# Quantum repeaters with individual rare-earth ions at telecommunication
wavelengths

F. Kimiaee Asadi  Institute for Quantum Science and Technology, and Department
of Physics & Astronomy, University of Calgary, 2500 University Drive NW,
Calgary, Alberta T2N 1N4, Canada  N. Lauk  Institute for Quantum Science and
Technology, and Department of Physics & Astronomy, University of Calgary, 2500
University Drive NW, Calgary, Alberta T2N 1N4, Canada  S. Wein  Institute for
Quantum Science and Technology, and Department of Physics & Astronomy,
University of Calgary, 2500 University Drive NW, Calgary, Alberta T2N 1N4,
Canada  N. Sinclair  Institute for Quantum Science and Technology, and
Department of Physics & Astronomy, University of Calgary, 2500 University
Drive NW, Calgary, Alberta T2N 1N4, Canada  C. O’Brien  Institute for Quantum
Science and Technology, and Department of Physics & Astronomy, University of
Calgary, 2500 University Drive NW, Calgary, Alberta T2N 1N4, Canada  C. Simon
Institute for Quantum Science and Technology, and Department of Physics &
Astronomy, University of Calgary, 2500 University Drive NW, Calgary, Alberta
T2N 1N4, Canada

###### Abstract

We present a quantum repeater scheme that is based on individual erbium and
europium ions. erbium ions are attractive because they emit photons at
telecommunication wavelength, while europium ions offer exceptional spin
coherence for long-term storage. Entanglement between distant erbium ions is
created by photon detection. The photon emission rate of each erbium ion is
enhanced by a microcavity with high Purcell factor, as has recently been
demonstrated. Entanglement is then transferred to nearby europium ions for
storage. Gate operations between nearby ions are performed using dynamically
controlled electric-dipole coupling. These gate operations allow entanglement
swapping to be employed in order to extend the distance over which
entanglement is distributed. The deterministic character of the gate
operations allows improved entanglement distribution rates in comparison to
atomic ensemble-based protocols. We also propose an approach that utilizes
multiplexing in order to enhance the entanglement distribution rate.

##  1 Introduction

Entanglement is a key requirement for many applications of quantum science.
These include, for example, quantum key distribution [Jennewein and
Higgins(2013)], global clock networks [Komar  _et al._(2014)Komar, Kessler,
Bishof, Jiang, Sørensen, Ye, and Lukin], long-baseline telescopes [Gottesman
_et al._(2012)Gottesman, Jennewein, and Croke], and the quantum internet
[Kimble(2008), Simon(2017)]. However, due to transmission loss the direct
transmission of entanglement over distances of more than several hundred
kilometers is practically impossible using current technology. The use of a
quantum repeater has been suggested to reduce (or eliminate) the impact of
loss in order to establish entanglement between distant locations. [Briegel
_et al._(1998)Briegel, Dür, Cirac, and Zoller]. In many quantum repeater
schemes, entanglement is first distributed between two locations that are
separated by a short distance, referred to as an elementary link. Then, the
range of entanglement is extended to successively longer distances by
performing entanglement swapping operations between the entangled states that
span each elementary link. Due to the availability and diversity of component
systems and the strong light-matter coupling offered by atomic ensembles, many
quantum repeater proposals use sources of entanglement, ensemble-based quantum
memories, and linear optics-based entanglement swapping operations [Sangouard
_et al._(2011)Sangouard, Simon, de Riedmatten, and Gisin]. However, the
success probability of linear optics-based entanglement swapping (without
auxiliary photons) cannot exceed 50%, which has a compounding effect for more
complex quantum networks. The use of auxiliary photons to improve the
entanglement swapping probability is possible [Grice(2011), Wein  _et
al._(2016)Wein, Heshami, Fuchs, Krovi, Dutton, Tittel, and Simon], but adds
complexity and compounds errors, thereby restricting their use in practice.
Single-emitter-based quantum repeaters, on the other hand, offer the
possibility to outperform ensemble-based repeaters by using deterministic
swapping operations [Sangouard  _et al._(2009)Sangouard, Dubessy, and Simon].
Impressive demonstrations of certain parts of a single-emitter quantum
repeater schemes have been performed using atom-cavity systems [Ritter  _et
al._(2012)Ritter, Nolleke, Hahn, Reiserer, Neuzner, Uphoff, Mucke, Figueroa,
Bochmann, and Rempe, Reiserer and Rempe(2015)], color centers in diamond
[Bernien  _et al._(2013)Bernien, Hensen, Pfaff, Koolstra, Blok, Robledo,
Taminiau, Markham, Twitchen, Childress _et al._, Hensen  _et al._(2015)Hensen,
Bernien, Dreau, Reiserer, Kalb, Blok, Ruitenberg, Vermeulen, Schouten,
Abellan, Amaya, Pruneri, Mitchell, Markham, Twitchen, Elkouss, Wehner,
Taminiau, and Hanson], trapped ions [Moehring  _et al._(2007)Moehring, Maunz,
Olmschenk, Younge, Matsukevich, Duan, and Monroe, Slodička  _et
al._(2013)Slodička, Hétet, Röck, Schindler, Hennrich, and Blatt], donor qubits
in silicon [Morse  _et al._(2017)Morse, Abraham, DeAbreu, Bowness, Richards,
Riemann, Abrosimov, Becker, Pohl, Thewalt _et al._], as well as quantum dots
[Delteil  _et al._(2016)Delteil, Sun, Gao, Togan, Faelt, and Imamoglu,
Stockill  _et al._(2017)Stockill, Stanley, Huthmacher, Clarke, Hugues, Miller,
Matthiesen, Le Gall, and Atatüre].

Over the years, crystals doped with rare earth (RE) ions have attracted
considerable attention for their use in electromagnetic signal processing
applications that range from quantum memories [Tittel  _et al._(2010)Tittel,
Afzelius, Chaneliere, Cone, Kröll, Moiseev, and Sellars, Bussières  _et
al._(2013)Bussières, Sangouard, Afzelius, de Riedmatten, Simon, and Tittel] to
biological imaging [Wu  _et al._(2009)Wu, Han, Milliron, Aloni, Altoe,
Talapin, Cohen, and Schuck]. Narrow optical and spin homogeneous linewidths,
their convenient wavelengths as well as their ability to be doped into solid-
state crystals are some of the most desired properties of RE ions. Compared to
nitrogen-vacancy centers in diamond [Chu  _et al._(2014)Chu, de Leon, Shields,
Hausmann, Evans, Togan, Burek, Markham, Stacey, Zibrov, Yacoby, Twitchen,
Loncar, Park, Maletinsky, and Lukin] and quantum dots [Brunner  _et
al._(2009)Brunner, Gerardot, Dalgarno, Wüst, Karrai, Stoltz, Petroff, and
Warburton], they also exhibit smaller spectral diffusion [Neuhauser  _et
al._(2000)Neuhauser, Shimizu, Woo, Empedocles, and Bawendi, de Sousa and Das
Sarma(2003), Xia  _et al._(2015)Xia, Kolesov, Wang, Siyushev, Reuter, Kornher,
Kukharchyk, Wieck, Villa, Yang, and Wrachtrup], and have a reduced sensitivity
to phonons.

Among the different RE ions, Er3+ is attractive due to its well-known optical
transition (around 1.5 $\mu$m) in the conventional telecommunication
wavelength window, in which absorption losses in optical fibers are minimal.
Another unique aspect of certain RE ions is the presence of hyperfine levels
that feature long lifetimes, which allows for long-term quantum state storage
[Lauritzen  _et al._(2010)Lauritzen, Minář, De Riedmatten, Afzelius,
Sangouard, Simon, and Gisin, Afzelius  _et al._(2010)Afzelius, Usmani, Amari,
Lauritzen, Walther, Simon, Sangouard, Minář, De Riedmatten, Gisin _et al._,
Heinze  _et al._(2013)Heinze, Hubrich, and Halfmann, Laplane  _et
al._(2017)Laplane, Jobez, Etesse, Gisin, and Afzelius]. In particular, a
coherence lifetime of six hours was reported in a europium-doped yttrium
orthosilicate crystal (151Eu3+:Y2SiO5) [Zhong  _et al._(2015a)Zhong, Hedges,
Ahlefeldt, Bartholomew, Beavan, Wittig, Longdell, and Sellars]. Motivated by
these properties, we propose and analyze a quantum repeater protocol that is
based on individual RE ions in which Er3+ ions are used to establish
entanglement over elementary links and 151Eu3+ ions are employed to store this
entanglement.

