from the MOT. The atoms have therefore spent $5~{}$ms $\leq
t_{if}\leq$ $35$ ms inside the fiber at the time of the measurement (see Fig.
8). Thus, and because we do not observe an increase of the temperature for
$t>1050~{}$ms, the measured temperature for these conditions ought represent
the steady-state temperature. In order to extract the temperature from this
data we proceed as follows: Using the measured FORT power, we calculate the
FORT potential and radial density according to Eq. (6) and (7). In good
approximation, $n(r)$ can be written as a Gaussian of $1/e$-width $\sigma_{a}$
Sulzbach  _et al._ (2019). After release from the FORT the density evolves as

$n(r,t)=\frac{N_{0}}{\pi\sigma_{t}^{2}}\exp\left(-\frac{r^{2}}{\sigma_{t}^{2}}\right),$
(1)

where $\sigma_{t}^{2}=\sigma_{a}^{2}+v_{th}^{2}t^{2}$ and
$v_{th}=\sqrt{2k_{B}T/m}$. The dependence of $n$ on the axial coordinate $z$
is irrelevant for this measurement and can therefore be neglected. Integration
of Eq. (1) over $0\leq r\leq r_{f}$, where $r_{f}$ is the fiber core radius
and normalization leads to the solid red curve in Fig. 3(a). We obtain a
temperature of $T=1.0(2)~{}$mK for a FORT depth of $T_{F}=3.8~{}$mK. The ratio
$T/T_{F}$ is larger than reported in Bajcsy  _et al._ (2011) but smaller than
in Yoon and Bajcsy (2019) for a similar fiber. We note that the temperature is
the only free parameter in our model after the FORT potential has been
characterized as shown in App.A.

Next, we performed TOF measurements with a variable delay after the atoms are
loaded for a short period of $3$ ms only. This enables measuring the
temperature as a function of time $t_{if}$ the atoms have spent inside the
fiber. In Fig. 3(d) we show the results for $t_{if}\sim 21.5$ ms. Comparison
to the simulation yields a temperature of $T=1.0(4)~{}$mK. The temperature
agrees well with the one obtained for maximum loading, where the atoms have
spent $>5~{}$ms inside the fiber. If, however, we measure the temperature at
the earliest possible time $t_{if}\sim 1.5$ ms after the loading is finished,
we obtain a temperature of $T=2.7(2)~{}$mK. Figure 3(b) depicts the results
for the temperatures at different times $t_{if}$ ($\circ$). The temperature is
initially much larger than the steady-state temperature ($\square$) from Fig.
3(a), but converges to this value with time spent inside the fiber.

We believe this decrease of temperature during the course of about 20-30 ms to
be the result of free evaporative cooling inside the FORT, as for instance
observed in Barrett  _et al._ (2001). As the atoms enter the FORT of finite
depth $T_{F}$, they are accelerated and heated to a maximum of $T\sim T_{F}$.
The trap is therefore overheated and a significant fraction of atoms has a
potential energy close to zero. Thus, any heating process present leads to
immediate loss of high-energy atoms from the trap and therefore cooling. The
rate of evaporation due to elastic collisions can be estimated Ketterle and
van Druten (1996) as $\Gamma_{ev}\sim 2$ s-1 for our estimated in-fiber atomic
density of $10^{12}~{}$cm-3 and trap depth $T_{F}$. This is too small to
explain the observed decrease in temperature. However, as discussed in Sec.
III.2.4, losses from the FORT are currently dominated by heating due to an
ellipticity of the FORT polarization Corwin  _et al._ (1999) leading to a loss
rate of 40 s-1. This rate matches quite well to the timescale of the here
observed decrease of temperature. Thus, experiments sensitive to the ensemble
temperature should be carried out with a sufficiently large delay after
loading of the fiber is finished to allow for thermalization.

We note that the observation of evaporation on such a short timescale briefly
after loading the FORT is facilitated by the ability to selectively probe the
atoms inside the HCPBGF. In free-space FORTs, probing is restricted to
timescales where the untrapped atoms have fallen out of the interaction volume
Barrett  _et al._ (2001).

####  III.1.2 Axially resolved measurements: velocity and acceleration

In deep, focused-beam FORTs, acceleration of the atoms due to radiation
pressure from the FORT or other laser beams can be typically neglected as it
is dominated by the steep gradient of the FORT potential. In fiber-based
FORTs, however, the atoms inside the fiber are basically untrapped along the
fiber axis and even modest photon scattering rates can lead to a significant
acceleration, affecting the propagation $z(t)$ inside the fiber. Moreover, as
the atomic transitions inside the HCPBGF saturate already at a power in the
range of a few nanowatts, any not properly blocked beam (e.g., the
$0^{\textrm{th}}$ order beams of the AOMs) can have a detrimental effect on
the total scattering rate. Thus, a careful study of the atomic motion is
advisable, showing, e.g., whether or not the loading efficiency is affected by
propagation effects. To this end, we recorded EIT spectra with anti-collinear
probe and control beams for pulsed loading of the fiber. Here, the spectra are
maximally susceptible to the atomic motion along the fiber axis as the two-
photon resonance is Doppler-shifted by $\sim 2k_{p,c}v_{a}$. This allows to
determine the average axial velocity $v_{a}(t)=v_{0}+a_{eff}t$ as a function
of time, and the effective acceleration $a_{eff}$. In the case of a cold
atomic cloud held at rest above the fiber tip, the most probable initial
velocity after entering the fiber can be estimated from the trap depth $T_{F}$
as $v_{0}=\sqrt{2k_{B}T_{F}/m}$. The position inside the HCPBGF is then simply
given by $z(t)=v_{0}t+\frac{1}{2}a_{eff}t^{2}$.

For an atomic cloud entering the HCPBGF at a downward velocity $v_{0}>0$ and
then experiencing solely a downward acceleration in the earth’s gravitational
field we expect to observe EIT spectra where the two-photon resonance, i.e.,
the EIT dip, is increasingly shifted to positive detunings with time $t_{if}$
inside the fiber for a downward propagating probe beam. This is confirmed by
EIT spectra taken at $t_{if}=35~{}$ms ($\circ$) and $t_{if}=65~{}$ms
($\square$) as shown in Fig. 4(a). The solid lines show calculated spectra
based on the theory in Blatt  _et al._ (2016).

Figure 4:  (a) EIT spectra with anti-collinear probe and control fields taken
at $t_{if}=35~{}$ms ($\circ$) and $t_{if}=65~{}$ms ($\square$) after the atoms
entered the HCPBGF. The solid lines show the results from a numerical
simulation with $d_{opt}=(2.2;1.3)$ (black; red),
$\Omega_{c}=1.2(1)\Gamma_{D_{1}}$, $v_{a}=(0.86;1.16)$ m/s (black; red),
$\gamma_{21}=(0.16;0.07)\Gamma_{D_{1}}$ (black; red), $T=1~{}$mK. (b) Measured
velocities of the atoms (positive when parallel to the direction of gravity)
vs. the time $t_{if}$ they have spent inside the HCPBGF. The estimated
uncertainty for determining the velocities is $\pm 0.05$ m/s. For regular
loading conditions ($\circ$), we obtain $v_{0}=0.70(6)~{}$m/s and
$a_{eff}=6.8(15)~{}$m/s2 from a linear least-squares fit. When the depumper is
turned on for only $200~{}\upmu$s just before the measurement, the effective
acceleration is larger ($\square$). The black solid (red dashed) line shows
the result of the rate equation model (see App. B) when the depumper is on
(off) during loading. Here, we used the measured depumper power,
$\Gamma_{F}^{(R)}=410~{}$s-1, and $v_{0}=0.74~{}$m/s which are both within the
experimentally determined uncertainties. If the acceleration were solely
determined by gravity, the gray dotted line would show the corresponding
dependence. When the notch filter used to suppress resonant scattering by the
FORT is removed and the depumper is continuously on during the loading process
($\diamond$), the atoms are effectively accelerated upwards. The blue dashed-
dotted line shows the least-squares linear fit yielding $v_{0}=0.77(8)$ m/s
and $a_{eff}=-23(2)~{}$m/s2.

We then extracted the velocity as a function of $t_{if}$ from a series of
measurements which is shown in Fig. 4(b) ($\circ$). A linear fit to the data
yields $v_{0}=0.70(6)~{}$m/s which agrees reasonably well with the expected
value due to a FORT depth of $T_{F}=3.8~{}$mK for this measurement. However,
the acceleration $a_{eff}=6.8(15)$m/s2 is smaller than expected from gravity
only (gray dotted line). Using this information, we can now calculate the
time-dependent position $z(t)$ of the atoms and, e.g., reconstruct for the
first time the density evolution inside such a fiber (see Sec. III.2.4).

