# Integrated Fiber-Mirror Ion Trap for Strong Ion-Cavity Coupling

B. Brandstätter<EMAIL_ADDRESS>Institut für
Experimentalphysik, Universität Innsbruck, 6020 Innsbruck, Austria  A. McClung
Institut für Experimentalphysik, Universität Innsbruck, 6020 Innsbruck,
Austria  Norman Bridge Laboratory of Physics 12-33, California Institute of
Technology, Pasadena, California 91125, USA  K. Schüppert  Institut für
Experimentalphysik, Universität Innsbruck, 6020 Innsbruck, Austria  B.
Casabone  Institut für Experimentalphysik, Universität Innsbruck, 6020
Innsbruck, Austria  K. Friebe  Institut für Experimentalphysik, Universität
Innsbruck, 6020 Innsbruck, Austria  A. Stute  Institut für Experimentalphysik,
Universität Innsbruck, 6020 Innsbruck, Austria  P.O. Schmidt  QUEST Institute
for Experimental Quantum Metrology, Physikalisch-Technische Bundesanstalt,
38116 Braunschweig, Germany  Institut für Quantenoptik, Leibniz Universität
Hannover, 30167 Hannover, Germany  C. Deutsch  Laboratoire Kastler Brossel,
ENS/UPMC-Paris 6/CNRS, 24 rue Lhomond, F-75005 Paris, France  Menlo Systems
GmbH, Am Klopferspitz 19a, 82152 Martinsried, Germany  J. Reichel  Laboratoire
Kastler Brossel, ENS/UPMC-Paris 6/CNRS, 24 rue Lhomond, F-75005 Paris, France
R. Blatt  Institut für Experimentalphysik, Universität Innsbruck, 6020
Innsbruck, Austria  Institut für Quantenoptik und Quanteninformation,
Österreichische Akademie der Wissenschaften, Otto-Hittmair-Platz 1, A-6020
Innsbruck, Austria.  T.E. Northup  Institut für Experimentalphysik,
Universität Innsbruck, 6020 Innsbruck, Austria

###### Abstract

We present and characterize fiber mirrors and a miniaturized ion-trap design
developed to integrate a fiber-based Fabry-Perot cavity (FFPC) with a linear
Paul trap for use in cavity-QED experiments with trapped ions. Our fiber-
mirror fabrication process not only enables the construction of FFPCs with
small mode volumes, but also allows us to minimize the influence of the
dielectric fiber mirrors on the trapped-ion pseudopotential. We discuss the
effect of clipping losses for long FFPCs and the effect of angular and lateral
displacements on the coupling efficiencies between cavity and fiber. Optical
profilometry allows us to determine the radii of curvature and ellipticities
of the fiber mirrors. From finesse measurements we infer a single-atom
cooperativity of up to $12$ for FFPCs longer than $200~{}\mu$m in length;
comparison to cavities constructed with reference substrate mirrors produced
in the same coating run indicates that our FFPCs have similar scattering
losses. We characterize the birefringence of our fiber mirrors, finding that
careful fiber-mirror selection enables us to construct FFPCs with degenerate
polarization modes. As FFPCs are novel devices, we describe procedures
developed for handling, aligning and cleaning them. We discuss experiments to
anneal fiber mirrors and explore the influence of the atmosphere under which
annealing occurs on coating losses, finding that annealing under vacuum
increases the losses for our reference substrate mirrors. X-ray photoelectron
spectroscopy measurements indicate that these losses may be attributable to
oxygen depletion in the mirror coating. Special design considerations enable
us to introduce an FFPC into a trapped ion setup. Our unique linear Paul trap
design provides clearance for such a cavity and is miniaturized to shield
trapped ions from the dielectric fiber mirrors. We numerically calculate the
trap potential in the absence of fibers. In the experiment additional
electrodes can be used to compensate distortions of the potential due to the
fibers. Home-built fiber feedthroughs connect the FFPC to external optics, and
an integrated nanopositioning system affords the possibility of retracting or
realigning the cavity without breaking vacuum.

##  I Introduction

Optical cavities can enhance the interaction between matter and light. In
quantum information experiments, high-finesse cavities act as an interface
between stationary and flying qubits, where flying qubits connect
computational nodes comprised of stationary qubits. Experimentally, atoms and
photons have proven to be promising candidates for the physical implementation
of stationary and flying qubits Zoller  _et al._ (2005), respectively. The
building blocks for an elementary quantum network have recently been
demonstrated using single neutral atoms in high-finesse cavities Ritter  _et
al._ (2012).

Trapped ions are promising candidates for quantum information processing, as
techniques for high-fidelity quantum operations are well established Leibfried
_et al._ (2003); Häffner, Roos, and Blatt (2008). Furthermore, trapped ions
offer a range of advantages for quantum networks: ions are stably trapped in
Paul traps for up to several days and can be well localized via ground-state
cooling. This localization allows the ions to be accurately positioned inside
a cavity mode for optimized coupling Stute  _et al._ (2012a). Several research
groups are currently working on the technological challenge of integrating
ions and cavities, and the coupling of ions to the field of a high-finesse
cavity has already been shown in a range of setups Guthöhrlein  _et al._
(2001); Mundt  _et al._ (2002); Russo  _et al._ (2009); Leibrandt  _et al._
(2009); Herskind  _et al._ (2009); Sterk  _et al._ (2012). Single photons have
been produced with a trapped ion inside a cavity Keller  _et al._ (2004);
Barros  _et al._ (2009), and entanglement between single ions and single
cavity photons has recently been demonstrated Stute  _et al._ (2012b).

For a high-fidelity ion-photon quantum interface, the coherent coupling
strength must be larger than the ion’s rate of spontaneous decay. In order to
maximize this coupling in cavity-QED systems, both the cavity length and the
cavity waist should be minimized. Such microcavities require mirror surfaces
with small radii of curvature. Additionally, the mirror substrates should have
low surface roughness to minimize losses in the mirror coating. These low-loss
coatings at optical wavelengths are dielectric. For an ion in vacuum close to
such dielectric mirrors, image charges and charge build-up on the mirrors
potentially distort the ion’s trapping potential.

The term fiber-based Fabry-Perot cavity (FFPC) describes two opposing optical
fiber tips, each with a mirror coating. At least one of the mirror surfaces is
concave and aligned relative to the other such that a stable standing wave
forms between them. FFPCs provide both a high coupling rate and a small
dielectric cross-section and are thus a promising way to integrate ions with
cavities Hunger  _et al._ (2010). A recent experiment provides the first
demonstration of an ion coupled to a fiber-cavity mode Steiner  _et al._
(2013).

In Sec. II, we discuss the development and optimization of FFPCs for ion
traps. We develop solutions for specific technological challenges in Sec. III,
including the annealing and baking of fiber mirrors. Finally, in Sec. IV, we
present a novel design for a linear Paul trap integrated with an FFPC. This
experimental system should enable us to reach the strong coupling regime
Kimble (2008) with a single calcium ion inside the high-finesse FFPC.

##  II Fiber-based Fabry-Perot cavities

Microfabricated optical cavities have several advantages over conventional
cavities in cavity-QED experiments. Microcavities offer access to smaller mode
volumes than have been demonstrated with macroscopic mirrors Hunger  _et al._
(2010) and thus higher interaction rates of atoms or solid-state emitters with
photons. Additionally, photons that exit the cavity are directly coupled into
a fiber. Furthermore, FFPCs provide flexibility in experimental setups, where
small mirrors are often easier to implement than centimeter-scale mirrors
fabricated on superpolished substrates. In the future it may be possible to
integrate microcavity arrays in scalable systems for quantum information
processing.

This range of advantages has motivated parallel development of microcavities
using various technologies. We identify three criteria for microcavity
development: (i) surfaces with small radii of curvature, (ii) surface
roughness low enough so that it does not contribute appreciably to the mirror
losses, and (iii) surfaces to which a low-loss mirror coating can be applied,
i.e., by ion-beam sputtering. Such surfaces are produced by silicon wet-
etching Trupke  _et al._ (2005), enclosing nitrogen bubbles in borosilicate
and polishing away the bubbles’ upper half Cui  _et al._ (2006), or by
transferring a coating produced on a microlens onto an optical fiber Steinmetz
_et al._ (2006); Muller  _et al._ (2009). All these approaches have been used
to produce cavities with moderate finesses of up to $6\times 10^{3}$. Recent
developments in the fabrication of glass microcavities by shaping surfaces
with controlled re-flow of borosilicate glassRoy and Barrett (2011) yielded
finesses of up to $3.2\times 10^{4}$. However, the best microcavity finesse of
$1.5\times 10^{5}$ has been measured recentlyMuller  _et al._ (2010) with
cavities constructed from coated, concave optical-fiber facets shaped by
CO2-laser ablation Hunger  _et al._ (2010).

In this process, a short pulse of focused CO2-laser light is absorbed in the
cleaved tip of a fiber and creates a depression by locally evaporating the
material. The created surface has a roughness of only $(0.2\pm 0.1)$ nm Hunger
_et al._ (2010). The parameters of the generated surface structures, such as
radius of curvature and diameter of the depression, are set by the pulse
duration, power, and beam waist of the CO2 laser. A highly reflective coating
is then applied to the shaped fiber surfaces by ion-beam sputtering in a high-
vacuum environment. FFPCs produced in this way are being used in atom-chip
setups, in which strong coupling to a BEC has been demonstrated Colombe  _et
al._ (2007). Currently, implementations of FFPCs with solid-state emitters
Muller  _et al._ (2010), ion traps Steiner  _et al._ (2013); Wilson  _et al._
(2011); VanDevender  _et al._ (2010), and neutral atoms are being developed in
several groups worldwide.

In this section, we present the recent development of FFPCs suitable for
integration with ion traps. We show that we can produce cavities of length up
to $350~{}\mu$m and finesse up to $1.1\times 10^{5}$. Furthermore, we
characterize the cavity losses due to surface roughness and the cavity
birefringence, and we describe technologies for cavity alignment and fiber-
mirror cleaning.

###  II.1 Development of FFPCs for ion traps

For neutral atoms, short cavities are favorable as they provide a small mode
volume and do not influence the trapping potential seen by the atom. To
implement FFPCs with ion traps, however, a sufficient separation between the
fibers and the ion is necessary so that the trapping potential is not
distorted by charges on the dielectric mirrors.

We report on the construction of FFPCs suitable for ion traps and on the
effects of increasing the cavity length, such as decreased finesse and
coupling efficiencies between fiber mode and cavity mode. Furthermore, we
present measurements of general interest when working with FFPCs: a direct
measurement of the scattering losses due to surface roughness, and a
characterization of the birefringence of fiber mirrors.

