# Efficient Collection of Single Photons Emitted from a Trapped Ion into a
Single Mode Fiber for Scalable Quantum Information Processing

Taehyun Kim, Peter Maunz, and Jungsang Kim  Fitzpatrick Institute for
Photonics, Electrical and Computer Engineering Department, Duke University,
Durham, North Carolina 27708, USA

###### Abstract

Interference and coincidence detection of two photons emitted by two remote
ions can lead to an entangled state which is a critical resource for scalable
quantum information processing. Currently, the success probabilities of
experimental realizations of this protocol are mainly limited by low coupling
efficiency of a photon emitted by an ion into a single mode fiber. Here, we
consider two strategies to enhance the collection probability of a photon
emitted from a trapped Yb+ ion, using analytic methods that can be easily
applied to other types of ion or neutral atoms. Our analysis shows that we can
achieve fiber coupling efficiency of over 30 % with an optical cavity made of
a flat fiber tip and a spherical mirror. We also investigate ways to increase
the fiber coupling efficiency using high numerical aperture optics, and show
that collection probability of over 15 % is possible with proper control of
aberration.

###### pacs:

32.80.Qk,42.50.Pq,42.15.Eq

##  I INTRODUCTION

Entanglement is a unique feature of quantum physics and it has found various
applications ranging from the experimental verification of the foundation of
quantum physics Einstein  _et al._ (1935) to many quantum information
protocols such as quantum teleportation Bennett  _et al._ (1993), quantum
cryptography Ekert (1991), quantum computation Gottesman and Chuang (1999);
Raussendorf and Briegel (2001), and long distance quantum communication
Briegel  _et al._ (1998). Most of these applications require a large number of
entangled states distributed over long distances, which can be generated using
one of two approaches. In the first approach, one can locally generate a pair
of entangled photons Kim  _et al._ (2006), distribute them to remote sites,
and map these flying qubits into stationary (memory) qubits Lloyd  _et al._
(2001). To efficiently transfer a flying qubit into a stationary qubit, the
spectral and spatial modes of the flying qubit have to be matched to the time-
inverted emission properties of the memory qubit, which is the most
significant challenge in this approach. In the second approach, one first
generates flying qubits that are entangled with a local stationary qubit at
each site, sends these photons to a common location, and then interferes these
two photons using a 50:50 beam splitter Duan  _et al._ (2001); Simon and
Irvine (2003). With 50 % chance, these photon pairs are projected into one of
two distinguishable Bell states, and once one of these two Bell states is
detected, the two stationary qubits at the remote sites are also projected to
a maximally entangled Bell state Pan  _et al._ (1998). The advantage of this
approach is that the spectral properties of the two photons are naturally
matched if one uses the same type of memory qubits. In order to achieve
interference of the two photons with high visibility, the photons have to be
indistinguishable. Therefore, we need to collect emitted photons into a single
spatial mode, such as a single mode fiber. Several experiments demonstrated
establishment of entanglement between remote trapped ions using this approach
Moehring  _et al._ (2007); Matsukevich  _et al._ (2008); Olmschenk  _et al._
(2009). While these experiments realized high fidelity entangled states, the
success probability of the probabilistic entanglement process is very small.
The success probability is mainly limited by the small photon collection
probability of about 0.4 % given by the small numerical aperture (NA) of the
collection optics and an insufficient overlap between the spatial mode of the
single mode fiber and the emission pattern of an ion. Because this protocol
requires one to collect photons from both sites and observe a coincidence, the
overall success probability scales quadratically with the single fiber
coupling efficiency, and was limited to only about $2\cdot 10^{-8}$ (see Table
2 in Sec. IV) Olmschenk  _et al._ (2009). To apply this system for large scale
quantum information processing or long distance quantum communication, it is
critical to dramatically enhance the collection probability of photons into a
single mode fiber.

In this paper, we show systematic approaches to increase the fiber coupling
efficiency of a single photon while preserving ion-photon entanglement. We
will analyze the design of an optimized optical cavity that can be integrated
with a surface ion trap, and evaluate various ways to achieve high NA
collection optics compatible with surface traps to enhance light collection
into a single mode fiber.

Figure 1: Ion-photon entanglement schemes in ${}^{171}{\rm Yb}^{+}$. (a)
Simplified level scheme of ${}^{171}{\rm Yb}^{+}$ with dipole-allowed
transitions. Zeeman levels are denoted by $|F,m_{F}\rangle$, where $F$ is the
total angular momentum including nuclear spin and $m_{F}$ is its projection
onto the quantization axis. ${}^{2}S_{1/2}|1,0\rangle$ and
${}^{2}S_{1/2}|0,0\rangle$ are first-order magnetic field-insensitive
hyperfine levels (clock states). (b) When the ion decays from
${}^{2}P_{1/2}|1,0\rangle$ to one of the Zeeman levels in ${}^{2}S_{1/2}|1,\pm
1\rangle$, the polarization of the emitted photon gets entangled with two
Zeeman levels Matsukevich  _et al._ (2008). (c) The frequency of the emitted
photon and the atomic clock states are entangled after a decay from
${}^{2}P_{1/2}|1,-1\rangle$ Moehring  _et al._ (2007). (d) Due to the
selection rules, excitation of ${}^{171}{\rm Yb}^{+}$ prepared in a
superposition of the clock states in ${}^{2}S_{1/2}$ generates a coherent
superposition of the excited states (two ${}^{2}P_{1/2}$ states with
$m_{F}=0$). Once excited state decays back to the original state, the
frequency of the emitted photon is entangled with clock states Olmschenk  _et
al._ (2009). In ${}^{171}{\rm Yb}^{+}$ system, ${}^{2}S_{1/2}|0,0\rangle$ is
generally used as an initial state due to its high preparation fidelity.
Therefore (c) and (d) scheme require additional microwave or optical Raman
transitions denoted by thick arrows.