As an interesting feature, the measured acceleration inside the fiber is
different from the case of free-falling atoms. Radiation pressure from the
FORT and depumper beams, which are both aligned anti-parallel to gravity and
are typically on during the loading and guiding period, are most likely the
source of a reduced acceleration. We therefore recorded spectra when the
depumper was switched on for only $200~{}\upmu$s just before the measurement
to avoid any influence on the atomic motion. The extracted velocity is shown
by the red square in Fig. 4(b). It is consistent with the one expected for a
purely gravitational acceleration within the error bars. To further
investigate the acceleration, we employed a population rate equation model
(see App. B), which then allows for calculation of the light pressure force
via the photon momentum, excited state scattering rate and population Metcalf
and van der Straten (1999). This model allows for consistently reproducing the
optical pumping by the FORT and depumper discussed in App. A as well as the
velocities shown in Fig. 4(b) by the black solid line (depumper always on) and
red dashed line (depumper pulsed for $200~{}\upmu$s). We suspect that the
discrepancy between the measured and simulated velocity without the depumper
($\square$, red dashed line) is due to a slightly higher initial velocity when
the depumper is not present, as it will be less-detuned from the cycling
transition $F=2\rightarrow F^{\prime}=3$ just outside of the HCPBGF where the
light-shift by the FORT is smaller and therefore slow the atoms down when it
is present. We thus conclude that the effective acceleration inside the fiber
is mainly determined by gravity and radiation pressure from the FORT and
depumper. Other sources–if present–play only a minor role.

Notes: (i) To further illustrate the importance of photon scattering on the
propagation dynamics we show in Fig. 4(b) by blue diamonds the results when a
notch filter, used to suppress resonant scattering by the FORT, is removed.
This leads to a reversal of the atoms’ propagation direction and prohibits a
proper filling of the fiber (see Sec. III.2.4). (ii) Although this EIT
configuration with anti-collinear probe/control beams allows for determination
of the thermal velocity distribution along the beams’ direction, we refrain
from a detailed analysis. As we will show in the next section, determination
of the ground state dephasing rate shows some irregularities which also affect
the precision of determining the temperature. Nonetheless, analysis of spectra
such as in Fig. 4(a) show that the temperature along the fiber axis is in the
same range of 1 mK as determined in Sec. III.1.1 for the radial direction.

####  III.1.3 Axially resolved measurements: dephasing

Information on the ground state dephasing rate $\gamma_{21}$ is vital for
experiments requiring long coherence times Xin  _et al._ (2019); Leong  _et
al._ (2020a); Li  _et al._ (2020). This rate is, e.g., affected by transit-
time-broadening, two-photon linewidth of the probe/control beams, and
inhomogeneous broadening due to the similar transverse dimensions of the
control and the atomic density distribution Blatt  _et al._ (2016). In
addition, it could depend on the spatial position due to a possible non-zero
magnetic field (gradient) which changes across the fiber length of $14~{}$cm,
leading to a spatially varying ground state splitting. EIT spectra taken with
collinear beams are minimally sensitive to atomic motion. Thus, spectra taken
at different times $t_{if}$ potentially allow for spatially-resolved
determination of the effective ground state dephasing rate $\gamma_{21}$.

An EIT spectrum for collinear probe/control beams taken at $t_{if}=15(1)$ is
shown in Fig. 5(a). As before, we adjusted the parameters of a numerical
simulation (solid line) until we reached best agreement with the experimental
data ($\circ$) Blatt  _et al._ (2016).

Figure 5:  (a) EIT spectra with collinear probe and control fields taken at
$t_{if}=15(1)~{}$ms ($\circ$) after the atoms entered the HCPBGF. The solid
line shows the result of a numerical simulation with $d_{opt}=2.0(1)$,
$\Omega_{c}=1.4(1)\Gamma_{D_{1}}$, $T=1~{}$mK, $v_{0}=1.0$ m/s,
$\gamma_{21}=0.07(2)\Gamma_{D_{1}}$. (b) Averaged dephasing rates
$\gamma_{21}$ vs. $t_{if}$ as extracted from multiple measurements such as in
(a). The error bars represent an estimated uncertainty of
$0.02\Gamma_{D_{1}}$.

Such spectra allow us to extract $\gamma_{21}$ as a function of $t_{if}$ (and
therefore position) inside the HCPBGF as shown in Fig. 5(b). The lower limit
that can be expected for our experiment is given by transit-time broadening
leading to $\gamma_{ttb}\sim v_{th}/d_{c}=0.013\Gamma_{D_{1}}$. Obviously, the
measured dephasing rates are significantly larger than this lower limit, as
already observed in Peters  _et al._ (2020). Additionally, for early as well
as later times $t_{if}$, $\gamma_{21}=0.08(2)\Gamma_{D_{1}}$ is roughly the
same. For $20$ ms $\leq t_{if}\leq 35$ ms, however, the dephasing rate
increases up to $\gamma_{21}=0.16(2)\Gamma_{D_{1}}$. We confirmed this
behavior during several measurement runs on different days and the data shown
here are averages of these multiple measurements. This only temporary increase
of the dephasing rate is again quite surprising as we would have expected a
steady increase for a spatially varying magnetic field (gradient). This issue
is yet undetermined, but again shows the importance of a detailed analysis.

The data and simulations presented above show that it is indeed possible to
gain spatially-resolved information along the fiber axis on the radial
distribution, axial position, and ground state dephasing rate for an atomic
ensemble loaded into a HCPBGF. This information can for instance be used to
understand and optimize the loading of atoms into a fiber as we will show in
the following section. We note that for these measurements to succeed with an
acceptable signal-to-noise ratio, we required to load around 5000-6000 atoms
for a pulsed loading of 2-3 ms duration.

###  III.2 Determination of the limits to the loading efficiency

In the following, we investigate the loading process of the HCPBGF as to
determine the current limits to the loading efficiency and prospects for
further enhancement. Contrary to other fiber loading techniques Bajcsy  _et
al._ (2011); Hilton  _et al._ (2018); Langbecker  _et al._ (2018), this has
not yet been reported using a dark funnel guide Blatt  _et al._ (2014).
Specifically, we investigate the transfer from the MOT to the HCPBGF, the
maximum number of atoms $N_{a}$ loaded into the HCPBGF, and their spatio-
temporal evolution.

####  III.2.1  MOT to HCPBGF atom transfer

In contrast to experiments loading laser-cooled atoms into Kagome-structured
hollow fibers of low numerical aperture, we first have to transfer the atoms
from the MOT near the HCPBGF tip before the FORT potential is deep enough to
guide atoms into the fiber. We do this by employing a DFR to maximize the
atomic density above the HCPBGF (see Sec. II.1). We therefore first
investigate the transfer of atoms from the MOT (centered $\sim 5.5$ mm above
the fiber tip) down to the HCPBGF by measuring the number of atoms trapped
above the fiber via fluorescence detection Townsend  _et al._ (1995) with a
calibrated CCD camera for different times during the transfer period $990$ ms
$\leq t\leq 1050~{}$ms (see Fig. 6).

Figure 6:  Measured number of atoms $N_{a}^{(MOT)}$ above the HCPBGF vs. time
$t^{\prime}$ after start of the transfer from the MOT to the HCPBGF. The atom
number above the HCPBGF is measured by fluorescence imaging. The error bars
correspond to the standard deviation of 20 successive measurements. The MOT
repumper is switched off near $t^{\prime}=0~{}$ms. The atom cloud reaches the
HCPBGF tip around $t^{\prime}=15~{}$ms. The MOT quadrupole field is switched
off at $t^{\prime}=45~{}$ms. The insets show false color images of the atom
cloud above the HCPBGF tip (yellow structure at lower end) at two different
timings.