####  II.1.1 Construction of long FFPCs

Figure 1: (a) Composite microscope photo of a fiber mirror, assembled from
multiple photos with different focal length. A highly reflective mirror
coating is fabricated on the entire fiber facet. At the mirror center, a light
reflection from the curved surface can be seen. (b) Photo of an FFPC. The
fibers are copper coated for ultra-high–vacuum compatibility and have a
cladding diameter of $200~{}\mu$m. The copper is etched back about
$400~{}\mu$m from the fiber facets. The glass cladding and the gray titanium
layer around the cladding, which starts about $100~{}\mu$m behind the facet,
can be seen. The two opposing mirrors form a Fabry-Perot cavity $200~{}\mu$m
in length.

Initial development of FFPCs focused on short cavities, such as the
$38.6~{}\mu$m FFPC used to strongly couple a BEC to a cavity field Colombe
_et al._ (2007). In order to construct cavities suited for ion-trap
experiments, we have developed technologies that allow us to increase the
length of FFPCs up to $350~{}\mu$m. These technologies include fabrication of
structures using higher CO2-laser powers and wider beam waists as well as the
use of non-standard $200~{}\mu$m-diameter fibers. Fig. 1 shows a composite
microscope picture of a coated fiber tip and a photo of an FFPC in our
laboratory.

Because of their effects on the trapping potential, fibers should remain far
from the ion. As we increase the separation $L$ between the two fiber mirrors
of a cavity, the spot size of the cavity mode at each mirror increases as a
function of $L$ and the mirror’s radius of curvature $r$. If the mirror
diameter $2\rho_{\mathrm{m}}$ is not much larger than the field diameter
$2w_{\mathrm{m}}$, the cavity mode is clipped at the mirror edge, reducing the
cavity finesse. For a conventional mirror, which has a spherical curvature
over its entire surface, $2\rho_{\mathrm{m}}$ is the physical diameter of the
mirror. However, a fiber mirror can be approximated as spherical only over the
length scale of the depression created in the CO2-ablation process. Thus,
$2\rho_{\mathrm{m}}$ corresponds to this length scale, which is bounded above
by the fiber diameter but is often much smaller due to limitations in the
ablation process.

In Ref. Hunger  _et al._ (2010), fiber mirrors were produced with radius of
curvature between $40~{}\mu$m and $2$ mm and $2\rho_{\mathrm{m}}$ between
$10~{}\mu$m and $45~{}\mu$m. To calculate the clipping losses associated with
a particular cavity geometry, the numerical methods of Fox and Li can be
usedFox and Li (1961); Siegman (1986). For example, if we require round-trip
clipping losses to be less than 10 ppm, then with spherical mirrors of $2$ mm
radius of curvature and $2\rho_{\mathrm{m}}=45~{}\mu$m, we are limited to
$L\leq 70~{}\mu$m. (Choosing radii of curvature in the near-confocal limit
would improve this bound but also reduce the atom-cavity coupling; the near-
planar assumption is a reasonable compromise and also robust to small
variations in cavity length.) This bound is incompatible with the target
lengths $L\gtrsim 150~{}\mu$m planned for our experimental system. (See Sec.
IV.2.1.)

One solution to minimize clipping losses is to produce larger mirror
structures on the fiber tips, that is, to modify the laser ablation process.
Specifically, we increase the beam waist at the fiber tip and use higher
CO2-laser power. Optimizing the laser ablation parameters is challenging due
to two competing processes: while the incident laser light evaporates fiber
material, mapping the Gaussian beam profile onto a concave depression, it also
induces sufficient heat to locally melt the fiber tip, producing a convex
structure due to surface tension. To avoid melting, heat needs to be conducted
away efficiently by either cooling the fiber or creating a heat sink. Instead
of standard $125~{}\mu$m-diameter fibers, we choose to use fibers of
$200~{}\mu$m diameter, where the additional glass functions as a heat sink.
However, the use of a non-standard fiber size means that fiber connectors and
tools for cleaving and splicing are more difficult to obtain.

Structures on the fiber tips are analyzed with an optical profilometerFog ,
and structure diameters $2\rho_{\mathrm{m}}$ and mirror depth $z$ are
extracted by fitting a polynomial to the profilometer data and finding its
turning points. The distance between the turning points is defined as
$2\rho_{\mathrm{m}}$. The radius of curvature $r$ of each fiber is
approximated via the fit of a circle to the surface. In Fig. 2, we show the
profilometer data as well as the fit for one fiber. The CO2-laser ablation
structures are not rotationally symmetric but have an elliptical shape due to
astigmatism of the CO2-laser beam Hunger  _et al._ (2010). We determine the
degree of ellipticity by identifying major and minor axes and comparing the
two radii of curvature. Note that $r$, $2\rho_{\mathrm{m}}$, and $z$ are mean
values of the fits to both axes.

Figure 2: Fiber surface measured by optical profilometry; compare also Fig. 3
in Ref. Hunger  _et al._ (2010). The depression on the fiber surface is
elliptical. Along the major and minor axes ($i=1,2$) of the structure, we fit
a polynomial and determine structure diameter $2\rho_{\mathrm{m}_{i}}$ and
structure depth $z_{i}$. $2\rho_{\mathrm{m}_{i}}$ and $z_{i}$ are defined at
the turning points of the polynomial. From the fit of a circle (note the
different axis scales of ordinate and abscissa), we extract the radius of
curvature $r_{i}$. Furthermore, we determine the ellipticity of the fiber and
the mean values of the fitted parameters, $r$, $2\rho_{\mathrm{m}}$, and $z$.
For the surface in this figure, these values are:
$2\rho_{\mathrm{m}_{i}}=(82,84)~{}\mu$m and $2\rho_{\mathrm{m}}=83~{}\mu$m,
$z_{i}=(2.6,2.8)~{}\mu$m and $z=2.7~{}\mu$m, and $r_{i}=(343,332)~{}\mu$m and
$r=338~{}\mu$m.

Using the technique described here, CO2-laser waists between $18~{}\mu$m and
$80~{}\mu$m and powers between $0.3$ W and $1.1$ W were used in Ref. Hunger
_et al._ (2010). In contrast, we modify the parameters to a beam waist of
$92~{}\mu$m and a laser power of $4.6$ W. (The pulse duration is $\sim 30$
ms.) As a result, we produce fiber-mirror structures with radius of curvature
between $180~{}\mu$m and $420~{}\mu$m and structure diameters of up to
$80~{}\mu$m.

In a single coating process, $76$ fibers produced with the CO2-laser
parameters specified above were coated with a highly reflective coating
centered around $860$ nm (ATFilms). On an alignment stage for test setups, we
construct and characterize the FFPCs. The mirrors were characterized at $844$
nm because a laser was available whose wavelength could be tuned over several
nanometers, facilitating the measurement of short cavity lengths. In Fig. 3,
we show the cavity finesse as a function of distance between the mirrors for
one FFPC. The cavity is set up with a single-mode fiber mirror as cavity input
and a multimode-fiber mirror as cavity output. The single-mode fiber mirror
has a diameter of $2\rho_{\mathrm{m}}=67~{}\mu$m and a radius of curvature of
$r=209~{}\mu$m; for the multimode fiber mirror, $2\rho_{\mathrm{m}}=80~{}\mu$m
and $r=355~{}\mu$m.

Figure 3: Fiber-cavity finesse at a wavelength of $844$ nm as a function of
the cavity length; compare also Fig. $9$ in Ref. Hunger  _et al._ (2010). The
points are measurement values from the fiber mirrors specified in the text;
the finesse decreases for longer cavities. Error bars represent one standard
deviation. The solid line shows the calculated finesse due to clipping losses
from the mirrors, where the mirrors are modeled as spheres with diameter
2$\rho_{\mathrm{m}}$ and radii of curvature given in the text. The grey dashed
line gives the cavity’s stability edge for these radii of curvature. The lack
of agreement between data and calculation demonstrates that additional loss
sources play a significant role.

The measured finesse declines gradually with mirror separation, from an
initial value of 71,600 at $85~{}\mu$m to roughly half of that at
$231~{}\mu$m. If the decrease in finesse were due to clipping losses, we would
expect a constant finesse for almost all cavity lengths, with a steep drop a
few microns before the stability boundary at $r=209~{}\mu$m. The clipping
losses from numerical calculations, which assume spherical mirrors, are also
plotted in Fig. 3 and do not agree with the data.

First, the most likely source of additional losses is a non-uniform thickness
of the coating layers: for steep mirror surfaces, ion-beam-sputtered layers
may be too thin Roy and Barrett (2011), shifting the coating towards
wavelengths shorter than the target value. As the mode-field diameter at each
mirror increases with increasing $L$, the mode may enter a region where the
coating is no longer suited for the measurement wavelength and transmission
losses are higher. This effect was observed in Ref. Roy and Barrett (2011), in
which lower finesses for cavities with smaller radii of curvature were
measured.

Second, for cavities at or beyond the stability boundary, that is, the final
two data points in Fig. 3, it is surprising that a nonzero finesse is
observed. For the radii of curvature determined from optical profilometry,
this is not a stable cavity configuration. A possible explanation is that the
assumption of a spherical mirror, while useful, is only an approximation.
Since the mode size at the mirror increases near the stability boundary, the
mode extends to regions where the curvature of the mirror deviates
significantly from a spherical fit. A more realistic model would assume a
Gaussian curvature for the mirrors, but it is difficult to use such a model to
calculate the expected losses in this regime accurately; the numerical
integration does not reduce to a fast Hankel transform as it does for
spherical mirrorsSiegman (1986).

We note that Fig. $9$ of Ref. Hunger  _et al._ (2010) presents finesse
measurements for short cavities that are compatible with the clipping-loss
model. However, for each cavity, only one data point with reduced finesse is
measured, and the theory curve corresponds to an average set of parameters
rather than the specific parameters of each set of mirrors. Thus, there is not
enough information to determine whether our measurements are consistent with
those of Ref. Hunger  _et al._ (2010). It would be interesting to explore the
hypothesis of coating layer uniformity by repeating these measurements over a
range of mirror curvatures.

####  II.1.2 Coupling efficiencies between cavity and fiber

In this section, we show a measurement of the transmitted intensity through an
FFPC as a function of the cavity length. The transmission is the product of
four terms: the fiber in-coupling efficiency, mode matching from the fiber
into the cavity, impedance matching of the cavity, and collection efficiency
of the output fiber. For free-space cavities, only the second and third terms
are relevant, and the input mode is matched to the mode of the cavity by beam
shaping and alignment. With FFPCs, in contrast, the cavity mirrors are built
into the in-coupling and out-coupling fibers, fixing this mode-matching
coefficient to a value determined by the cavity and the fiber parameters.
Because of this key difference, it is interesting to consider mode-matching
from a single-mode fiber to an FFPC.