Our analysis mainly focuses on ${}^{171}{\rm Yb}^{+}$ whose simplified level
scheme is shown in Figure 1 (a) Olmschenk  _et al._ (2007), but the
conclusions can be applied to other atomic systems. There are several ways to
entangle a photon’s degree of freedom with internal electronic states of a
trapped ${}^{171}{\rm Yb}^{+}$ ion. If Zeeman levels are denoted by
$|F,m_{F}\rangle$, where $F$ is the total angular momentum and $m_{F}$ is its
projection onto the quantization axis, one can excite the Yb ion to
${}^{2}P_{1/2}|1,0\rangle$ and collect the photon from the $\sigma^{\pm}$
transition as shown in Figure 1 (b). The polarization of the photon will then
be entangled with the two ground state Zeeman levels ${}^{2}S_{1/2}|1,\pm
1\rangle$ Matsukevich  _et al._ (2008). Alternatively, one can collect a
photon generated by $\sigma$-transitions from ${}^{2}P_{1/2}|1,-1\rangle$ as
shown in Figure 1 (c) or by $\pi$-transitions from a superposition of
${}^{2}P_{1/2}|1,0\rangle$ and ${}^{2}P_{1/2}|0,0\rangle$ to the atomic clock
states composed of ${}^{2}S_{1/2}|1,0\rangle$ and ${}^{2}S_{1/2}|0,0\rangle$
as shown in Figure 1 (d). The frequency of the photon is then entangled with
the clock state qubits. The spatial emission pattern of photons from
$\sigma^{\pm}$ and $\pi$-transitions and their spectral properties together
with the ability to filter out photons from unwanted transitions will dictate
strategies that can be adopted to enhance the photon collection efficiency.
For remote entanglement generation, we will analyze the approach using an
optical cavity in Section II, and an approach using high NA optics (either a
reflector or a lens) in Section III. Section IV estimates the entanglement
generation rate we can achieve based on these approaches, and conclusions are
presented in Section V.

##  II Cavity enhanced photon collection

A small high finesse cavity can be used to collect light from a single atom by
induced emission into the cavity mode. Although efficient light collection
using an optical cavity has been demonstrated for neutral atoms McKeever  _et
al._ (2004); Hijlkema  _et al._ (2007), additional challenges have to be met
for trapped ions: readout transitions of trapped ions are typically in the
ultraviolet (UV) and the ion trap has to fit in a small cavity with UV-
dielectric coatings. In addition, trapped ions have to be shielded from the
charges which can accumulate on dielectric surfaces Harlander  _et al._
(2010).

The collection probability of a single photon from an excited atom into a
single mode fiber is dictated by several factors. First, the excitation has to
leave the coupled atom-cavity system via the cavity decay channel rather than
spontaneous emission, characterized by the photon extraction probability
$P_{cavity}$. While some photons leave the cavity through the desired output
channel, other photons are either lost due to scattering and absorption in the
mirrors or transmitted through the undesired mirror(s). We define $r_{t}$ as
the fraction of the desired transmission over the total cavity decay. Finally,
we consider overlap between the mode of the cavity and that of the single mode
fiber, which will be called mode matching factor $\epsilon$. The overall
probability $P_{fiber}$ of extracting the photon through the single mode fiber
is then given by

$P_{fiber}=\epsilon\cdot r_{t}\cdot P_{cavity}.$ (1)

In this section, we will investigate how to maximize each factor within the
physical constraints. We will mainly consider a cavity geometry shown in
Figure 2 Kim and Kim (2009) and optimize system parameters to maximize the
collection efficiency under the given constraints. This discussion will also
identify components that deserve further investigation to improve the
performance of the system in the future, and provide a metric to compare
different cavity systems.

Figure 2: Geometry of an optical cavity integrated with surface trap. Optical
cavity is composed of a small spherical mirror and a fiber in a ferrule whose
tip is coated with high reflecting dielectric layers. $l$ is cavity length,
$w_{0}$ is beam waist at the fiber tip, and $r_{ion}$ is beam radius at the
trapping position of the ion. Surface trap can be patterned with Au on the
fiber/ferrule surface whose diameter is $>$2 mm.

When an atomic system is coupled to a cavity mode, a single photon can be
generated either by fast excitation with resonant $\pi$-pulse followed by
spontaneous emission Moehring  _et al._ (2007); Matsukevich  _et al._ (2008);
Olmschenk  _et al._ (2009); Bochmann  _et al._ (2008), or by stimulated Raman
adiabatic passage (STIRAP) process McKeever  _et al._ (2004); Hijlkema  _et
al._ (2007). In this section, we will also compare these two methods in terms
of collection probability $P_{fiber}$ and repetition rate.

###  II.1 Fast excitation with resonant $\pi$-pulse: Design principles

In this subsection, we consider the case where the ion is excited by an
ultrafast pulse whose duration is short compared to any time constants of the
system, such that the initial state is an excited state of the ion. When an
excited ion is placed inside an optical cavity, the spontaneous emission rate
into the cavity mode can be enhanced by the Purcell factor, $2C+1$ Mundt  _et
al._ (2002), where $C=g^{2}/2\kappa\gamma$ is the cooperativity, $g$ is the
coherent ion-field coupling constant, $\kappa$ is the cavity-field decay rate,
and $2\gamma$ is the spontaneous emission rate.

The parameter $g^{2}$ is inversely proportional to the cavity mode volume and
$\kappa=\pi\cdot FSR/\mathcal{F}$, where $FSR$ and $\mathcal{F}$ are the free
spectral range and the finesse of the cavity, respectively. Therefore, we need
a high finesse cavity with small mode volume to achieve large cooperativity.
Small mode volume cavities can be realized by many different approaches such
as microdisk Gayral  _et al._ (1999), microsphere Braginsky  _et al._ (1989),
microtoroid Armani  _et al._ (2003), and photonic crystals Painter  _et al._
(1999). Integration with ion traps requires a cavity with empty mode volume,
so we will consider only traditional open cavities formed by two (curved)
mirrors. Cavity mode shape between two spherical mirrors is Gaussian and the
mode volume is $\pi w_{0}^{2}l/4$, where $w_{0}$ is the beam waist of the
Gaussian mode function and $l$ is length of the cavity. The cooperativity is
reduced to
$C=3\mathcal{F}\cdot\text{Br}\cdot\lambda^{2}/(\pi^{3}r_{ion}^{2})$, where Br
is a branching ratio, $\lambda$ is wavelength of transition, and $r_{ion}$ is
the radius of the mode function at ion location. In this system, we need high
$\mathcal{F}$ and small $r_{ion}$ to maximize $C$.