One disadvantage of RE ions is their weak light-matter coupling, which has
mostly precluded their use as single quantum emitters. However, optical
detection and addressing of single RE ions has recently been shown by multiple
groups [Zhong  _et al._(2018)Zhong, Kindem, Bartholomew, Rochman, Craiciu,
Verma, Nam, Marsili, Shaw, Beyer _et al._, Dibos  _et al._(2018)Dibos, Raha,
Phenicie, and Thompson, Kolesov  _et al._(2012)Kolesov, Xia, Reuter, Stöhr,
Zappe, Meijer, Hemmer, and Wrachtrup, Yin  _et al._(2013)Yin, Rancic, de Boo,
Stavrias, McCallum, Sellars, and Rogge, Utikal  _et al._(2014)Utikal,
Eichhammer, Petersen, Renn, Götzinger, and Sandoghdar]. Moreover, very
recently Dibos et al. [Dibos  _et al._(2018)Dibos, Raha, Phenicie, and
Thompson] demonstrated an enhancement in the emission rate of a single Er3+
ion in Y2SiO5 by a factor of more that 300 using a silicon nanophotonic
cavity. Strong coupling of ensembles of Nd ions was previously demonstrated
using nanophotonic cavities fabricated from Y2SiO5 [Zhong  _et
al._(2015b)Zhong, Kindem, Miyazono, and Faraon] and yttrium orthovanadate
[Zhong  _et al._(2017)Zhong, Kindem, Bartholomew, Rochman, Craiciu, Miyazono,
Bettinelli, Cavalli, Verma, Nam, Marsili, Shaw, Beyer, and Faraon] hosts. In
light of these results, for our scheme we propose to couple single RE ions to
a high-finesse cavity in order to enhance the light-matter coupling and thus
to increase the collection efficiency as well as the indistinguishability of
the emitted single photons.

The paper is organized as follows. In Sec. 2 we introduce our proposal and
discuss the required components as well as the underlying principles. The
entanglement distribution rates and possible implementations are discussed in
Secs. 3 and 4. We conclude and provide an outlook in Sec. 5.

##  2 Quantum repeater protocol

The goal of our repeater scheme is to generate entanglement between nodes that
are separated by a long distance by swapping entanglement that is established
between nodes that are separated by smaller distances. Each node consists of a
Y2SiO5 crystal containing an Er3+ ion that features an enhanced emission rate
of its telecommunication transition due to a high-finesse cavity, and a nearby
151Eu3+ ion that acts as a quantum memory due to its long spin coherence
lifetime. Entanglement is transferred, or gates are performed, between nearby
atoms using electric dipole-dipole coupling [Longdell and Sellars(2004),
Longdell  _et al._(2004)Longdell, Sellars, and Manson]. Our proposal has some
similarity to a scheme that involves the excitation of electron spins in
nitrogen-vacancy centers to generate entanglement between nodes, nuclear spins
of carbon atoms for storage, and gates that are performed using a magnetic
dipole-dipole coupling [Reiserer  _et al._(2016)Reiserer, Kalb, Blok, van
Bemmelen, Taminiau, Hanson, Twitchen, and Markham]. We emphasize that our
(electric dipole) approach to implementing gates is quite different than
previous approaches, as are many of the physical properties of the system (in
addition to the emission wavelength).

In the first step, entanglement between pairs of neighboring Er3+ ions is
created (see Fig.1 (a)) which illustrates this step for the i-th and i+1-th
ions) by first creating local entanglement between the spin state of each ion
and a spontaneously-emitted single photon. Then, by performing a joint Bell-
state measurement (BSM) on photons that are spontaneously-emitted by
neighboring Er3+ ions, the spin state of these ions is projected onto a
maximally-entangled state. This procedure is the same as that used to create
entanglement between two remote nitrogen-vacancy centers in diamond [Barrett
and Kok(2005), Bernien  _et al._(2013)Bernien, Hensen, Pfaff, Koolstra, Blok,
Robledo, Taminiau, Markham, Twitchen, Childress _et al._].

Figure 1: In each cavity, the blue and black dots represent individual 151Eu3+
and Er3+ ions, respectively. In order to distribute entanglement to the nodes
that are situated at each end of a long channel, first a) entanglement between
two neighboring Er3+ ions is created via the detection of two photons; b) the
quantum state of each Er3+ ion is then mapped to a nearby 151Eu3+ ion using an
electric dipole interaction; c) using two-photon detection, Er3+ ions are used
again to establish entanglement between the other neighboring Er3+ ions.
Finally, we extend the entanglement distance by performing entanglement
swapping using an electric dipole interaction between nearby ions (red
circle). d) Flow-chart that depicts our scheme. e) Simplified energy level
structure of Er3+ and 151Eu3+ ions in the presence of an external magnetic
field. The electronic Zeeman levels of Er3+, and nuclear hyperfine levels of
151Eu3+, are split. We encode qubit states $\ket{\uparrow}$ and
$\ket{\downarrow}$ in the $m_{s}=-\frac{1}{2}$ and $m_{s}=\frac{1}{2}$
$\left(m_{I}=-\frac{3}{2}\text{ and }m_{I}=\frac{3}{2}\right)$ levels for the
Er3+ (151Eu3+) ion, respectively.

Next, the quantum state of each Er3+ ion is mapped to a nearby 151Eu3+ ion for
storage by exploiting a non-vanishing permanent electric dipole moment, a
common feature of many RE ions that are doped into solids. This mapping is
achieved using a mutual electric dipole-dipole interaction between close-lying
Er3+ and 151Eu3+ ions. The small nuclear magnetic moment of 151Eu3+ results in
a magnetic-dipole coupling that is orders of magnitude smaller than the
electric-dipole coupling, thus our scheme allows for much shorter gate
durations in comparison to those based on magnetic interactions [Reiserer  _et
al._(2016)Reiserer, Kalb, Blok, van Bemmelen, Taminiau, Hanson, Twitchen, and
Markham]. Another advantage of the proposed scheme is the ability to
dynamically control the interaction optically by bringing the ion to the
excited state and back to the ground state. This allows for the realization of
deterministic two-qubit gate operations.

The mapping allows the Er3+ ions to be re-initialized so that new elementary
links can be created between them. Fig.1(c) illustrates this processes for the
$\mathrm{(i-1)}$th and $\mathrm{(i)}$th nodes. Immediately after generating
entanglement between (the other) neighboring Er3+ ions, the entanglement
distance is extended by performing entanglement swapping between each of the
closely-lying Er3+ and 151Eu3+ ions. As a result, the outer nodes becoming
entangled. Fig. 1 (d) depicts a flow chart of our scheme. Photon loss and
errors might cause some steps in the protocol to fail.

As will be discussed in Sec. II A–C, the generation of entanglement between
Er3+ ions, the mapping of entanglement to the 151Eu3+ ions, and the
entanglement swapping steps, all rely on the detection of single photons that
are spontaneously emitted from the Er3+ ions. However, the lifetime of the
4I15/2$\leftrightarrow$ 4I13/2 telecommunication transition of Er3+:Y2SiO5 is
relatively long $T_{1}=11.4$ ms [Liu and Jacquier(2006)] with radiative
lifetime being even longer $T_{rad}=54$ ms [McAuslan  _et al._(2009)McAuslan,
Longdell, and Sellars], which necessitates the need for a high-finesse cavity
to enhance the emission rate. Nonetheless, the cavity will also significantly
increase the probability of collecting the emitted photons as compared to
emission into free space. We emphasize that the repetition rate of the
protocol (i.e., the number of attempts of the protocol per unit time) is
limited by the communication time between distant nodes, which means that
there is no need for very fast emission. Next we discuss each of the steps of
the protocol in detail.

###  2.1  Entanglement generation

Our scheme considers several remote cavities that each contain a single pair
of close-lying Er3+ and 151Eu3+ ions. Note that there may be other RE ions
within each cavity, but we assume that we can address a single such close-
lying pair. See also section4 below. An externally-applied magnetic field
splits the degenerate electronic ground levels of Er3+ via the Zeeman effect.
We refer to the resultant $m_{s}=-\frac{1}{2}$ and $m_{s}=\frac{1}{2}$ Zeeman
levels as the qubit states $\ket{\uparrow}$ and $\ket{\downarrow}$,
respectively (see Fig. 1 (e)).

To generate entanglement between distant Er3+ ions that are separated by a
long distance $L_{0}$, we follow the scheme of Barret and Kok [Barrett and
Kok(2005), Bernien  _et al._(2013)Bernien, Hensen, Pfaff, Koolstra, Blok,
Robledo, Taminiau, Markham, Twitchen, Childress _et al._] (see also Fig. 1
(d)). First, each Er3+ ion is prepared in one of the qubit states (here
$m_{s}=-\frac{1}{2}$; denoted $\ket{\uparrow}$). After this initialization
step, a $\frac{\pi}{2}$ microwave (MW) pulse rotates each Er3+ ion into a
superposition of $\ket{\uparrow}$ and $\ket{\downarrow}$ states. The
application of a brief laser pulse that is resonant with the $\ket{\uparrow}$
$\leftrightarrow$ $\ket{e_{2}}$ transition, followed by spontaneous emission,
will entangle each qubit state with the emitted photon number. That is, when
the qubit state is $\ket{\uparrow}$ $(\ket{\downarrow})$ there will be $1$
$(0)$ emitted photon(s). The spontaneously-emitted photons are then directed
to a beam splitter located in between the ions using optical fibers. A single-
photon detection at one of the two beam splitter output ports projects the
Er3+ ions onto an entangled state.