The mode overlap $\epsilon$ is defined as the overlap between the TEM00 mode
of the cavity and the spatial mode of the single-mode fiber. This overlap
depends on the radius of curvature of the mirror, on the core diameter of the
fiber, and on the cavity length. In addition, either an offset of the mirror
center from the fiber core or an angle between mirror and fiber core causes a
mode mismatch which cannot be corrected. Ref. Hunger  _et al._ (2010) shows
that mode matching can be as high as 85% for a short FFPC but calculates that
it declines for longer cavities. Assuming that the mirror surface is
orthogonal to the fiber core ($\theta=0$) and that there is no offset between
the core and the mirror center ($d=0$), the coupling efficiency $\epsilon_{a}$
is given by Joyce and DeLoach (1984)

$\epsilon_{a}=\frac{4}{(\frac{w_{\mathrm{f}}}{w_{\mathrm{c}}}+\frac{w_{\mathrm{c}}}{w_{\mathrm{f}}})^{2}+\frac{s^{2}}{z_{\mathrm{R_{f}}}z_{\mathrm{R_{c}}}}}$
(1)

with waists $w_{\mathrm{f}}$, $w_{\mathrm{c}}$ and Rayleigh lengths
$z_{\mathrm{R_{f}}}$, $z_{\mathrm{R_{c}}}$ of the beam exiting the fiber and
of the cavity mode, respectively. The distance from the waist of the mode
exiting the fiber to the cavity waist is denoted by $s$.

A tilt of the fiber with respect to the mirror by an angle $\theta$ reduces
the overlap by a factor of $e^{-(\theta/\theta_{e})^{2}}$ for small values of
$\theta$, with the angular tolerance $\theta_{e}$. Similarly, a displacement
of the fiber core from the cavity axis reduces $\epsilon$ by a factor of
$e^{-(d/d_{e})^{2}}$, where analytic expressions for angular tolerance
$\theta_{e}$ and displacement tolerance $d_{e}$ can be found in Ref. Joyce and
DeLoach (1984). The total mode overlap is then given by
$\epsilon=\epsilon_{a}e^{-(d/d_{e})^{2}}e^{-(\theta/\theta_{e})^{2}}$.

Fig. 4(a) and Fig. 4(b) show the calculated mode overlap for a single-mode
fiber of $6~{}\mu$m core diameter and an FFPC with $r_{1}=209~{}\mu$m and
$r_{2}=355~{}\mu$m. Cavity lengths up to $209~{}\mu$m are plotted, considering
non-zero values for $\theta$ and $d$. For $\theta=0^{\circ}$ and $d=0~{}\mu$m,
the mode matching between fiber and cavity increases steeply for short cavity
lengths, has a maximum of $0.84$ at length $54~{}\mu$m, and decreases to half
that value by $200~{}\mu$m. Thus, we see that with proper alignment, it is
possible to build long cavities with reasonable mode matching. As $\theta$ and
$d$ increase, the maximum value for $\epsilon$ drops, but $\epsilon$ becomes
relatively insensitive to cavity length. The range of values for $\theta$ and
$d$ plotted in Fig. 4 reflects estimates of realistic errors in the fiber-
mirror fabrication procedure. Over this range, and for all cavity lengths
shown, these errors cause $\epsilon$ to decrease by almost an order of
magnitude.

The theory predicts a steep decrease in mode matching as the cavity length
approaches the smaller radius of curvature of the two mirrors, $209~{}\mu$m.
Therefore, when mode matching is important, cavity lengths close to the
stability boundary should be avoided. As we have seen in Sec. II.1.1, however,
the stability boundary of a fiber cavity does not correspond to a calculation
based on spherical mirror parameters.

Figure 4:  (a) Coupling efficiency from a single-mode fiber to an FFPC
calculated from Eq. 1, taking into account non-zero values of $\theta$ without
displacement ($d=0~{}\mu$m). (b) Coupling efficiency for small displacements
$d$ ($\theta=0^{\circ}$). (c) Measurement of the transmission through an FFPC
as a function of the cavity length, referenced to the first data point. The
error bars represent one standard deviation.

In Fig. 4(c), we show a measurement of the transmission of the FFPC discussed
in Sec. II.1.1. Note that due to the large core diameter and acceptance angle
of multimode fibers, collection efficiency is unity for our cavity parameters
Hunger  _et al._ (2010). We measure the transmission through the cavity and
the fibers normalized to the transmission of the first point as a function of
the cavity length. The transmission first decreases as a function of cavity
length and then remains constant at around $20\%$ of the initial value for
cavities longer than $150~{}\mu$m. Only the relative transmission is measured
because an absolute transmission is difficult to calibrate and does not
provide additional information on how mode overlap scales with length Hunger
_et al._ (2010); Hood, Kimble, and Ye (2001). In order to extract the mode-
matching efficiency from this transmission measurement and compare it to Figs.
4(a) and (b), we would have to determine independently whether the decrease in
transmission is due to increasing impedance mismatch or increasing mode
mismatch.

For experiments in which optimal transmission through long fiber cavities is
important, the coupling can be improved by minimizing cavity scatter and
absorption losses, therefore optimizing the impedance matching. The mode
matching can be maximized by minimizing $\theta$ and $d$. An interesting
possibility for improving the mode matching would be to tailor the single-mode
fiber mode, e.g., by expanding the fiber core at the tip.

####  II.1.3 Surface-loss measurement

We measure the scattering losses in the mirror coatings due to surface
roughness of the CO2-laser shaped fibers, and we find no such losses within a
$1$ ppm measurement error. This measurement is the first direct comparison of
the losses of fiber-mirror coatings (including losses induced by surface
roughness) with losses of identical coatings fabricated on fused silica
substrates. For this purpose, fibers and substrates were coated together in
the same fabrication run.

The losses of highly reflective mirror coatings depend critically on the
surface quality of the mirror substrate: surface roughness of the substrate
material results in scattering losses of the mirror. In order to reduce the
scattering losses to $\sim 1$ ppm at near-infrared wavelengths, the surface
roughness needs to be less than $1$ Å rms Hunger  _et al._ (2010). Mirror
substrates of surface roughness less than $1$ Å rms are referred to as
superpolished. With superpolished substrates, we assume that all scattering
and absorption losses come from point defects in the coating.

As CO2-laser ablation is a novel technique for producing curved mirror
substrates, it is of great interest to determine the quality of the shaped
surfaces and the induced scattering losses. Fiber surface roughness $\sigma$
has previously been measured with an atomic force microscope and linked to the
scattering losses $S$ via $S\approx(4\pi\sigma/\lambda)^{2}$, where
$\sigma=(0.2\pm 0.1)$ nm, corresponding to $S=(10\pm 10)$ ppm for near-
infrared light at $\lambda=780$ nm Hunger  _et al._ (2010).

In contrast, we compare fiber mirrors with reference mirrors produced on
superpolished substrates. We measure reference-mirror cavities to have a
finesse of $(1.10\pm 0.04)\times 10^{5}$, in comparison to a finesse of
$(1.14\pm 0.05)\times 10^{5}$ for the fiber-mirror cavities. For identical
mirrors, the total losses per mirror
$\mathcal{L}_{\mathrm{tot}}=\mathcal{T+L}$ , the sum of transmission
$\mathcal{T}$ and losses $\mathcal{L}$, are calculated from the finesse via
$\mathcal{L}_{\mathrm{tot}}=\pi/\mathcal{F}$. The reference substrates and the
fiber mirrors thus have identical total losses of
$\mathcal{L}_{\mathrm{tot}}=(28\pm 1)$ ppm. In an additional measurement (Sec.
III.1), we determine the total losses of the same reference substrates after
annealing the mirrors. This measurement of losses of $(17\pm 1)$ ppm agrees
with $15$ ppm target transmission of the coating and $2$ ppm scatter and
absorption losses from the coating.

We conclude that fiber surface roughness does not cause any additional
scattering losses. This result suggests that it may be possible to construct
FFPCs with finesses as high as those achieved with state-of-the-art
superpolished mirrors Rempe  _et al._ (1992).

####  II.1.4 Birefringence of fiber mirrors

We investigate the birefringent splitting of orthogonal polarization modes in
FFPCs. We observe that FFPCs exhibit significant birefringence, whereas
cavities built with mirrors produced on superpolished substrates fabricated in
the same coating process do not exhibit measurable birefringence. Furthermore,
we observe that the birefringent splitting of the FFPCs varies as a function
of the cavity alignment.

For experiments in which quantum information is encoded in photon
polarization, it is advantageous for the modes of orthogonal linear
polarization to be degenerate in the cavity Ritter  _et al._ (2012); Stute
_et al._ (2012b). Therefore, it is of interest to control the birefringence of
the cavity mirrors. Typically, the birefringence in a cavity of mirrors
fabricated on superpolished substrates is induced via stress inside the
mirrors due to the mounting Lynn (2003). In contrast, fiber mirrors are
mounted a few millimeters from the mirror surface, a length scale much greater
than the surface diameter. Therefore, we assume that the stress is intrinsic
to the coating and is created in the fabrication process of the coating.

To measure the birefringent splitting, we measure the transmission curve of an
FFPC while driving the cavity and sweeping its frequency. To calibrate the
abscissa, we modulate the driving laser to produce frequency sidebands with a
known splitting. We repeatedly rotate and realign the fibers and measure the
relative detuning between the polarization modes. The observed splittings
range from smaller than the FWHM of the cavity resonance ($\approx 30$ MHz) up
to a few gigahertz. This result is comparable to the birefringent splitting of
$200$ MHz measured in Ref. Hunger  _et al._ (2010). Furthermore, when aligning
the cavity to support different TEM modes, we observe that these TEM modes
have different birefringent splittings.

To quantify this last observation, we have modified the experimental setup
because the TEM mode of the cavity is not preserved inside the output
multimode fiber. The multimode fiber is replaced by a mirror fabricated on a
superpolished substrate that was coated in the same coating run as the fiber.
The TEM mode is then imaged with a CCD camera, while a photodiode measures the
cavity transmission curve. We measure a splitting of $37$ MHz for TEM11 and
$122$ MHz for TEM01 (Fig. 5), demonstrating that the birefringent splitting is
dependent on the TEM mode and thus on the specific mirror region that the
cavity mode samples.

Figure 5:  Birefringent splitting of the orthogonal polarization modes inside
the cavity described in the text. Birefringent splitting of (a) $37$ MHz for
TEM11 and of (b) $122$ MHz for TEM01. The $100$ MHz sidebands for the
frequency calibration are indicated with dashed lines. The insets show CCD-
camera images of the cavity modes.

We conclude that birefringence is not homogeneous over the mirror coating on
the fiber. Thus, simply rotating two birefringent fiber mirrors with respect
to one another will not necessarily eliminate cavity birefringence. However,
by careful selection of fibers and proper alignment, we have built cavities
which satisfy a target birefringent splitting, in our case, degenerate
polarization modes.