As long as the MOT repumper is on ($t^{\prime}\lesssim 0~{}$ms) the number of
atoms above the fiber increases. After the MOT repumper is off, leaving only
the DFR on, the atom cloud transfer to the HCPBGF occurs without significant
losses as long as the cloud has not reached the fiber tip ($0~{}$ms $\leq
t^{\prime}\leq 15~{}$ms). Once the lower end of the cloud has reached the
HCPBGF tip at $t^{\prime}\sim 15~{}$ms, however, the number of atoms decreases
quickly due to the losses in the regions where the HCPBGF mount blocks the
trapping light. At $t^{\prime}=45~{}$ms, where the MOT quadrupole field
confining the atoms is switched off, basically no atoms are left above the
HCPBGF. These results show that the magnetic offset field settings, which
determine the position where the cloud is held during loading, have to be
carefully optimized, or losses from the dark funnel guide will become
relevant. On the other hand, these magnetic field settings can also be used to
quickly change the number of atoms loaded into the HCPBGF while keeping all
other parameters the same. For this peak atom number of
$N_{a}^{(MOT)}=7(1)\times 10^{6}$ in the MOT we detected $N_{a}=1.5(1)\times
10^{5}$ inside the fiber. The loading efficiency from MOT to HCPBGF-based FORT
is therefore roughly $\eta=2.1(3)~{}\%$. The uncertainty of $\eta$ is based on
the shot-to-shot fluctuations and does not account for an incorrect
calibration of the fluorescence imaging setup. Therefore we expect the
uncertainty to be actually larger than $0.3~{}\%$. The efficiency is
nonetheless roughly in agreement with our previously estimated loading
efficiency Blatt  _et al._ (2014) and of similar magnitude as in Hilton  _et
al._ (2018).

####  III.2.2 Dependence of the loading efficiency on the FORT power

Given that the MOT to HCPBGF loading efficiency is only a few percent, we
might expect that the loading efficiency increases for a deeper FORT potential
when the ensemble temperature above the HCPBGF is kept constant. However, we
found that this assumption is only partially correct.

In Fig. 7(a) we measured $N_{a}$ as a function of FORT power $P_{F}$ ($\propto
T_{F}$) during the loading phase. Here we show the results for the case where
the trapping detuning is kept constant at $-2.5\Gamma_{D_{2}}$ ($\square$) and
when the detuning is ramped down to $-5\Gamma_{D_{2}}$ during the loading
process ($\circ$) to optimize sub-Doppler cooling above the HCPBGF.

Figure 7:  (a) Measured $N_{a}$ vs. relative FORT power $P_{F}/P_{F}^{max}$
with ($\circ$) and without ($\square$) sub-Doppler cooling above the HCPBGF.
The solid lines are linear fits to the corresponding experimental data for
$P_{F}/P_{F}^{max}\leq 0.7$. (b) Number of atoms $N_{a}$ loaded into the
HCPBGF vs. the power $P_{DFR}$ of the DFR (of a single beam) for different
depumper powers. The error bars in the vertical direction correspond to an
estimated $8\%$ uncertainty. The depumper peak Rabi frequency inside the
HCPBGF was around $1.4\Gamma_{D_{2}}$ ($\circ$), $2.9\Gamma_{D_{2}}$
($\square$), and $7.6\Gamma_{D_{2}}$ ($\diamond$), respectively.

In both cases the atom number initially increases nearly linearly with the
FORT power as expected. For $P_{F}/P_{F}^{max}>0.7$, however, the atom number
saturates around $N_{a}\sim 1.6\times 10^{5}$ in the case where sub-Doppler
cooling is performed. The solid lines are linear fits to the corresponding
experimental data for $P_{F}/P_{F}^{max}\leq 0.7$ and are extrapolated until
$P_{F}/P_{F}^{max}=1$. Without sub-Doppler cooling, the linear region extends
further, but also here $N_{a}$ saturates near $N_{a}\sim 1.0\times 10^{5}$. We
interpret these results as follows: As the maximum atom number inside the
HCPBGF seems to saturate at different values for the two conditions, and this
maximum value can change from day to day (going sometimes up to $N_{a}\geq
2\times 10^{5}$), we believe that this limit is due to the density and
temperature of the atoms above the fiber. Loading more atoms inside the MOT,
better compression and/or cooling during the transfer should therefore lead to
a better loading efficiency and is only a technical issue. We note that a
similar saturation behavior has also been observed for a different loading
technique using a Kagome fiber Hilton  _et al._ (2018).

####  III.2.3 Dependence of the loading efficiency on the repumper power and
configuration

The DFR Blatt  _et al._ (2014) is at the heart of our loading method as it
guides the atoms and enhances their density near the fiber tip. This
distinguishes our loading technique from others and therefore deserves a
closer look.

First, we determine how much the DFR affects the loading efficiency. To this
end, we compare the loading using a DFR to the case of a free-falling cloud as
well as simple cooling and compression during the transfer. For a free-falling
cloud, without any additional compression, cooling or guiding until the fiber
tip, we were not able to detect any atoms inside the HCPBGF, in contrast to
Bajcsy  _et al._ (2011). When we keep the MOT repumper on during the HCPBGF
loading phase and optimize the MOT repumper power to $180~{}\upmu$W for a
maximum $N_{a}$ while the DFR is still blocked, we loaded $N_{a}=6.6(7)\times
10^{4}$ atoms. This corresponds to a compression and cooling of the cloud
during the transfer as it is shifted towards the fiber. Density-limiting
collisions are here not suppressed. If, however, we guide the atoms in the
dark funnel by switching off the MOT repumper at the beginning of the HCPBGF
loading phase ($t=1000~{}$ms) while keeping the DFR on at a power of
$25(1)~{}\upmu$W per beam, $N_{a}$ increases by a factor of $\sim 2.7$. This
demonstrates the effectiveness of our guiding method based on the DFR. It also
nicely matches the results by Kuppens et al. who studied the loading of a
free-space dipole trap and reported a factor of two improvement in the loading
efficiency when using a shadow in the repumper Kuppens  _et al._ (2000).

Next, we consider the dependence of $N_{a}$ on the DFR and depumper power. The
latter one is relevant as it serves to confine the atoms in $F=1$. Inside the
HCPBGF the depumper compensates scattering by the FORT whereas outside of the
fiber it compensates scattering by repumper stray light Anderson  _et al._
(1994). The results are shown in Fig. 7(b) for three different depumper
powers. For all datasets $N_{a}$ first increases with DFR power $P_{DFR}$,
reaches a maximum and then decreases again. We achieve a maximum $N_{a}$ for
$P_{DFR}\sim 25~{}\upmu$W and a depumper Rabi frequency of $2.9\Gamma_{D_{2}}$
($\square$). The position of the maximum shifts to larger $P_{DFR}$ when the
depumper gets stronger. However, the maximum $N_{a}$ for the strongest
depumper ($\diamond$) is lower than for optimum conditions ($\square$). An
exact interpretation of this behavior is difficult, as the depumper beam
serves to pump the atoms both inside and outside of the HCPBGF into $F=1$
where the intensities are very different due to the diverging beams above the
fiber. Also, the loading process temporally coincides with propagation of
atoms already inside the fiber and thus a temporal modulation of the depumper
power for the spatially separated regions is not possible. An order-of-
magnitude estimation however shows that even for the weakest depumper in Fig.
7(b) its Rabi frequency is at least comparable to stray light by the DFR over
the whole MOT-HCPBGF distance. Therefore, optical pumping into $F=1$ is in
principle efficient due to the long pumping times. For regions close to the
fiber tip, where the depumper is most crucial, we therefore expect the
depumper to effectively confine the population in the ground state $F=1$. Near
the fiber tip, however, light-shifts induced by the depumper itself might
become relevant that could explain the decrease of the maximum $N_{a}$ for a
stronger depumper ($\diamond$). A more detailed modeling of the loading would
be required for a detailed understanding, which is, however, beyond the scope
of this work. Nonetheless, by carefully adjusting the DFR and depumper powers
we can achieve optimum loading conditions into the HCPBGF.

####  III.2.4 Temporal evolution of the loading process

Understanding the dynamics of the loading process is crucial to optimizing the
loading efficiency from the MOT into the HCPBGF. As the atoms enter the deep
FORT inside the fiber their density increases and density-dependent losses
become relevant. The maximum $N_{a}$ that can be loaded therefore depends on
the relative loading and loss rates.

To study the loading process in time we modified the sequence described above
by adjusting the switch-off time of the quadrupole field and the timings of
all laser beams corresponding to the time of each measurement shown in Fig. 8
($\circ$). Here, a time of $t^{\prime}=0$ ms corresponds to $t=1000$ ms after
the MOT loading sequence has started, i.e., loading of the FORT begins.