Using these cavity parameters, one can express the standard formula for
$P_{cavity}$ Cui and Raymer (2005) as a function of the cavity length ($l$)
and the beam radius at ion location ($r_{ion}$):

$\displaystyle P_{cavity}$ $\displaystyle=$
$\displaystyle\left(\frac{\kappa}{\kappa+\gamma}\right)\left(\frac{2C}{2C+1}\right)$
(2) $\displaystyle=$
$\displaystyle\left(\frac{1}{1+l/l_{c}}\right)\left(\frac{1}{1+r_{ion}^{2}/r_{c}^{2}}\right),$

where $l_{c}$ is a characteristic cavity length defined as $l_{c}=\pi
c/(2\gamma\mathcal{F})$ and $r_{c}$ is a characteristic beam radius defined as
$r_{c}=\sqrt{6\lambda^{2}\mathcal{F}\cdot\text{Br}/\pi^{3}}$. Eq. (2) implies
that if either $l$ or $r_{ion}$ is much larger than the corresponding
characteristic lengths, the overall $P_{cavity}$ will decrease rapidly. The
characteristic cavity length (beam radius) decreases (increases) as the
finesse of the cavity increases. This indicates that for high finesse cavities
one should use a short cavity but the beam radius can be larger, and the
design shifts to longer cavities with smaller beam radius as the finesse of
the cavity decreases.

Cavity finesse $\mathcal{F}$ is defined by
$2\pi\sqrt{1-\mathcal{L}}/\mathcal{L}$, where
$\mathcal{L}=T_{f}+T_{e}+L_{f}+L_{e}$ is the total loss of the cavity, $T_{f}$
and $L_{f}$ ($T_{e}$ and $L_{e}$) are transmission and passive loss through
dielectric coating on the fiber tip (the other end mirror) in Figure 2,
respectively. Passive loss is due to scattering loss mainly determined by
surface roughness Trupke  _et al._ (2005) and absorption in the dielectric
coating. We can define $\mathcal{F}_{0}\equiv
2\pi\sqrt{1-(L_{f}+L_{e})}/(L_{f}+L_{e})$, to denote the maximum achievable
finesse limited by the uncontrollable passive loss. Then the actual finesse
can be approximated to $\mathcal{F}\simeq\mathcal{F}_{0}(1-r_{t})$ where
$r_{t}=T_{f}/\mathcal{L}$. Assuming the mode matching factor $\epsilon$ is
constant, one can see that $P_{fiber}$ defined in Eq. (1) is maximized when

$r_{t}=r_{t0}\equiv\frac{1}{1+1/\sqrt{(1+2C_{0})(1+l/l_{c0})}}$ (3)

to a value

$P_{fiber,max}=\frac{C_{0}}{1+C_{0}+l/2l_{c0}+\sqrt{(1+2C_{0})(1+l/l_{c0})}},$
(4)

where $C_{0}\equiv
3\mathcal{F}_{0}\cdot\text{Br}\cdot\lambda^{2}/(\pi^{3}r_{ion}^{2})$ and
$l_{c0}\equiv\pi c/(2\gamma\mathcal{F}_{0})$ are the cooperativity and
characteristic length with maximum possible finesse $\mathcal{F}_{0}$,
respectively. From Eq. (4), one can see that $C_{0}$ should be maximized while
$l$ should be minimized in order to maximize the fiber coupling efficiency.

Finally, fiber collection efficiency is further reduced by mismatch between
cavity mode ($E$) and fiber mode ($G_{fiber}$) at the fiber tip characterized
by the mode matching factor

$\epsilon=\frac{|\int_{A}da\vec{E}^{*}\cdot\vec{G}_{fiber}|^{2}}{\int_{A}da|\vec{E}|^{2}\int_{A}da|\vec{G}_{fiber}|^{2}},$
(5)

where $G_{fiber}$ and $E$ are the transverse Gaussian mode shapes of fiber and
cavity, respectively. The optimal $w_{0}$ of the fiber cavity should be
determined to maximize the coupling $P_{fiber}$ to an ion at height $h_{ion}$,
which may not necessarily match with the fiber mode. If one uses additional
optics to couple the cavity mode into a fiber mode rather than integrate the
fiber mode directly underneath the flat cavity mirror, the mode matching
factor given by Eq. (5) can be improved substantially, approaching a value of
1.

###  II.2 Cavity design example

The optimum design of the cavity-ion trap system is often constrained by the
ability to achieve high performance components. In our system, the dominating
constraint is the finesse $\mathcal{F}_{0}$ and the radius of curvature (RoC)
for the curved cavity mirror one can achieve, limited by the coating quality
and polishing capability. Compared to near-infrared wavelengths used in
neutral atom experiments McKeever  _et al._ (2004); Hijlkema  _et al._ (2007),
the passive loss of cavity mirror coating is significantly higher in the
ultraviolet range relevant for most trapped ions. Scattering loss from a rough
surface is given by $1-\exp[-(4\pi\sigma/\lambda)^{2}]$ Trupke  _et al._
(2005) where $\sigma$ is RMS surface roughness. For example, we observed
$\approx$ 150 parts per million (ppm) loss at 369.5 nm on a super-polished
substrate whose RMS surface roughness is measured to be $<$0.1 nm,
corresponding to estimated scattering loss of $<$12 ppm. We can attribute most
of the loss in this case to absorption loss in the coating material in the UV
wavelength range. Since the RMS surface roughness of the fiber tip and the
mirror substrate with small RoC is generally larger than that of a super-
polished mirror substrate with large RoC, it is reasonable to assume that the
uncontrolled loss in our mirrors is $L_{f}+L_{e}\approx$1500 ppm,
corresponding to $\mathcal{F}_{0}\approx 4200$.

Next, we have to pick the radius of the cavity mode $r_{ion}$ at the ion
location and the cavity length $l$. In order to maximize $C_{0}$, one should
find a minimum value of $r_{ion}$. Since the ion cannot be trapped at the beam
waist in a fiber tip cavity shown in Fig. (2), the optimal waist $w_{0}$ is
determined by the distance between the ion and the waist which is located on
the flat fiber tip. When this distance is given by $h_{ion}$, $r_{ion}$ can be
minimized if $w_{0}=\sqrt{h_{ion}\cdot\lambda/\pi}$. In a surface trap, the
minimum ion height from the surface is constrained by considerations such as
the exposed dielectric material Harlander  _et al._ (2010), anomalous heating
Labaziewicz  _et al._ (2008) and the scattering of laser light from the
surface, and $h_{ion}$ = 50 $\mu$m is a good compromise. In this case, the
optimal waist is $w_{0}$ = 2.4 $\mu$m, corresponding to $r_{ion}$ = 3.4
$\mu$m.