###  II.2 Ion-cavity system

We now focus on the relevant rates of our experimental ion-cavity system
considering ${}^{40}\mathrm{Ca}^{+}$ and the FFPC parameters from the
measurements shown above. Until very recently, experimental ion-cavity systems
have used cavities constructed with superpolished-substrate mirrors
Guthöhrlein  _et al._ (2001); Mundt  _et al._ (2002); Russo  _et al._ (2009);
Leibrandt  _et al._ (2009); Herskind  _et al._ (2009); in addition, an ion has
now been coupled to an FFPC Steiner  _et al._ (2013). All of these systems
operate in a regime in which the coherent coupling rate $g$ between a single
ion and a photon is smaller than the rate of at least one incoherent process,
such as scattering from the ion or loss from the cavity. To increase the
coherent coupling strength $g$ between ion and photon, the cavity mode waist
$w_{0}$ and the cavity length $L$ should be minimized, as for a dipole
coupling

$g=\frac{\lambda}{\pi w_{0}}\sqrt{\frac{3c\gamma_{c}}{L}},$ (2)

where $\gamma_{c}$ is the spontaneous emission rate between the states coupled
via the cavity, and $w_{0}$ is calculated via the length $L$ and the radius of
curvature $r$ of the mirrors. For a cavity in which both mirrors have the same
radius of curvature,

$w_{0}^{2}=\frac{\lambda}{2\pi}\sqrt{L(2r-L)}.$ (3)

Laser machining of fiber mirrors produces radii of curvature that are two
orders of magnitude smaller than radii of curvature produced via
superpolishing techniques Hunger  _et al._ (2010). Thus, short cavity lengths
and small cavity waists are inherent to FFPCs, allowing for high coupling
rates. Our setup is comprised of a linear Paul trap, described in Sec. IV.2.1,
and an FFPC. In the setup, the cavity axis is perpendicular to the trap axis.
Ions trapped along this axis would have a separation of around $5~{}\mu$m.
Thus, the number of ions that can be coupled to the same antinode of the
cavity depends on the size of the cavity waist. For a mode waist $w_{0}$ of
$5~{}\mu$m, it is possible to have two ions displaced symmetrically from the
maximum of the cavity mode. Both ions are then coupled to the cavity with
$88\%$ of the maximum strength.

Our cavity mirrors are chosen for optimum finesse at either the
$4P_{1/2}-3D_{3/2}$ or the $4P_{3/2}-3D_{5/2}$ transition of
${}^{40}\mathrm{Ca}^{+}$, which have wavelengths of $866$ nm and $854$ nm,
respectively. The $3D_{3/2}$ and $3D_{5/2}$ states are metastable states, and
the $3D_{5/2}$ state is used for quantum-information processing Häffner, Roos,
and Blatt (2008). Here, we calculate the system parameters, that is, the
coherent coupling rate $g$ and cavity decay rate $\kappa$, for the FFPC
characterized in Sec. II.1.1. The finesse and the cavity length have been
measured, and $w_{0}$ and $g$ are calculated from the radius of curvature
measured interferometrically and are therefore approximate values. $\kappa$ is
calculated via the finesse $\mathcal{F}$ by

$\kappa=\frac{c\pi}{2\mathcal{F}L},$ (4)

and the single-atom cooperativity $C$ is given by

$C=\frac{g^{2}}{2\kappa\gamma}.$ (5)

Table 1:  Cavity-QED system parameters for various setups of neutral-atom and ion experiments using FFPCs and cavities consisting of superpolished mirror substrates. These parameters are compared to our ${}^{40}\mathrm{Ca}^{+}$–FFPC system with the FFPC characterized in Sec. II.1.1; given the measured values of cavity length $L$ and finesse $\mathcal{F}$, we calculate the mode waist $w_{0}$, cavity decay rate $\kappa$, coherent coupling rate $g$ and single-atom cooperativity $C$. experiment |  $L$ ($\mu$m) |  $w_{0}$ ($\mu$m) | $\mathcal{F}$ |  $\kappa$
(MHz$/2\pi$) |  $g$ (MHz$/2\pi$) |  $\gamma$ (MHz$/2\pi$) | $C$
---|---|---|---|---|---|---|---
neutral Cs Hood  _et al._ (2000) | $10.9$ | $14.0$ | $480,000$ | $14.1$ | $110$ | $2.6$ | $164$
neutral Rb \- FFPC Colombe  _et al._ (2007) | $38.6$ | $3.9$ | $37,000$ | $53$ | $215$ | $3$ | $145$
$\text{Ca}^{+}$ ions Keller  _et al._ (2004) | $8000.0$ | $37.0$ | $78,000$ | $1.2$ | $0.92$ | $11.2$ | $0.03$
$\text{Ca}^{+}$ ions Stute  _et al._ (2012a) | $19960.0$ | $13.2$ | $77,000$ | $0.05$ | $1.43$ | $11.2$ | $1.8$
$\text{Yb}^{+}$ ions - FFPC Steiner  _et al._ (2013) | $230.0$ | $6.6$ | $1,000$ | $320$ | $6$ | $2$ | $0.03$
current setup | $85.0$ | $5.1$ | $72,000$ | $12$ | $41$ | $11.2$ | $6.1$
$\text{Ca}^{+}$ ions - FFPC | $131.0$ | $5.4$ | $64,000$ | $9$ | $31$ | $11.2$ | $4.8$
| $206.0$ | $3.2$ | $45,000$ | $8$ | $41$ | $11.2$ | $9.3$

The parameters of our ion-cavity system are compared in Tab. 1 to a selection
of single-atom cavity-QED experiments. With neutral Cs and Rb atoms,
cooperativities over $10^{2}$ have been demonstrated by using short, high-
finesse cavities Hood  _et al._ (2000) and by using an FFPC to obtain a small
mode waist Colombe  _et al._ (2007). In contrast, ion-trap experiments have
been limited to $C\lesssim 1$, primarily because relatively long cavities have
been necessary to avoid distortion of the trapping potential Keller  _et al._
(2004); Stute  _et al._ (2012b). Here, we see that fiber mirrors offer a
promising route towards much shorter cavities and smaller mode waists, as in
the first demonstration of an ion-trap FFPC Steiner  _et al._ (2013).

In our system, the atomic decay rates $\gamma$ of the $4P_{3/2}$ state and the
$4P_{1/2}$ state are $2\pi\times 11.4$ MHz and $2\pi\times 11.2$ MHz,
respectively, including decay channels to both $S$ and $D$ manifolds. From
Tab. 1, we see that the coherent coupling rate $g$ is larger than $\kappa$ and
$\gamma$ for the range of possible cavity lengths of this FFPC. Longer
cavities exhibit smaller $\kappa$, although the finesse decreases with
increasing cavity length (Fig. 3). Additionally, the sharp decrease in cavity
waist gives increasingly larger $g$ as the near-concentric limit is
approached. These trends contribute to higher cooperativities as the cavity
length is increased.

###  II.3 Practical techniques for FFPCs

FFPCs differ from conventional cavities in several ways. The mirror is
fabricated on the fiber, so that different techniques are required to align a
cavity. Furthermore, due to the small size of both fibers and mirrors,
technologies for mounting and cleaning as well as for annealing of the fiber
mirrors are necessary. In this section, we describe new methods developed in
our work with FFPCs. Experiments annealing fiber mirrors are presented
separately in Sec. III together with measurements of annealing mirrors
fabricated on superpolished substrates.

####  II.3.1 Fiber preparation

We work with copper-coated single-mode and multimode fibers of non-standard
$200~{}\mu$m diameterIVG . Here, we summarize techniques for fiber
preparation.

Etching copper coating from fibers: Fibers need to be stripped properly before
they are cleaved or connectorized. We etch away the copper with a $25\%$
nitric acid (HNO3) solution or a $20\%$ iron(III) chloride (FeCl3) solution at
$50^{\circ}$C until the coating is no longer visible. This process takes only
a few minutes. The first method is faster but also requires more precaution in
handling the chemicals.

Removing titanium from fibers: After the copper is etched off, the glass
fibers are still coated with a thin layer of titanium, which is used as an
adhesive between glass and copper. This layer is not insulating, and if the
fibers have contact with trap electrodes, they cause a short circuit.
Furthermore, the titanium layer shifts the ion’s trapping potential when it is
brought close to the trap center. We find that the titanium can be scratched
away gently with diamond paste, which is then rinsed off thoroughly with
solvents.

Mounting fibers: To protect the fibers from dirt or damage, they need to be
mounted properly during every stage of the experimental process, e.g., in the
coating chamber, during testing or in the experimental setup. In the coating
device, the fiber holders need to be vacuum compatible. We clamp each fiber
with a screw inside an aluminum cylinder. The fiber tip protrudes $0.5$ mm
from the cylinder for the coating. We encase the fiber in a Teflon sleeve so
that the screw does not damage it. The same holders are used for fiber storage
before and after coating. However, to set up a test cavity, the fibers should
not be clamped rigidly with screws as it impairs their optical transmission.
Instead, we fix the fibers with a magnet in stainless-steel v-grooves. In the
experimental setup, the fibers are then glued with UHV epoxyEPO  onto Pyrex
v-grooves.

Connectorizing fibers: In the testing process, we often switch between fibers
and thus prefer slide-on slide-off bare-fiber adapters to connectors that need
to be glued. The bare-fiber adapters are custom-made for the $200~{}\mu$m
cladding diameterBul .

Splicing non-standard $200~{}\mu$m diameter fibers to standard $125~{}\mu$m
diameter fibers: To be able to use standard fiber tools, it is useful to work
with $125~{}\mu$m diameter fibers. We find that it is possible to splice the
$200~{}\mu$m diameter fibers to standard $125~{}\mu$m diameter fibers of the
same core diameter with a commercial fiber splicerVyt  with negligible losses.
In the splicing process, we account for the larger diameter of the
$200~{}\mu$m fiber by shifting the splice filament such that it preferentially
heats the larger fiber.

####  II.3.2 Alignment of long cavities

To obtain a cavity-transmission signal, it is sufficient to align the fibers
by eye to form a very short cavity of about $30~{}\mu$m in length and sweep
the length across a free spectral range. However, as the CO2-laser ablation
does not produce perfect surfaces — generally, the center of the depression is
offset from the center of the fiber facet, and the mirror surface is not
exactly orthogonal to the fiber core — the cavity must be aligned further via
optimization of the cavity transmission signal. For aligning FFPCs, one fiber
is fixed while the second fiber is mounted on a six-axis nano-positioning
systemTho . To build longer FFPCs, the distance between the mirrors is then
increased stepwise while optimizing the alignment by maximizing the
transmission signal. Using this technique, we are able to build cavities of
lengths up to $350~{}\mu$m.

We observe that cavities of up to $\sim 100~{}\mu$m in length are robust to
changes in mirror position or angle. As the fiber mirrors are separated
further, however, very small changes misalign the cavity even though the
mirrors’ radii of curvature suggest that the cavity is still far away from the
edge of the stability region. In this case, the size of the mirror is the
limiting factor for misalignment. As a consequence, for building long
cavities, care must be taken that the fibers are mounted very stably.