Figure 8:  Measured $N_{a}$ inside the HCPBGF ($\circ$) vs. time $t^{\prime}$
after the start of the transfer from the MOT. The MOT quadrupole field
strength is schematically shown by the gray dotted line. The error bars
correspond to a $10\%$ measurement uncertainty of $N_{a}$. Numerical results
of the evolution based on Eqns. (3)-(4) for $a_{eff}=g=+9.81~{}$m/s2,
$\Gamma_{L}=\beta_{L}=0$ are shown by the gray dashed-dotted line and for
$a_{eff}=+6.8~{}$m/s2, $v_{0}=0.8$ m/s, $\Gamma_{L}=40(5)~{}$s-1 and
$\beta_{L}=1(1)\times 10^{-11}~{}$cm3 s-1 by the blue solid line. The loading
rate $R(t)$ is taken from the data shown in the inset with an amplitude
scaling factor of $S_{f}=2$ to account for losses during the initial $5~{}$ms
inside the fiber. The red dashed line shows the results when an acceleration
of $a_{eff}=-23~{}$m/s2 inside the HCPBGF is assumed (see text). Inset:
$N_{a}$ vs. $t^{\prime}$ for a loading time of 1 ms. The error bars correspond
to an uncertainty in the measurement of 1300 atoms. The red solid line shows
the idealized evolution of $N_{a}$ used for the simulations.

At $t^{\prime}\sim 15$ ms the number of atoms inside the HCPBGF rises rapidly
with a rate of roughly $4000$ ms-1. According to images taken with a CCD
camera (see Fig. 6 inset) this time coincides with the time the lower part of
the atom cloud reaches the fiber tip. During the next $\sim 20$ ms the rate
decreases while $N_{a}$ reaches its maximum value at $t^{\prime}\sim 50~{}$ms.
This agrees well with the data in Fig. 6 showing the evolution of atoms above
the fiber. The maximum $N_{a}$ is reached around 5 ms after the quadrupole
field has been switched off. For times $t^{\prime}>50~{}$ms $N_{a}$ decreases
during the course of around 80 ms.

To understand this behavior we first need to determine the HCPBGF loading
rate. This rate obviously depends on the evolution of the atomic density above
the fiber where the FORT potential is deep enough to guide atoms into the
HCPBGF. We estimate the region from which atoms are guided into the HCPBGF to
be in the range of $200~{}\upmu$m (axially) and $50~{}\upmu$m (radially),
respectively. As these values are much smaller than the (initial) dimensions
of the atomic cloud above the HCPBGF $(l_{z},l_{r})\sim(3000~{}\upmu$m$\times
300~{}\upmu$m) the measured atom number inside the HCPBGF is proportional to
the local density $\rho_{tip}$ of the atom cloud at the fiber tip. In our
case, the axial velocity of the cloud near the fiber tip can be estimated as
$\sim 0.33~{}$m/s. Thus, by loading the HCPBGF for only a period of 1-3 ms (by
pulsing the FORT, see above) the atom cloud moves much less than its length.
Measuring $N_{a}$ as a function of time $t^{\prime}$ therefore yields a value
proportional to the local atomic density $\rho_{tip}(t^{\prime})$ above the
HCPBGF. The result of such measurement is shown in the inset of Fig. 8
($\circ$). $N_{a}(t^{\prime})$ increases from around 10 ms to 25 ms as the
cloud approaches the fiber. Then, the atom number stays roughly constant
(neglecting the data point at $t^{\prime}=34~{}$ms) for the next $\sim 25$ ms
as the cloud is held near the fiber tip. After the MOT quadrupole field is
switched off at $45~{}$ms $N_{a}(t^{\prime})$ decreases quickly for
$t^{\prime}>48~{}$ms. We thus estimate the peak loading rate as $\sim
4700~{}$ms-1 which is in reasonable agreement with the initial loading rate
obtained from Fig. 8.

The data shown in the inset of Fig. 8 can now be used to simulate how $N_{a}$
is expected to evolve in time. To this end we used an idealized atom
number/density evolution shown by the red solid line. In regular focused-beam
FORTs of large $N_{a}$ the evolution of trapped atoms could be modeled
according to

$\frac{dN_{a}}{dt}=R^{\prime}(t)-\Gamma_{L}^{\prime}N_{a}-\beta_{L}^{\prime}N_{a}^{2},$
(2)

where $R^{\prime}(t)$ is the time-dependent loading rate,
$\Gamma_{L}^{\prime}$ describes losses due to collisions with background gas
as well as heating and $\beta_{L}^{\prime}$ describes two-body collisions,
respectively Kuppens  _et al._ (2000). In the case of our extended fiber-based
FORT, however, Eq. (2) will not suffice. In addition, we have to take
propagation of the atoms through the fiber as well as their acceleration (see
Sec. III.1.2) into account. We therefore employ the following continuity
equation with loss terms and consider a changing atomic velocity due to an
effective acceleration $a_{eff}$ as measured in Sec. III.1.2

$\displaystyle\frac{\partial}{\partial t}n+\frac{\partial}{\partial
z}\left(n\cdot v_{a}\right)$ $\displaystyle=$
$\displaystyle-\Gamma_{L}n-\beta_{L}n^{2},$ (3)
$\displaystyle\frac{\partial}{\partial t}v_{a}$ $\displaystyle=$
$\displaystyle a_{eff}.$ (4)

Here, the unprimed rates $\Gamma_{L}$, $\beta_{L}$ refer to the loss of
density instead of atom number. Using the loading rate $R^{\prime}(t)$ as
shown in the inset of Fig. 8 by the red line as initial condition at the fiber
input (after conversion into a density rate $r(t)$ using the known FORT
potential and atomic temperature), as well as an initial velocity of
$v_{0}=+0.8~{}$m/s [see Fig. 4(b)] we can solve Eqns. (3)-(4) either
numerically or analytically (see App. C) yielding

$n(z,t)={e^{-\Gamma_{L}T(z)}\over 1+{\beta_{L}r(t-T(z))\over
v_{0}\Gamma_{L}}(1-e^{-\Gamma_{L}T(z)})}{r(t-T(z))\over v_{0}},$ (5)

where the time $T(z)$ is defined in the appendix. Finally, the density is
converted into an atom number $N_{a}$ by integration over the axial and radial
dimensions. Since the data in the inset of Fig. 8 was obtained $5~{}$ms after
the atoms entered the fiber, losses are present during this period that have
to be accounted for by a scaling factor $S_{f}$ in the initial loading rate
for the simulations. This mainly affects the peak $N_{a}$ that can be loaded.

The gray dashed-dotted line in Fig. 8 shows the result for ideal conditions
with an acceleration of +9.81 m/s2 due to gravity and vanishing loss rates
$\Gamma_{L},\,\beta_{L}$. Initially, the agreement with the measured data is
reasonable. For longer times, however, a huge discrepancy in terms of maximum
$N_{a}$ and decay is obvious that illustrates the importance of losses and
heating Kuppens  _et al._ (2000); Corwin  _et al._ (1999). The result which
matches our experimental data ($\circ$) best is shown by the blue solid line
with $\Gamma_{L}=40(5)$ s-1 and $\beta_{L}=1(1)\times 10^{-11}$ cm3 s-1, and
$S_{f}=2$. An animation of the corresponding spatio-temporal density evolution
can be found here. As we can see, even for optimum conditions only about
$30~{}\%$ of the fiber are filled with atoms and the axial density is far from
homogeneous. The rate $\Gamma_{L}$ clearly dominates the losses, which is
about 14 and 5 times larger than reported by Okaba et al. Okaba  _et al._
(2014) and Hilton et al. Hilton  _et al._ (2018), respectively, for Kagome-
structured fibers. It is also significantly larger than reported by Kuppens et
al. for a free-space trap, where they found losses being dominated by
$\beta_{L}\sim 4\times 10^{-11}$ cm3 s-1 while $\Gamma_{L}$ was found to be
near zero Kuppens  _et al._ (2000). Due to the large $\Gamma_{L}$ in our case,
we can only determine the upper limit of as $\beta_{L}<2\times 10^{-11}$ cm3
s-1. This value is roughly in agreement with the lower limits reported in
Hilton  _et al._ (2018) and Kuppens  _et al._ (2000) and could currently be
limited by light-assisted collisions due to the depumper.