A possible loss in the fiber can lead to a situation where both Er3+ ions emit
a photon but only one photon is detected while the other is lost. In this
case, the ions are left in a product state rather than an entangled state. To
exclude this possibility, immediately after the first excitation-emission step
of each Er3+ ion, a $\pi$ MW pulse inverts each qubit state. Then a second
excitation pulse is applied. The detection of two consecutive single photons
at the beam splitter will leave the qubits in an entangled state:

$\ket{\psi}_{\text{Er}}^{\pm}=\frac{1}{\sqrt{2}}(\ket{\uparrow\downarrow}\pm\ket{\downarrow\uparrow}).$
(1)

Here the + (-) sign corresponds to the case in which the same (different)
detector(s) received a photon.

###  2.2 Controlled logic

After successful entanglement of the two distant Er3+ ions we transfer this
entanglement to the neighboring 151Eu3+ ions for a long-term storage.
Efficient entanglement mapping between neighboring rare-earth ions can be
employed by performing CNOT gate operations and single qubit rotations and
read-out. Here we first explain the underlying mechanism and general scheme
for how to implement a CNOT gate in our system, and in Sec.2.3 we will discuss
in more detail how to swap entanglement between the ions using this gate.

Figure 2: General pulse sequence of $\pi$-rotations to perform a controlled
logic gate between nearby Er3+ and 151Eu3+ ions. The numbers indicate the
sequential time ordering of the pulses. For this gate, the Er3+ spin state
acts as the control qubit and the 151Eu3+ spin state acts as the target. A
$\pi$ pulse excites the Er3+ ion if it is in the state $\ket{\uparrow}$. When
this occurs, pulses 2, 3 and 4 are not resonant with the 151Eu3+ ion, leaving
its state unaffected. Pulse 5 then brings the Er3+ ion back to its original
state. On the other hand, when Er3+ is in the state $\ket{\downarrow}$, pulses
1 and 5 will be ineffective and hence it will remain in the ground state.
Instead pulses 2, 3 and 4 will now be resonant with the 151Eu3+ ion and
(optically) swap its spin-state

Due to a lack of site symmetry when doped into a crystal, a RE ion can have a
permanent electric dipole moment that is different depending on whether the
ion is in its ground or optically excited state. The difference in the
permanent dipole moments affects the optical transition frequency of other
nearby RE ions via the Stark effect due to a modified local electric field
environment. It is possible to dynamically control the shift in the transition
frequency of one ion by optically exciting its neighboring ion. Based on this
interaction, we can perform a controlled-NOT (CNOT) operation between nearby
RE ions [Ohlsson  _et al._(2002)Ohlsson, Mohan, and Kröll]. The modification
of the transition frequency $\Delta\nu$ of a 151Eu3+ ion by an Er3+ ion due to
the mutual electric dipole-dipole interaction is given by [Altner  _et
al._(1996)Altner, Zumofen, Wild, and Mitsunaga]:

$\Delta\nu=\frac{\Delta\mu_{\text{Er}}\Delta\mu_{\text{Eu}}}{4\pi\epsilon\epsilon_{0}hr^{3}}\left(\left(\hat{\mu}_{\text{Er}}\\!\cdot\\!\hat{\mu}_{\text{Eu}}\right)-3\left(\hat{\mu}_{\text{Er}}\\!\cdot\\!\hat{r}\right)\left(\hat{\mu}_{\text{Eu}}\\!\cdot\\!\hat{r}\right)\right),$
(2)

where $r$ is the distance between the ions, and $\Delta\mu$ is the change of
the permanent electric dipole moment of each ion, $h$ is the Planck constant,
$\epsilon_{0}$ is vacuum permittivity, and $\epsilon$ is the dielectric
constant.

To perform a CNOT gate between the nearby Er (control) ion and Eu (target)
ion, a sequence of five $\pi$ pulses is applied (see Fig.(2)). First a $\pi$
pulse is applied to the Er3+ ion on resonance with the $\ket{\uparrow}$
$\leftrightarrow$ $\ket{e_{1}}$ transition. From here, two cases must be
considered: either the state of the Er3+ ion is $\ket{\uparrow}$ or it is
$\ket{\downarrow}$.

If the Er3+ ion is in state $\ket{\uparrow}$, it will be excited by pulse 1.
This changes the permanent electric dipole moment of the Er3+ ion, and thus
its local electric field. Consequently, the transition frequency of the nearby
151Eu3+ ion will be shifted by $\Delta\nu$. For the case that this frequency
shift is large enough (that is, the 151Eu3+ ion is sufficiently close to the
Er3+ ion), the 151Eu3+ ion will be unaffected by pulses 2, 3 and 4, thereby
remaining in its initial ground state (see Sec. 4.2 for more discussion).
Finally, pulse 5 will bring Er3+ ion back to its initial state. Hence, in this
case, the pulse sequence does not modify the initial state of the ion pair
system.

On the other hand, if the Er3+ ion is initially in $\ket{\downarrow}$, then
pulses 1 and 5 will have no effect on the Er3+ ion. Since the Er3+ ion will
not be excited, optical pulses 2, 3 and 4 will be resonant with the
transitions of the 151Eu3+ ion, and the pulse sequence will optically flip the
two spin states of the 151Eu3+ ion.

###  2.3 Entanglement mapping and distribution

Both the ground and excited states of 151Eu3+ have three doubly-degenerate
nuclear hyperfine levels. With the application of a magnetic field, each
doublet $\left(m_{I}=\pm\frac{1}{2},\pm\frac{3}{2},\pm\frac{5}{2}\right)$ will
be split. For our proposal, we denote the $m_{I}=-\frac{3}{2}$ and
$\frac{3}{2}$ hyperfine levels [Zhong  _et al._(2015a)Zhong, Hedges,
Ahlefeldt, Bartholomew, Beavan, Wittig, Longdell, and Sellars] as the 151Eu3+
ion qubit states, $\ket{\uparrow}$ and $\ket{\downarrow}$, respectively (see
Fig. 1 (e)). To map the state of each Er3+ ion onto a nearby Eu ion, we first
perform a CNOT gate between them. In our scheme, the 151Eu3+ ion is initially
prepared in one of the ground state levels (here $\ket{\uparrow}$) using
optical pumping. In this special case, pulse 4 does not affect the system and
can be neglected. This reduces the total gate time and improves the state
mapping fidelity (see Sec. 4.2).

After considering the phases that are acquired by performing the CNOT gate on
the ions in neighboring nodes (labeled here as 1 and 2 instead of
$\mathrm{(i)}$ and $\mathrm{(i+1)}$), the final state is

$\displaystyle\ket{\psi}_{\text{Er,Eu}}^{\pm}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\ket{\uparrow\downarrow}_{\text{Er}_{1},\text{Er}_{2}}\ket{\uparrow\downarrow}_{\text{Eu}_{1},\text{Eu}_{2}}\right.$
$\displaystyle\left.\pm
e^{i\phi}\ket{\downarrow\uparrow}_{\text{Er}_{1},\text{Er}_{2}}\ket{\downarrow\uparrow}_{\text{Eu}_{1},\text{Eu}_{2}}\right),$
(3)

where

$\displaystyle\phi$
$\displaystyle=(\omega_{\downarrow,\text{Er}_{2}}-\omega_{\downarrow,\text{Er}_{1}})\tau+(\omega_{\text{Eu}_{2}}-\omega_{\text{Eu}_{1}})\tau_{2}$
$\displaystyle+(\omega_{\downarrow,\text{Eu}_{2}}-\omega_{\downarrow,\text{Eu}_{1}})(\tau_{3}+\tau_{4})$
$\displaystyle+k_{\downarrow\text{Eu}_{1}}x_{\text{Eu}_{1}}-k_{\downarrow\text{Eu}_{2}}x_{\text{Eu}_{2}},$
(4)

and $x$ is the distance the photon travels between each 151Eu3+ ion and the
beam splitter,
$k_{\downarrow\text{Eu}}=\frac{\omega_{\downarrow,\text{Eu}}}{c}$ is the
wavenumber, $\tau_{j}$ is the time elapsed after application of the
$j^{\text{th}}$ pulse and $\tau$ is the total time duration that is needed to
perform a CNOT gate. Since these phases are known, they can be compensated by
applying local operations on the ions.

To conclude the mapping step, a $\frac{\pi}{2}$ microwave (MW) pulse is
applied to rotate each Er3+ qubit. This is followed by a state measurement of
both nearest-neighbor Er3+ ions, see also Sec. IV below. This projects the
151Eu3+ ions onto an entangled state. Depending on the outcome of these
measurements, the entangled state between remote 151Eu3+ ions would be
$\ket{\psi^{+}}$ or $\ket{\psi^{-}}$. For example, in the case that we begin
with $\ket{\psi}_{\text{Er}}^{+}$ (given in Eq. 1), after performing the gate,
MW rotation, and measurement, if both Er ions are found in the state
$\ket{\uparrow}$, the entangled state between remote 151Eu3+ ions is
$\frac{1}{\sqrt{2}}\left(\ket{\uparrow\downarrow}_{\text{Eu}_{1},\text{Eu}_{2}}+\ket{\downarrow\uparrow}_{\text{Eu}_{1},\text{Eu}_{2}}\right)$.