####  II.3.3 Cleaning fiber mirrors

To clean ultra-low loss mirrors fabricated on a superpolished substrate, the
mirror is rotated on a spin cleaner and the surface is swabbed with water,
acetone and isopropyl alcohol during rotation Northup (2008). Fiber mirrors,
however, are too delicate to swab. The high-temperature gradient of the
CO2-ablation process makes the fiber tip brittle, and we find that any stress
or pressure usually breaks the tip.

One obvious strategy has been to keep the fiber mirrors as clean as possible
and shield them from contamination. Unfortunately, even in a clean
environment, the fiber mirrors sometimes decrease in finesse as they
accumulate dust. To address this problem, we have developed a cleaning
procedure for ultra-low loss fiber mirrors. We use spectrophotometric grade
solvents, heated to $50^{\circ}$C, to clean the mirror fiber tips in an
ultrasonic bath for two minutes, first in acetone and then in methanol.
Immediately after taking the fiber out of the methanol, we use clean helium to
dry the mirror surface for at least half a minute.

There are a few cases in which this method does not recover the initial
finesse. However, we typically see full recovery of the finesse by cleaning
fiber mirrors with this procedure.

##  III Annealing mirrors

Ultralow-loss mirrors at optical wavelengths are routinely employed in quantum
optics experiments. Using ion-beam sputtering, mirrors can be fabricated with
total losses (transmission, absorption, scatter) as low as $1.6$ ppm in the
near infrared Rempe  _et al._ (1992). In order to achieve such low losses in
dielectric mirror coatings, it is a standard procedure to anneal the coatings
after fabrication. Annealing leads to homogenization of the oxide layers and
improves the stoichiometry of non-perfect oxides Atanassova, Dimitrova, and
Koprinarova (1995), reducing coating losses typically by $10$ ppm. This
procedure is thus a key step in the process of manufacturing ultra-low-loss
mirrors.

The recent development of fiber cavities raises the question of whether
annealing fiber mirrors is possible. Since the surface roughness of
CO2-laser–ablated fiber tips is comparable to that of superpolished mirror
substrates (Sec. II.1.3), the finesse of fiber cavities can in principle reach
the record values achieved with mirrors fabricated on substrates. To reach
this high finesse, however, annealing would be essential.

Our initial efforts to anneal fiber mirrors have been unsuccessful. We
attribute some of the difficulties to possible chemical reactions of the
fiber-coating material with oxygen in the air. For this reason, we have
investigated annealing under vacuum. Furthermore, knowledge about the effects
of baking mirrors under vacuum is essential for all experiments in which low-
loss mirrors are placed under ultra-high vacuum (UHV), which requires a vacuum
bake. The typical temperatures for a vacuum bake are lower than annealing
temperatures, but the same chemical processes are at work. In various
experiments, degradation of cavity mirrors under vacuum has been observed
(Ref. Cetina  _et al._ (2013); Sterk  _et al._ (2012) and references therein),
but evidence of changes in mirrors under vacuum has been mostly anecdotal.
Because the cavities are part of complex experimental systems, in which
repeated bake-outs are impractical, a careful study of these effects has not
yet been undertaken.

In order to study annealing and baking under vacuum systematically, we use
reference mirrors which have been produced in the same coating run as the
fiber mirrors (Sec. II.1). These reference mirrors are fabricated on fused
silica substrates of half-inch and $7.75$ mm diameters. They are coated with a
highly reflective coating comprised of $37$ alternating layers of Ta2O5 and
SiO2, where the inner- and outermost layers are Ta2O5. The layers are
deposited using ion-beam sputtering, and each layer has a $\lambda/4$
thickness for peak reflectivity at $\lambda=860$ nm. Using these mirrors, we
systematically measure effects from annealing under air and vacuum in a clean
and controlled system.

In this section, annealing refers to a $90$ minute ramp from room temperature
to $450^{\circ}$C, a $90$ minute bake at $450^{\circ}$C, and a $90$ minute
ramp down to room temperature. Vacuum annealing consists of placing the
mirrors in a clean stainless-steel chamber, which is then pumped to pressures
lower than $10^{-5}$ mbar by a turbo pump, after which the temperature ramp is
started. For annealing under air, the mirrors are placed inside clean glass
Petri dishes. Care is taken that the mirrors are properly cleaned before any
finesse measurement Northup (2008).

###  III.1 Repeated annealing under vacuum and under air

Figure 6:  Finesse after annealing at $450^{\circ}$C under alternating air and
vacuum environments. Annealing under vacuum shows repeatable losses in cavity
finesse, and annealing under air repeatable gains up to a maximum finesse of
$(1.80\pm 0.03)\times 10^{5}$ for mirror pair $1$ (points). Mirror pair $2$
(open circles) establishes the maximum finesse after repeated annealing under
air. The error bars represent one standard deviation of the measurement
uncertainty.

With a first pair of coated mirrors, we constructed a cavity with a finesse of
$(1.05\pm 0.09)\times 10^{5}$ prior to annealing. After an initial test in
which the mirror pair was annealed under vacuum, the finesse had degraded to
$(3.9\pm 0.6)\times 10^{4}$. To investigate this unexpected result, we
conducted a series of measurements, in which we alternated between annealing
under vacuum and air. Annealing these mirrors under air resulted in a recovery
of the finesse, that is, a decrease of the losses that had been induced by
vacuum annealing. In fact, the new finesse of $(1.4\pm 0.1)\times 10^{5}$ was
higher than the initial value, indicating that annealing had removed intrinsic
coating losses as expected. Two subsequent measurements showed that the losses
when annealing under vacuum and gains when annealing under air are repeatable,
and the maximum finesse for this pair of mirrors is $(1.80\pm 0.03)\times
10^{5}$; these data are summarized in Fig. 6.

With a second pair of reference mirrors, we reproduce the initial finesse of
the first pair: $(1.06\pm 0.06)\times 10^{5}$. Annealing directly under air as
the only step yields a finesse of $(1.75\pm 0.06)\times 10^{5}$, implying that
the maximum finesse of this coating is independent of previous annealing
cycles. Repeated annealing under air established the maximum finesse for this
pair of mirrors. This finesse corresponds to total losses
$\mathcal{L}_{\mathrm{tot}}$ of $17$ ppm. We attribute 2ppm Rempe  _et al._
(1992) to scattering and absorption losses and $15$ ppm to transmission,
consistent with the target transmission of the coating run.

Our finding that the change in finesse depends on the annealing environment
suggests that annealing affects the chemical composition of the mirror
coating. We hypothesize that during a vacuum bake, oxygen escapes from the
outermost Ta2O5 layer, which leads to defects in the coating. Subsequent
annealing under air gives the surface the possibility to regain the oxygen,
thus removing these defects of the chemical structure. In order to test this
theory of oxygen depletion, we conducted a series of X-ray photoelectron
spectroscopy measurements.

###  III.2 X-ray photoelectron spectroscopy measurements

X-ray photoelectron spectroscopy (XPS) is used to quantitatively determine the
chemical composition of the Ta2O5 layer on the surface of the mirror coatings.
We acquire the XPS data with a Thermo Multilab $2000$ utilizing monochromatic
Al K$\alpha$ radiation at $1486.6$ eV. The atomic composition of the samples
is obtained from XPS survey scans taken with an overall resolution of $2.0$
eV. The oxygen and tantalum content are determined from the O ($1$s) and the
Ta ($4$d) lines, respectively, measured with a higher resolution of $0.1$ eV.
Measured intensity ratios are converted into atomic percentages using the
theoretical photoionization cross-sections of Scofield Scofield (1976), also
taking into account the energy-dependent transmission of the electron-energy
analyzer Klauser  _et al._ (2010).

####  III.2.1 XPS measurements of mirrors annealed under air and under vacuum

We acquire XPS spectra from two mirrors and compare their chemical
composition. Prior to the measurement, one mirror was annealed in air, while
the other was annealed in vacuum. To calculate the amount of oxygen and
tantalum from the XPS spectra, we subtract the background and integrate over
the O ($1$s) and Ta ($4$d$5$) photoelectron lines. When weighted by the
Scofield sensitivity factors, which represent the emission probability of an
electron, these integrals give the relative proportions of the elements in the
material. The sensitivity factor is $15.64$ for the Ta ($4$d$5$) line and
$2.93$ for the O ($1$s) line.

Using this method, we compare the chemical compositions of the two mirrors.
The oxygen concentration of the mirror annealed under vacuum is $(0.9\pm
0.7)\%$ lower than the oxygen concentration of the mirror annealed under air.
This difference would constitute a loss of every $90$th oxygen atom from the
surface of the mirror annealed under vacuum. The large error bars are due to
the relative uncertainty of $0.5\%$ between measurements on the same
apparatus. To resolve the effect more clearly, we conduct a second experiment
based on the measurement of a single mirror over time.

####  III.2.2 Continuous XPS measurement during vacuum annealing

To observe the effect of oxygen loss from the surface directly, we perform
real-time XPS measurements during the process of annealing in vacuum on a
mirror that has previously been annealed in air. The mirror is placed inside
the XPS vacuum chamber and a reference XPS measurement is taken. Between
subsequent XPS measurements, the mirror temperature is increased stepwise up
to $608^{\circ}$C over $200$ minutes. This procedure gives an exact chemical
analysis of the mirror surface at each step of the annealing process.
Integrating the area under the oxygen and tantalum peaks, we calculate the
oxygen content in the surface of the mirror. The insets of Fig. 7 show the O
($1$s) and the Ta ($4$d) lines of one of the XPS spectra which we use for this
analysis. The temperature of the mirror in this measurement is measured with a
pyrometer (IMPAC) with a measurement uncertainty of $20^{\circ}$C.

Figure 7:  Oxygen content in the Ta2O5 layer on the mirror surface. The
temperature of the mirror coating is increased stepwise and XPS measurements
are taken at each temperature. The mirror has been annealed in air before the
measurement. The solid line is taken during the heating process, the dotted
line during the cool-down of the substrate. The uncertainty of the temperature
measurement is $20^{\circ}$C. The relative uncertainty of the oxygen content
is $0.5$%. The insets show sample XPS spectra of the O (1s) and the Ta (4d)
lines (solid), including the background (dashed), at one temperature setting.

Figure 7 shows the results of these measurements, in which the oxygen content
decreases as the temperature increases. The atomic percentage of oxygen of the
mirror before annealing is $78.2$%; at $405^{\circ}$C it drops to $75.7$%. The
oxygen content briefly recovers between $450^{\circ}$C and $550^{\circ}$C,
suggesting a phase transition Bansal (1994) or outgassing of oxygen. At
$608^{\circ}$C, the oxygen content reaches its lowest point of $75.7$%. The
mirror cool-down lasts $90$ minutes, during which the oxygen content neither
decreases nor increases significantly. We note that the absolute uncertainty
of the measurement is around $10$%. The entire annealing process results in a
$(2.5\pm 0.7)\%$ drop in oxygen content, supporting the hypothesis of oxygen
depletion from the Ta2O5 layer.