The loss rate $\Gamma_{L}$ is determined by collisions of trapped atoms with
background gas inside the fiber as well as heating. In both studies using
Kagome-structured fibers Hilton  _et al._ (2018); Okaba  _et al._ (2014),
collisional losses due to residual gases inside the fiber were assumed to
dominate $\Gamma_{L}$ while heating was neglected. Heating by photon
scattering of the FORT can be estimated for our trap as $\dot{T}\sim
10~{}\upmu$K s-1 Grimm  _et al._ (2000) and is therefore negligible on the
considered timescale. However, heating induced losses due to a possible
elliptical polarization of the FORT, as discussed by Corwin et al. Corwin  _et
al._ (1999), resulting in a hopping between Zeeman levels of different
potential depths, can be estimated as around $57(23)$ s-1. This fits quite
well to the loss rate determined above from Fig. 8. Here we assumed an
ellipticity of the FORT inside the HCPBGF of 0.9, a hopping rate of $690(280)$
s-1 according to the results in App. A, potentials for states $F=1,\,m_{F}=+1$
and $F=2,\,m_{F}=+2$ according to Eq. (6), and initial and final temperatures
of $T_{i}=1~{}$mK and $T_{f}=4.0~{}$mK, respectively. The ellipticity of the
FORT is estimated and based on the fact that though our fiber can maintain
linear polarizations quite well at the output for a certain input alignment,
inside the fiber the polarization is slightly elliptical due to a circular
birefringence component Peters  _et al._ (2020).

Notes: (i) The agreement between numerical and experimental results is quite
good despite the simplicity of our model. (ii) The loss rate $\Gamma_{L}$, can
be explained well by heating and the rate $\beta_{L}$ is of similar magnitude
as in other FORTs. It also explains well the time required to reach the
steady-state temperature as shown in Fig. 3(b). (iii) Currently, $N_{a}$ is
limited by evaporation from the FORT. In-fiber cooling (during the loading and
guiding process) might therefore lead to a larger loading efficiency. (iv)
Optical pumping between the hyperfine ground states via a Raman coupling by
the broadband FORT (see App. A) significantly affects the acceleration for our
loading scheme. Employing a narrowband FORT is therefore crucial when using
such system for interferometry Xin  _et al._ (2018). (v) As was shown in Sec.
III.1.2, the atoms experience an acceleration of $a=-23~{}$m/s inside the
fiber and reverse their direction when resonant scattering by the FORT is not
suppressed. This was the case in our previous work Peters  _et al._ (2020).
Suppression of resonant scattering surprisingly did not affect the loading
efficiency. We therefore tried to simulate these conditions. However, the
theoretical model used to simulate the temporal and spatial density evolution
can only be used when the atoms are traveling into a single direction. To
model the case of direction reversal we had to employ a cruder model which
divides the initial loading rate into slices, each assigned with a distinct
velocity that would change in each time step according to the acceleration.
The results are shown in Fig. 8 by the red dashed line. Interestingly, the
results are quite similar to the case of a falling cloud without reversal of
direction. This is due to the fact that the atoms are pushed out of the fiber
only at late times where losses, dominated by heating and collisions, have
already significantly reduced $N_{a}$ as illustrated in an animation of the
spatio-temporal density evolution. Only up to $15~{}\%$ of the fiber are
filled with atoms under these conditions and the density is temporarily
increased significantly.

We conclude that the loading efficiency is currently limited by the number of
atoms and the density above the HCPBGF as well as the loss rates inside the
fiber. Using a deeper FORT, however, will not lead to a further improvement.
Especially the loss rate $\Gamma_{L}$ seems to be dominated by heating inside
the slightly elliptical FORT. As this ellipticity cannot be avoided for our
fiber, in-fiber cooling seems to be the most effective way to reduce the
losses. According to the simulation, the loading efficiency could be easily
increased by a factor beyond 1.8 for loss rates such as reported in Sesko  _et
al._ (1989).

##  IV Summary and Outlook

We presented thorough experimental investigations, accompanied by theoretical
calculations to carefully characterize and optimize the loading of laser-
cooled atoms from a MOT into a FORT inside a hollow-core fiber. Although
optical access in such a fiber is restricted to the input and output ports, we
demonstrated spatially-resolved probing of the atoms inside the fiber with a
resolution of a few millimeters using EIT and a time-of-flight technique. We
observed for the first time a decrease of the atomic ensemble temperature
during the course of around 20 ms after entering the fiber, reaching a steady-
state value of about $1/4$ of the trap depth. Also, we determined an in-fiber
acceleration slightly smaller than expected from gravity which could be
explained by radiation pressure. Finally, we determined a peak number density
of $\sim 10^{12}~{}$cm-3 and loss rates of 40(5) s-1, dominated by heating,
and $1(1)\times 10^{-11}~{}$cm3 s-1 due to two-body collisions. Using these
results, we were able to reconstruct the spatio-temporal density evolution
inside such fiber for the first time. We found good agreement between the
experimental data and the simple theoretical models used.

Although our results for our fiber-guided FORT basically confirm the
expectations for free-space FORTs, there are also some significant
differences. Most notably, propagation of the atoms inside the extended fiber
has to be taken into account, which is not required for standard focused-beam
FORTs. Although the loading efficiency is currently not limited by propagation
through our fiber, it has to be considered for shorter fibers or lower loss
rates.

Our results allow for optimizing the loading efficiency into such hollow-core
fiber and thus have direct implications for applications. First of all, the
results on the loss rates show that there is significant room for improvement
on our record $d_{opt}=1000$ achieved so far Blatt  _et al._ (2014). An even
larger $d_{opt}$ would be required for experiments on many-body physics with
light Chang  _et al._ (2008); Noh and Angelakis (2017), which seems to be
feasible. Secondly, as the loading efficiency is in part limited by heating
and density-dependent losses, in-fiber cooling Leong  _et al._ (2020b); Naik
_et al._ (2020) will have to be applied with care. On the one hand, cooling
will lead to a reduction of inhomogenous broadening and compensation of
heating. On the other hand, the atomic number density will also increase if
the FORT potential is kept constant, leading to increased losses. Thus,
careful modeling and characterization of the loading will be required for
achieving optimum conditions, as we have done in our present work. Finally,
our results on the atomic velocity inside the fiber show that a careful
analysis is required when such a platform is used for atom interferometry Xin
_et al._ (2018).

###### Acknowledgements.

We thank M. Schlosser for helpful discussions on the FORT bandwidth and
comments on the manuscript, C.-Y. Hsu and M. Coelle for assistance with time-
of-flight measurements, and the group of T. Walther for providing us with a
home-made ultra-low noise laser diode driver with high modulation bandwidth.
This project has received funding from the European Union’s Horizon 2020
research and innovation programme under the Marie Sklodowska Curie grant
agreement No. 765075.

##  Appendix A Characterization of the FORT

We here summarize the results of the characterization of the FORT guiding the
atoms into the HCPBGF. Precise knowledge of the FORT potential is required for
a quantitative analysis of the measurements shown in the main part of the
paper, as it affects the atomic temperature, peak number density and its
distribution, resonance shifts, scattering rate, and collisions Grimm  _et
al._ (2000).

For far-detuned traps, where the FORT detuning is much larger than the
hyperfine and fine structure splitting, the following radial potential is
applicable for the ground state Grimm  _et al._ (2000); Corwin  _et al._
(1999)

$\displaystyle U_{F}(r)=-\frac{\pi c^{2}}{2}$
$\displaystyle\left[\frac{\Gamma_{D_{2}}\left(2+g_{F}m_{F}\sqrt{1-\epsilon}\right)}{\omega_{D_{2}}^{3}\left(\omega_{D_{2}}-\omega\right)}\right.$
(6)
$\displaystyle\left.+\frac{\Gamma_{D_{1}}\left(1-g_{F}m_{F}\sqrt{1-\epsilon}\right)}{\omega_{D_{1}}^{3}\left(\omega_{D_{1}}-\omega\right)}\right]I(r).$

Here $\Gamma_{D_{1,2}}$ and $\omega_{D_{1,2}}$ are the excited state decay
rates and corresponding transition frequencies, of the D1,2 lines, $g_{F}$ is
the Landé factor, $m_{F}$ is the magnetic quantum number, $\omega$ is the
frequency of the FORT field, and $I(r)$ is its radial intensity profile. The
ellipticity $\epsilon$ of the polarization is defined as
$\epsilon=(P_{max}-P_{min})/(P_{max}+P_{min})$, where $P_{max,min}$ are the
maximum/minimum transmitted powers through a rotated linear polarizer.

The FORT potential directly determines the atomic level shifts via
$\Delta_{ls}(r)=U_{F}(r)/\hbar$, with the reduced Planck constant $\hbar$, and
also the thermalized atomic number density distribution Grimm  _et al._ (2000)
(see also Sec. III.1.1)

$n(r)=n_{0}\,\exp{\left(-\frac{U_{F}(r)-U_{F}(0)}{k_{B}T}\right)},$ (7)

where $n_{0}$ is the peak density and $T<T_{F}$ is the ensemble temperature
inside the deep FORT. Therefore, spectroscopy of the atoms while the FORT is
on will yield information on the FORT depth, density distribution and
temperature Peyronel  _et al._ (2012); Hilton  _et al._ (2020). We confirmed
experimentally via light-shift spectroscopy Hilton  _et al._ (2020) that Eqns.
(6)-(7) would predict the correct spectroscopic shifts for the measured FORT
power, wavelength and beam width.