Once entanglement is established in neighboring elementary links, we perform a
joint measurement on both ions at each intermediate node to distribute
entanglement (see Fig. 1 (c)). In our scheme, it is possible to perform
entanglement swapping deterministically using the permanent electric dipole-
dipole interaction.

To perform the desired entanglement swapping, first a CNOT gate is applied in
which Er3+ serves as the control qubit and 151Eu3+ as the target qubit
(similarly as before). Here, both ions are in a superposition state and so all
5 pulses of the CNOT gate sequence are required. Then the Er3+ ion is measured
in the diagonal (X) basis. Next, another CNOT operation is performed but now
the target and control qubits are exchanged. Finally the Er3+ ion is measured
in the logical (Z) basis. This ‘reverse’ CNOT gate is performed in order to
avoid directly measuring the spin state of 151Eu3+ ion optically (the optical
lifetime of 151Eu3+ is longer than the Er3+ spin coherence lifetime). Fig.(2)
shows the pulse sequence needed to perform the first CNOT gate between the
ions.

Based on the outcomes of the measurements on each Er3+ ion, the outer nodes
will be projected onto one of the four Bell states. To be more precise, when
performing entanglement swapping between $\mathrm{Er_{i}}$ and
$\mathrm{Eu_{i}}$, if only one of the state measurements of $\mathrm{Er_{i}}$
is $\ket{\uparrow}$, the state of $\mathrm{Er_{i-1}}$ and $\mathrm{Eu_{i+1}}$
will project onto the $\ket{\psi^{+}}$ or the $\ket{\psi^{-}}$ Bell state. On
the other hand, if both measurements of $\mathrm{Er_{i}}$ are $\ket{\uparrow}$
or both $\ket{\downarrow}$, the state will project onto the $\ket{\phi^{+}}$
or the $\ket{\phi^{-}}$ Bell state.

To verify or utilize the entanglement that is generated between the distant
ions, a measurement must be performed on the endpoint nodes. Our protocol
results in an entangled state of the Er3+ ion at the first node with the
151Eu3+ ion at the last node. A measurement of the Er3+ ion can be performed
by direct optical excitation. A measurement of the 151Eu3+ ion spin can be
performed by using the nearby Er3+ as a readout ion. This is done by mapping
the 151Eu3+ state to Er3+ using gate operations (also see section 4.3 for
further details).

##  3 Entanglement generation rates and multiplexing

In our scheme, the success probability of generating an entangled state
between two neighboring Er3+ ions that are separated by a distance $L_{0}$ is
$p_{t}=\frac{1}{2}\eta_{t}^{2}p^{2}\eta_{d}^{2}$, where
$\eta_{t}=e^{-\frac{L_{0}}{2L_{att}}}$ is the transmission probability of a
photon through optical fiber and $L_{att}\approx 22$ km (which corresponds to
a loss of 0.2 dB$/$km), $\eta_{d}$ is the detection efficiency, and $p$ is the
success probability of emitting a single photon into a cavity mode. Similarly,
the success probability of mapping the state of an Er3+ ion onto a nearby
151Eu3+ ion is $p_{m}\approx p\eta_{d}$, because it requires one optical read-
out of each Er3+ ion. Therefore, the average time to generate entanglement in
an elementary link and perform the state mapping steps is

$\langle
T\rangle_{L_{0}}=\frac{L_{0}}{c}\left(\frac{1}{p_{t}p_{m}^{2}}\right),$ (5)

where $c=2\times 10^{8}\mathrm{\frac{m}{s}}$ is the speed of light in fiber.

The average time that is required to distribute entanglement over two
neighboring elementary links, which corresponds to a distance $L=2L_{0}$, can
be estimated as follows. First, entangled pairs of 151Eu3+ ions are generated
over an elementary link and, once successful, Er3+ ions are then generated in
a neighboring elementary link. Thus, the probability of establishing
entanglement for both links is

$p_{0}=\left(\frac{1}{P_{A}}+\frac{1}{P_{B}}\right)^{-1},$ (6)

where $P_{A}=p_{t}$ and $P_{B}=p_{t}p_{m}^{2}$ represent the success
probability of generating entanglement between Er3+ and 151Eu3+ ions,
respectively. Then, entanglement is extended by performing a BSM between the
close-lying Er3+ and 151Eu3+ ions at the center node. The success probability
for the entanglement swapping step is $p_{s}\approx p^{2}\eta_{d}^{2}$ because
it requires two optically-induced spin read-outs of an Er3+ ion. Consequently,
the average time to distribute entanglement over a distance $2L_{0}$ is

$\braket{T}_{2L_{0}}=\frac{L_{0}}{c}\frac{1}{p_{0}p_{s}}.$ (7)

Accordingly, the average time to distribute an entangled pair over a distance
$L=2^{n}L_{0}$, with $n$ denoting the number of nesting levels, is [Sangouard
_et al._(2009)Sangouard, Dubessy, and Simon, Sangouard  _et
al._(2011)Sangouard, Simon, de Riedmatten, and Gisin]:

$\braket{T}_{L}=\left(\frac{3}{2}\right)^{n-1}\frac{L_{0}}{c}\frac{1}{p_{0}p_{s}^{n}}.$
(8)

The factor $3/2$ for each of the next nesting levels, which is a good
approximation for the exact result [Sangouard  _et al._(2011)Sangouard, Simon,
de Riedmatten, and Gisin], can be understood in following way.

In contrast to the first nesting level, a successful entanglement distribution
for higher nesting levels does not require waiting for a success in one link
before the establishment of entanglement in another link can be attempted;
rather, the establishment of entanglement can be attempted in both links
simultaneously. For example, if a goal is to distribute entanglement over the
distance $4L_{0}$, entanglement must be generated in two neighboring links
each of length $2L_{0}$ before entanglement swapping is performed. If the
average waiting time for a success in one link of length $2L_{0}$ is
$\braket{T}_{2L_{0}}$, entanglement will be established in one of two links
after $\braket{T}_{2L_{0}}/2$. Then, after another $\braket{T}_{2L_{0}}$ time
duration, entanglement will be established in the other link. Hence, the
average time to establish entanglement in the two neighboring links is
$3\braket{T}_{2L_{0}}/2$. The same arguments hold for the next nesting levels,
resulting in a factor $\left(\frac{3}{2}\right)^{n-1}$ for $n$ nesting levels.

Multiplexing can be employed to significantly enhance the rate of entanglement
distribution. Referring to the encoding of several individual qubits, each
into their own distinguishable modes, multiplexing has been utilized for some
quantum repeater proposals [Sangouard  _et al._(2011)Sangouard, Simon, de
Riedmatten, and Gisin, Simon  _et al._(2007)Simon, de Riedmatten, Afzelius,
Sangouard, Zbinden, and Gisin, Collins  _et al._(2007)Collins, Jenkins,
Kuzmich, and Kennedy, Sinclair  _et al._(2014)Sinclair, Saglamyurek,
Mallahzadeh, Slater, George, Ricken, Hedges, Oblak, Simon, Sohler, and
Tittel]. As outlined in Fig. 3, we consider an array of $m$ cavities in which
each cavity emits a photon that features a distinguishable resonance frequency
from the rest. This can be accomplished using frequency translation (see Sec.
IV for more details). Using a coupler device, each photon is directed into a
common fiber and is directed towards a BSM station that is composed of
similarly-designed decouplers and single-photon detectors. Benefiting from the
distribution of entanglement into $m$ parallel modes, the probability that at
least one entangled state is distributed over the entire channel is
$1-(1-P_{t})^{m}$, which can be made unity for a sufficiently high $m$. Here,
$P_{t}=(2/3)^{n-1}p_{0}p_{s}^{n}$ is the success probability of distributing
entanglement over a distance $L$ using a qubit that is encoded into a single
mode. Further details of this set-up are described in Sec.4.4.

Figure 3: Our multiplexed scheme consists of $2^{n}$+1 nodes that span the
total channel distance $L$. Each node consists of an array of $m$ cavities
that emit photons which feature differing carrier frequencies. The carrier
frequency of each photon is determined by a frequency translation device. A
coupling element (COUPLER) ensures that each photon traverses a common channel
to a Bell-state measurement station that consists of a beam splitter,
decoupling element (DECOUPLER), and 2$m$ single photon detectors. This set-up
allows $m$ entanglement generation protocols to be operated in parallel.