For a cavity created between a flat mirror and a concave mirror, the beam
waist of the Gaussian mode is located at the flat mirror surface. The length
of the cavity $l$ is related to the Rayleigh range $z_{R}=\pi
w_{0}^{2}/\lambda$ of the Gaussian mode as $z_{R}^{2}=({\rm RoC}-l)l$. To trap
the ion inside the cavity volume, we require $l>h_{ion}=z_{R}$ and therefore
$l$ should be chosen to be very close to the RoC. A practical mirror one can
polish using conventional super-polishing technique has a RoC $\geq$ 5 mm, so
we choose $l\approx{\rm RoC}-z_{R}^{2}/{\rm RoC}\approx$ 5 mm. The cavity
length must be aligned to be about 0.5 $\mu$m shorter than the RoC of the
mirror. Laser ablation can create a smooth curved surface with RoC between 40
$\mu$m and 2 mm with 0.2 nm RMS roughness Hunger  _et al._ (2010), but the
aperture diameter of a mirror fabricated in this way is limited to
$<100~{}\mu$m which is too small for near-concentric cavity geometry.

Using these parameters, we obtain $C_{0}\approx 4.8$ and $l_{c0}\approx$ 1.8
mm. The transmission of the out coupling mirror is determined by setting
$r_{t}\approx 86\%$ according to Eq. (3), corresponding to $T_{f}\approx$
9,500 ppm. This choice determines critical cavity parameters as
$\mathcal{F}=570$, $C=$ 0.65, $l_{c}$=13 mm and $r_{c}=$ 3.9 $\mu$m. Measured
beam waist of a typical UV fiber at 369.5 nm is about 1.5 $\mu$m and in Figure
3 the change of the $P_{fiber}$ as a function of the cavity length $l$ for an
ion height $h_{ion}$ = 50 $\mu$m is shown. The mode matching efficiency
$\epsilon$ is about 82 % if $w_{0}$ is optimized for maximum $P_{cavity}$
($w_{0}$ = 2.4 $\mu$m), and $P_{cavity}$ is reduced when $\epsilon$ is 100 %.
Therefore, there is an optimal length that maximizes $P_{fiber}$, near which
one can achieve more than 30 % of coupling efficiency compared to 0.4 %
Olmschenk  _et al._ (2009).

If additional optics is used to couple the cavity mode into a single mode
fiber (instead of using fiber tip as the flat mirror of the cavity), we can
achieve $\epsilon\approx$1 and a maximum fiber coupling efficiency of over 35
%. A comparison with a cavity constructed using two concave mirrors is
presented in Appendix A.

Figure 3: Probability of collecting photons with a fiber cavity into the mode
of a single mode fiber as a function of cavity length ($l$). As the cavity
length approaches the RoC of the spherical mirror, the beam waist at the fiber
tip shrinks. $T_{f}$ = 9,500 ppm and $T_{e}+L_{f}+L_{e}$= 1500 ppm are
assumed. The radius $r_{ion}$ is smallest for a cavity length of 4.9995 mm
while best mode overlap between cavity and fiber is achieved for a length of
4.9999 mm.

In summary, we expect to reach $(g,\kappa,\gamma)/2\pi\approx(18,26,10)$ MHz
using a fiber tip cavity. With pulsed excitation, one can achieve more than 30
% coupling efficiency into the single mode fiber if the minimum available RoC
is about 5 mm, and the average rate of single photon emission process is on
the order of $\text{max}(1/g,1/\kappa)$.

###  II.3 Excitation by stimulated Raman adiabatic passage

In an atomic system where the initial state $|i\rangle$ and the final states
$|f_{1}\rangle$, $|f_{2}\rangle$ are different as shown in Fig. 1 (b) and can
be connected by a resonantly enhanced Raman transition, the STIRAP process can
be used to generate a single photon in the cavity mode McKeever  _et al._
(2004); Hijlkema  _et al._ (2007).

For this process, a laser beam is resonant with one branch of the Raman
transition ($|i\rangle\Leftrightarrow|e\rangle$) with Rabi frequency
$\Omega_{P}$, and the cavity vacuum stimulates the emission of the photon on
the other branch ($|f\rangle\Leftrightarrow|e\rangle$) with vacuum Rabi
frequency $2g$. As $\Omega_{P}$ is slowly increased from zero, the ion
adiabatically follows the dark state of the Hamiltonian starting from
$|i\rangle$ to $|f\rangle$. Because the excited state $|e\rangle$ is not
populated, spontaneous emission does not occur. The photon collection
probability $P_{cavity}$ is $2C/(2C+1)$. If used together with the fiber
cavity geometry discussed above, $P_{cavity}\approx 0.56$ and
$P_{fiber}\approx 0.40$ can be reached.

If one is interested in an ideal single photon source, STIRAP is a better
choice because of its larger photon collection efficiency per trial.
Furthermore, in contrast to the fast excitation, the pulse shape of the
emitted photon can be controlled by $\Omega_{P}(t)$. For the protocol to
generate entangled ion pair based on entanglement swapping between a photon
pair, successful generation rate of the photon pair is the relevant figure of
merit, and is proportional to $P_{fiber}^{2}\cdot R_{rep}$, where $R_{rep}$ is
repetition rate of excitation process. Therefore in addition to the
availability of the necessary atomic states, the optimal excitation method
depends on the details of the experimental setup, such as the time required to
prepare the initial state as well as cavity QED parameters.

To estimate the time required to complete a single STIRAP photon generation,
we need to add the time necessary to prepare the initial state $|i\rangle$ as
well as the time for the adiabatic evolution and/or emission. Adiabaticity
requires that $\dot{\Omega}_{P}\ll g^{2}$, and if $\Omega_{P}$ increases
linearly over time $T$, $T\gg 1/g$. In the strong coupling regime
($g\gg\kappa,\gamma$), slow ramping of $\Omega_{P}$ guarantees that the
quantum state of the system will be adiabatically transferred from
$|i,n=0\rangle$ to $|f,n=1\rangle$ without spontaneous emission where $n$ is
the number of photons in the cavity mode. The single photon then leaves the
cavity with a rate of $\kappa$. The Rabi rate $\Omega_{P}$ should remain at
its maximum value until the single photon leaves the cavity. Therefore with
strong coupling, the total photon generation time includes
$\mathcal{O}(1/\kappa)$ as well as time necessary for adiabatic process ($\gg
1/g$). In the weak coupling regime, the cavity decay process will occur much
faster than the adiabatic transfer, and therefore the minimum time is mainly
limited by $T\gg 1/g$. Because Yb ion-cavity system lies in weak coupling
regime, STIRAP process will require much more time to extract a photon
compared to fast excitation case. In the rest of this paper, we will mainly
consider fast excitation to compare the cavity result with high NA optics in
the following section.