An important set of parameters of a cylindrically symmetric FORT are the trap
frequencies $\omega_{r,z}$ Grimm  _et al._ (2000). Already small disturbances
of the trapping potential occurring at frequencies
$\omega_{mod}=2\omega_{r,z}/n$ with $n=1,2,...$ will lead to parametric
heating and losses from the trap Savard  _et al._ (1997); Friebel  _et al._
(1998). This is crucial in our experiment, as we intentionally modulate the
FORT on/off for probing the atoms without inhomogeneous broadening induced by
the FORT Bajcsy  _et al._ (2011); Blatt  _et al._ (2016); Peters  _et al._
(2020). This modulation frequency should thus be chosen such that parametric
heating and loss does not occur. When the FORT is inside a HCPBGF, the axial
trap frequency cannot be defined due to the constant potential along the fiber
axis leaving only the radial frequency where the potential is approximately
harmonic. To investigate the FORT trap frequency we followed the method of
Friebel et al. Friebel  _et al._ (1998). We modulated the trapping potential
sinusoidally with different modulation frequencies $\omega_{mod}$ and then
measured the remaining number of atoms left in the FORT. The experimentally
observed resonances are slightly red-shifted, but otherwise agree well with
the calculated radial resonance frequencies at
$\omega_{mod}=(\omega_{r},2\omega_{r})$, where $\omega_{r}=2\pi\times 51$ kHz.
Interestingly, we observed an additional less-pronounced resonance near
$4\omega_{r}$. This resonance might be due to the occupation of high-lying
vibrational levels where the harmonic approximation of the FORT potential is
not valid anymore, as known for optical lattices Roati  _et al._ (2001);
Jáuregui  _et al._ (2001).

Photon scattering potentially results in heating and loss as well as unwanted
population dynamics. As to avoid hyperfine-changing collisions Sesko  _et al._
(1989) that could lead to additional losses from the FORT inside the HCPBGF
($h/k_{b}\cdot 6.835~{}$GHz$=328$ mK $\gg T_{F}$) we keep a depumper beam,
tuned near the transition $5^{2}S_{1/2}\,F=2\rightarrow
5^{2}P_{3/2}\,F^{\prime}=2$, on whenever the FORT is on. This confines the
population in $F=1$ inside and outside of the HCPBGF. Population
redistribution between the two hyperfine ground states should therefore only
be due to the FORT and depumper. We thus measured the population $N_{a}(t)$ in
$F=1$ as a function of time after the depumper had been switched off while
only the FORT was on. As the FORT couples both ground states off-resonantly to
the excited states, optical pumping occurs with an effective relaxation rate
$\gamma$, leading to total depolarization of the population according to
$N_{a}(t)=N_{1}+N_{2}\cdot\exp(-\gamma t)$. A fit to the measured data yields
$\gamma=690(280)$ s-1, $N_{1}=0.34(8),N_{2}=0.66(8)$. Next, we measured the
power spectrum of the FORT laser system with an optical spectrum analyzer and
calculated the spontaneous FORT scattering rate according to

$\begin{split}\Gamma_{F}=\frac{\pi
c^{2}}{2\hbar}\displaystyle\int\left(\frac{2\Gamma_{D_{2}}^{2}}{\omega_{D_{2}}^{3}\left(\omega_{D_{2}}-\omega\right)^{2}}+\frac{\Gamma_{D_{1}}^{2}}{\omega_{D_{1}}^{3}\left(\omega_{D_{1}}-\omega\right)^{2}}\right)\\\
I^{\prime}(\omega)d\omega,\end{split}$ (8)

where $I^{\prime}(\omega)=dI^{\prime}(\omega)/d\omega$ is the intensity per
frequency interval $d\omega$, we assumed linear polarization and that the
hyperfine structure is not resolved. We obtained $\Gamma_{F}\sim 186~{}$s-1,
i.e., smaller than $\gamma$. Care must be taken when comparing the FORT
scattering rate $\Gamma_{F}$ to the population relaxation rate $\gamma$. As
was first pointed out by Cline et al., the relaxation rate can be orders of
magnitude smaller than the total spontaneous scattering rate, due to an
interference between the D1,2 lines suppressing Raman scattering Cline  _et
al._ (1994). When we consider this interference effect, we obtain a relaxation
rate of $\gamma_{rel}=0.07\Gamma_{F}$ between $F=1,2$, i.e., about fifty times
smaller than measured. To solve this discrepancy, we note that our FORT system
is comprised of two independent and unstabilized laser diodes. As a result,
the FORT field intensity shows fluctuations, whose spectral components can
drive Raman transitions between $F=1,2$ without radiation of spontaneously
emitted photons. We will show and discuss the validity of this assumption in
Sec. III.1.2.

##  Appendix B Simulation of optical pumping and acceleration by the FORT and
depumper beams

We here present the model used to simulate the data on the axial velocity
shown in Fig. 4(b). Both FORT and depumper are basically always switched on
during loading of the HCPBGF. Therefore, optical pumping by both fields has to
be considered as it affects the population redistribution between the two
ground states $F=1,2$ (see App. A) as well as the acceleration $a=F/m$. Here,
$F=\hbar k\Gamma\rho_{ee}$ is the force on an atom induced by absorption of
photons followed by spontaneous emission at rate $\Gamma$ and $\rho_{ee}$ is
the population in the excited state Metcalf and van der Straten (1999).

In order to calculate the relevant state-resolved populations in our multi-
level system, we numerically solved the following population rate equations,
where the index $i$ corresponds to the following hyperfine states in ascending
energetic order (ground states $F=1,2$, D2 line $F^{\prime}=1,2,3$). In this
model we assume that population is mainly redistributed between $F=1,2$ by
Raman scattering at rate $\Gamma_{F}^{(R)}=690(280)~{}$s-1 ($\equiv\gamma$
within the measurement uncertainty, see App. A) while spontaneous scattering
of the FORT at rate $\Gamma_{F}=186~{}$s-1 leads predominantly to Rayleigh
scattering due to an interference effect between the D1,2 lines Cline  _et
al._ (1994) and therefore does not affect the populations. We thus only need
to consider the ground state populations as well as the excited states coupled
by the depumper at the D2 line. $\Omega_{d}(\delta_{i})$ are the depumper Rabi
frequencies depending on the detuning from levels $F^{\prime}=1,2,3$. The
branching ratios are taken from Steck . Furthermore, we assumed an unpolarized
FORT and depumper, i.e., we neglected the Zeeman substructure and only
considered stimulated emission by the near-resonant depumper. The population
rate equations then read

$\displaystyle\dot{N}_{1}$ $\displaystyle=$
$\displaystyle\Gamma_{D_{2}}\left(\frac{5}{6}N_{3}+\frac{1}{2}N_{4}\right)-\frac{5}{8}\Gamma_{F}^{(R)}N_{1}+\frac{3}{8}\Gamma_{F}^{(R)}N_{2},$
(9) $\displaystyle\dot{N}_{2}$ $\displaystyle=$
$\displaystyle\Gamma_{D_{2}}\left(\frac{1}{6}N_{3}+\frac{1}{2}N_{4}+1N_{5}\right)+\frac{5}{8}\Gamma_{F}^{(R)}N_{1}-\frac{3}{8}\Gamma_{F}^{(R)}N_{2},$
$\displaystyle-\Omega_{d}(\delta_{1})\left(N_{2}-N_{3}\right)-\Omega_{d}(\delta_{2})\left(N_{2}-N_{4}\right)-\Omega_{d}(\delta_{3})\left(N_{2}-N_{5}\right),$
$\displaystyle\dot{N}_{3}$ $\displaystyle=$
$\displaystyle-\Gamma_{D_{2}}N_{3}+\Omega_{d}(\delta_{1})\left(N_{2}-N_{3}\right),$
(11) $\displaystyle\dot{N}_{4}$ $\displaystyle=$
$\displaystyle-\Gamma_{D_{2}}N_{4}+\Omega_{d}(\delta_{2})\left(N_{2}-N_{4}\right),$
(12) $\displaystyle\dot{N}_{5}$ $\displaystyle=$
$\displaystyle-\Gamma_{D_{2}}N_{5}+\Omega_{d}(\delta_{3})\left(N_{2}-N_{5}\right).$
(13)

The total FORT scattering rate $\Gamma_{F}$ has a direct influence on the
change of atomic momentum. The Raman pumping rate $\Gamma_{F}^{(R)}$ only
affects the momentum if the depumper is on, as the net momentum transfer of
$\pm h/c\cdot 6.8~{}$GHz is zero over many pumping cycles. The rate equations
(9)-(13) can be either solved numerically, where the effective acceleration
$a_{eff}=g-F/m$ of the atoms is then given by

$a_{eff}=g-\frac{\hbar
k}{m}\left(\Gamma_{F}+\Gamma_{D_{2}}\sum_{i=3}^{5}N_{i}\right),$ (14)

or analytically by using the stationary solutions. Then, the acceleration can
be determined as

$a_{eff}=g-\frac{\hbar
k}{m}\left(\Gamma_{F}+\frac{5}{8}\Gamma_{F}^{(R)}p_{0}\right),$ (15)

where the first term is due to downward acceleration by gravity and the other
terms due to the upward directed light pressure by the FORT and depumper
beams. The parameter $p_{0}$ is between 1.5 and 2 and depends on the detuning
of the depumper from the transition $F=1\rightarrow F^{\prime}=2$. For the
previously used FORT scattering rates we thus obtain $a_{eff}\sim 5~{}$m/s2,
close to the measured value.