The entanglement distribution rate of our scheme is plotted as a function of
distance for $n=3$ in Fig. 4 and compared to that employing the well-known
DLCZ scheme [Duan  _et al._(2001)Duan, Lukin, Cirac, and Zoller], which uses
ensemble-based memories, as well as that using direct transmission with a
single-photon source which produces photons at 10 GHz. This 10 GHz photon rate
is an optimistic rate for the direct transmission of photons. The rate that we
assume for a single photon source is much faster than the rate for the
proposed repeater because they have different limitations. Even though the
photon rate of our scheme (which depends on the cavity characteristics and the
optical lifetime of the Er3+ ion) is much lower, the time scale for the
repeater is actually determined by the communication time L${}_{0}/c$ which is
even longer. Note that the direct transmission scheme can also be interpreted
as the Pirandola bound [Pirandola  _et al._(2017)Pirandola, Laurenza,
Ottaviani, and Banchi] for a repetition rate of 10/1.44$=$6.9 GHz. The
performance of our protocol and the DLCZ scheme with $m=100$ multiplexed
channels is also shown Fig. 4. For more information on other approaches see
Ref. [Sangouard  _et al._(2011)Sangouard, Simon, de Riedmatten, and Gisin]. In
this review paper, the multiplexed DLCZ outperformed the other repeater
protocols considered (see Fig. 18 of ref.[Sangouard  _et al._(2011)Sangouard,
Simon, de Riedmatten, and Gisin]). As will be discussed in Sec. IV B, the use
of even more than $m=100$ parallel spectral channels is possible.

Fig. 4 shows that our use of deterministic gates is advantageous for the rate.
The scaling of the DLCZ scheme with distance is slightly better than that of
our scheme due to the requirement of detecting only one photon for each
elementary entanglement creation step (rather than two photons for our
scheme). However, this comes at the significant expense of requiring phase
stabilization for the long-distance fiber links.

Figure 4: A comparison of the entanglement distribution rate for various
schemes. Direct transmission scheme using a 10 GHz single-photon source (E) is
compared with original DLCZ protocol (A) and our scheme (B) using 8 elementary
links, each of length $L_{0}$. Also shown are the rates that correspond to the
multiplexed versions of DLCZ (C) as well as our scheme (D) with $m=100$ using
the same number of elementary links. We assume that the detection efficiency
and the success probability of emitting a single photon by an ion is
$p=\eta_{d}=0.9$, and in the case of DLCZ, a storage efficiency of
$\eta_{m}=0.9$. The repetition rate for each repeater protocol is set by the
communication time $L_{0}/c$.

##  4 Implementation

For our scheme, we consider ion beam-milled Y2SiO5 photonic crystal cavity
systems that have been weakly-doped with Er3+ and 151Eu3+ ions [Zhong  _et
al._(2015b)Zhong, Kindem, Miyazono, and Faraon, Zhong  _et al._(2017)Zhong,
Kindem, Bartholomew, Rochman, Craiciu, Miyazono, Bettinelli, Cavalli, Verma,
Nam, Marsili, Shaw, Beyer, and Faraon, Thiel  _et al._(2012)Thiel, Babbitt,
and Cone]. Co-doped crystals may be grown from the melt [Thiel  _et
al._(2012)Thiel, Babbitt, and Cone], or individual RE ions may be implanted
into single Y2SiO5 crystals [Kornher  _et al._(2016)Kornher, Xia, Kolesov,
Kukharchyk, Reuter, Siyushev, Stöhr, Schreck, Becker, Villa, Wieck, and
Wrachtrup]. After a milling step, the output of the cavity can be coupled to
an optical waveguide using, e.g., microscopy [Davanco  _et al._(2017)Davanco,
Liu, Sapienza, Zhang, De Miranda Cardoso, Verma, Mirin, Nam, Liu, and
Srinivasan], bonding [Murray  _et al._(2015)Murray, Ellis, Meany, Floether,
Lee, Griffiths, Jones, Farrer, Ritchie, Bennett, and Shields] or a pick-and-
place technique [Kim  _et al._(2017)Kim, Aghaeimeibodi, Richardson, Leavitt,
Englund, and Waks], with the latter having been used to heterogeneously
interface InAs/InP quantum dots with Si waveguides. Despite the lack of on-
demand control of the position and relative orientation of each ion [Kornher
_et al._(2016)Kornher, Xia, Kolesov, Kukharchyk, Reuter, Siyushev, Stöhr,
Schreck, Becker, Villa, Wieck, and Wrachtrup], a suitable Er-Eu ion pair can
be identified by performing laser-induced fluorescence spectroscopy. This
involves exciting each Er3+ (151Eu3+) ion with a laser and measuring the
spectral shift of the resultant fluorescence from other 151Eu3+ (Er3+) ions
due to the aforementioned electric-dipole coupling (see Sec. 2.3) [Thiel  _et
al._(2012)Thiel, Babbitt, and Cone]. Note that measurements using a
0.02%Er:1%Eu:Y2SiO5 bulk crystal revealed the optical transition frequency of
sets of Er3+ ions that lie within approximately one nanometer from adjacent
Eu3+ ions [Thiel  _et al._(2012)Thiel, Babbitt, and Cone]. These results
suggest that the transition frequencies of suitable Er-Eu pairs can be rapidly
distinguished from other spectator ions.

The magnetic field applied in the $D_{1}-D_{2}$ plane at 135 degrees relative
to the $D_{1}$ axis results in the decay of the excited Er3+ ion back into the
initial spin state via spontaneous emission with a probability of higher than
90%[Hastings-Simon  _et al._(2008)Hastings-Simon, Lauritzen, Staudt, van
Mechelen, Simon, de Riedmatten, Afzelius, and Gisin], and a Zeeman-level
lifetime of about 130ms was measured for a magnetic field of 1.2 mT at 2.1 K
temperature[Hastings-Simon  _et al._(2008)Hastings-Simon, Lauritzen, Staudt,
van Mechelen, Simon, de Riedmatten, Afzelius, and Gisin]. At large external
magnetic fields of 1 T or more and temperatures below 3 K, one-phonon direct
process is the dominant spin-relaxation mechanism and the temperature and
magnetic field dependence of the relaxation rate could be approximated by
[Böttger  _et al._(2006)Böttger, Thiel, Sun, and Cone]:

$R(B)=R_{0}+\alpha_{D}g^{3}B^{5}\text{coth}(\frac{g\mu_{B}B}{2kT})$ (9)

where $\alpha_{D}$ is the anisotropic constant, $\mu_{B}$ is the Bohr
magneton, and $k$ is the Boltzmann constant. The field independent
contribution $R_{0}$ can be attributed to cross-relaxations with paramagnetic
impurities in the crystal and hence depends strongly on the crystal purity.
Extrapolating from Refs [Böttger  _et al._(2006)Böttger, Thiel, Sun, and Cone,
Hastings-Simon  _et al._(2008)Hastings-Simon, Lauritzen, Staudt, van Mechelen,
Simon, de Riedmatten, Afzelius, and Gisin], the spin relaxation time would be
about 40ms at 20mK for an external magnetic field of 1T at the 135∘ in
$D_{1}-D_{2}$ plane. At this magnetic field, however, the splitting of the
Zeeman levels would be too large to address it with microwave pulses and
optical Raman pulses should be applied instead.

For an ensemble-doped Er3+:Y2SiO5 the spin coherence lifetime at mK
temperatures and external magnetic fields of few hundred mT can be as short as
$\sim 7\hskip 2.84526pt\mu$s [Probst  _et al._(2015)Probst, Rotzinger,
Ustinov, and Bushev]. However, for an ensemble-doped Er3+:Y2SiO5 crystal spin
flip-flop processes are one of the main sources of decoherence, the spin
coherence lifetime of a single Er3+ ion in Y2SiO5 is largely determined by
spin-spin magnetic dipole interactions. The magnetic moments of the
constituent spins of Y2SiO5 are small: $-0.137\mu_{N}$, $-0.5\mu_{N}$, and
$-1.89\mu_{N}$ for $\mathrm{{}^{89}Y}$, $\mathrm{{}^{29}Si}$, and
$\mathrm{{}^{17}O}$ respectively. Compared to yttrium ions,
$\mathrm{{}^{29}Si}$ and $\mathrm{{}^{17}O}$ have low isotopic natural
abundances, so we assume the contribution of these isotopes to be negligible
and only consider the Er-Y interactions. In a large enough magnetic field (a
few hundred milli-tesla or more) compared to the Er-Y coupling strength, the
magnetic moment of Er3+ will detune the closest-proximity Y ions from
resonance with those further away. This well-known effect referred to as the
“frozen core” has been observed and results in weaker decoherence of the Er3+
ion by nearby Y ions [Guillot-Noël  _et al._(2007)Guillot-Noël, Vezin,
Goldner, Beaudoux, Vincent, Lejay, and Lorgeré, Fraval  _et al._(2004)Fraval,
Sellars, Morrison, and Ferris, Macfarlane and Shelby(1987)].

Hence, for a single Er3+ in Y2SiO5, the presence of a strong magnetic field
may increase the spin coherence lifetime into the milliseconds range. To
further increase the spin coherence lifetime, dynamical decoupling immediately
after the second optical excitation step is necessary. While this has not been
demonstrated in Er3+:Y2SiO5, it is a widely-employed method, and has been used
to extend coherence lifetimes in Ce-doped yttrium aluminum garnet [Siyushev
_et al._(2014)Siyushev, Xia, Reuter, Jamali, Zhao, Yang, Duan, Kukharchyk,
Wieck, Kolesov _et al._] as well as nitrogen-vacancy centers [Reiserer  _et
al._(2016)Reiserer, Kalb, Blok, van Bemmelen, Taminiau, Hanson, Twitchen, and
Markham].