The discrepancy between this result and our earlier measurement of Sec.
III.2.1 might be due to oxygen reuptake when the vacuum-baked mirror was in
air before the XPS measurement. Both measurements show that we can attribute
the lower finesse of the vacuum-annealed mirrors to the loss of oxygen of the
Ta2O5-layer on the mirror surface.

###  III.3 Baking under vacuum

Up to now, we have only presented measurements of mirror annealing at
temperatures of $450^{\circ}$C and higher. The depletion of oxygen observed at
these temperatures suggests that this effect also takes place — in a moderate
form — when baking mirrors at standard temperatures for a vacuum bake-out,
typically $200^{\circ}$C to $300^{\circ}$C. The XPS measurement of Sec.
III.2.2 shows a linear decrease of oxygen when heating the mirror from room
temperature up to $405^{\circ}$C in vacuum. At a temperature of $160^{\circ}$C
one percent of the oxygen is already lost from the surface, and at
$300^{\circ}$C $1.8\%$ of the oxygen is lost.

We expect that if we bake mirrors under vacuum conditions at different
temperatures, one would find decreasing mirror finesses as the temperature of
the bake increases. This measurement would presumably show the same
temperature dependence of oxygen depletion following a vacuum bake as the XPS
measurements. We can estimate the mirror losses by linearly extrapolating the
two annealing measurements from Fig. 6 to lower temperatures. The first
annealing under vacuum was performed with non-annealed mirrors, while the
second time, these mirrors had been pre-annealed under air. According to these
measurements, we would expect $32$ ppm and $8$ ppm of additional losses at a
baking temperature of $300^{\circ}$C for the non-annealed mirrors and the
annealed mirrors, respectively. To understand whether this difference in
losses can be attributed to the pre-annealing, it would be interesting to
repeat the measurements for several mirror pairs.

A pre-annealed test mirror, however, baked under vacuum conditions at
$300^{\circ}$C, showed a decrease of the finesse from $1.9\times 10^{5}$ to
$4\times 10^{4}$ after baking. These $62$ ppm of additional mirror losses are
higher than expected from the annealing results. Furthermore, the mirror
finesse could not be recovered by successive air bakes, suggesting that the
mirror was damaged in the baking process. Lacking additional undamaged test
mirrors, we did not perform further vacuum bakes at moderate temperatures.
However, a systematic study of vacuum bakes over a range of temperatures would
provide valuable information for cavity setups in UHV.

###  III.4 Discussion

Annealing under vacuum decreases the mirror finesse rather than increasing it.
As a consequence, fiber mirrors should not be annealed under vacuum.

Tests of annealing fiber mirrors under air have not been successful so far.
Even when a clean annealing environment is established, the fiber mirrors seem
to get dirty after baking in air and cannot be cleaned successfully. We
suspect that contamination from the copper coating of the fiber damages the
mirror coating. A way to remove this source of contamination is the use of
chemically more inert fiber coating materials such as gold.

Furthermore, we expect the mirror losses to increase under a vacuum bake even
at moderate temperatures. The XPS measurements of Sec. III.2.2 show that the
oxygen decreases linearly with increasing baking temperature, and thus the
amount of defects in the mirror increases. Vacuum baking should therefore be
done at the lowest possible temperatures, although baking at higher
temperatures under oxygen atmosphere might be a solution.

##  IV Experimental ion-trap apparatus with an FFPC

In our combined FFPC ion-trap setup, the fibers sit on UHV-compatible
positioners enabling in-vacuum cavity alignment. In addition, these
positioners allow us to pull back the fibers from the trap center, so that the
ion trap can be tested without the influence of the fibers on the ion’s
trapping potential. The ion trap is a modified version of the linear Paul trap
presented in Refs. Gulde (2003); Riebe (2005). Here, we describe in detail the
ion trap, the integration of the FFPC into the trap setup, and the underlying
design considerations.

###  IV.1 Experimental design considerations

Ions are trapped quasi-permanently and well isolated from environmental
perturbations in RF Paul traps Paul (1990) under UHV conditions. Possible ion-
trap designs include surface-electrode traps Chiaverini  _et al._ (2005),
segmented linear traps based on microchip technology Schulz  _et al._ (2008),
‘endcap’ Schrama  _et al._ (1993); Wilson  _et al._ (2011) or stylus ion traps
Maiwald  _et al._ (2009), and linear blade traps Gulde (2003); Riebe (2005).
Two criteria for selecting a specific design are low ion heating rate and deep
trap depth, both of which contribute to long ion lifetimes in the trap. The
heating rate increases with decreasing ion-electrode distance Turchette  _et
al._ (2000), while for comparable trap dimensions and applied RF voltages, the
trap depth in three-dimensional traps is considerably deeper than in two-
dimensional traps. Linear blade traps, with heating rates as low as a few
quanta per second and trap depths on the order of tens of eV, are known to
have long ion lifetimes Rohde  _et al._ (2001); Benhelm  _et al._ (2008).

When considering the implementation of dielectric mirrors into an ion trap,
one should keep in mind the effects of dielectrics on the ion. Charges on
dielectrics in vacuum are quasi-permanent and distort the ion-trap potential.
They can be produced by UV light via photoelectron ionization Harlander  _et
al._ (2010) in a way that is not well understood and difficult to model. The
best strategy is to avoid any charging of dielectrics and to minimize the
influence of possible charges on the ion. As a solution, dielectric mirrors
are either placed far away from the trap Mundt  _et al._ (2002); Herskind  _et
al._ (2009); Russo  _et al._ (2009); Leibrandt  _et al._ (2009); Guthöhrlein
_et al._ (2001) or the dielectric components are well shielded Steiner  _et
al._ (2013); Brady  _et al._ (2011); Wilson  _et al._ (2011); VanDevender  _et
al._ (2010); Kim, Maunz, and Kim (2011).

Therefore, when integrating an FFPC into an ion trap, the following
restrictions should be respected: the ion-trap potential should be as deep as
possible, and the trap geometry should be such that it shields the ion from
any charges on the fibers. Furthermore, exposure of the fibers to UV light
should be minimized in order to keep them from accumulating charges. In case
the fibers become charged, the trap design should be flexible enough to
compensate for those charges, i.e., through the application of compensation
voltages. Here, we describe a setup that combines these features.

###  IV.2 Ion trap and vacuum chamber

####  IV.2.1 Miniaturized linear Paul trap

We choose a miniaturized linear Paul trap similar to the standard design
described in Refs. Gulde (2003); Riebe (2005); see Fig. 8. Four blade-shaped
electrodes operated with radio frequency (RF) and ground (GND) voltages
confine the ion radially, and two tip electrodes with positive voltage add
confinement along the trap axis. The trap has a deep trapping potential of
several electron volts inherent to three-dimensional traps, but in contrast to
traps of similar design Gulde (2003); Riebe (2005); Schulz  _et al._ (2008),
it is miniaturized in order to make its dimensions comparable to those of the
FFPCs, thus shielding the ions from charges on the fibers. The distance
between opposing blade tips on the diagonal is $340~{}\mu$m, which means that
the minimum ion-electrode distance is only $170~{}\mu$m; in contrast, in the
design of Ref. Riebe (2005) the ion-electrode distance is $800~{}\mu$m. The
distance along the trap axis between the two tip electrodes is $2.8$ mm, about
half the length of previous designs. These axial electrodes have $300~{}\mu$m
diameter holes for optical access along the trap axis. Another significant
change is that the angle between neighboring blade electrodes is not
$90^{\circ}$. In order to provide space for the fibers, the two angles between
the blades shielding the fibers are increased to $120^{\circ}$. As a result,
the other two angles are $60^{\circ}$. This change does not alter the trap
depth significantly. The trap has four additional rod-like electrodes,
parallel to the trap axis, $1.7$ mm from the trap center, and of $200~{}\mu$m
diameter. These electrodes allow for compensation of ions’ micromotion. The
rods are supplied with independent voltages, enabling compensation for charges
on the dielectric fibers. In Sec. IV.4, we show a simulation of the ion-trap
potential.

Figure 8: (a,b) Miniaturized ion-trap design. Red: two radio-frequency (RF)
and two ground (GND) blades of the linear Paul trap; the distance between two
opposing blades on the diagonal is $340~{}\mu$m (the distances between
neighboring blades are $290~{}\mu$m and $150~{}\mu$m). Blue: endcap
electrodes, $2.8$ mm apart. Green: four compensation electrodes, $1.7$ mm from
the trap center. White: Ceramic (MACOR) mount. Yellow: Holes in which
alignment rods are temporarily inserted. The plane for optical access,
including laser cooling, manipulation, and fluorescence detection, is
perpendicular to the fiber-cavity axis. $300~{}\mu$m wide holes in the endcaps
provide additional optical access in this plane. (c) Photo of the ion-trap
center. (d) CCD camera image of a linear string of ions.

Precise machining and positioning of the trap electrodes is necessary for a
trap of such small electrode separations. The stainless-steel blade electrodes
are aligned and mounted via two precision-machined glass-ceramic (MACOR)
holders. The fabrication tolerance for the ceramic mounts is less than
$50~{}\mu$m. The dimensions of the holders are then measured after machining,
and the blade electrodes are subsequently electron-discharge machined to fit
the holders exactly. Precision alignment of the blades with respect to the
ceramic mount and to one another is done with alignment rods, which are later
removed. After mounting the electrodes, we measured the dimensions of the ion
trap using a microscope. We find that the inaccuracy in blade-to-blade
separation is less than $30~{}\mu$m.

As the ion-electrode distance is very small, we expect higher ion-heating
rates in comparison with larger traps, so that it may be difficult to work
with ions in the motional ground state. However, the advantages of this design
are manifold: the trap has a deep trapping potential, while the electrode
distances are comparable to those of microfabricated traps and to the size of
the FFPC. Small traps do not need RF voltages as high as those of large traps
and are driven by simple RF resonators Gandolfi  _et al._ (2012). The blade
separation along the FFPC axis is only $150~{}\mu$m, thus shielding the ions.
The small diameter of the holes in the tip electrodes helps us to align laser
beams on the ion. We have successfully loaded strings of ions and single ions
in the ion trap (Fig. 8(d)).

####  IV.2.2 Vacuum vessel

To minimize collisions of ions with background gas, the trap needs to be
mounted under ultra-high vacuum. The chamber is designed to optimize vacuum
conditions, optical access, and stability requirements. The implementation of
the FFPC leaves only one plane of optical access available for lasers and
collection of fluorescence from the ions. Therefore, we chose an octagonal
vacuum chamber, which provides optical access from eight sides in this plane.
Fig. 9 shows a technical drawing of the experimental chamber. The axis
orthogonal to both the FFPC and the trap axes is used for fluorescence
detection. Here, within two inverted viewports, high NA objectives are
installed which allow for efficient light collection and thus fast detection
of the state of the ion with a camera and a PMT.