##  Appendix C Analytic derivation of the density evolution inside the fiber

We consider the propagation of an ensemble inside a fiber where atoms are
moving with positive acceleration into a single direction, i.e., there is no
change of direction. The atomic number density $n(z,t)$ inside the fiber at
time $t$ in position $z$ is determined by the atomic number density at the
beginning $n(z=0,t)$ at the previous time $t-T(z)$, where $T(z)$ is the time
to travel from $z=0$ to $z$ with the acceleration $a_{eff}$. This time is a
solution of the equation

$z=v_{0}T+\frac{1}{2}a_{eff}T^{2},$ (16)

where $v_{0}$ is the initial velocity. The solution of this equation reads

$T(z)={z\over v_{0}}{2\over\sqrt{1+2a_{eff}z/v_{0}^{2}}+1}.$ (17)

If the fiber is loaded with the density rate $r(t)$, then the density
$n(z=0,t)$ is given by

$n(z=0,t)=r(t)/v_{0}.$ (18)

Thus, the density $n(z,t)$ at time $t$ and position $z$ is equal to the
density $n(z=0,t-T(z))$ but subject to collisional losses during the time
$T(z)$. The differential equation for this density $\nu(\tau)=n(z=0,t-\tau)$
reads

${d\nu(\tau)\over d\tau}=-\Gamma_{L}\nu(\tau)-\beta_{L}\nu^{2}(\tau).$ (19)

with the initial condition $\nu(\tau=0)=r(t-T(z))/v_{0}$. The analytical
solution of this equation is given by

$\displaystyle\nu(\tau)$ $\displaystyle=$
$\displaystyle{\Gamma_{L}e^{-\Gamma_{L}\tau}\nu(\tau=0)\over\Gamma_{L}+\beta_{L}\nu(\tau=0)(1-e^{-\Gamma_{L}\tau})}$
$\displaystyle=$ $\displaystyle{e^{-\Gamma_{L}\tau}\over
1+{\beta_{L}r(t-T)\over v_{0}\Gamma_{L}}(1-e^{-\Gamma_{L}\tau})}{r(t-T)\over
v_{0}}$

and the corresponding solution for $n(z,t)$ is

$n(z,t)={e^{-\Gamma_{L}T(z)}\over 1+{\beta_{L}r(t-T(z))\over
v_{0}\Gamma_{L}}(1-e^{-\Gamma_{L}T(z)})}{r(t-T(z))\over v_{0}}.$ (21)

The total atom number $N_{a}(t)$ is then given by the integral of $n(z,t)$
along the radial and axial directions.

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# Classes of exact static spheroidal Einstein-Maxwell solutions.

S.M.KOZYREV Scientific center gravity wave studies ”Dulkyn”, PB 595, Kazan,
420111, Russian Federation. _Email_<EMAIL_ADDRESS>
###### Abstract

In this paper we study the spheroidal cases of static charged fluid
configurations in general relativity. We consider the effect of the
anisotropic stresses of electromagnetic field on the shape of static charged
self-graviting objects. It is shown that electromagnetic fields can have
significant effect on the structure and properties of self-graviting objects.

##  1 Introduction

Exact solutions of the Einstein-Maxwell field equations are of crucial
importance in relativistic astrophysics. These solutions may be utilised to
model a charged relativistic star as they are matchable to the Reissner-
Nordstrom [1], [2] exterior at the boundary. A wide spread assumption in the
study of stellar structure is that the shape of star can be modeled as a
spherical symmetry object. This approach has been used extensively in the
study of star, star system and galaxies. However, in many systems, deviation
from spherical symmetry may play an important role in determining of them
properties. Physical situation where unspherical shape may be relevant are
very diverse. On the other hand, self-graviting objects resulting from the
coupling electromagnetic field to gravity are a system where anisotropic
pressure occurs naturally.

Anisotropy appears as an extra assumption on the behavior of electromagnetic
field and on the shape of equilibrium configuration. Since we still do not
have a formulation of the possible anisotropic stresses is emerging in these
or other contexts, we take the approach of finding several exact solutions
representing physical situations, modelled by ellipsoid of revolution.
Solutions to the equation in spheroidal coordinates have application to a wide
range of problems in physics [5]. Our goals hear is to find exact spheroidal
solution, offering an analysis of the change in the physical properties of the
stellar and galaxy models due to presence of electromagnetic field.

##  2 Einstein-Maxwell Equations and Static spheroidal configurations.

In this paper we study static, spheroidal solutions of the Einstein-Maxwell
system featuring a spinless charge configurations.The vacuum Einstein-Maxwell
equations, in geometrized units such that $c=8\pi
G=\mu_{0}=\varepsilon_{0}=1$, can be written as [3]

$R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=T_{\mu\nu},$ (1) $F^{\mu\nu}_{;\nu}=0,$ (2)

with the electromagnetic energy-mementum tensor given by

$\displaystyle T_{\mu\nu}$ $\displaystyle=$ $\displaystyle
F_{\mu\eta}F^{\eta}_{\nu}-\frac{1}{4}g_{\mu\nu}F_{\eta\zeta}F^{\eta\zeta},$
(3)

where

$\displaystyle F_{\mu\nu}$ $\displaystyle=$ $\displaystyle
A_{\nu,\mu}-A_{\mu,\nu},$ (4)

is the electromagnetic field tensor and $A_{\mu}$ is the electromagnetic four
potential.

To start with, note that by using coordinate freedom inherent in general
relativity any static spheroidal geometry can by put into form where are only
two independent metric components typically functions of the coordinates
$\xi$. As we have already mentioned we consider the ansatz static spheroidal
space-time.

The two-dimensional elliptic coordinate system is defined from the set of all
ellipses and all hyperbolas with a common set of two focal points. We denote
the separation of the two focal points by $2c$.

Oblate spheroidal coordinates are derived from elliptic coordinates by
rotating the elliptical coordinate system about the perpendicular bisector of
the focal points. The coordinates are often labelled $\eta$, $\xi$ and
$\theta$ with the transformation to Cartesian coordinates given by

$\displaystyle x=c\eta\xi sin(\theta),$ (5) $\displaystyle
y=c\sqrt{\left(\xi^{2}-1\right)\left(1-\eta^{2}\right)},$ (6) $\displaystyle
z=c\eta\xi sin(\theta)cos(\theta).$ (7)

Similarly, one can obtain the prolate spheroidal coordinates by rotating it
about the parallel bisector.

$\displaystyle x=c\eta\xi,$ (8) $\displaystyle
y=c\sqrt{\left(\xi^{2}-1\right)\left(1-\eta^{2}\right)}cos(\theta),$ (9)
$\displaystyle
z=c\sqrt{\left(\xi^{2}-1\right)\left(1-\eta^{2}\right)}sin(\theta).$ (10)

Let the spacetime ansatz be described by the spheroidal metric given by

$\displaystyle ds^{2}=-B\left(\xi\right)dt^{2}+A\left(\xi\right)d\Omega^{2}.$
(11)

For this static spheroidal ansatz of space-time we take the electromagnetic
potential as

$\displaystyle A_{\mu}$ $\displaystyle=$ $\displaystyle(\psi,0,0,0),$ (12)

where it is assumed that the electric potential $\psi$ depends on $\xi$ only.