Spin polarization of up to $90\%$ in Er3+:Y2SiO5 has been realized by using
stimulated emission and spin-mixing methods [Lauritzen  _et
al._(2008)Lauritzen, Hastings-Simon, de Riedmatten, Afzelius, and Gisin]. The
efficiency of spin polarization is determined by a competition between decay
of the ground-level population relative to the optical pumping efficiency of
the ion. Since the latter will be enhanced due to the cavity-induced decay
rate, we expect a near-unity Er3+-spin polarization for our scheme. Since the
ground state lifetime of 151Eu3+ is several hours, spin polarization can be
achieved by performing continuous optical pumping of all but one of the ground
states.

In Y2SiO5, RE ions can occupy two crystallographically inequivalent sites with
$C_{1}$ symmetry. Due to this lack of symmetry, the orientation of the dipole
moments and their magnitudes, and hence their dipole-dipole interaction
strengths, are unknown. Previous measurements of the linear Stark shift of
Er3+:Y2SiO5 [Minář  _et al._(2009)Minář, Lauritzen, de Riedmatten, Afzelius,
Simon, and Gisin] allow the calculation of the projection of the electric-
dipole moment difference onto the direction of the externally applied electric
field to be approximately $0.84\times 10^{-31}$Cm [Macfarlane(2007)]. The
dipole moment difference for 151Eu3+:Y2SiO5 can be as high as
$\Delta\mu_{Eu}=0.81\times 10^{-31}$Cm [Wesenberg  _et al._(2007)Wesenberg,
Mølmer, Rippe, and Kröll], resulting in the shift of the transition frequency
of $10$ and $0.01$ MHz for $r_{ij}=1$ and $10$ nm, respectively. As a
comparison, the magnetic dipole-dipole interaction between Er3+ magnetic
moment, which can be as high as $\mu_{Er}=14.65\ \mu_{B}$ [Sun  _et
al._(2008)Sun, Böttger, Thiel, and Cone], and 151Eu magnetic moment with its
intrinsic value of $\mu_{Eu}=3.42\ \mu_{N}$ [Pichanick  _et
al._(1960)Pichanick, Sandars, and Woodgate] is much weaker and amounts to 363
and 0.363 kHz for $r_{ij}=1$ and $10$ nm, respectively.

###  4.1 Cavity

The cavity serves three main purposes in this proposal. It improves the
quantum efficiency, enhances the single-photon indistinguishability, and
increases the rate of Er3+ emission. To achieve these benefits, the cavity
must provide a significant Purcell enhancement to the
$\ket{e_{2}}\longrightarrow\ket{\uparrow}$ transition of Er3+.

In the context of RE ions, Purcell factors of several hundred have been
achieved [Dibos  _et al._(2018)Dibos, Raha, Phenicie, and Thompson, Zhong  _et
al._(2015b)Zhong, Kindem, Miyazono, and Faraon], and up to $10^{3}$ seems to
be a reasonable goal [O’Brien  _et al._(2016)O’Brien, Zhong, Faraon, and
Simon]. The Purcell factor can be written: $P=(\gamma/\gamma_{r})C$, where
$1/\gamma$ is the excited state lifetime, $\gamma_{r}$ is the radiative decay
rate, and $C$ is the cavity cooperativity. The cavity-enhanced quantum
efficiency (probability of emitting a photon into the cavity mode) is then
given by: $p=\eta P/(1+\eta P)$, where $\eta=\beta\gamma_{r}/\gamma$ is the
Er3+ spin-conserving quantum efficiency, and $\beta$ is the probability of an
excited ion to relax into the initial spin state via the spontaneous emission.
For Er3+:Y2SiO5, $\beta=0.9$, $\gamma=2\pi\times 14$ Hz, and
$\gamma_{r}=2\pi\times 3$ Hz [McAuslan  _et al._(2009)McAuslan, Longdell, and
Sellars], resulting in $\eta=0.19$. For $P=100$, a cavity quantum efficiency
of $p=0.95$ is possible. Increasing the Purcell factor to $P=1000$ allows
$p=0.995$.

The single-photon indistinguishability is a metric that quantifies the quality
of interference between photons that originate from the same quantum emitter.
Without a cavity, the single-photon indistinguishabilty can be defined as
$I_{1}=T_{2}/2T_{1}$ [Grange  _et al._(2015)Grange, Hornecker, Hunger, Poizat,
Gérard, Senellart, and Auffèves]. For an ensemble of 0.0015% Er3+:Y2SiO5 we
have $T_{1}=11.4$ms [Liu and Jacquier(2006)] and optical $T_{2}$ is around
200$\mu$s at 1 T and a temperature of a few K [Böttger  _et al._(2009)Böttger,
Thiel, Cone, and Sun]. This would imply $I_{1}=0.009$ without a cavity, which
would require significant spectral filtering to achieve successful
entanglement generation. With a cavity, the single-photon indistinguishability
can be approximated by: $I_{1}=(1+\eta P)/(\zeta+1+\eta P)$, where
$\zeta=2T_{1}/T_{2}^{\star}=2T_{1}/T_{2}-1$ is the dephasing ratio. Therefore,
with Purcell factors $P=1000$ and $P=20,000$ the single-photon
indistinguishability would be $I_{1}=0.63$ and $I_{1}=0.97$, respectively. In
a large magnetic field, however, $T_{2}=4$ ms was measured at a few K [Böttger
_et al._(2009)Böttger, Thiel, Cone, and Sun]. For an ensemble-doped
Er3+:Y2SiO5 spin flip-flop processes are the dominant decoherence mechanism
and applying a large magnetic field can freeze these processes. As a result we
have a long optical $T_{2}$. The flip-flop process can also be suppressed by
reducing the Er3+ concentration. Therefore in our system, where we assume very
low dopant concetration, we still can expect to have optical coherence times
of the order of a few ms even at lower magnetic fields (hundreds of mT). With
$P=100$ and $T_{2}=4$ms, $I_{1}=0.82$ is possible, which could be further
improved by attempting to spectrally filter the narrow $2\pi\times 1.4$ kHz
bandwidth photons. With $P=1000$, $I_{1}=0.98$ could be achieved without
spectrally filtering the $2\pi\times 14$ kHz photons (corresponding to a
duration of 11 $\mu$s).

###  4.2 CNOT gate

To perform the CNOT gate, it is necessary for Er3+ to remain excited for a
time that is long enough to apply three $\pi$-pulses to 151Eu3+. This implies
that each Er3+ ion must be excited to a different Zeeman level ($\ket{e_{1}}$)
than the level that is coupled to the cavity ($\ket{e_{2}}$). This can be done
if the Zeeman splitting between $\ket{e_{1}}$ and $\ket{e_{2}}$ is much larger
than the cavity linewidth. In this section, for Er3+ we use
$\ket{e}_{\text{Er}}=\ket{e_{1}}$.

To quantify the fidelity of the CNOT gate and state transfer procedures, we
use a model nine-level system where both Er3+ and 151Eu3+ are represented by
three-level systems with two lower-energy spin states ($\ket{\uparrow}$,
$\ket{\downarrow}$) and a single excited state ($\ket{e}$) (see Fig. 2). We
use the dipole-interaction Hamiltonian:

$H=\Delta\nu\sigma^{+}_{\text{Er}\uparrow}\sigma^{-}_{\text{Er}\uparrow}\sigma^{+}_{\text{Eu}\uparrow}\sigma^{-}_{\text{Eu}\uparrow}+\sum_{k,l}\frac{\Omega_{k,l}}{2}\left(\sigma^{+}_{k,l}+\sigma^{-}_{k,l}\right),$
(10)

where $\Delta\nu$ is the dipole interaction strength and $\Omega_{k,l}$ is the
Rabi frequency for transition $\\{k,l\\}$ for $l\in\\{\uparrow,\downarrow\\}$,
$k\in\\{\text{Er},\text{Eu}\\}$. Without loss of generality, we choose
$\Delta\nu>0$ and $\Omega_{k,l}>0$. The master equation is

$\displaystyle\dot{\rho}$
$\displaystyle=-i[H,\rho]+\sum_{k,l}\gamma_{k,l}\mathcal{D}(\sigma_{k,l}^{-})\rho+\frac{\gamma^{\star}_{k}}{2}\mathcal{D}(\sigma^{+}_{k,l}\sigma^{-}_{k,l})\rho$
(11) $\displaystyle+\sum_{k}\frac{\chi_{k}}{2}\mathcal{D}(\sigma_{z,k})\rho,$

where
$\mathcal{D}(\sigma)\rho=\sigma\rho\sigma^{\dagger}-\sigma^{\dagger}\sigma\rho/2-\rho\sigma^{\dagger}\sigma/2$.
The operators are defined as $\sigma_{k,l}^{+}=\ket{e}\bra{l}_{k}$,
$\sigma_{k,l}^{-}=(\sigma_{k,l}^{+})^{\dagger}$, and
$\sigma_{z,k}=(\ket{\uparrow}\bra{\uparrow}_{k}-\ket{\downarrow}\bra{\downarrow}_{k})/2$.
The rate $\gamma_{k,l}$ is the decay rate for transition $\\{k,l\\}$,
$\gamma^{\star}_{k}$ is the optical pure dephasing rate, and $\chi_{k}$ is the
spin decoherence rate for ion $k$.