The FFPC fibers are fed into vacuum with a home-made fiber feedthrough
comprising stainless-steel tubes brazed into a CF-flange. The inner diameter
of the tubes is $0.5$ mm, and the fibers are glued into the tubes with vacuum
epoxyEPO . In contrast to commercial fiber feedthroughs, a home-made
feedthrough is advantageous as it is compatible with any fiber type, including
non-standard cladding diameters.

The trap is mounted together with the FFPC on the top flange of the vacuum
chamber, which also supports all electrical feedthroughs, the fiber
feedthrough, and the calcium oven. A vibration-isolating materialDuP  is
sandwiched between that top flange and the trap mount. The vacuum chamber sits
within a hole in an optical breadboard, which allows us to use short mounting
posts for optical components, thus providing improved stability.

Figure 9:  Technical drawing of the vacuum vessel and the ion-trap fiber-
cavity apparatus.

###  IV.3 Integration of fibers

The FFPC needs to be aligned with respect to the ion trap such that the center
of the ion trap overlaps with the waist of the cavity mode. Furthermore, the
fiber mirrors need to be aligned precisely to form a Fabry-Perot resonator,
and the cavity length has to be actively stabilized.

In the FFPC experiment of Ref. Colombe  _et al._ (2007), it was possible to
shift the trapping potential for neutral atoms with respect to the cavity
using currents on the chip. The alignment of the fiber mirrors was done before
the system was placed under ultra-high vacuum: one fiber was glued on a cavity
mount carefully positioned with respect to the atom chip, and the second fiber
was then aligned to form an FFPC with the first fiber and glued with UHV
epoxy. While the glue cured, the fiber was continuously aligned by optimizing
the cavity transmission signal.

We have tested the method described above to implement an FFPC into the ion
trap. We find that with short cavities of about $70~{}\mu$m in length, this
method is successful. Unfortunately, long cavities need to be aligned with
considerably higher precision, and small drifts of the fiber mirrors lead to
misalignment of the cavity. Although it is possible to cure the glue while
keeping the FFPC aligned, the cavity signal degrades over time after the glue
has set. Furthermore, the cavity signal disappears when the cavity assembly is
moved or rotated. These effects may be caused by slight temperature changes or
by tension in the cavity mount, and we expect that vacuum baking would also
contribute to misalignment. It may be possible to stabilize the fiber mirror
position passively well enough to maintain alignment, but instead we decided
on an active technique to mount the FFPC inside vacuum.

Each fiber is mounted on a three-axis nanopositioning system compatible with
ultra-high vacuum. Along the cavity axis, the fibers can be translated by up
to $8$ mmSMA (a). Along the other two axes, smaller positioners provide $4$ mm
of traveling rangeSMA (b). The positioners operate via the slip-stick
principle and have a minimum step size of $50$ nm, but can be moved with sub-
nanometer resolution by charging the piezo actuators. The three-axis system
for each fiber has dimensions of $(17\times 22\times 21)$ mm. Fig. 10(a) shows
a test setup of an FFPC aligned with the positioning system.

Figure 10: FFPC positioning system. (a) 3D nanopositioning setup for each
fiber. (b) Each fiber is glued to a glass v-groove and mounted on a shear-mode
piezoelectric crystal which allows active length stabilization of the cavity.
The piezoelectric crystals are glued to insulating MACOR spacers which are
screwed onto the nanopositioning stages.

Each fiber is glued to a Pyrex v-groove, which sits on a shear-mode
piezoelectric actuator; see Fig.10(b). The additional actuator is needed to
stabilize the cavity length actively as the bandwidth of the positioning
system is too low. These actuators are fixed to the positioners, and the whole
assembly is mounted on the same holder as the ion trap. Fig. 11 shows a photo
of the ion-trap setup together with the FFPC aligned via the micropositioning
system.

Figure 11: Photo of the linear Paul trap with an integrated FFPC. The trap
axis is horizontal, and the fiber-positioning systems are visible above and
below the trap. The calcium oven points towards the trap center from the right
side. Vacuum-compatible coaxial cables connect positioners and trap electrodes
to vacuum feedthroughs.

The in-vacuum positioning system offers a range of advantages for the setup.
First, it provides the option of realigning the cavity under vacuum in case of
misalignment due to baking or transport of the vacuum chamber. Second, it is
possible to pull the fibers back by almost a centimeter from the trapping
region, allowing the trapping of ions without dielectrics close to the
trapping region. Also, if the fibers are out of the trapping region during ion
loading, the trap blades shield the fibers, and the fiber mirrors do not get
coated with calcium. Furthermore, the fibers can be moved towards the trap
center iteratively while compensating for charges on the dielectric mirrors
via the compensation electrodes. Finally, the positioners allow us to change
the mirror separation of the FFPC inside vacuum and thus build cavities of
variable cavity length and waist, resulting in an adjustable coupling
parameter $g$.

###  IV.4 Simulations

The trap potential of the Paul trap is characterized in numerical simulations
using CPOCPO  and Matlab. CPO solves the electromagnetic field equations for
each electrode, which are then combined with Matlab to give the net potential
over the trapping region. The fibers are included in the simulations as
dielectric cylinders. These simulations were used to determine trap depth and
frequencies and to optimize the trap geometry.

Figure 12:  (a) Simulation of the trap potential of the miniaturized linear
Paul trap. The radio-frequency (RF) and ground (GND) blades as well as the
dielectric fibers are indicated in black. The potential in the plane
perpendicular to the trap axis is plotted in eV. The fibers are separated by
$200~{}\mu$m. (b) Trap potential without fibers. (c) Trap potential of a
symmetric trap with $90^{\circ}$ angles between RF and ground blades and blade
separations of $1.6$ mm.

We calculate the trap potential of the trap described in Sec. IV.2.1 for the
following parameters: RF amplitude of $130$ V, RF frequency of $2\pi\times 35$
MHz, and tip electrodes at $200$ V. As we cannot predict the amount of charge
on the fibers, we assume it to be zero in the simulations. Fig. 12(a) shows
the potential in the radial direction. The radial trap frequencies are
$2\pi\times 9.9$ MHz and $2\pi\times 9.6$ MHz, the axial trap frequency is
$1.9$ MHz, and the trap depth is $1.3$ eV.

In Fig. 12(b), the potential of the same trap without fibers is shown. In the
absence of the dielectric fibers, the trap depth is now $2$ eV instead of
$1.3$ eV.

Changing the blade angle from the asymmetric case of $120^{\circ}$ and
$60^{\circ}$ between RF and ground blades to angles of $90^{\circ}$ results in
a increase of trap depth of the radio frequency potential by a factor of
three. At the same time, however, the influence of the DC endcap potential
decreases, and therefore the overall trap depth does not change significantly.
The radial symmetry of the trap is no longer broken and the radial trap
frequencies are degenerate.

The trap implemented in the experimental setup has blade separations that are
five times smaller than those in the trap of Ref. Riebe (2005). To maintain a
stable trapping configuration in the smaller trap, either the radio-frequency
amplitude has to be decreased or its frequency has to be increased. Both
result in a lower trap depth: the depth of the miniaturized trap is an order
of magnitude smaller. Fig. 12(c) shows the potential of the trap from Ref.
Riebe (2005), in which the blade separations along the diagonal are $1.6$ mm,
all angles between RF and ground blades are $90^{\circ}$, and the trap depth
is $20$ eV.

The simulations confirm that the trapping potential of the trap implemented in
the setup has suitable trap depth and trap frequencies. Although not as deep
as a conventional Paul trap, it is nevertheless an order of magnitude deeper
than typical planar ion traps, giving this geometry a clear advantage over
planar designs.

##  V Summary and outlook

We have built an ion-trap apparatus with an integrated FFPC of tunable length
and thus of tunable coupling parameter $g$. In the process, we have set up and
tested a miniature linear Paul trap for 40Ca+. Furthermore, we have extended
the parameter regime for CO2-laser shaped fiber tips in order to build high-
finesse FFPCs of up to $350~{}\mu$m in length, compatible with the integration
of an ion trap. We have performed several experiments to characterize the
properties of FFPCs, which we expect will further the development of this new
technology. We describe several techniques for handling mirror-coated fibers
and discuss methods to improve the fiber-mirror finesse.

Our apparatus is intended for performing cavity QED experiments. In
particular, we would like to demonstrate strong coupling between an ion and a
photon, in which $g\gg\kappa,\gamma$ Kimble (2008). We have calculated that
our system enters this regime for a cavity in near-concentric configuration.
Similar to the cavity described in Ref. Stute  _et al._ (2012a), the fiber
cavity can be tuned into resonance with the $P$ to $D$ transition in 40Ca+,
where $D$ is a metastable state used as a qubit state. With a drive laser
coupling the electronic ground state $S$ and the intermediate state $P$, a
vacuum-stimulated Raman transition from $S$ to $D$ produces single photons
inside the cavity. An ion coupled to an FFPC would enable a coherent
interaction rate $g$ that dominates over the ion’s spontaneous decay rate
$\gamma$, providing access to a new experimental regime for ion-trap cavity
QED experiments.

###### Acknowledgements.

The authors would like to thank Ramin Lalezari from ATFilms for valuable
discussions on mirror coatings; Frederik Klauser from the Institute of
Physical Chemistry of the University of Innsbruck for his aid with the XPS
measurements; Stefan Haslwantler, Johannes Ghetta, Ben Ames, and Jasleen
Lugani for valuable discussions and aid in the construction of the FFPCs and
the experimental apparatus; and the mechanical workshop of the Institute of
Experimental Physics at the University of Innsbruck. We gratefully acknowledge
support from the Austrian Science Fund (FWF): Project. Nos. F4003 and F4019,
the European Research Council through the CRYTERION Project, the European
Commission via the Atomic QUantum TEchnologies (AQUTE) Integrating Project,
and the Institut für Quanteninformation GmbH.