##  III Collection of photons with high numerical aperture optics

Figure 4: Definitions of coordinate systems and symbols with respect to
spherical mirror. (a) Ion is located at the origin of coordinate system, and
polar (cylindrical) coordinates are used to describe the optical field before
(after) reflection. $\theta$ is the polar angle with respect to
$\hat{z}$-axis, and $\phi$ is azimuthal angle from $\hat{x}$-axis. (b)
$\theta_{max}$ is the maximum half opening angle corresponding to the mirror
aperture. $h$ is a distance to the analysis plane from the center of sphere.
Surface ion trap can be fabricated on the substrate area surronding mirror.

As an alternative method to collect light from a single trapped ion, we
consider the use of high NA collection optics whose geometric aberrations are
well-corrected to achieve diffraction limited coupling performance into a
single mode fiber. Reflective optics is frequently used as an alternative to
refractive optics to realize a large NA Maiwald  _et al._ (2009); Noek  _et
al._ (2010); Shu  _et al._ (2009, 2010); Sondermann  _et al._ (2007); Luo  _et
al._ (2009); Lindlein  _et al._ (2007). A parabolic mirror can perfectly
collimate light emitted from a point source, and has been studied extensively
Sondermann  _et al._ (2007); Luo  _et al._ (2009); Lindlein  _et al._ (2007).
In this section we will investigate the properties of light reflected by a
spherical mirror and numerically calculate the collection probability into a
single mode fiber. Although the analysis is presented for reflective optics,
the general conclusions apply to other approaches if similar NA and aberration
control can be realized.

###  III.1 Dipole emission and Gaussian mode

Distinguishing light from $\pi$ and $\sigma$ transitions is essential for
achieving high fidelities in remote entanglement protocols. Therefore, the
modes of photons from $\pi$ and $\sigma$ transitions have to be orthogonal
after collection. For large numerical aperture, this is only possible if we
choose $\vec{\it{B}}$ parallel to the optical axis $\hat{z}$. The optical
fields of photons emitted by the ion via the three possible transitions are
given by Luo  _et al._ (2009)

$\displaystyle\vec{E}_{\Delta l=1,\Delta m=0}$ $\displaystyle=$
$\displaystyle\frac{ie^{ikr}}{r}\sqrt{\frac{3}{8\pi}}\sin\theta\hat{\theta},~{}~{}$
$\displaystyle\vec{E}_{\Delta l=1,\Delta m=\pm 1}$ $\displaystyle=$
$\displaystyle\frac{ie^{ikr}}{r}e^{\pm
i\phi}\sqrt{\frac{3}{16\pi}}(\pm\cos\theta\hat{\theta}+i\hat{\phi}).$ (6)

Once it is reflected by a curved surface, spherical waves from an ion become
paraxial waves, and the reflected optical field will be described in
cylindrical coordinates shown in Figure 4 (a). If the curved surface is
parabolic and the trapped ion is located at the focus of paraboloid, all the
reflected rays propagate parallel to the $\hat{z}$-axis and the traveling
wavefronts are flat.

For a spherical mirror, most of the reflected rays propagate with radially
varying phase delay and non-zero angle with respect to the $\hat{z}$-axis due
to spherical aberration, even if the ion is trapped at the nominal focus
(midpoint between the center of sphere and the vertex of the spherical
surface). In this case, rays cross each other and the optical field
distribution in $\rho$-coordinate changes as it propagates along the
$\hat{z}$-direction [Figure 4 (b)]. The probability of collecting a photon
into a single mode fiber can be estimated by numerically calculating the mode
matching factor [Eq. (5)] between the reflected optical field distribution
($\vec{R}_{l,m}$) incident on the fiber and the Gaussian fiber mode
($\vec{G}_{fiber}$) by setting $\vec{E}=\vec{R}_{l,m}$.

Luo et al. Luo  _et al._ (2009) analyzed the case of a parabolic mirror and
calculated the overlap of the reflected mode with a Gaussian mode. They showed
that the collection probability of a photon emitted by the $\pi$-transition
($\Delta l=1,\Delta m=0$) vanishes due to the rotational cylindrical symmetry
of polarization, while the collection efficiency for $\sigma$-transition
($\Delta l=1,\Delta m=\pm 1$) approaches 50 % as the mirror aperture
($\rho_{0}$) becomes much larger than the focal length ($f$) of the parabolic
mirror. Provided the axis of single mode fiber is collinear with the
quantization axis, a single mode fiber can be used as transition-selective
filter which maps photons generated by $\sigma^{+}$ and
$\sigma^{-}$-transitions into two orthogonal circular polarizations
($(\hat{x}\mp i\hat{y})/\sqrt{2}$ ) respectively, and rejects photons
generated by the $\pi$ transition. For the frequency qubit shown in Fig. 1
(d), it would be interesting to collect photons from $\pi$-transitions and
reject $\sigma$ transition photons Olmschenk  _et al._ (2009). We can
implement this orthogonal filter by adding an additional $\phi$-dependent
phase ($e^{i\phi}$) to the optical field after it is reflected by the mirror
[Eq. (33) and (35) of Ref. Luo  _et al._ (2009)]. A phase that depends on the
azimuthal angle can be added using computer generated holograms Heckenberg
_et al._ (1992); Arlt  _et al._ (1998), diffractive optics Khonina  _et al._
(2000); Kennedy  _et al._ (2002), phase plates Beijersbergen  _et al._ (1994);
Sueda  _et al._ (2004), or spatial light modulators Leach  _et al._ (2004);
Ren  _et al._ (2010). Therefore we assume that whenever we intend to collect
light from $\pi$-transitions, an additional phase factor $e^{i\phi}$ is added
to the (near-) collimated field.

Figure 5: Maximum collection probability of photons emitted by a
$\sigma$-transition ($\Delta l=1,\Delta m=\pm 1$) and a $\pi$-transition
($\Delta l=1,\Delta m=0$). For the $\pi$-transition calculation, we assume
that phase $e^{i\phi}$ is added to the collimated field. We also plotted the
amount of light collimated by parabolic mirror before being collected by
single mode fiber. $\rho_{0}$ is the radius of mirror aperture and $f$ is the
focal length of the parabolic mirror. Inset: Spherical probability
distribution of photons emitted by different transitions with respect to the
quantization axis. Distance from the origin is proportional to the probability
of photons propagating in the corresponding angle.

The collection probabilities of light from a single ion onto a large area
detector (referred to as “collimated case”) and into a single mode fiber
(“fiber coupled case”) are shown in Figure 5 as a function of the mirror
aperture radius ($\rho_{0}$) for both $\sigma$\- and $\pi$-transitions. When
$\rho_{0}$ is smaller than $f$, the fiber coupling efficiency of
$\sigma$-transition photons is much larger than that of $\pi$-transition
photons due to the dipole emission pattern [inset in Figure 5]. When
$\rho_{0}$ is much larger than $f$, spatial overlap with fiber mode becomes
less sensitive to the details of the emission pattern but the spatially
varying polarization distribution limits the overall coupling efficiencies of
both transitions near 50 %.