To solve for the Er-Eu state after applying the five-pulse CNOT gate sequence,
we first assume that each $\pi$-pulse is a square pulse and that they are
applied to the system sequentially with no time delay between pulses. In this
case, the time taken to apply each $\pi$-pulse is given by
$T_{k,l}=\pi/\Omega_{k,l}$. We also assume that $\Omega_{k,l}$ and $\Delta\nu$
are much larger than any dissipative rate so that we can treat the dissipation
perturbatively.

We then define the zero-order (reversible) superoperator $\mathcal{L}_{0}$
where $\mathcal{L}_{0}\rho=-i[H,\rho]$. Then we define the first-order
(irreversible) perturbation superoperator $\mathcal{L}_{1}$ as
$\mathcal{L}_{1}\rho=\dot{\rho}-\mathcal{L}_{0}\rho$. From this, we define the
rotation superoperator $\mathcal{R}_{k,l}(\theta)$ corresponding to the pulse
where $\Omega_{k,l}\neq 0$ and $\Omega_{k^{\prime},l^{\prime}}=0$ for all
$k^{\prime}\neq k$ and $l^{\prime}\neq l$. This superoperator is given by

$\mathcal{R}_{k,l}(\theta)=e^{\mathcal{L}_{0}\frac{\theta}{\Omega_{k,l}}}\left[1+\int_{0}^{\frac{\theta}{\Omega_{k,l}}}d\tau
e^{-\mathcal{L}_{0}\tau}\mathcal{L}_{1}e^{\mathcal{L}_{0}\tau}\right],$ (12)

which is accurate to first-order in
$\gamma_{k^{\prime},l^{\prime}},\gamma^{\star}_{k^{\prime}},\chi_{k^{\prime}}\ll\Omega_{k,l},\Delta\nu$
for all $k^{\prime}$ and $l^{\prime}$.

To simulate the entanglement swapping between 151Eu3+ and Er3+, we begin with
an initial superposition state $\rho_{0}$ where
$\ket{\psi}=(\ket{\uparrow}+\ket{\downarrow})_{\text{Er}}(\ket{\uparrow}+\ket{\downarrow})_{\text{Eu}}$
and $\rho_{0}=\ket{\psi}\bra{\psi}$. Then after applying the CNOT pulse
scheme, the final state is

$\rho=\mathcal{R}_{\text{Er},\uparrow}(\theta)\mathcal{R}_{\text{Eu},\uparrow}(\theta)\mathcal{R}_{\text{Eu},\downarrow}(\theta)\mathcal{R}_{\text{Eu},\uparrow}(\theta)\mathcal{R}_{\text{Er},\uparrow}(\theta)\rho_{0},$

where ideally $\theta=\pi$; however, we set $\theta=\pi+\epsilon$ to simulate
a $\pi$-pulse with a small rotation deviation $|\epsilon|\ll\pi$ caused by a
non-ideal pulse area. The total time taken to apply this sequence is
$T_{\text{gate}}=2T_{\text{Er},\uparrow}+2T_{\text{Eu},\uparrow}+T_{\text{Eu},\downarrow}$.

For a high fidelity gate, the pulse durations must be small so that the gate
is fast and qubit states do not dephase during the sequence. This favors
larger Rabi frequencies. On the other hand, it is also necessary that the
three Eu pulses do not excite the 151Eu3+ spin state if Er3+ is excited. One
way to achieve this is to set $\Omega_{\text{Eu},l}$ to satisfy the detuning
condition $\Omega_{\text{Eu},l}^{2}/\Delta\nu^{2}\ll 1$ so that there is
little chance for off-resonant excitation. However, since $\Delta\nu$ cannot
be arbitrarily large (the separation between Er and Eu cannot be arbitrarily
small), the detuning criteria necessitates a very slow gate, which cannot
provide a high fidelity.

Alternatively, the detuning condition can be circumvented if
$\Omega_{\text{Eu},l}$ is chosen so that the Eu pulses perform an effective
$2\pi$ rotation on the Eu spin state when Er is excited, but still perform a
$\pi$-pulse when Er is not excited. This can be accomplished by requiring that
$\Omega_{\text{Eu},l}$ satisfies the effective Rabi frequency relation
$\sqrt{\Delta\nu^{2}+\Omega_{\text{Eu},l}^{2}}=2\Omega_{\text{Eu},l}$. This
sets $\Omega_{\text{Eu},l}=\Delta\nu/\sqrt{3}$. As a consequence of fixing the
Rabi frequency, an accurate characterization of the dipole interaction
strength for each Er-Eu pair is necessary to achieve a high fidelity. This is
because any mischaracterization $\delta\nu$ from the true value
$\Delta\nu-\delta\nu$ will cause a deviation from the desired $2\pi$ rotation.
To account for this, we also consider a perturbation of the fidelity for
deviation $\Delta\nu\rightarrow\Delta\nu-\delta\nu$ where we assume
$|\delta\nu|\ll\Delta\nu\propto\Omega_{k,l}$.

The effective $2\pi$ pulse leaves a relative phase between the 151Eu3+ ground-
state spins of $\varphi=-\pi(2-\sqrt{3})/2$. If Er3+ is in $\ket{\downarrow}$,
then only $\pi$-rotations are applied to 151Eu3+. Hence, in this case, there
is no acquired relative phase from the detuned pulses. However, if the state
is $\ket{\uparrow\downarrow}$, 151Eu3+ will be affected by the third pulse
performing an effective $2\pi$ rotation, and so $\ket{\uparrow\downarrow}$
will acquire a relative phase of $\varphi$. Likewise, if the state is
$\ket{\uparrow\uparrow}$, 151Eu3+ will be affected by the second and fourth
pulses performing an effective $2\pi$ rotation, and so
$\ket{\uparrow\uparrow}$ will acquire a relative phase of $2\varphi$. Hence,
in the absence of dissipation and using perfect square $\pi$-pulses, the
expected final state is

$\ket{\psi_{f}}=\frac{1}{2}\left(\ket{\downarrow\downarrow}+\ket{\downarrow\uparrow}+e^{i\varphi}\ket{\uparrow\downarrow}+e^{i2\varphi}\ket{\uparrow\uparrow}\right)$
(13)

Since the expected acquired phase $\varphi=-\pi(2-\sqrt{3})/2$ is known, and
independent of the dipole interaction strength $\Delta\nu$, it can be tracked
or corrected and so we use $\ket{\psi_{f}}$ as the final state when
calculating the fidelity.

We use the above expressions for $\rho$ and $\ket{\psi_{f}}$ to compute the
fidelity $F_{\text{CNOT}}=|\bra{\psi_{f}}\rho\ket{\psi_{f}}|$. For simplicity,
we choose to set
$\Omega_{\text{Er},l}=\Omega_{\text{Eu},l}=\Omega=\Delta\nu/\sqrt{3}$;
however, $\Omega_{\text{Er},l}$ is not restricted by the dipole interaction
strength and could be made larger than $\Omega$ to further decrease the total
gate time and increase fidelity. The solution $F_{\text{CNOT}}$ to first-order
in $\gamma_{k,l},\gamma^{\star}_{k},\chi_{k}\ll\Omega\propto\Delta\nu$ for all
$k,l$ and second-order in $\epsilon\ll\pi$ and $\xi=\delta\nu/\Delta\nu\ll 1$
is

$F_{\text{CNOT}}\simeq
1-T_{\text{CNOT}}\Gamma-\epsilon^{2}-\frac{13\pi}{16}\epsilon\xi-\frac{43\pi^{2}}{128}\xi^{2}$
(14)

Here $T_{\text{CNOT}}=5\pi/\Omega=5\pi\sqrt{3}/\Delta\nu$ is the total gate
time and $\Gamma$ is the effective dissipation rate:

$\Gamma\simeq\frac{1}{80}\left[31\gamma_{\text{Er}}+17\gamma^{\star}_{\text{Er}}+8\chi_{\text{Er}}+11\gamma_{\text{Eu}}+8\gamma^{\star}_{\text{Eu}}+17\chi_{\text{Eu}}\right],$
(15)

where we define $\gamma_{k}=\gamma_{k\uparrow}+\gamma_{k\downarrow}$. To
obtain the reverse CNOT fidelity, it is only necessary to swap the dissipative
parameters for Er3+ and 151Eu3+.