## References

  * Zoller  _et al._ (2005) P. Zoller, T. Beth, D. Binosi, R. Blatt, H. Briegel, D. Bruss, T. Calarco, J. I. Cirac, D. Deutsch, J. Eisert, A. Ekert, C. Fabre, N. Gisin, P. Grangiere, M. Grassl, S. Haroche, A. Imamoglu, A. Karlson, J. Kempe, L. Kouwenhoven, S. Kr ll, G. Leuchs, M. Lewenstein, D. Loss, N. L tkenhaus, S. Massar, J. E. Mooij, M. B. Plenio, E. Polzik, S. Popescu, G. Rempe, A. Sergienko, D. Suter, J. Twamley, G. Wendin, R. Werner, A. Winter, J. Wrachtrup, and A. Zeilinger, Eur. Phys. J. D 36, 203 (2005).
  * Ritter  _et al._ (2012) S. Ritter, C. Nölleke, C. Hahn, A. Reiserer, A. Neuzner, M. Uphoff, M. Mücke, E. Figueroa, J. Bochmann, and G. Rempe, Nature 484, 195 (2012).
  * Leibfried  _et al._ (2003) D. Leibfried, B. DeMarco, V. Meyer, D. Lucas, M. Barrett, J. Britton, W. M. Itano, B. Jelenkovic, C. Langer, T. Rosenband, and D. J. Wineland, Nature 422, 412 (2003).
  * Häffner, Roos, and Blatt (2008) H. Häffner, C. Roos, and R. Blatt, Phys. Rep. 469, 155 (2008).
  * Stute  _et al._ (2012a) A. Stute, B. Casabone, B. Brandstätter, D. Habicher, P. O. Schmidt, T. E. Northup, and R. Blatt, Appl. Phys. B 107, 1145 (2012a).
  * Guthöhrlein  _et al._ (2001) G. R. Guthöhrlein, M. Keller, K. Hayasaka, W. Lange, and H. Walther, Nature 414, 49 (2001).
  * Mundt  _et al._ (2002) A. B. Mundt, A. Kreuter, C. Becher, D. Leibfried, J. Eschner, F. Schmidt-Kaler, and R. Blatt, Phys. Rev. Lett. 89, 103001 (2002).
  * Russo  _et al._ (2009) C. Russo, H. G. Barros, A. Stute, F. Dubin, E. S. Phillips, T. Monz, T. E. Northup, C. Becher, T. Salzburger, H. Ritsch, P. O. Schmidt, and R. Blatt, Appl. Phys. B 95, 205 (2009).
  * Leibrandt  _et al._ (2009) D. R. Leibrandt, J. Labaziewicz, V. Vuletić, and I. L. Chuang, Phys. Rev. Lett. 103, 103001 (2009).
  * Herskind  _et al._ (2009) P. Herskind, A. Dantan, J. Marler, M. Albert, and M. Drewsen, Nat. Phys. 5, 494 (2009).
  * Sterk  _et al._ (2012) J. D. Sterk, L. Luo, T. A. Manning, P. Maunz, and C. Monroe, Phys. Rev. A 85, 062308 (2012).
  * Keller  _et al._ (2004) M. Keller, B. Lange, K. Hayasaka, W. Lange, and H. Walther, Nature 431, 1075 (2004).
  * Barros  _et al._ (2009) H. G. Barros, A. Stute, T. E. Northup, C. Russo, P. O. Schmidt, and R. Blatt, New J. Phys. 11, 103004 (2009).
  * Stute  _et al._ (2012b) A. Stute, B. Casabone, P. Schindler, T. Monz, P. O. Schmidt, B. Brandstätter, T. E. Northup, and R. Blatt, Nature 485, 482 (2012b).
  * Hunger  _et al._ (2010) D. Hunger, T. Steinmetz, Y. Colombe, C. Deutsch, T. W. Hänsch, and J. Reichel, New J. Phys. 12, 065038 (2010).
  * Steiner  _et al._ (2013) M. Steiner, H. M. Meyer, C. Deutsch, J. Reichel, and M. Köhl, Phys. Rev. Lett. 110, 043003 (2013).
  * Kimble (2008) H. J. Kimble, Nature 453, 1023 (2008).
  * Trupke  _et al._ (2005) M. Trupke, E. Hinds, S. Eriksson, E. Curtis, Z. Moktadir, E. Kukharenka, and M. Kraft, Appl. Phys. Lett. 87, 211106 (2005).
  * Cui  _et al._ (2006) G. Cui, J. M. Hannigan, R. Loeckenhoff, F. M. Matinaga, M. G. Raymer, S. Bhongale, M. Holland, S. Mosor, S. Chatterjee, H. M. Gibbs, and G. Khitrova, Opt. Express 14, 2289 (2006).
  * Steinmetz  _et al._ (2006) T. Steinmetz, Y. Colombe, D. Hunger, T. W. Hänsch, A. Balocchi, R. J. Warburton, and J. Reichel, Appl. Phys. Lett. 89, 111110 (2006).
  * Muller  _et al._ (2009) A. Muller, E. B. Flagg, M. Metcalfe, J. Lawall, and G. S. Solomon, Appl. Phys. Lett. 95, 173101 (2009).
  * Roy and Barrett (2011) A. Roy and M. D. Barrett, Appl. Phys. Lett. 99, 171112 (2011).
  * Muller  _et al._ (2010) A. Muller, E. B. Flagg, J. R. Lawall, and G. S. Solomon, Opt. Lett. 35, 2293 (2010).
  * Colombe  _et al._ (2007) Y. Colombe, T. Steinmetz, G. Dubois, F. Linke, D. Hunger, and J. Reichel, Nature 450, 272 (2007).
  * Wilson  _et al._ (2011) A. Wilson, H. Takahashi, A. Riley-Watson, F. Orucevic, P. Blythe, A. Mortensen, D. R. Crick, N. Seymour-Smith, E. Brama, M. Keller, and W. Lange,  (2011), arxiv:1101.5877.
  * VanDevender  _et al._ (2010) A. P. VanDevender, Y. Colombe, J. Amini, D. Leibfried, and D. J. Wineland, Phys. Rev. Lett. 105, 023001 (2010).
  * Fox and Li (1961) A. G. Fox and T. Li, Bell Syst. Tech. J 40, 453 (1961).
  * Siegman (1986) A. E. Siegman,  _Lasers_ (University Science Books, Sausalito, CA, 1986).
  * (29) Fogale: Microsurf3D.
  * Joyce and DeLoach (1984) W. B. Joyce and B. C. DeLoach, Appl. Opt. 23 (1984).
  * Hood, Kimble, and Ye (2001) C. J. Hood, H. J. Kimble, and J. Ye, Phys. Rev. A 64, 033804 (2001).
  * Rempe  _et al._ (1992) G. Rempe, R. J. Thompson, H. J. Kimble, and R. Lalezari, Opt. Lett. 17, 363 (1992).
  * Lynn (2003) T. W. Lynn, Ph.D. thesis, California Institute of Technology, Pasadena, 2003 (2003).
  * Hood  _et al._ (2000) C. J. Hood, T. W. Lynn, A. C. Doherty, A. S. Parkins, and H. J. Kimble, Science 287, 1447 (2000).
  * (35) IVG fiber: Cu$800-200-$custom and Cu$50-200$.
  * (36) EPO-TEK $353$ND and EPO-TEK $301$.
  * (37) Bullet: NBG-FC230.
  * (38) Vytran: FFS-2000.
  * (39) Thorlabs: Nanomax$602$D/M.
  * Northup (2008) T. E. Northup, Ph.D. thesis, California Institute of Technology, Pasadena, 2008 (2008).
  * Atanassova, Dimitrova, and Koprinarova (1995) E. Atanassova, T. Dimitrova, and J. Koprinarova, Appl. Surf. Sci. 84, 193 (1995).
  * Cetina  _et al._ (2013) M. Cetina, A. Bylinskii, L. Karpa, D. Gangloff, K. M. Beck, Y. Ge, M. Scholz, A. T. Grier, I. Chuang, and V. Vuletic, “One-dimesional array of ion chains coupled to an optical cavity,”  (2013), arxiv:1302.2904.
  * Scofield (1976) J. Scofield, J. Electron Spectrosc. Rel. Phenom. 8 (2), 129 (1976).
  * Klauser  _et al._ (2010) F. Klauser, S. Ghodbane, R. Boukherroub, S. Szunerits, D. Steinmüller-Nethl, E. Bertel, and N. Memmel, Diam. Relat. Mater. 19, 474 (2010).
  * Bansal (1994) N. P. Bansal, J. Mat. Sci. 29, 5065 (1994).
  * Gulde (2003) S. T. Gulde, Ph.D. thesis, Leopold-Franzens-Universität Innsbruck, Innsbruck, 2003 (2003).
  * Riebe (2005) M. Riebe, Ph.D. thesis, University of Innsbruck, Innsbruck, 2005 (2005).
  * Paul (1990) W. Paul, Rev. Mod. Phys. 62, 531 (1990).
  * Chiaverini  _et al._ (2005) J. Chiaverini, R. B. Blakestad, J. Britton, J. D. Jost, C. Langer, D. Leibfried, R. Ozeri, and D. J. Wineland, Quantum Inf. Comput. 5, 419 (2005).
  * Schulz  _et al._ (2008) S. A. Schulz, U. G. Poschinger, F. Ziesel, and F. Schmidt-Kaler, New J. Phys. 10, 045007 (2008).
  * Schrama  _et al._ (1993) C. Schrama, E. Peik, W. Smith, and H. Walther, Opt. Commun. 101, 32 (1993).
  * Maiwald  _et al._ (2009) R. Maiwald, D. Leibfried, J. Britton, J. C. Bergquist, G. Leuchs, and D. J. Wineland, Nature Phys. 5, 551 554 (2009).
  * Turchette  _et al._ (2000) Q. A. Turchette, Kielpinski, B. E. King, D. Leibfried, D. M. Meekhof, C. J. Myatt, M. A. Rowe, C. A. Sackett, C. S. Wood, W. M. Itano, C. Monroe, and D. J. Wineland, Phys. Rev. A 61, 063418 (2000).
  * Rohde  _et al._ (2001) H. Rohde, S. T. Gulde, C. F. Roos, P. A. Barton, D. Leibfried, J. Eschner, F. Schmidt-Kaler, and R. Blatt, J. Opt. B 3, S34 (2001).
  * Benhelm  _et al._ (2008) J. Benhelm, G. Kirchmair, C. F. Roos, and R. Blatt, Nat. Phys. 4, 463 (2008).
  * Harlander  _et al._ (2010) M. Harlander, M. Brownnutt, W. H nsel, and R. Blatt, New J. Phys. 12, 093035 (2010).
  * Brady  _et al._ (2011) G. Brady, A. Ellis, D. Moehring, D. Stick, C. Highstrete, K. Fortier, M. Blain, R. Haltli, A. Cruz-Cabrera, R. Briggs, J. Wendt, T. Carter, S. Samora, and S. Kemme, Appl. Phys. B 103, 801 (2011).
  * Kim, Maunz, and Kim (2011) T. Kim, P. Maunz, and J. Kim, Phys. Rev. A 84, 063423 (2011).
  * Gandolfi  _et al._ (2012) D. Gandolfi, M. Niedermayr, M. Kumph, M. Brownnutt, and R. Blatt, Rev. Sci. Instrum. 83, 084705 (2012).
  * (60) DuPont: Kalrez.
  * SMA (a) (a), sMARACT: SLC-20.
  * SMA (b) (b), sMARACT: SL-06.
  * (63) CPO Ltd. Charged Particle Optics programs - www.electronoptics.com.