Figure 6: Matching collimated ray distribution to Gaussian beam with finite
wavefront radius of curvature. For clarity, figures are not drawn to scale.

The integration of an ion trap with large aperture parabolic mirrors has been
considered Maiwald  _et al._ (2009), In this work, we will concentrate our
analysis on the collection probability one can achieve with a spherical
mirror, which can be integrated with a surface trap, as shown schematically in
Figure 4 (b) Noek  _et al._ (2010). As spherical mirrors feature substantial
aberrations at large NA, this analysis also provides strategies for
effectively managing aberrations present in most practical collection optics.
When the ion is trapped at the nominal focus of the spherical mirror, the rays
collimated by a spherical mirror are not perfectly parallel to the optical
axis but converge initially and eventually diverge after crossing the optical
axis (Figure 6). The resulting far-field wavefronts are close to spherical
with residual optical path difference (OPD) as shown in Fig. 6. To estimate
the probability of coupling this mode to an optical fiber, we calculate the
mode overlap to a Gaussian mode with spherical wavefronts:

$\vec{G}(\rho,\phi,z)=\frac{1}{w}\sqrt{\frac{2}{\pi}}\exp\left(i\psi\right)\exp\left(-\frac{\rho^{2}}{w^{2}(z)}\right)\exp\left(\frac{ik\rho^{2}}{2R(z)}\right)(\alpha\hat{x}+\beta\hat{y}),$
(7)

where $R(z)$ is the radius of curvarture (RoC) of the wavefront, $w(z)$ is the
beam radius at the measurement plane, and $\psi=\arctan(-z/2z_{R})$ is Gouy
phase. The residual OPD between the wavefront of collimated light field and
the spherical wavefronts of the Gaussian mode limits the maximum coupling
efficiency one can achieve with this method. The coupling efficiency can be
recovered to values similar to the ideal parabolic case if the residual OPD is
reduced to below the diffraction limit.

###  III.2 Design examples

The residual OPD (expressed in units of wavelength $\lambda$ = 369.5 nm) of
the rays reflecting off a micromirror with RoC $R$ = 160 $\mu$m, measured at a
location $h$ = 50 mm compared to a best-matching Gaussian mode is plotted in
Figure 7. The ion is assumed to be located at the focal point of the spherical
mirror. The two horizontal gray dashed lines show the limits of Rayleigh
criterion representing the diffraction limit (OPD $\leq\pm\lambda/4$), below
which residual OPD does not lead to reduced coupling efficiency Barakat
(1965). The black dashed line shows the residual OPD for a spherical
micromirror with opening angle ($\theta_{max}$) of 32∘, where 10.5 % of
$\sigma$-transition light falls within the aperture and is collimated by the
mirror. In this case, the OPD remains within the Rayleigh criterion, and a
numerical calculation estimates that 6.2 % of photons from
$\sigma$-transitions can be matched to a single mode fiber. In contrast, the
light field generated by the $\pi$-transition has little overlap with a
Gaussian mode for $\theta_{max}=32^{\circ}$ due to the emission pattern.

Figure 7: Example of residual OPD expressed in units of wavelength
($\lambda=369.5$ nm). The RoC of matching wavefront is chosen to maximize the
mode overlap rather than minimizing the maximum OPD. According to Rayleigh
wavefront criteria Barakat (1965), maximum OPD should be less than $\lambda/4$
to avoid the effect of non-zero OPD and this range is marked by two dashed
lines. The dark dashed line and dotted line shows the residual OPD (measured
at a location $h$ = 50 mm) for the rays reflected by a spherical mirror ($R$ =
160 $\mu$m) with $\theta_{max}$ = 32∘ and $\theta_{max}$ = 48∘, respectively.
In the inset, thickness variation of phase plate is plotted, designed to
compensate OPD shown by the dotted line. 500$\mu m$-thick fused silica
substrate is assumed in this design. With this phase correcting plate placed
at $h$ = 50 mm, residual OPD at $h$ = 50 mm is dramatically suppressed. Solid
line is OPD of phase corrected wavefront measured at $h$ = 100 mm, which shows
that OPD barely changes for additional propagation once OPD is compensated.

When $\theta_{max}$ is extended to $48^{\circ}$ to increase the amount of
collimated light from $\pi$-transition to 7.3 %, the maximum OPD becomes
larger than $\lambda$ due to spherical aberrations (dotted line in Figure 7)
and the maximum coupling efficiency is still limited to 0.4 %. The residual
OPD must be compensated in order to recover the fiber coupling efficiency. If
the distribution of the OPD is known, one can utilize a phase plate
Beijersbergen  _et al._ (1994); Sueda  _et al._ (2004) or a spatial light
modulator Leach  _et al._ (2004); Ren  _et al._ (2010) to introduce
compensating OPD to cancel out the residual OPD. Inset of Figure 7 shows an
example of a phase plate which can be fabricated on a 500 $\mu$m-thick fused
silica substrate to compensate the residual OPD for the $\pi$-transition case.
Using this phase corrector plate, the coupling efficiency into the fiber mode
can be enhanced to 3.0 %. We used $h$ = 50 mm as a typical location to place
phase correcting optics, but the general conclusion is independent of where
the mode matching is calculated. We also explored optimization of ion height
along the optical axis, and concluded that no substantial improvement can be
achieved.

Figure 8: Fiber coupling efficiency of a photon emitted by $\sigma$\- and
$\pi$-transition using a spherical mirror with a phase corrector plate when
$R=160\mu m$. We also plotted fiber coupling efficiency by parabolic mirror
for comparison. This plot shows that most of aberration of spherical mirror
can be corrected by phase plate when $\theta_{max}<50^{\circ}$.

Figure 8 shows the probability of coupling emitted photons into a single mode
fiber using a spherical mirror and an optimized phase corrector plate for each
value of $\theta_{max}$. With optimum phase correction, the collection
probability as a function of the opening angle $\theta_{max}$ of a spherical
mirror grows similar to that of a parabolic mirror until $\theta_{max}$
reaches $\sim 50^{\circ}$ where matching of slightly diverging rays to a
Gaussian mode starts to fail. Beyond this limit, we need to correct the angle
of collimated rays in addition to the phase which is beyond the scope of this
paper. Likewise, $\pi$-transition coupling efficiency could be improved to
over 3.0 % from 0.4 %, approaching the parabolic mirror limit if all
aberrations are properly compensated.