For a dipole interaction strength of $\Delta\nu=2\pi\times 46$ kHz
corresponding to an Er-Eu separation of about 6 nm, the CNOT gate time can be
as small as $T_{\text{CNOT}}=94\hskip 2.84526pt\mu$s. To estimate the
fidelity, we use the parameters $\gamma_{\text{Er}}=2\pi\times 3$ Hz,
$\gamma^{\star}_{\text{Er}}=2\pi\times 8$ Hz, $\gamma_{\text{Eu}}=2\pi\times
1.3$ Hz, and $\gamma^{\star}_{\text{Eu}}=2\pi\times 19$ Hz estimated from Ref.
[McAuslan  _et al._(2009)McAuslan, Longdell, and Sellars]; also
$\chi_{\text{Er}}\simeq 2\pi\times 80$ Hz and $\chi_{\text{Eu}}\simeq 0$. In
this case, for a small $\pi$-pulse over-rotation of $\epsilon\simeq\pi/64$ and
a dipole interaction strength over-estimation of $2\%$ ($\xi\simeq 0.02$), the
fidelity is $F_{\text{CNOT}}=0.986$. The reverse CNOT fidelity is slightly
smaller due to the spin dephasing of Er: $F_{\text{R-CNOT}}=0.980$.

This method can also be used to compute the fidelity of state transfer from
Er3+ to 151Eu3+. In this case, we remove the fourth pulse
($\mathcal{R}_{\text{Eu},\uparrow}$) and use the initial state
$\ket{\psi}=(\ket{\uparrow}+\ket{\downarrow})_{\text{Er}}\ket{\uparrow}_{\text{Eu}}$.
The expected final state is
$\ket{\psi_{f}}=\ket{\downarrow\downarrow}+e^{i2\varphi}\ket{\uparrow\uparrow}$.
In this case, the fidelity is similar to the CNOT gate (Eq. (14)):

$F_{\text{ST}}\simeq
1-T_{\text{ST}}\Gamma-\frac{5}{8}\epsilon^{2}-\frac{3\pi}{16}\epsilon\xi-\frac{21\pi^{2}}{256}\xi^{2}$
(16)

where $T_{\text{ST}}=4\pi/\Omega=4\pi\sqrt{3}/\Delta\nu$. The effective
dissipation rate was found to be very similar to Eq. (15), but depends less
strongly on 151Eu3+ dissipation:

$\Gamma\simeq\frac{1}{80}\left[29\gamma_{\text{Er}}+16\gamma^{\star}_{\text{Er}}+9\chi_{\text{Er}}+11\gamma_{\text{Eu}}+7\gamma^{\star}_{\text{Eu}}+9\chi_{\text{Eu}}\right].$
(17)

Using the same parameters as above, the fidelity of state transfer is
$F_{\text{ST}}=0.989$.

###  4.3 State measurement

The spin readout of Er3+ is performed by optical excitation. Since only the
$\ket{\uparrow}-\ket{e_{2}}$ transition is coupled to the cavity, optical
excitation will result in a presence or absence of a photon emission depending
on the state of the Er3+ ion. The measurement of the 151Eu3+ ion can be
performed through its nearby Er3+ ion as a readout ion, provided that the Er3+
ion is initially prepared in the $\ket{\uparrow}$ state. Performing spin
readout of the 151Eu3+ ion in the Z basis is then achieved by optically
exciting the 151Eu3+ ion ($\ket{\uparrow}$ to $\ket{e}$ transition) followed
by exciting the Er3+ ion ($\ket{\uparrow}$ to $\ket{e_{1}}$ transition). The
Er3+ ion will excite to $\ket{e_{1}}$ (remain in ($\ket{\uparrow}$) if the
state of the 151Eu3+ ion is $\ket{\downarrow}$ ($\ket{\uparrow}$) due to the
permanent electric dipole-dipole interaction which shifts the Er3+ optical
resonance. To readout in the X basis, a $\frac{\pi}{2}$ MW pulse should be
applied to the 151Eu3+ ion in order to rotate the ground-state spins before
the optical excitation step.

State measurement of each Er3+ ion requires the detection of an emitted
photon. Due to the Purcell effect, as discussed in Sec. 4.1, the emission of a
single photon from the ion is highly preferential into the cavity mode. For a
high quality cavity, the probability that the photon emits into the cavity
tends to unity. Therefore, the detection probability will be limited by
coupling losses and single photon detectors, which can have detection
efficiency as high as $95\%$, as has been demonstrated in superconducting
detectors [Lita  _et al._(2008)Lita, Miller, and Nam, Marsili  _et
al._(2013)Marsili, Verma, Stern, Harrington, Lita, Gerrits, Vayshenker, Baek,
Shaw, Mirin _et al._]. To do better than this limit, it is necessary to pump
the Er3+ ion into a cycling transition such that many photons will be emitted
by the cavity, and eventually detected. Using such a cycling transition,
detection probability can be as high as $98.7\%$ [O’Brien  _et
al._(2016)O’Brien, Zhong, Faraon, and Simon]. The detection efficiency is not
$100\%$ in this case because there is a small chance for the ion to decay into
a different state than the initial state, thus ending the photon cycling
[Bartholomew  _et al._(2016)Bartholomew, Ahlefeldt, and Sellars]. This chance
grows linearly with the number of cycles before detection.

###  4.4 Spectrally-multiplexed implementation

Our spectral multiplexing scheme relies on the possibility that many spectral
channels can be operated in parallel. This requires that, in one node,
different cavities emit photons that feature different carrier frequencies.
This can be accomplished by frequency translation. Noise-free translation over
tens of gigahertz [Sinclair  _et al._(2014)Sinclair, Saglamyurek, Mallahzadeh,
Slater, George, Ricken, Hedges, Oblak, Simon, Sohler, and Tittel, Grimau
Puigibert  _et al._(2017)Grimau Puigibert, Aguilar, Zhou, Marsili, Shaw,
Verma, Nam, Oblak, and Tittel] can be achieved by using voltage-swept,
commercially-available, waveguide electro-optic modulators that can be
optically-coupled to the output port of each cavity. After frequency
translation, the output can be coupled to a common spatial mode (e.g. a
waveguide or fiber) by using a tunable ring resonator filter that features
resonance linewidths as narrow as 1 MHz [Yang  _et al._(2018)Yang, Oh, Lee,
Yang, Yi, Shen, Wang, and Vahala]. Arrayed waveguides or fiber-Bragg gratings
may also be used, however they are bulky and their resonance linewidths are
currently not at the MHz level. The modulators and filters may be fabricated
on a single chip, offering the possibility of low loss and up to 104 spectral
modes. The Bell-state measurement station consists of a beamsplitter, two sets
of ring resonator filters which are identical to that used at the nodes, and
an array of superconducting nanowire-based photon detectors, chosen due to
their combination of high-efficiency and low noise properties [Marsili  _et
al._(2013)Marsili, Verma, Stern, Harrington, Lita, Gerrits, Vayshenker, Baek,
Shaw, Mirin _et al._].

##  5 Conclusion

Our proposal for a quantum repeater which is based on individual RE ions
promises the deterministic establishment of high-fidelity entanglement over
long-distances at a rate which exceeds that corresponding to the direct
transmission of photons. Our scheme utilizes some of the most desirable
features of RE-ion-doped crystals, specifically emission within the low-loss
telecommunications window (Er3+) and the hours-long nuclear spin coherence
lifetime (151Eu3+:Y2SiO5) that is needed to perform long-distance transmission
and swapping of entanglement. Moreover, control logic gates between close-
lying individual 151Eu3+ and Er3+ ions allow the quasi-deterministic swapping
of entanglement by means of a permanent electric dipole-dipole interaction.
The multiplexed version of our scheme improves the entanglement distribution
rate by at least a factor of 100 over that of the single-mode version of our
repeater.

Looking forward, it is interesting to consider the possibility of employing
individual 167Er3+ ions instead of Er-Eu ion pairs for a telecommunication
wavelength quantum repeater. In the presence of strong magnetic fields,
167Er3+:Y2SiO5 features a nuclear spin coherence lifetime in the one-second
range [Rančić  _et al._(2017)Rančić, Hedges, Ahlefeldt, and Sellars], allowing
the possibility of entanglement generation and storage using the same ion,
pairs (or small ensembles) of Er3+ ions. One of the main challenges for future
work in this direction is to devise a scheme whereby individual Er3+ ions may
be addressed and coupled within a single cavity. This could be achieved by
using magnetic dipole-dipole interactions in a similar spirit to what has been
demonstrated using nitrogen vacancy centers and carbon spins in diamond
[Reiserer  _et al._(2016)Reiserer, Kalb, Blok, van Bemmelen, Taminiau, Hanson,
Twitchen, and Markham], or by using the cavity mode to mediate the
interaction. Another interesting direction is the possibility of long-term
storage using host-ion spins such as yttrium in Y2SiO5 [Car  _et
al._(2018)Car, Veissier, Louchet-Chauvet, Le Gouët, and Chanelière].

## ACKNOWLEDGMENTS

The authors would like to thank M. Afzelius, P. Barclay, P. A. Bushev, C. W.
Thiel and W. Tittel for helpful discussions. This work was supported by the
Natural Sciences and Engineering Research Council (NSERC) through its
Discovery Grant program, the CREATE grant ’Quanta’, and through graduate
scholarships, by the University of Calgary through an Eyes High postdoctoral
fellowship, by Alberta Innovates Technology Futures (AITF) through graduate
scholarships, and by the Defense Advanced Research Projects Agency (DARPA)
through the Quiness program (Contract No. W31P4Q-13-1-0004).

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