We adopt coordinates that allow us to write spheroidal geometry in prolate
form

$\displaystyle
d\Omega^{2}=c^{2}\frac{\xi^{2}-\eta^{2}}{\xi^{2}-1}d\xi^{2}+c^{2}\frac{\xi^{2}-\eta^{2}}{1-\eta^{2}}d\eta^{2}+c^{2}(\xi^{2}-1)(1-\eta^{2})d\theta^{2},$
(13)

and in oblate form

$\displaystyle
d\Omega^{2}=c^{2}\frac{\xi^{2}-\eta^{2}}{\xi^{2}-1}d\xi^{2}+c^{2}\frac{\xi^{2}-\eta^{2}}{1-\eta^{2}}d\eta^{2}+(c\xi\eta)^{2}d\theta^{2},$
(14)

where $A,B$ are function of $\xi$ only and $\xi\geq 1$, $-1\leq\eta\leq 1$,
$0\leq\theta\leq 2\pi$.

###  2.1 Prolate spheroidal configurations.

After a bit of algebra, the field equations (1) - (2) are explicitly given in
forms of the metric (11) in prolate case.

$\displaystyle\frac{A^{\prime
2}}{4A^{2}}+\frac{A^{\prime}B^{\prime}}{2AB}=-\frac{\varepsilon^{2}}{2(1-\xi^{2})A}=T_{11},$
(15)

$\displaystyle\frac{\xi^{2}-1}{2(\eta^{2}-1)}\left[-\frac{A^{\prime\prime}}{A}-\frac{B^{\prime\prime}}{B}+\frac{A^{\prime
2}}{A^{2}}+\frac{B^{\prime
2}}{2B^{2}}\right]=\frac{\varepsilon^{2}}{2(\xi^{2}-1)(\eta^{2}-1)A}=T_{22},$
(16)

$\displaystyle\frac{A^{\prime}}{A}+\frac{B^{\prime}}{B}=0=T_{12},$ (17)

$\displaystyle\frac{(\xi^{2}-1)(\eta^{2}-1)}{2(\xi^{2}-\eta^{2})}\left[-\frac{A^{\prime\prime}}{A}-\frac{B^{\prime\prime}}{B}+\frac{A^{\prime
2}}{A^{2}}+\frac{B^{\prime
2}}{2B^{2}}\right]=-\frac{\varepsilon^{2}(\eta^{2}-1)}{2(\xi^{2}-\eta^{2})A}=T_{33},$
(18)

$\displaystyle\frac{B(\xi^{2}-1)}{c(\xi^{2}-\eta^{2})}\left[-\frac{A^{\prime\prime}}{A^{2}}+\frac{3A^{\prime
2}}{4A^{3}}-\frac{2\xi}{\xi^{2}-1}\frac{A^{\prime}}{A^{2}}\right]=\frac{\varepsilon^{2}B}{2c(\xi^{2}-1)(\xi^{2}-\eta^{2})A^{2}}=T_{00},$
(19)

where prime (’) denoting derivative with respect to the $\xi$ coordinate.

After a simple integration, from (15) - (19) we obtain

$A=\frac{1}{8}\left[a_{0}\pm\varepsilon\ln\left(\frac{\xi+1}{\xi-1}\right)\right]^{2},$
$B=\frac{b_{0}}{A},\\\ $ (20)

where $a_{0},b_{0}$ arbitrary constants.

###  2.2 Oblate spheroidal configurations.

Replacing the line element in the field equations, the oblate set is

$\displaystyle\frac{A^{\prime
2}}{4A^{2}}+\frac{A^{\prime}B^{\prime}}{2AB}=-\frac{\varepsilon^{2}}{2\xi^{2}(\xi^{2}-1)A}=T_{11},$
(21)

$\displaystyle\frac{\xi^{2}-1}{2(\eta^{2}-1)}\left[-\frac{A^{\prime\prime}}{A}-\frac{B^{\prime\prime}}{B}+\frac{A^{\prime
2}}{A^{2}}+\frac{B^{\prime
2}}{2B^{2}}\right]=-\frac{\varepsilon^{2}}{2\xi^{2}(\eta^{2}-1)A}=T_{22},$
(22)

$\displaystyle\frac{A^{\prime}}{A}+\frac{B^{\prime}}{B}=0=T_{12},$ (23)

$\displaystyle\frac{\xi^{2}\eta^{2}(\xi^{2}-1)}{2(\xi^{2}-\eta^{2})}\left[\frac{A^{\prime\prime}}{A}+\frac{B^{\prime\prime}}{B}-\frac{A^{\prime
2}}{A^{2}}-\frac{B^{\prime 2}}{2B^{2}}\right]$
$\displaystyle=-\frac{\varepsilon^{2}\eta^{2}}{2(\xi^{2}-\eta^{2})A}=T_{33},$
(24)

$\displaystyle\frac{B(\xi^{2}-1)}{c(\xi^{2}-\eta^{2})}\left[-\frac{A^{\prime\prime}}{A^{2}}+\frac{3A^{\prime
2}}{4A^{3}}-\frac{(2\xi^{2}-1)}{\xi(\xi^{2}-1)}\frac{A^{\prime}}{A^{2}}\right]=\frac{B\varepsilon^{2}}{2c\xi^{2}(\xi^{2}-\eta^{2})A^{2}}=T_{00},$
(25)

Then the solutions of the gravitational field equations take in oblate case
the form

$A=2\left(a_{0}\pm\varepsilon\arctan\sqrt{\frac{\xi^{2}-1}{\xi^{2}+1}}\right)^{2},$
$B=\frac{b_{0}}{A},\\\ $ (26)

where $a_{0},b_{0}$ arbitrary constants.

###  2.3 Analysis

To see that all these metrics is asymptotically flat Minkowski it is enough to
show that the metric components behave in an appropriate way at large
$\xi$-coordinate values, e.g., $g_{\mu\nu}=\eta_{\mu\nu}+O(1/\xi)$ as
$\xi\rightarrow\infty$. By inspection of the coefficients, we verify that this
is so ($a_{0}=1,b_{0}=1$).

In fact, in the present approach, it is easy to show that, in the case of
absent the electromagnetic field $\varepsilon=0$, Einstein’s field equations
yield only the flat space

$A=1,$ $B=1,\\\ $ (27) $\phi=1.$

Therefore, we see that it is possible to explain the shape of spheroidal
configurations by electromagnetic or other fields [4]. This seems to be a
remarkable result, although in a way it should be anticipated since the
directional components of ”equation of state” of electromagnetic field are
anisotropic in the oblate and prolate cases. However, in this case there is a
contribution from the electromagnetic field that makes $T_{\mu\nu}$ nonzero.
On the other hand, it seems natural that we have obtained an ”equation of
state” that describes vacuum, since we do not have matter.

##  3 Discussion

In this article we delineated the qualitative features one would expect from
spheroidal object. It is demonstrated that our model can successfully predict
the spheroidal configuration in terms of a self-gravitating spacetime solution
to the Einstein field equations and reproduce the not spherically-symmetric
shape in terms of the non-trivial energy density and anisotropic pressure of
the electromagnetic field which was absent in the context of empty space.

Hence the approach followed in this paper has proved to be a fruitful avenue
for generating new exact solutions for describing the spacetimes of charged
configurations.

We believe that following this hypotheses the shape of galaxy and rotation
curve may be explained by action of electromagnetic or other fields. The
solution presented here could be a first approximation at the galactic space-
time provided the presence of any physical fields. Therefore, it is necessary
to study how these results modify the standard method of interpretation
rotation data. Further investigation into the nature solutions with view to
separating the real rotational effects from the electromagnetic, scalar or
other fields anisotropy might be rewarding.

##  4 Acknowledgements

I am grateful to S.V. Sushkov and R.A. Daishev for the helpful discussions.
The work was supported in part by the Institute of Applied Problems.

## References

  * [1] H .Reissner, ”Uber die Eigengravitation des elektrischen Feldes nach der Einsteinschen Theorie”. Annalen der Physik 50: 106 120.(1916)
  * [2] G. Nordstrom, ”On the Energy of the Gravitational Field in Einstein’s Theory”. Verhandl. Koninkl. Ned. Akad. Wetenschap., Afdel. Natuurk., Amsterdam 26: 1201 1208.(1918)
  * [3] R. C. Tolman Relativity Thermodynamics And Cosmology., Oxford University Press, Oxford, (1934)
  * [4] S.M. Kozyrev, New static spheroidal solution in Jordan-Brands-Dicke theory., arXiv:1012.1097 [gr-qc] 2010.
  * [5] C. Flammer. Spheroidal Wave Functions. Stanford University Press, 1957.