Use of a microfabricated mirror integrated under a surface trap has been
proposed for efficient collection of emitted photons for qubit state detection
Noek  _et al._ (2010). In that approach, however, the aberrations of the
microfabricated mirror cannot be controlled well enough to produce a flat
wavefront. One important advantage of using a microfabricated mirror compared
to a conventional macroscopic mirror Shu  _et al._ (2010) is that the
geometric aberrations scale as the dimensions of the optics are reduced Brady
and Hagen (2009). The residual OPD is reduced while the wavelength ($\lambda$)
remains the same eventually achieving diffraction limit. Table 1 summarizes
photon coupling efficiency using a single spherical mirror with the same large
NA ($\theta_{max}$ = 48∘) without any additional aberration correction. Last
column for RoC = 16 $\mu$m shows that there will be a significant improvement
if the dimensions of the mirror and the position of the ion are shrunk by a
factor of 10. For such micro optics, the detailed shape of the reflector has
little impact on the collection efficiency. On the other hand, if the
dimensions of mirror are increased by a factor of 10, the collection
probability will vanish unless there is proper phase compensating optics
(first column). Therefore the microfabricated mirror with RoC $\ll$ 100 $\mu$m
can increase the fiber collection efficiency drastically. However due to
technical limitations of the ion height, it is impractical to use mirrors with
RoC much smaller than $\sim$ 100 $\mu$m at present day.

Mirror RoC ($\mu$m) | 1600 | 160 | 16
---|---|---|---
Transition | $\sigma$ | $\pi$ | $\sigma$ | $\pi$ | $\sigma$ | $\pi$
Collimated light by mirror (%) | 21.2 | 7.3 | 21.2 | 7.3 | 21.2 | 7.3
Fiber coupling probability (%) |  $<$0.1 |  $<$0.1 | 6.2 | 0.4 | 15.8 | 2.8
Table 1: Estimation of fiber coupling efficiency of $\sigma$ and $\pi$
transition light by ideal spherical mirrors of different dimensions.
$\theta_{max}$ is set to 48∘. In our simulation, we added an ideal hyperbolic
lens in the beam path, which can transform the matched Gaussian mode into UV
fiber mode ($w_{0}\sim$ 1.54 $\mu$m) and performed full ray tracing from the
ion position to the fiber plane to evaluate the mode overlap integral.

###  III.3 Alternative approaches

The analysis in this section shows that efficient coupling of light into a
single mode fiber can be achieved provided one can correct the aberration of a
large NA collection optics to conform to the wavefront of a Gaussian beam, to
below the diffraction limit. Practical means to realize such high NA optics
with well-corrected aberration include (1) design of multi-element refractive
lenses, (2) fabrication of paraboloidal mirrors (using e.g. diamond turning
technology), (3) spherical mirror with corrector plates to compensate for the
residual aberration, and (4) Fresnel reflectors or diffractive optical
components.

Fresnel lenses have been investigated as a way to collect light with large NA
without spherical aberration Cruz-Cabrera  _et al._ (2007); Streed  _et al._
(2011) because it adds the necessary phase shift to a continuous wavefront.
Using the similar design principle, we can consider integrating Fresnel
reflector with an ion trap in place of the micro mirror shown in Figure 4 (b).
One critical consideration for Fresnel diffractive optical elements is the
diffraction efficiency that depends on the grating periods Cruz-Cabrera  _et
al._ (2007). The Fresnel reflector deserves more careful analysis for use in
aberratioin-free fiber coupling applications.

##  IV Application to remote entanglement generation

Based on our estimation of photon coupling efficiency into a single mode
fiber, we can calculate the expected entangled ion pair generation rate
considering practical experimental constraints. Table 2 compares the expected
entanglement generation rate of various approaches with existing experimental
results. Matsukevich et al. Matsukevich  _et al._ (2008) generated entangled
ion pairs by using polarization qubit [Figure 1 (b)]. Because state
preparation step requires only optical pumping into ${}^{2}S_{1/2}|0,0\rangle$
without any qubit rotation Olmschenk  _et al._ (2007), the same procedure can
be repeated at 520 kHz repetition rate, and the entangled pair generation rate
is analyzed in the first column of Table 2. When the remote entanglement is
established by coincident detection of a photon pair, the ionic qubit is
encoded in the Zeeman substates ${}^{2}S_{1/2}|1,\pm 1\rangle$ which are
susceptible to magnetic field fluctuation and therefore these two states
should be transferred to atomic clock states ${}^{2}S_{1/2}|\pm 1,0\rangle$ by
microwave transitions or Raman transitions. The photonic polarization qubit is
also susceptible to decoherence from birefringence in the vacuum window,
downstream optics and single mode fibers. Therefore this method requires
careful control of potential decoherence mechanisms to achieve high fidelity.
Olmschenk et al. Olmschenk  _et al._ (2009) used the first-order magnetic
field-insensitive hyperfine clock state qubit accompanied by $\pi$ transition
as shown in Figure 1 (d), and collected emitted photons from a direction
orthogonal to the quantization axis to maximize collection of photons
generated by $\pi$ transition [inset of Figure 5]. To generate ion-photon
entangled state, the ion should be prepared in a superposition state before
excitation, which requires rotation of clock states using microwave transition
and limits the overall repetition rate to 75 kHz. The analysis of this
experiment is listed in the second column of Table 2. The limitation of this
collection geometry is the increased overlap between different polarizations
from $\sigma$ and $\pi$ transitions, which reduces fidelity as NA is increased
Olmschenk  _et al._ (2009).

Numerical aperture | 0.23 | 0.6 | cavity
---|---|---|---
Optic axis vs. B-field | $\parallel$ | $\perp$ | $\parallel$ | $\parallel$ | $\parallel$ | $\parallel$
Type of atomic transition | $\sigma$ | $\pi$ | $\sigma$ | $\pi$ | $\sigma$ | $\sigma$
Photonic qubit | Pol. | Freq. | Pol. | Freq. | Freq. | Pol.
Detector efficiency | 0.2 | 0.2 | 0.3 | 0.3 | 0.3 | 0.3
Decay via right transition | 2/3 | 1/3 | 2/3 | 1/3 | 2/3 | n.a.
Collected fraction | 0.0198 | 0.0198 | 0.136 | 0.028 | 0.136 | 0.337
Mode overlap | 0.82 | 0.82 | 0.85 | 0.32 | 0.85 | 0.95
Misc. efficiency | 0.26 | 0.25 | 0.63 | 0.62 | 0.63 | 0.63
Single Photon collection efficiency (%) | 0.0554 | 0.0269 | 1.46 | 0.0547 | 1.46 | 6.04
Bell state identification | 0.25 | 0.25 | 0.5 | 0.5 | 0.5 | 0.5
Coincidence efficiency | $7.7\times 10^{-8}$ | $1.8\times 10^{-8}$ | $1.1\times 10^{-4}$ | $1.5\times 10^{-7}$ | $1.1\times 10^{-4}$ | $1.8\times 10^{-3}$
Repetition rate (kHz) | 520 | 75 | 500 | 75 | 75 | 500
Entanglement rate (Hz) | 0.04 | 0.0014 | 53 | 0.011 | 8 | 913
Table 2: Comparison of entanglement generation rate based on different setup.
Theoretically maximum two types of Bell states encoded in a photon pair can be
unambiguously identified from the others, and we include this identification
efficiency on the 5th row. When atom decays to the ground state, some of the
decay channels are undesired and we included the fraction of desired decay
channel in 7th row. Collected fraction corresponds to the fraction of
collimated light when the photons are collected with reflective or refractive
optics and $r_{t}\cdot P_{cavity}$ for the case of the cavity system.
Experimental condition of each column is described in the main text.

From the third to the fifth column in Table 2 shows the expected entangled ion
pair generation rates when the numerical aperture is increased to 0.6 using
either reflective optics or refractive optics discussed in Section III. We
considered several practical improvements that can be made over the existing
experiments. Although previous experiments Moehring  _et al._ (2007);
Matsukevich  _et al._ (2008); Olmschenk  _et al._ (2009) detected only the
singlet state out of four Bell states, one of triplet states can also be
distinguished from the other two. Therefore we assume that probability of Bell
state measurement is 1/2 rather than 1/4. In previous experiments, various
losses in photon collection efficiency caused by non-ideal optical components
further reduced the overall efficiency by a factor of $\sim$0.25. We estimate
that we can use better optical components to increase this efficiency to more
than 60 % based on realistic analysis. Third column analyzes experimental
setup similar to the first column Matsukevich  _et al._ (2008) when the
numerical aperture is increased to 0.6. This setup can achieve the highest
expected generation rate among the three cases, but requires decoherence
control similar to the first column setup. Forth column lists values we can
achieve using similar setup to the second column Olmschenk  _et al._ (2009),
but we assume that optic axis of collection optics is parallel to the
quantization axis compared to orthogonal setup used in the second column. As
was discussed in section III, if a phase plate is used in the parallel
configuration, only the $\pi$ transition can be coupled into a single mode
fiber and the fidelity will remain high even with very large numerical
aperture. However, NA of 0.6 is not large enough to show significant
improvement due to non-ideal emission pattern of $\pi$-transition in this
collection direction. Fifth column estimates the generation rate which can be
achieved with a scheme in Figure 1 (c) Moehring  _et al._ (2007). This
approach has advantages of high fidelity due to frequency qubit and $\sigma$
transition which allows high collection efficiency. However, to achieve near
unity excitation probability, the initial state should be prepared to either a
superposition of the clock states or one of ${}^{2}S_{1/2}|1,\pm 1\rangle$,
which limits the repetition rate around 75 kHz.

The last column shows the entanglement generation rate which can be achieved
by an optical cavity discussed in Section II. In schemes using optical
cavities, the FSR of cavity should be matched to either 12.6 GHz or 14.7 GHz
to collect photons encoding frequency qubits. These constraints lead to
minimum cavity length of $\sim$10 mm, precision control of RoC of the
spherical mirror down to $\mu$m level, and an upper limit on the cavity
finesse of $\sim$ 700 to cover 19.6 MHz linewidth of the ${}^{2}P_{1/2}$
state. Therefore frequency qubit is not compatible with optical cavity
schemes. However, we can achieve highest entanglement generation rate compared
to other approaches using polarization qubit or time-bin qubit. In this case,
the remaining challenge will be the minimization of decoherence similar to all
the other schemes involving the polarization qubit.

Based on the above analysis, collection of photons generated by
$\sigma$-transition along quantization axis is the optimal choice for both the
large NA system and the cavity system. Cavity system prefers polarization
qubit as the photonic qubit of choice, but it can achieve the highest
entanglement generation rate. For large NA system, frequency qubit has lower
generation rate than polarization qubit mainly due to slow initial state
preparation. In principle, this limitation can be overcome with fast microwave
transition or Raman transition. A custom lens system designed for optimal
chamber configuration can achieve a NA of more than 0.6 with diffraction-
limited performance. Even though the maximum collection probability with free
space optics is smaller, free space optics has the advantage that it can be
realized with off-the-shelf technology today. Therefore, $\sigma$-transition-
based frequency qubit system with large numerical aperture lens can be a good
compromise between high fidelity and high throughput.

##  V Conclusions

In this paper, we discussed various ways to enhance collection of photons
emitted by a single atomic ion into a single-mode optical fiber. Based on this
analysis, we conclude that the generation rate of entangled ion pairs through
entanglement swapping of photons can be dramatically enhanced beyond the
current values using either a high NA collection optics or a carefully
designed optical cavity. Such enhancement is expected to enable the
realization of scalable quantum multi-processor and quantum repeaters based on
nodes of trapped ions connected via photonic network.

This research was funded by Office of the Director of National Intelligence
and Intelligence Advanced Research Projects Activity through Army Research
Office.

##  Appendix A Cavity with two concave mirrors

In this Appendix, we estimate the maximum collection efficiency one can
achieve with a cavity made of two curved mirrors with an ion trapped in the
center. Compared to the fiber cavity, ion can be trapped at the beam waist of
the cavity mode, and we can estimate the lower bound and the upper bound of
$P_{cavity}$. In the near-concentric limit, we estimate the optimal $w_{0}$ to
be $0<w_{0}<\sqrt{\lambda/\pi}\sqrt[4]{\text{RoC}\cdot\lambda/2}$, which
corresponds to $1>1/(1+(w_{0}/r_{c})^{2})>0.87$. Combined with
$1/(1+\text{2RoC}/l_{c})\approx 0.53$ which is possible with symmetric
geometry, we can estimate that $0.53>P_{cavity}>0.46$ which is comparable to
$P_{cavity}\approx$ 0.41 of the fiber cavity. Mode matching has to be
performed to couple these photons into a single mode fiber, and it is not easy
to integrate this geometry with surface traps.

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